kinds of hypothesis in statistics

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Lesson 10 of 24 By Avijeet Biswal

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In today’s data-driven world, decisions are based on data all the time. Hypothesis plays a crucial role in that process, whether it may be making business decisions, in the health sector, academia, or in quality improvement. Without hypothesis and hypothesis tests, you risk drawing the wrong conclusions and making bad decisions. In this tutorial, you will look at Hypothesis Testing in Statistics.

What Is Hypothesis Testing in Statistics?

Hypothesis Testing is a type of statistical analysis in which you put your assumptions about a population parameter to the test. It is used to estimate the relationship between 2 statistical variables.

Let's discuss few examples of statistical hypothesis from real-life - 

• A teacher assumes that 60% of his college's students come from lower-middle-class families.
• A doctor believes that 3D (Diet, Dose, and Discipline) is 90% effective for diabetic patients.

Now that you know about hypothesis testing, look at the two types of hypothesis testing in statistics.

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Importance of Hypothesis Testing in Data Analysis

Here is what makes hypothesis testing so important in data analysis and why it is key to making better decisions:

Avoiding Misleading Conclusions (Type I and Type II Errors)

One of the biggest benefits of hypothesis testing is that it helps you avoid jumping to the wrong conclusions. For instance, a Type I error could occur if a company launches a new product thinking it will be a hit, only to find out later that the data misled them. A Type II error might happen when a company overlooks a potentially successful product because their testing wasn’t thorough enough. By setting up the right significance level and carefully calculating the p-value, hypothesis testing minimizes the chances of these errors, leading to more accurate results.

Making Smarter Choices

Hypothesis testing is key to making smarter, evidence-based decisions. Let’s say a city planner wants to determine if building a new park will increase community engagement. By testing the hypothesis using data from similar projects, they can make an informed choice. Similarly, a teacher might use hypothesis testing to see if a new teaching method actually improves student performance. It’s about taking the guesswork out of decisions and relying on solid evidence instead.

Optimizing Business Tactics

In business, hypothesis testing is invaluable for testing new ideas and strategies before fully committing to them. For example, an e-commerce company might want to test whether offering free shipping increases sales. By using hypothesis testing, they can compare sales data from customers who received free shipping offers and those who didn’t. This allows them to base their business decisions on data, not hunches, reducing the risk of costly mistakes.

Hypothesis Testing Formula

Z = ( x̅ – μ0 ) / (σ /√n)

• Here, x̅ is the sample mean,
• μ0 is the population mean,
• σ is the standard deviation,
• n is the sample size.

How Hypothesis Testing Works?

An analyst performs hypothesis testing on a statistical sample to present evidence of the plausibility of the null hypothesis. Measurements and analyses are conducted on a random sample of the population to test a theory. Analysts use a random population sample to test two hypotheses: the null and alternative hypotheses.

The null hypothesis is typically an equality hypothesis between population parameters; for example, a null hypothesis may claim that the population means return equals zero. The alternate hypothesis is essentially the inverse of the null hypothesis (e.g., the population means the return is not equal to zero). As a result, they are mutually exclusive, and only one can be correct. One of the two possibilities, however, will always be correct.

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Null Hypothesis and Alternative Hypothesis

The Null Hypothesis is the assumption that the event will not occur. A null hypothesis has no bearing on the study's outcome unless it is rejected.

H0 is the symbol for it, and it is pronounced H-naught.

The Alternate Hypothesis is the logical opposite of the null hypothesis. The acceptance of the alternative hypothesis follows the rejection of the null hypothesis. H1 is the symbol for it.

Let's understand this with an example.

A sanitizer manufacturer claims that its product kills 95 percent of germs on average. 

To put this company's claim to the test, create a null and alternate hypothesis.

H0 (Null Hypothesis): Average = 95%.

Alternative Hypothesis (H1): The average is less than 95%.

Another straightforward example to understand this concept is determining whether or not a coin is fair and balanced. The null hypothesis states that the probability of a show of heads is equal to the likelihood of a show of tails. In contrast, the alternate theory states that the probability of a show of heads and tails would be very different.

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Hypothesis Testing Calculation With Examples

Let's consider a hypothesis test for the average height of women in the United States. Suppose our null hypothesis is that the average height is 5'4". We gather a sample of 100 women and determine their average height is 5'5". The standard deviation of population is 2.

To calculate the z-score, we would use the following formula:

z = ( x̅ – μ0 ) / (σ /√n)

z = (5'5" - 5'4") / (2" / √100)

z = 0.5 / (0.045)

We will reject the null hypothesis as the z-score of 11.11 is very large and conclude that there is evidence to suggest that the average height of women in the US is greater than 5'4".

Steps in Hypothesis Testing

Hypothesis testing is a statistical method to determine if there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. Here’s a breakdown of the typical steps involved in hypothesis testing:

Formulate Hypotheses

• Null Hypothesis (H0): This hypothesis states that there is no effect or difference, and it is the hypothesis you attempt to reject with your test.
• Alternative Hypothesis (H1 or Ha): This hypothesis is what you might believe to be true or hope to prove true. It is usually considered the opposite of the null hypothesis.

Choose the Significance Level (α)

The significance level, often denoted by alpha (α), is the probability of rejecting the null hypothesis when it is true. Common choices for α are 0.05 (5%), 0.01 (1%), and 0.10 (10%).

Select the Appropriate Test

Choose a statistical test based on the type of data and the hypothesis. Common tests include t-tests, chi-square tests, ANOVA, and regression analysis. The selection depends on data type, distribution, sample size, and whether the hypothesis is one-tailed or two-tailed.

Collect Data

Gather the data that will be analyzed in the test. To infer conclusions accurately, this data should be representative of the population.

Calculate the Test Statistic

Based on the collected data and the chosen test, calculate a test statistic that reflects how much the observed data deviates from the null hypothesis.

Determine the p-value

The p-value is the probability of observing test results at least as extreme as the results observed, assuming the null hypothesis is correct. It helps determine the strength of the evidence against the null hypothesis.

Make a Decision

Compare the p-value to the chosen significance level:

• If the p-value ≤ α: Reject the null hypothesis, suggesting sufficient evidence in the data supports the alternative hypothesis.
• If the p-value > α: Do not reject the null hypothesis, suggesting insufficient evidence to support the alternative hypothesis.

Report the Results

Present the findings from the hypothesis test, including the test statistic, p-value, and the conclusion about the hypotheses.

Perform Post-hoc Analysis (if necessary)

Depending on the results and the study design, further analysis may be needed to explore the data more deeply or to address multiple comparisons if several hypotheses were tested simultaneously.

Types of Hypothesis Testing

To determine whether a discovery or relationship is statistically significant, hypothesis testing uses a z-test. It usually checks to see if two means are the same (the null hypothesis). Only when the population standard deviation is known and the sample size is 30 data points or more, can a z-test be applied.

A statistical test called a t-test is employed to compare the means of two groups. To determine whether two groups differ or if a procedure or treatment affects the population of interest, it is frequently used in hypothesis testing.

3. Chi-Square 

You utilize a Chi-square test for hypothesis testing concerning whether your data is as predicted. To determine if the expected and observed results are well-fitted, the Chi-square test analyzes the differences between categorical variables from a random sample. The test's fundamental premise is that the observed values in your data should be compared to the predicted values that would be present if the null hypothesis were true.

ANOVA , or Analysis of Variance, is a statistical method used to compare the means of three or more groups. It’s particularly useful when you want to see if there are significant differences between multiple groups. For instance, in business, a company might use ANOVA to analyze whether three different stores are performing differently in terms of sales. It’s also widely used in fields like medical research and social sciences, where comparing group differences can provide valuable insights.

Hypothesis Testing and Confidence Intervals

Both confidence intervals and hypothesis tests are inferential techniques that depend on approximating the sample distribution. Data from a sample is used to estimate a population parameter using confidence intervals. Data from a sample is used in hypothesis testing to examine a given hypothesis. We must have a postulated parameter to conduct hypothesis testing.

Bootstrap distributions and randomization distributions are created using comparable simulation techniques. The observed sample statistic is the focal point of a bootstrap distribution, whereas the null hypothesis value is the focal point of a randomization distribution.

A variety of feasible population parameter estimates are included in confidence ranges. In this lesson, we created just two-tailed confidence intervals. There is a direct connection between these two-tail confidence intervals and these two-tail hypothesis tests. The results of a two-tailed hypothesis test and two-tailed confidence intervals typically provide the same results. In other words, a hypothesis test at the 0.05 level will virtually always fail to reject the null hypothesis if the 95% confidence interval contains the predicted value. A hypothesis test at the 0.05 level will nearly certainly reject the null hypothesis if the 95% confidence interval does not include the hypothesized parameter.

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Simple and Composite Hypothesis Testing

Depending on the population distribution, you can classify the statistical hypothesis into two types.

Simple Hypothesis: A simple hypothesis specifies an exact value for the parameter.

Composite Hypothesis: A composite hypothesis specifies a range of values.

A company is claiming that their average sales for this quarter are 1000 units. This is an example of a simple hypothesis.

Suppose the company claims that the sales are in the range of 900 to 1000 units. Then this is a case of a composite hypothesis.

One-Tailed and Two-Tailed Hypothesis Testing

The One-Tailed test, also called a directional test, considers a critical region of data that would result in the null hypothesis being rejected if the test sample falls into it, inevitably meaning the acceptance of the alternate hypothesis.

In a one-tailed test, the critical distribution area is one-sided, meaning the test sample is either greater or lesser than a specific value.

In two tails, the test sample is checked to be greater or less than a range of values in a Two-Tailed test, implying that the critical distribution area is two-sided.

If the sample falls within this range, the alternate hypothesis will be accepted, and the null hypothesis will be rejected.

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Right Tailed Hypothesis Testing

If the larger than (>) sign appears in your hypothesis statement, you are using a right-tailed test, also known as an upper test. Or, to put it another way, the disparity is to the right. For instance, you can contrast the battery life before and after a change in production. Your hypothesis statements can be the following if you want to know if the battery life is longer than the original (let's say 90 hours):

• The null hypothesis is (H0 <= 90) or less change.
• A possibility is that battery life has risen (H1) > 90.

The crucial point in this situation is that the alternate hypothesis (H1), not the null hypothesis, decides whether you get a right-tailed test.

Left Tailed Hypothesis Testing

Alternative hypotheses that assert the true value of a parameter is lower than the null hypothesis are tested with a left-tailed test; they are indicated by the asterisk "<".

Suppose H0: mean = 50 and H1: mean not equal to 50

According to the H1, the mean can be greater than or less than 50. This is an example of a Two-tailed test.

In a similar manner, if H0: mean >=50, then H1: mean <50

Here the mean is less than 50. It is called a One-tailed test.

Type 1 and Type 2 Error

A hypothesis test can result in two types of errors.

Type 1 Error: A Type-I error occurs when sample results reject the null hypothesis despite being true.

Type 2 Error: A Type-II error occurs when the null hypothesis is not rejected when it is false, unlike a Type-I error.

Suppose a teacher evaluates the examination paper to decide whether a student passes or fails.

H0: Student has passed

H1: Student has failed

Type I error will be the teacher failing the student [rejects H0] although the student scored the passing marks [H0 was true]. 

Type II error will be the case where the teacher passes the student [do not reject H0] although the student did not score the passing marks [H1 is true].

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Practice Problems on Hypothesis Testing

Here are the practice problems on hypothesis testing that will help you understand how to apply these concepts in real-world scenarios:

A telecom service provider claims that customers spend an average of ₹400 per month, with a standard deviation of ₹25. However, a random sample of 50 customer bills shows a mean of ₹250 and a standard deviation of ₹15. Does this sample data support the service provider’s claim?

Solution: Let’s break this down:

• Null Hypothesis (H0): The average amount spent per month is ₹400.
• Alternate Hypothesis (H1): The average amount spent per month is not ₹400.
• Population Standard Deviation (σ): ₹25
• Sample Size (n): 50
• Sample Mean (x̄): ₹250

1. Calculate the z-value:

z=250-40025/50 −42.42

2. Compare with critical z-values: For a 5% significance level, critical z-values are -1.96 and +1.96. Since -42.42 is far outside this range, we reject the null hypothesis. The sample data suggests that the average amount spent is significantly different from ₹400.

Out of 850 customers, 400 made online grocery purchases. Can we conclude that more than 50% of customers are moving towards online grocery shopping?

Solution: Here’s how to approach it:

• Proportion of customers who shopped online (p): 400 / 850 = 0.47
• Null Hypothesis (H0): The proportion of online shoppers is 50% or more.
• Alternate Hypothesis (H1): The proportion of online shoppers is less than 50%.
• Sample Size (n): 850
• Significance Level (α): 5%

z=p-PP(1-P)/n

z=0.47-0.500.50.5/850  −1.74

2. Compare with the critical z-value: For a 5% significance level (one-tailed test), the critical z-value is -1.645. Since -1.74 is less than -1.645, we reject the null hypothesis. This means the data does not support the idea that most customers are moving towards online grocery shopping.

In a study of code quality, Team A has 250 errors in 1000 lines of code, and Team B has 300 errors in 800 lines of code. Can we say Team B performs worse than Team A?

Solution: Let’s analyze it:

• Proportion of errors for Team A (pA): 250 / 1000 = 0.25
• Proportion of errors for Team B (pB): 300 / 800 = 0.375
• Null Hypothesis (H0): Team B’s error rate is less than or equal to Team A’s.
• Alternate Hypothesis (H1): Team B’s error rate is greater than Team A’s.
• Sample Size for Team A (nA): 1000
• Sample Size for Team B (nB): 800

p=nApA+nBpBnA+nB

p=10000.25+8000.3751000+800 ≈ 0.305

z=​pA−pB​p(1-p)(1nA+1nB)

z=​0.25−0.375​0.305(1-0.305) (11000+1800) ≈ −5.72

2. Compare with the critical z-value: For a 5% significance level (one-tailed test), the critical z-value is +1.645. Since -5.72 is far less than +1.645, we reject the null hypothesis. The data indicates that Team B’s performance is significantly worse than Team A’s.

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Applications of Hypothesis Testing

Apart from the practical problems, let's look at the real-world applications of hypothesis testing across various fields:

Medicine and Healthcare

In medicine, hypothesis testing plays a pivotal role in assessing the success of new treatments. For example, researchers may want to find out if a new exercise regimen improves heart health. By comparing data from patients who followed the program to those who didn’t, they can determine if the exercise significantly improves health outcomes. Such rigorous testing allows medical professionals to rely on proven methods rather than assumptions.

Quality Control and Manufacturing

In manufacturing, ensuring product quality is vital, and hypothesis testing helps maintain those standards. Suppose a beverage company introduces a new bottling process and wants to verify if it reduces contamination. By analyzing samples from the new and old processes, hypothesis testing can reveal whether the new method reduces the risk of contamination. This allows manufacturers to implement improvements that enhance product safety and quality confidently.

Education and Learning

In education and learning, hypothesis testing is a tool to evaluate the impact of innovative teaching techniques. Imagine a situation where teachers introduce project-based learning to boost critical thinking skills. By comparing the performance of students who engaged in project-based learning with those in traditional settings, educators can test their hypothesis. The results can help educators make informed choices about adopting new teaching strategies.

Environmental Science

Hypothesis testing is essential in environmental science for evaluating the effectiveness of conservation measures. For example, scientists might explore whether a new water management strategy improves river health. By collecting and comparing data on water quality before and after the implementation of the strategy, they can determine whether the intervention leads to positive changes. Such findings are crucial for guiding environmental decisions that have long-term impacts.

Marketing and Advertising

In marketing, businesses use hypothesis testing to refine their approaches. For instance, a clothing brand might test if offering limited-time discounts increases customer loyalty. By running campaigns with and without the discount and analyzing the outcomes, they can assess if the strategy boosts customer retention. Data-driven insights from hypothesis testing enable companies to design marketing strategies that resonate with their audience and drive growth.

Limitations of Hypothesis Testing

Hypothesis testing has some limitations that researchers should be aware of:

• It cannot prove or establish the truth: Hypothesis testing provides evidence to support or reject a hypothesis, but it cannot confirm the absolute truth of the research question.
• Results are sample-specific: Hypothesis testing is based on analyzing a sample from a population, and the conclusions drawn are specific to that particular sample.
• Possible errors: During hypothesis testing, there is a chance of committing type I error (rejecting a true null hypothesis) or type II error (failing to reject a false null hypothesis).
• Assumptions and requirements: Different tests have specific assumptions and requirements that must be met to accurately interpret results.

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After reading this tutorial, you would have a much better understanding of hypothesis testing, one of the most important concepts in the field of Data Science . The majority of hypotheses are based on speculation about observed behavior, natural phenomena, or established theories.

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1. What is hypothesis testing in statistics with example?

Hypothesis testing is a statistical method used to determine if there is enough evidence in a sample data to draw conclusions about a population. It involves formulating two competing hypotheses, the null hypothesis (H0) and the alternative hypothesis (Ha), and then collecting data to assess the evidence. An example: testing if a new drug improves patient recovery (Ha) compared to the standard treatment (H0) based on collected patient data.

2. What is H0 and H1 in statistics?

In statistics, H0​ and H1​ represent the null and alternative hypotheses. The null hypothesis, H0​, is the default assumption that no effect or difference exists between groups or conditions. The alternative hypothesis, H1​, is the competing claim suggesting an effect or a difference. Statistical tests determine whether to reject the null hypothesis in favor of the alternative hypothesis based on the data.

3. What is a simple hypothesis with an example?

A simple hypothesis is a specific statement predicting a single relationship between two variables. It posits a direct and uncomplicated outcome. For example, a simple hypothesis might state, "Increased sunlight exposure increases the growth rate of sunflowers." Here, the hypothesis suggests a direct relationship between the amount of sunlight (independent variable) and the growth rate of sunflowers (dependent variable), with no additional variables considered.

4. What are the 3 major types of hypothesis?

The three major types of hypotheses are:

• Null Hypothesis (H0): Represents the default assumption, stating that there is no significant effect or relationship in the data.
• Alternative Hypothesis (Ha): Contradicts the null hypothesis and proposes a specific effect or relationship that researchers want to investigate.
• Nondirectional Hypothesis: An alternative hypothesis that doesn't specify the direction of the effect, leaving it open for both positive and negative possibilities.

5. What software tools can assist with hypothesis testing?

Several software tools offering distinct features can help with hypothesis testing. R and RStudio are popular for their advanced statistical capabilities. The Python ecosystem, including libraries like SciPy and Statsmodels, also supports hypothesis testing. SAS and SPSS are well-established tools for comprehensive statistical analysis. For basic testing, Excel offers simple built-in functions.

6. How do I interpret the results of a hypothesis test?

Interpreting hypothesis test results involves comparing the p-value to the significance level (alpha). If the p-value is less than or equal to alpha, you can reject the null hypothesis, indicating statistical significance. This suggests that the observed effect is unlikely to have occurred by chance, validating your analysis findings.

7. Why is sample size important in hypothesis testing?

Sample size is crucial in hypothesis testing as it affects the test’s power. A larger sample size increases the likelihood of detecting a true effect, reducing the risk of Type II errors. Conversely, a small sample may lack the statistical power needed to identify differences, potentially leading to inaccurate conclusions.

8. Can hypothesis testing be used for non-numerical data?

Yes, hypothesis testing can be applied to non-numerical data through non-parametric tests. These tests are ideal when data doesn't meet parametric assumptions or when dealing with categorical data. Non-parametric tests, like the Chi-square or Mann-Whitney U test, provide robust methods for analyzing non-numerical data and drawing meaningful conclusions.

9. How do I choose the proper hypothesis test?

Selecting the right hypothesis test depends on several factors: the objective of your analysis, the type of data (numerical or categorical), and the sample size. Consider whether you're comparing means, proportions, or associations, and whether your data follows a normal distribution. The correct choice ensures accurate results tailored to your research question.

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About the Author

Avijeet is a Senior Research Analyst at Simplilearn. Passionate about Data Analytics, Machine Learning, and Deep Learning, Avijeet is also interested in politics, cricket, and football.

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S.3 hypothesis testing.

In reviewing hypothesis tests, we start first with the general idea. Then, we keep returning to the basic procedures of hypothesis testing, each time adding a little more detail.

The general idea of hypothesis testing involves:

• Making an initial assumption.
• Collecting evidence (data).
• Based on the available evidence (data), deciding whether to reject or not reject the initial assumption.

Every hypothesis test — regardless of the population parameter involved — requires the above three steps.

Example S.3.1

Is normal body temperature really 98.6 degrees f section  .

Consider the population of many, many adults. A researcher hypothesized that the average adult body temperature is lower than the often-advertised 98.6 degrees F. That is, the researcher wants an answer to the question: "Is the average adult body temperature 98.6 degrees? Or is it lower?" To answer his research question, the researcher starts by assuming that the average adult body temperature was 98.6 degrees F.

Then, the researcher went out and tried to find evidence that refutes his initial assumption. In doing so, he selects a random sample of 130 adults. The average body temperature of the 130 sampled adults is 98.25 degrees.

Then, the researcher uses the data he collected to make a decision about his initial assumption. It is either likely or unlikely that the researcher would collect the evidence he did given his initial assumption that the average adult body temperature is 98.6 degrees:

• If it is likely , then the researcher does not reject his initial assumption that the average adult body temperature is 98.6 degrees. There is not enough evidence to do otherwise.
• either the researcher's initial assumption is correct and he experienced a very unusual event;
• or the researcher's initial assumption is incorrect.

In statistics, we generally don't make claims that require us to believe that a very unusual event happened. That is, in the practice of statistics, if the evidence (data) we collected is unlikely in light of the initial assumption, then we reject our initial assumption.

Example S.3.2

Criminal trial analogy section  .

One place where you can consistently see the general idea of hypothesis testing in action is in criminal trials held in the United States. Our criminal justice system assumes "the defendant is innocent until proven guilty." That is, our initial assumption is that the defendant is innocent.

In the practice of statistics, we make our initial assumption when we state our two competing hypotheses -- the null hypothesis ( H 0 ) and the alternative hypothesis ( H A ). Here, our hypotheses are:

• H 0 : Defendant is not guilty (innocent)
• H A : Defendant is guilty

In statistics, we always assume the null hypothesis is true . That is, the null hypothesis is always our initial assumption.

The prosecution team then collects evidence — such as finger prints, blood spots, hair samples, carpet fibers, shoe prints, ransom notes, and handwriting samples — with the hopes of finding "sufficient evidence" to make the assumption of innocence refutable.

In statistics, the data are the evidence.

The jury then makes a decision based on the available evidence:

• If the jury finds sufficient evidence — beyond a reasonable doubt — to make the assumption of innocence refutable, the jury rejects the null hypothesis and deems the defendant guilty. We behave as if the defendant is guilty.
• If there is insufficient evidence, then the jury does not reject the null hypothesis . We behave as if the defendant is innocent.

In statistics, we always make one of two decisions. We either "reject the null hypothesis" or we "fail to reject the null hypothesis."

Errors in Hypothesis Testing Section  

Did you notice the use of the phrase "behave as if" in the previous discussion? We "behave as if" the defendant is guilty; we do not "prove" that the defendant is guilty. And, we "behave as if" the defendant is innocent; we do not "prove" that the defendant is innocent.

This is a very important distinction! We make our decision based on evidence not on 100% guaranteed proof. Again:

• If we reject the null hypothesis, we do not prove that the alternative hypothesis is true.
• If we do not reject the null hypothesis, we do not prove that the null hypothesis is true.

We merely state that there is enough evidence to behave one way or the other. This is always true in statistics! Because of this, whatever the decision, there is always a chance that we made an error .

Let's review the two types of errors that can be made in criminal trials:

Table S.3.2 shows how this corresponds to the two types of errors in hypothesis testing.

Note that, in statistics, we call the two types of errors by two different  names -- one is called a "Type I error," and the other is called  a "Type II error." Here are the formal definitions of the two types of errors:

There is always a chance of making one of these errors. But, a good scientific study will minimize the chance of doing so!

Making the Decision Section  

Recall that it is either likely or unlikely that we would observe the evidence we did given our initial assumption. If it is likely , we do not reject the null hypothesis. If it is unlikely , then we reject the null hypothesis in favor of the alternative hypothesis. Effectively, then, making the decision reduces to determining "likely" or "unlikely."

In statistics, there are two ways to determine whether the evidence is likely or unlikely given the initial assumption:

• We could take the " critical value approach " (favored in many of the older textbooks).
• Or, we could take the " P -value approach " (what is used most often in research, journal articles, and statistical software).

In the next two sections, we review the procedures behind each of these two approaches. To make our review concrete, let's imagine that μ is the average grade point average of all American students who major in mathematics. We first review the critical value approach for conducting each of the following three hypothesis tests about the population mean $\mu$:

In Practice

• We would want to conduct the first hypothesis test if we were interested in concluding that the average grade point average of the group is more than 3.
• We would want to conduct the second hypothesis test if we were interested in concluding that the average grade point average of the group is less than 3.
• And, we would want to conduct the third hypothesis test if we were only interested in concluding that the average grade point average of the group differs from 3 (without caring whether it is more or less than 3).

Upon completing the review of the critical value approach, we review the P -value approach for conducting each of the above three hypothesis tests about the population mean \(\mu\). The procedures that we review here for both approaches easily extend to hypothesis tests about any other population parameter.

13 Different Types of Hypothesis

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There are 13 different types of hypothesis. These include simple, complex, null, alternative, composite, directional, non-directional, logical, empirical, statistical, associative, exact, and inexact.

A hypothesis can be categorized into one or more of these types. However, some are mutually exclusive and opposites. Simple and complex hypotheses are mutually exclusive, as are direction and non-direction, and null and alternative hypotheses.

Below I explain each hypothesis in simple terms for absolute beginners. These definitions may be too simple for some, but they’re designed to be clear introductions to the terms to help people wrap their heads around the concepts early on in their education about research methods .

Types of Hypothesis

Before you Proceed: Dependent vs Independent Variables

A research study and its hypotheses generally examine the relationships between independent and dependent variables – so you need to know these two concepts:

• The independent variable is the variable that is causing a change.
• The dependent variable is the variable the is affected by the change. This is the variable being tested.

Read my full article on dependent vs independent variables for more examples.

Example: Eating carrots (independent variable) improves eyesight (dependent variable).

1. Simple Hypothesis

A simple hypothesis is a hypothesis that predicts a correlation between two test variables: an independent and a dependent variable.

This is the easiest and most straightforward type of hypothesis. You simply need to state an expected correlation between the dependant variable and the independent variable.

You do not need to predict causation (see: directional hypothesis). All you would need to do is prove that the two variables are linked.

Simple Hypothesis Examples

2. Complex Hypothesis

A complex hypothesis is a hypothesis that contains multiple variables, making the hypothesis more specific but also harder to prove.

You can have multiple independent and dependant variables in this hypothesis.

Complex Hypothesis Example

In the above example, we have multiple independent and dependent variables:

• Independent variables: Age and weight.
• Dependent variables: diabetes and heart disease.

Because there are multiple variables, this study is a lot more complex than a simple hypothesis. It quickly gets much more difficult to prove these hypotheses. This is why undergraduate and first-time researchers are usually encouraged to use simple hypotheses.

3. Null Hypothesis

A null hypothesis will predict that there will be no significant relationship between the two test variables.

For example, you can say that “The study will show that there is no correlation between marriage and happiness.”

A good way to think about a null hypothesis is to think of it in the same way as “innocent until proven guilty”[1]. Unless you can come up with evidence otherwise, your null hypothesis will stand.

A null hypothesis may also highlight that a correlation will be inconclusive . This means that you can predict that the study will not be able to confirm your results one way or the other. For example, you can say “It is predicted that the study will be unable to confirm a correlation between the two variables due to foreseeable interference by a third variable .”

Beware that an inconclusive null hypothesis may be questioned by your teacher. Why would you conduct a test that you predict will not provide a clear result? Perhaps you should take a closer look at your methodology and re-examine it. Nevertheless, inconclusive null hypotheses can sometimes have merit.

Null Hypothesis Examples

4. Alternative Hypothesis

An alternative hypothesis is a hypothesis that is anything other than the null hypothesis. It will disprove the null hypothesis.

We use the symbol H A or H 1 to denote an alternative hypothesis.

The null and alternative hypotheses are usually used together. We will say the null hypothesis is the case where a relationship between two variables is non-existent. The alternative hypothesis is the case where there is a relationship between those two variables.

The following statement is always true: H 0 ≠ H A .

Let’s take the example of the hypothesis: “Does eating oatmeal before an exam impact test scores?”

We can have two hypotheses here:

• Null hypothesis (H 0 ): “Eating oatmeal before an exam does not impact test scores.”
• Alternative hypothesis (H A ): “Eating oatmeal before an exam does impact test scores.”

For the alternative hypothesis to be true, all we have to do is disprove the null hypothesis for the alternative hypothesis to be true. We do not need an exact prediction of how much oatmeal will impact the test scores or even if the impact is positive or negative. So long as the null hypothesis is proven to be false, then the alternative hypothesis is proven to be true.

5. Composite Hypothesis

A composite hypothesis is a hypothesis that does not predict the exact parameters, distribution, or range of the dependent variable.

Often, we would predict an exact outcome. For example: “23 year old men are on average 189cm tall.” Here, we are giving an exact parameter. So, the hypothesis is not composite.

But, often, we cannot exactly hypothesize something. We assume that something will happen, but we’re not exactly sure what. In these cases, we might say: “23 year old men are not on average 189cm tall.”

We haven’t set a distribution range or exact parameters of the average height of 23 year old men. So, we’ve introduced a composite hypothesis as opposed to an exact hypothesis.

Generally, an alternative hypothesis (discussed above) is composite because it is defined as anything except the null hypothesis. This ‘anything except’ does not define parameters or distribution, and therefore it’s an example of a composite hypothesis.

6. Directional Hypothesis

A directional hypothesis makes a prediction about the positivity or negativity of the effect of an intervention prior to the test being conducted.

Instead of being agnostic about whether the effect will be positive or negative, it nominates the effect’s directionality.

We often call this a one-tailed hypothesis (in contrast to a two-tailed or non-directional hypothesis) because, looking at a distribution graph, we’re hypothesizing that the results will lean toward one particular tail on the graph – either the positive or negative.

Directional Hypothesis Examples

7. Non-Directional Hypothesis

A non-directional hypothesis does not specify the predicted direction (e.g. positivity or negativity) of the effect of the independent variable on the dependent variable.

These hypotheses predict an effect, but stop short of saying what that effect will be.

A non-directional hypothesis is similar to composite and alternative hypotheses. All three types of hypothesis tend to make predictions without defining a direction. In a composite hypothesis, a specific prediction is not made (although a general direction may be indicated, so the overlap is not complete). For an alternative hypothesis, you often predict that the even will be anything but the null hypothesis, which means it could be more or less than H 0 (or in other words, non-directional).

Let’s turn the above directional hypotheses into non-directional hypotheses.

Non-Directional Hypothesis Examples

8. Logical Hypothesis

A logical hypothesis is a hypothesis that cannot be tested, but has some logical basis underpinning our assumptions.

These are most commonly used in philosophy because philosophical questions are often untestable and therefore we must rely on our logic to formulate logical theories.

Usually, we would want to turn a logical hypothesis into an empirical one through testing if we got the chance. Unfortunately, we don’t always have this opportunity because the test is too complex, expensive, or simply unrealistic.

Here are some examples:

• Before the 1980s, it was hypothesized that the Titanic came to its resting place at 41° N and 49° W, based on the time the ship sank and the ship’s presumed path across the Atlantic Ocean. However, due to the depth of the ocean, it was impossible to test. Thus, the hypothesis was simply a logical hypothesis.
• Dinosaurs closely related to Aligators probably had green scales because Aligators have green scales. However, as they are all extinct, we can only rely on logic and not empirical data.

9. Empirical Hypothesis

An empirical hypothesis is the opposite of a logical hypothesis. It is a hypothesis that is currently being tested using scientific analysis. We can also call this a ‘working hypothesis’.

We can to separate research into two types: theoretical and empirical. Theoretical research relies on logic and thought experiments. Empirical research relies on tests that can be verified by observation and measurement.

So, an empirical hypothesis is a hypothesis that can and will be tested.

• Raising the wage of restaurant servers increases staff retention.
• Adding 1 lb of corn per day to cows’ diets decreases their lifespan.
• Mushrooms grow faster at 22 degrees Celsius than 27 degrees Celsius.

Each of the above hypotheses can be tested, making them empirical rather than just logical (aka theoretical).

10. Statistical Hypothesis

A statistical hypothesis utilizes representative statistical models to draw conclusions about broader populations.

It requires the use of datasets or carefully selected representative samples so that statistical inference can be drawn across a larger dataset.

This type of research is necessary when it is impossible to assess every single possible case. Imagine, for example, if you wanted to determine if men are taller than women. You would be unable to measure the height of every man and woman on the planet. But, by conducting sufficient random samples, you would be able to predict with high probability that the results of your study would remain stable across the whole population.

You would be right in guessing that almost all quantitative research studies conducted in academic settings today involve statistical hypotheses.

Statistical Hypothesis Examples

• Human Sex Ratio. The most famous statistical hypothesis example is that of John Arbuthnot’s sex at birth case study in 1710. Arbuthnot used birth data to determine with high statistical probability that there are more male births than female births. He called this divine providence, and to this day, his findings remain true: more men are born than women.
• Lady Testing Tea. A 1935 study by Ronald Fisher involved testing a woman who believed she could tell whether milk was added before or after water to a cup of tea. Fisher gave her 4 cups in which one randomly had milk placed before the tea. He repeated the test 8 times. The lady was correct each time. Fisher found that she had a 1 in 70 chance of getting all 8 test correct, which is a statistically significant result.

11. Associative Hypothesis

An associative hypothesis predicts that two variables are linked but does not explore whether one variable directly impacts upon the other variable.

We commonly refer to this as “ correlation does not mean causation ”. Just because there are a lot of sick people in a hospital, it doesn’t mean that the hospital made the people sick. There is something going on there that’s causing the issue (sick people are flocking to the hospital).

So, in an associative hypothesis, you note correlation between an independent and dependent variable but do not make a prediction about how the two interact. You stop short of saying one thing causes another thing.

Associative Hypothesis Examples

• Sick people in hospital. You could conduct a study hypothesizing that hospitals have more sick people in them than other institutions in society. However, you don’t hypothesize that the hospitals caused the sickness.
• Lice make you healthy. In the Middle Ages, it was observed that sick people didn’t tend to have lice in their hair. The inaccurate conclusion was that lice was not only a sign of health, but that they made people healthy. In reality, there was an association here, but not causation. The fact was that lice were sensitive to body temperature and fled bodies that had fevers.

12. Causal Hypothesis

A causal hypothesis predicts that two variables are not only associated, but that changes in one variable will cause changes in another.

A causal hypothesis is harder to prove than an associative hypothesis because the cause needs to be definitively proven. This will often require repeating tests in controlled environments with the researchers making manipulations to the independent variable, or the use of control groups and placebo effects .

If we were to take the above example of lice in the hair of sick people, researchers would have to put lice in sick people’s hair and see if it made those people healthier. Researchers would likely observe that the lice would flee the hair, but the sickness would remain, leading to a finding of association but not causation.

Causal Hypothesis Examples

13. Exact vs. Inexact Hypothesis

For brevity’s sake, I have paired these two hypotheses into the one point. The reality is that we’ve already seen both of these types of hypotheses at play already.

An exact hypothesis (also known as a point hypothesis) specifies a specific prediction whereas an inexact hypothesis assumes a range of possible values without giving an exact outcome. As Helwig [2] argues:

“An “exact” hypothesis specifies the exact value(s) of the parameter(s) of interest, whereas an “inexact” hypothesis specifies a range of possible values for the parameter(s) of interest.”

Generally, a null hypothesis is an exact hypothesis whereas alternative, composite, directional, and non-directional hypotheses are all inexact.

See Next: 15 Hypothesis Examples

This is introductory information that is basic and indeed quite simplified for absolute beginners. It’s worth doing further independent research to get deeper knowledge of research methods and how to conduct an effective research study. And if you’re in education studies, don’t miss out on my list of the best education studies dissertation ideas .

[1] https://jnnp.bmj.com/content/91/6/571.abstract

[2] http://users.stat.umn.edu/~helwig/notes/SignificanceTesting.pdf

• Chris Drew (PhD) https://helpfulprofessor.com/author/chris-drew-phd-2/ 10 Reasons you’re Perpetually Single
• Chris Drew (PhD) https://helpfulprofessor.com/author/chris-drew-phd-2/ 20 Montessori Toddler Bedrooms (Design Inspiration)
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2 thoughts on “13 Different Types of Hypothesis”

Wow! This introductionary materials are very helpful. I teach the begginers in research for the first time in my career. The given tips and materials are very helpful. Chris, thank you so much! Excellent materials!

You’re more than welcome! If you want a pdf version of this article to provide for your students to use as a weekly reading on in-class discussion prompt for seminars, just drop me an email in the Contact form and I’ll get one sent out to you.

When I’ve taught this seminar, I’ve put my students into groups, cut these definitions into strips, and handed them out to the groups. Then I get them to try to come up with hypotheses that fit into each ‘type’. You can either just rotate hypothesis types so they get a chance at creating a hypothesis of each type, or get them to “teach” their hypothesis type and examples to the class at the end of the seminar.

Cheers, Chris

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• Indian J Crit Care Med
• v.23(Suppl 3); 2019 Sep

An Introduction to Statistics: Understanding Hypothesis Testing and Statistical Errors

Priya ranganathan.

1 Department of Anesthesiology, Critical Care and Pain, Tata Memorial Hospital, Mumbai, Maharashtra, India

2 Department of Surgical Oncology, Tata Memorial Centre, Mumbai, Maharashtra, India

The second article in this series on biostatistics covers the concepts of sample, population, research hypotheses and statistical errors.

How to cite this article

Ranganathan P, Pramesh CS. An Introduction to Statistics: Understanding Hypothesis Testing and Statistical Errors. Indian J Crit Care Med 2019;23(Suppl 3):S230–S231.

Two papers quoted in this issue of the Indian Journal of Critical Care Medicine report. The results of studies aim to prove that a new intervention is better than (superior to) an existing treatment. In the ABLE study, the investigators wanted to show that transfusion of fresh red blood cells would be superior to standard-issue red cells in reducing 90-day mortality in ICU patients. 1 The PROPPR study was designed to prove that transfusion of a lower ratio of plasma and platelets to red cells would be superior to a higher ratio in decreasing 24-hour and 30-day mortality in critically ill patients. 2 These studies are known as superiority studies (as opposed to noninferiority or equivalence studies which will be discussed in a subsequent article).

SAMPLE VERSUS POPULATION

A sample represents a group of participants selected from the entire population. Since studies cannot be carried out on entire populations, researchers choose samples, which are representative of the population. This is similar to walking into a grocery store and examining a few grains of rice or wheat before purchasing an entire bag; we assume that the few grains that we select (the sample) are representative of the entire sack of grains (the population).

The results of the study are then extrapolated to generate inferences about the population. We do this using a process known as hypothesis testing. This means that the results of the study may not always be identical to the results we would expect to find in the population; i.e., there is the possibility that the study results may be erroneous.

HYPOTHESIS TESTING

A clinical trial begins with an assumption or belief, and then proceeds to either prove or disprove this assumption. In statistical terms, this belief or assumption is known as a hypothesis. Counterintuitively, what the researcher believes in (or is trying to prove) is called the “alternate” hypothesis, and the opposite is called the “null” hypothesis; every study has a null hypothesis and an alternate hypothesis. For superiority studies, the alternate hypothesis states that one treatment (usually the new or experimental treatment) is superior to the other; the null hypothesis states that there is no difference between the treatments (the treatments are equal). For example, in the ABLE study, we start by stating the null hypothesis—there is no difference in mortality between groups receiving fresh RBCs and standard-issue RBCs. We then state the alternate hypothesis—There is a difference between groups receiving fresh RBCs and standard-issue RBCs. It is important to note that we have stated that the groups are different, without specifying which group will be better than the other. This is known as a two-tailed hypothesis and it allows us to test for superiority on either side (using a two-sided test). This is because, when we start a study, we are not 100% certain that the new treatment can only be better than the standard treatment—it could be worse, and if it is so, the study should pick it up as well. One tailed hypothesis and one-sided statistical testing is done for non-inferiority studies, which will be discussed in a subsequent paper in this series.

STATISTICAL ERRORS

There are two possibilities to consider when interpreting the results of a superiority study. The first possibility is that there is truly no difference between the treatments but the study finds that they are different. This is called a Type-1 error or false-positive error or alpha error. This means falsely rejecting the null hypothesis.

The second possibility is that there is a difference between the treatments and the study does not pick up this difference. This is called a Type 2 error or false-negative error or beta error. This means falsely accepting the null hypothesis.

The power of the study is the ability to detect a difference between groups and is the converse of the beta error; i.e., power = 1-beta error. Alpha and beta errors are finalized when the protocol is written and form the basis for sample size calculation for the study. In an ideal world, we would not like any error in the results of our study; however, we would need to do the study in the entire population (infinite sample size) to be able to get a 0% alpha and beta error. These two errors enable us to do studies with realistic sample sizes, with the compromise that there is a small possibility that the results may not always reflect the truth. The basis for this will be discussed in a subsequent paper in this series dealing with sample size calculation.

Conventionally, type 1 or alpha error is set at 5%. This means, that at the end of the study, if there is a difference between groups, we want to be 95% certain that this is a true difference and allow only a 5% probability that this difference has occurred by chance (false positive). Type 2 or beta error is usually set between 10% and 20%; therefore, the power of the study is 90% or 80%. This means that if there is a difference between groups, we want to be 80% (or 90%) certain that the study will detect that difference. For example, in the ABLE study, sample size was calculated with a type 1 error of 5% (two-sided) and power of 90% (type 2 error of 10%) (1).

Table 1 gives a summary of the two types of statistical errors with an example

Statistical errors

In the next article in this series, we will look at the meaning and interpretation of ‘ p ’ value and confidence intervals for hypothesis testing.

Source of support: Nil

Conflict of interest: None

9.1 Null and Alternative Hypotheses

The actual test begins by considering two hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

H 0 , the — null hypothesis: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.

H a —, the alternative hypothesis: a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 .

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are reject H 0 if the sample information favors the alternative hypothesis or do not reject H 0 or decline to reject H 0 if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in H 0 and H a :

H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

Example 9.1

H 0 : No more than 30 percent of the registered voters in Santa Clara County voted in the primary election. p ≤ 30 H a : More than 30 percent of the registered voters in Santa Clara County voted in the primary election. p > 30

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25 percent. State the null and alternative hypotheses.

Example 9.2

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are the following: H 0 : μ = 2.0 H a : μ ≠ 2.0

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : μ __ 66
• H a : μ __ 66

Example 9.3

We want to test if college students take fewer than five years to graduate from college, on the average. The null and alternative hypotheses are the following: H 0 : μ ≥ 5 H a : μ < 5

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : μ __ 45
• H a : μ __ 45

Example 9.4

An article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third of the students pass. The same article stated that 6.6 percent of U.S. students take advanced placement exams and 4.4 percent pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6 percent. State the null and alternative hypotheses. H 0 : p ≤ 0.066 H a : p > 0.066

On a state driver’s test, about 40 percent pass the test on the first try. We want to test if more than 40 percent pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : p __ 0.40
• H a : p __ 0.40

Collaborative Exercise

Bring to class a newspaper, some news magazines, and some internet articles. In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.

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What is Hypothesis Testing? Types and Methods

• Soumyaa Rawat
• Jul 23, 2021

Hypothesis Testing  

Hypothesis testing is the act of testing a hypothesis or a supposition in relation to a statistical parameter. Analysts implement hypothesis testing in order to test if a hypothesis is plausible or not. 

In data science and statistics , hypothesis testing is an important step as it involves the verification of an assumption that could help develop a statistical parameter. For instance, a researcher establishes a hypothesis assuming that the average of all odd numbers is an even number. 

In order to find the plausibility of this hypothesis, the researcher will have to test the hypothesis using hypothesis testing methods. Unlike a hypothesis that is ‘supposed’ to stand true on the basis of little or no evidence, hypothesis testing is required to have plausible evidence in order to establish that a statistical hypothesis is true. 

Perhaps this is where statistics play an important role. A number of components are involved in this process. But before understanding the process involved in hypothesis testing in research methodology, we shall first understand the types of hypotheses that are involved in the process. Let us get started! 

Types of Hypotheses

In data sampling, different types of hypothesis are involved in finding whether the tested samples test positive for a hypothesis or not. In this segment, we shall discover the different types of hypotheses and understand the role they play in hypothesis testing.

Alternative Hypothesis

Alternative Hypothesis (H1) or the research hypothesis states that there is a relationship between two variables (where one variable affects the other). The alternative hypothesis is the main driving force for hypothesis testing. 

It implies that the two variables are related to each other and the relationship that exists between them is not due to chance or coincidence. 

When the process of hypothesis testing is carried out, the alternative hypothesis is the main subject of the testing process. The analyst intends to test the alternative hypothesis and verifies its plausibility.

Null Hypothesis

The Null Hypothesis (H0) aims to nullify the alternative hypothesis by implying that there exists no relation between two variables in statistics. It states that the effect of one variable on the other is solely due to chance and no empirical cause lies behind it. 

The null hypothesis is established alongside the alternative hypothesis and is recognized as important as the latter. In hypothesis testing, the null hypothesis has a major role to play as it influences the testing against the alternative hypothesis. 

(Must read: What is ANOVA test? )

Non-Directional Hypothesis

The Non-directional hypothesis states that the relation between two variables has no direction. 

Simply put, it asserts that there exists a relation between two variables, but does not recognize the direction of effect, whether variable A affects variable B or vice versa. 

Directional Hypothesis

The Directional hypothesis, on the other hand, asserts the direction of effect of the relationship that exists between two variables. 

Herein, the hypothesis clearly states that variable A affects variable B, or vice versa. 

Statistical Hypothesis

A statistical hypothesis is a hypothesis that can be verified to be plausible on the basis of statistics. 

By using data sampling and statistical knowledge, one can determine the plausibility of a statistical hypothesis and find out if it stands true or not. 

(Related blog: z-test vs t-test )

Performing Hypothesis Testing  

Now that we have understood the types of hypotheses and the role they play in hypothesis testing, let us now move on to understand the process in a better manner. 

In hypothesis testing, a researcher is first required to establish two hypotheses - alternative hypothesis and null hypothesis in order to begin with the procedure. 

To establish these two hypotheses, one is required to study data samples, find a plausible pattern among the samples, and pen down a statistical hypothesis that they wish to test. 

A random population of samples can be drawn, to begin with hypothesis testing. Among the two hypotheses, alternative and null, only one can be verified to be true. Perhaps the presence of both hypotheses is required to make the process successful. 

At the end of the hypothesis testing procedure, either of the hypotheses will be rejected and the other one will be supported. Even though one of the two hypotheses turns out to be true, no hypothesis can ever be verified 100%. 

(Read also: Types of data sampling techniques )

Therefore, a hypothesis can only be supported based on the statistical samples and verified data. Here is a step-by-step guide for hypothesis testing.

Establish the hypotheses

First things first, one is required to establish two hypotheses - alternative and null, that will set the foundation for hypothesis testing. 

These hypotheses initiate the testing process that involves the researcher working on data samples in order to either support the alternative hypothesis or the null hypothesis. 

Generate a testing plan

Once the hypotheses have been formulated, it is now time to generate a testing plan. A testing plan or an analysis plan involves the accumulation of data samples, determining which statistic is to be considered and laying out the sample size. 

All these factors are very important while one is working on hypothesis testing.

Analyze data samples

As soon as a testing plan is ready, it is time to move on to the analysis part. Analysis of data samples involves configuring statistical values of samples, drawing them together, and deriving a pattern out of these samples. 

While analyzing the data samples, a researcher needs to determine a set of things -

Significance Level - The level of significance in hypothesis testing indicates if a statistical result could have significance if the null hypothesis stands to be true.

Testing Method - The testing method involves a type of sampling-distribution and a test statistic that leads to hypothesis testing. There are a number of testing methods that can assist in the analysis of data samples. 

Test statistic - Test statistic is a numerical summary of a data set that can be used to perform hypothesis testing.

P-value - The P-value interpretation is the probability of finding a sample statistic to be as extreme as the test statistic, indicating the plausibility of the null hypothesis. 

Infer the results

The analysis of data samples leads to the inference of results that establishes whether the alternative hypothesis stands true or not. When the P-value is less than the significance level, the null hypothesis is rejected and the alternative hypothesis turns out to be plausible. 

Methods of Hypothesis Testing

As we have already looked into different aspects of hypothesis testing, we shall now look into the different methods of hypothesis testing. All in all, there are 2 most common types of hypothesis testing methods. They are as follows -

Frequentist Hypothesis Testing

The frequentist hypothesis or the traditional approach to hypothesis testing is a hypothesis testing method that aims on making assumptions by considering current data. 

The supposed truths and assumptions are based on the current data and a set of 2 hypotheses are formulated. A very popular subtype of the frequentist approach is the Null Hypothesis Significance Testing (NHST). 

The NHST approach (involving the null and alternative hypothesis) has been one of the most sought-after methods of hypothesis testing in the field of statistics ever since its inception in the mid-1950s. 

Bayesian Hypothesis Testing

A much unconventional and modern method of hypothesis testing, the Bayesian Hypothesis Testing claims to test a particular hypothesis in accordance with the past data samples, known as prior probability, and current data that lead to the plausibility of a hypothesis. 

The result obtained indicates the posterior probability of the hypothesis. In this method, the researcher relies on ‘prior probability and posterior probability’ to conduct hypothesis testing on hand. 

On the basis of this prior probability, the Bayesian approach tests a hypothesis to be true or false. The Bayes factor, a major component of this method, indicates the likelihood ratio among the null hypothesis and the alternative hypothesis. 

The Bayes factor is the indicator of the plausibility of either of the two hypotheses that are established for hypothesis testing.  

(Also read - Introduction to Bayesian Statistics ) 

To conclude, hypothesis testing, a way to verify the plausibility of a supposed assumption can be done through different methods - the Bayesian approach or the Frequentist approach. 

Although the Bayesian approach relies on the prior probability of data samples, the frequentist approach assumes without a probability. A number of elements involved in hypothesis testing are - significance level, p-level, test statistic, and method of hypothesis testing. 

(Also read: Introduction to probability distributions )

A significant way to determine whether a hypothesis stands true or not is to verify the data samples and identify the plausible hypothesis among the null hypothesis and alternative hypothesis. 

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Home » What is a Hypothesis – Types, Examples and Writing Guide

What is a Hypothesis – Types, Examples and Writing Guide

Table of Contents

Definition:

Hypothesis is an educated guess or proposed explanation for a phenomenon, based on some initial observations or data. It is a tentative statement that can be tested and potentially proven or disproven through further investigation and experimentation.

Hypothesis is often used in scientific research to guide the design of experiments and the collection and analysis of data. It is an essential element of the scientific method, as it allows researchers to make predictions about the outcome of their experiments and to test those predictions to determine their accuracy.

Types of Hypothesis

Types of Hypothesis are as follows:

Research Hypothesis

A research hypothesis is a statement that predicts a relationship between variables. It is usually formulated as a specific statement that can be tested through research, and it is often used in scientific research to guide the design of experiments.

Null Hypothesis

The null hypothesis is a statement that assumes there is no significant difference or relationship between variables. It is often used as a starting point for testing the research hypothesis, and if the results of the study reject the null hypothesis, it suggests that there is a significant difference or relationship between variables.

Alternative Hypothesis

An alternative hypothesis is a statement that assumes there is a significant difference or relationship between variables. It is often used as an alternative to the null hypothesis and is tested against the null hypothesis to determine which statement is more accurate.

Directional Hypothesis

A directional hypothesis is a statement that predicts the direction of the relationship between variables. For example, a researcher might predict that increasing the amount of exercise will result in a decrease in body weight.

Non-directional Hypothesis

A non-directional hypothesis is a statement that predicts the relationship between variables but does not specify the direction. For example, a researcher might predict that there is a relationship between the amount of exercise and body weight, but they do not specify whether increasing or decreasing exercise will affect body weight.

Statistical Hypothesis

A statistical hypothesis is a statement that assumes a particular statistical model or distribution for the data. It is often used in statistical analysis to test the significance of a particular result.

Composite Hypothesis

A composite hypothesis is a statement that assumes more than one condition or outcome. It can be divided into several sub-hypotheses, each of which represents a different possible outcome.

Empirical Hypothesis

An empirical hypothesis is a statement that is based on observed phenomena or data. It is often used in scientific research to develop theories or models that explain the observed phenomena.

Simple Hypothesis

A simple hypothesis is a statement that assumes only one outcome or condition. It is often used in scientific research to test a single variable or factor.

Complex Hypothesis

A complex hypothesis is a statement that assumes multiple outcomes or conditions. It is often used in scientific research to test the effects of multiple variables or factors on a particular outcome.

Applications of Hypothesis

Hypotheses are used in various fields to guide research and make predictions about the outcomes of experiments or observations. Here are some examples of how hypotheses are applied in different fields:

• Science : In scientific research, hypotheses are used to test the validity of theories and models that explain natural phenomena. For example, a hypothesis might be formulated to test the effects of a particular variable on a natural system, such as the effects of climate change on an ecosystem.
• Medicine : In medical research, hypotheses are used to test the effectiveness of treatments and therapies for specific conditions. For example, a hypothesis might be formulated to test the effects of a new drug on a particular disease.
• Psychology : In psychology, hypotheses are used to test theories and models of human behavior and cognition. For example, a hypothesis might be formulated to test the effects of a particular stimulus on the brain or behavior.
• Sociology : In sociology, hypotheses are used to test theories and models of social phenomena, such as the effects of social structures or institutions on human behavior. For example, a hypothesis might be formulated to test the effects of income inequality on crime rates.
• Business : In business research, hypotheses are used to test the validity of theories and models that explain business phenomena, such as consumer behavior or market trends. For example, a hypothesis might be formulated to test the effects of a new marketing campaign on consumer buying behavior.
• Engineering : In engineering, hypotheses are used to test the effectiveness of new technologies or designs. For example, a hypothesis might be formulated to test the efficiency of a new solar panel design.

How to write a Hypothesis

Here are the steps to follow when writing a hypothesis:

Identify the Research Question

The first step is to identify the research question that you want to answer through your study. This question should be clear, specific, and focused. It should be something that can be investigated empirically and that has some relevance or significance in the field.

Conduct a Literature Review

Before writing your hypothesis, it’s essential to conduct a thorough literature review to understand what is already known about the topic. This will help you to identify the research gap and formulate a hypothesis that builds on existing knowledge.

Determine the Variables

The next step is to identify the variables involved in the research question. A variable is any characteristic or factor that can vary or change. There are two types of variables: independent and dependent. The independent variable is the one that is manipulated or changed by the researcher, while the dependent variable is the one that is measured or observed as a result of the independent variable.

Formulate the Hypothesis

Based on the research question and the variables involved, you can now formulate your hypothesis. A hypothesis should be a clear and concise statement that predicts the relationship between the variables. It should be testable through empirical research and based on existing theory or evidence.

Write the Null Hypothesis

The null hypothesis is the opposite of the alternative hypothesis, which is the hypothesis that you are testing. The null hypothesis states that there is no significant difference or relationship between the variables. It is important to write the null hypothesis because it allows you to compare your results with what would be expected by chance.

Refine the Hypothesis

After formulating the hypothesis, it’s important to refine it and make it more precise. This may involve clarifying the variables, specifying the direction of the relationship, or making the hypothesis more testable.

Examples of Hypothesis

Here are a few examples of hypotheses in different fields:

• Psychology : “Increased exposure to violent video games leads to increased aggressive behavior in adolescents.”
• Biology : “Higher levels of carbon dioxide in the atmosphere will lead to increased plant growth.”
• Sociology : “Individuals who grow up in households with higher socioeconomic status will have higher levels of education and income as adults.”
• Education : “Implementing a new teaching method will result in higher student achievement scores.”
• Marketing : “Customers who receive a personalized email will be more likely to make a purchase than those who receive a generic email.”
• Physics : “An increase in temperature will cause an increase in the volume of a gas, assuming all other variables remain constant.”
• Medicine : “Consuming a diet high in saturated fats will increase the risk of developing heart disease.”

Purpose of Hypothesis

The purpose of a hypothesis is to provide a testable explanation for an observed phenomenon or a prediction of a future outcome based on existing knowledge or theories. A hypothesis is an essential part of the scientific method and helps to guide the research process by providing a clear focus for investigation. It enables scientists to design experiments or studies to gather evidence and data that can support or refute the proposed explanation or prediction.

The formulation of a hypothesis is based on existing knowledge, observations, and theories, and it should be specific, testable, and falsifiable. A specific hypothesis helps to define the research question, which is important in the research process as it guides the selection of an appropriate research design and methodology. Testability of the hypothesis means that it can be proven or disproven through empirical data collection and analysis. Falsifiability means that the hypothesis should be formulated in such a way that it can be proven wrong if it is incorrect.

In addition to guiding the research process, the testing of hypotheses can lead to new discoveries and advancements in scientific knowledge. When a hypothesis is supported by the data, it can be used to develop new theories or models to explain the observed phenomenon. When a hypothesis is not supported by the data, it can help to refine existing theories or prompt the development of new hypotheses to explain the phenomenon.

When to use Hypothesis

Here are some common situations in which hypotheses are used:

• In scientific research , hypotheses are used to guide the design of experiments and to help researchers make predictions about the outcomes of those experiments.
• In social science research , hypotheses are used to test theories about human behavior, social relationships, and other phenomena.
• I n business , hypotheses can be used to guide decisions about marketing, product development, and other areas. For example, a hypothesis might be that a new product will sell well in a particular market, and this hypothesis can be tested through market research.

Characteristics of Hypothesis

Here are some common characteristics of a hypothesis:

• Testable : A hypothesis must be able to be tested through observation or experimentation. This means that it must be possible to collect data that will either support or refute the hypothesis.
• Falsifiable : A hypothesis must be able to be proven false if it is not supported by the data. If a hypothesis cannot be falsified, then it is not a scientific hypothesis.
• Clear and concise : A hypothesis should be stated in a clear and concise manner so that it can be easily understood and tested.
• Based on existing knowledge : A hypothesis should be based on existing knowledge and research in the field. It should not be based on personal beliefs or opinions.
• Specific : A hypothesis should be specific in terms of the variables being tested and the predicted outcome. This will help to ensure that the research is focused and well-designed.
• Tentative: A hypothesis is a tentative statement or assumption that requires further testing and evidence to be confirmed or refuted. It is not a final conclusion or assertion.
• Relevant : A hypothesis should be relevant to the research question or problem being studied. It should address a gap in knowledge or provide a new perspective on the issue.

Advantages of Hypothesis

Hypotheses have several advantages in scientific research and experimentation:

• Guides research: A hypothesis provides a clear and specific direction for research. It helps to focus the research question, select appropriate methods and variables, and interpret the results.
• Predictive powe r: A hypothesis makes predictions about the outcome of research, which can be tested through experimentation. This allows researchers to evaluate the validity of the hypothesis and make new discoveries.
• Facilitates communication: A hypothesis provides a common language and framework for scientists to communicate with one another about their research. This helps to facilitate the exchange of ideas and promotes collaboration.
• Efficient use of resources: A hypothesis helps researchers to use their time, resources, and funding efficiently by directing them towards specific research questions and methods that are most likely to yield results.
• Provides a basis for further research: A hypothesis that is supported by data provides a basis for further research and exploration. It can lead to new hypotheses, theories, and discoveries.
• Increases objectivity: A hypothesis can help to increase objectivity in research by providing a clear and specific framework for testing and interpreting results. This can reduce bias and increase the reliability of research findings.

Limitations of Hypothesis

Some Limitations of the Hypothesis are as follows:

• Limited to observable phenomena: Hypotheses are limited to observable phenomena and cannot account for unobservable or intangible factors. This means that some research questions may not be amenable to hypothesis testing.
• May be inaccurate or incomplete: Hypotheses are based on existing knowledge and research, which may be incomplete or inaccurate. This can lead to flawed hypotheses and erroneous conclusions.
• May be biased: Hypotheses may be biased by the researcher’s own beliefs, values, or assumptions. This can lead to selective interpretation of data and a lack of objectivity in research.
• Cannot prove causation: A hypothesis can only show a correlation between variables, but it cannot prove causation. This requires further experimentation and analysis.
• Limited to specific contexts: Hypotheses are limited to specific contexts and may not be generalizable to other situations or populations. This means that results may not be applicable in other contexts or may require further testing.
• May be affected by chance : Hypotheses may be affected by chance or random variation, which can obscure or distort the true relationship between variables.

About the author

Muhammad Hassan

Researcher, Academic Writer, Web developer

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Hypothesis tests #

Formal hypothesis testing is perhaps the most prominent and widely-employed form of statistical analysis. It is sometimes seen as the most rigorous and definitive part of a statistical analysis, but it is also the source of many statistical controversies. The currently-prevalent approach to hypothesis testing dates to developments that took place between 1925 and 1940, especially the work of Ronald Fisher , Jerzy Neyman , and Egon Pearson .

In recent years, many prominent statisticians have argued that less emphasis should be placed on the formal hypothesis testing approaches developed in the early twentieth century, with a correspondingly greater emphasis on other forms of uncertainty analysis. Our goal here is to give an overview of some of the well-established and widely-used approaches for hypothesis testing. We will also provide some perspectives on how these tools can be effectively used, and discuss their limitations. We will also discuss some new approaches to hypothesis testing that may eventually come to be as prominent as these classical approaches.

A falsifiable hypothesis is a statement, or hypothesis, that can be contradicted with evidence. In empirical (data-driven) research, this evidence will always be obtained through the data. In statistical hypothesis testing, the hypothesis that we formally test is called the null hypothesis . The alternative hypothesis is a second hypothesis that is our proposed explanation for what happens if the null hypothesis is wrong.

Test statistics #

The key element of a statistical hypothesis test is the test statistic , which (like any statistic) is a function of the data. A test statistic takes our entire dataset, and reduces it to one number. This one number ideally should contain all the information in the data that is relevant for assessing the two hypotheses of interest, and exclude any aspects of the data that are irrelevant for assessing the two hypotheses. The test statistic measures evidence against the null hypothesis. Most test statistics are constructed so that a value of zero represents the lowest possible level of evidence against the null hypothesis. Test statistic values that deviate from zero represent greater levels of evidence against the null hypothesis. The larger the magnitude of the test statistic, the stronger the evidence against the null hypothesis.

A major theme of statistical research is to devise effective ways to construct test statistics. Many useful ways to do this have been devised, and there is no single approach that is always the best. In this introductory course, we will focus on tests that starting with an estimate of a quantity that is relevant for assessing the hypotheses, then proceed by standardizing this estimate by dividing it by its standard error. This approach is sometimes referred to as “Wald testing”, after Abraham Wald .

Testing the equality of two proportions #

As a basic example, let’s consider risk perception related to COVID-19. As you will see below, hypothesis testing can appear at first to be a fairly elaborate exercise. Using this example, we describe each aspect of this exercise in detail below.

The data and research question #

The data shown below are simulated but are designed to reflect actual surveys conducted in the United States in March of 2020. Partipants were asked whether they perceive that they have a substantial risk of dying if they are infected with the novel coronavirus. The number of people stating each response, stratified on age, are shown below (only two age groups are shown):

Each subject’s response is binary – they either perceive themselves to be high risk, or not to be at high risk. When working with this type of data, we are usually interested in the proportion of people who provide each response within each stratum (age group). These are conditional proportions, conditioning on the age group. The numerical values of the conditional proportions are given below:

There are four conditional proportions in the table above – the proportion of younger people who perceive themselves to be at higher risk, 0.110=25/(25+202); the proportion of younger people who do not perceive themselves to be at high risk, 0.890=202/(25+202); the proportion of older people who perceive themselves to be at high risk 0.195=30/(30+124); and the proportion of older people who do not perceive themselves to be at high risk, 0.805=124/(30+124).

The trend in the data is that younger people perceive themselves to be at lower risk of dying than older people, by a difference of 0.195-0.110=0.085 (in terms of proportions). But is this trend only present in this sample, or is it generalizable to a broader population (say the entire US population)? That is the goal of conducting a statistical hypothesis test in this setting.

The population structure #

Corresponding to our data above is the unobserved population structure, which we can denote as follows

The symbols \(p\) and \(q\) in the table above are population parameters . These are quantitites that we do not know, and wish to assess using the data. In this case, our null hypothesis can be expressed as the statement \(p = q\) . We can estimate \(p\) using the sample proportion \(\hat{p} = 0.110\) , and similarly estimate \(q\) using \(\hat{q} = 0.195\) . However these estimates do not immediately provide us with a way of expressing the evidence relating to the hypothesis that \(p=q\) . This is provided by the test statistic.

A test statistic #

As noted above, a test statistic is a reduction of the data to one number that captures all of the relevant information for assessing the hypotheses. A natural first choice for a test statistic here would be the difference in sample proportions between the two age groups, which is 0.195 - 0.110 = 0.085. There is a difference of 0.085 between the perceived risks of death in the younger and older age groups.

The difference in rates (0.085) does not on its own make a good test statistic, although it is a good start toward obtaining one. The reason for this is that the evidence underlying this difference in rates depends also on the absolute rates (0.110 and 0.195), and on the sample sizes (227 and 154). If we only know that the difference in rates is 0.085, this is not sufficient to evaluate the hypothesis in a statistical manner. A given difference in rates is much stronger evidence if it is obtained from a larger sample. If we have a difference of 0.085 with a very large sample, say one million people, then we should be almost certain that the true rates differ (i.e. the data are highly incompatiable with the hypothesis that \(p=q\) ). If we have the same difference in rates of 0.085, but with a small sample, say 50 people per age group, then there would be almost no evidence for a true difference in the rates (i.e. the data are compatiable with the hypothesis \(p=q\) ).

To address this issue, we need to consider the uncertainty in the estimated rate difference, which is 0.085. Recall that the estimated rate difference is obtained from the sample and therefore is almost certain to deviate somewhat from the true rate difference in the population (which is unknown). Recall from our study of standard errors that the standard error for an estimated proportion is \(\sqrt{p(1-p)/n}\) , where \(p\) is the outcome probability (here the outcome is that a person perceives a high risk of dying), and \(n\) is the sample size.

In the present analysis, we are comparing two proportions, so we have two standard errors. The estimated standard error for the younger people is \(\sqrt{0.11\cdot 0.89/227} \approx 0.021\) . The estimated standard error for the older people is \(\sqrt{0.195\cdot 0.805/154} \approx 0.032\) . Note that both standard errors are estimated, rather than exact, because we are plugging in estimates of the rates (0.11 and 0.195). Also note that the standard error for the rate among older people is greater than that for younger people. This is because the sample size for older people is smaller, and also because the estimated rate for older people is closer to 1/2.

In our previous discussion of standard errors, we saw how standard errors for independent quantities \(A\) and \(B\) can be used to obtain the standard error for the difference \(A-B\) . Applying that result here, we see that the standard error for the estimated difference in rates 0.195-0.11=0.085 is \(\sqrt{0.021^2 + 0.032^2} \approx 0.038\) .

The final step in constructing our test statistic is to construct a Z-score from the estimated difference in rates. As with all Z-scores, we proceed by taking the estimated difference in rates, and then divide it by its standard error. Thus, we get a test statistic value of \(0.085 / 0.038 \approx 2.24\) .

A test statistic value of 2.24 is not very close to zero, so there is some evidence against the null hypothesis. But the strength of this evidence remains unclear. Thus, we must consider how to calibrate this evidence in a way that makes it more interpretable.

Calibrating the evidence in the test statistic #

By the central limit theorem (CLT), a Z-score approximately follows a normal distribution. When the null hypothesis holds, the Z-score approximately follows the standard normal distribution (recall that a standard normal distribution is a normal distribution with expected value equal to 0 and variance equal to 1). If the null hypothesis does not hold, then the test statistic continues to approximately follow a normal distribution, but it is not the standard normal distribution.

A test statistic of zero represents the least possible evidence against the null hypothesis. Here, we will obtain a test statistic of zero when the two proportions being compared are identical, i.e. exactly the same proportions of younger and older people perceive a substantial risk of dying from a disease. Even if the test statistic is exactly zero, this does not guarantee that the null hypothesis is true. However it is the least amount of evidence that the data can present against the null hypothesis.

In a hypothesis testing setting using normally-distrbuted Z-scores, as is the case here (due to the CLT), the standard normal distribution is the reference distribution for our test statistic. If the Z-score falls in the center of the reference distribution, there is no evidence against the null hypothesis. If the Z-score falls into either tail of the reference distribution, then there is evidence against the null distribution, and the further into the tails of the reference distribution the Z-score falls, the greater the evidence.

The most conventional way to quantify the evidence in our test statistic is through a probability called the p-value . The p-value has a somewhat complex definition that many people find difficult to grasp. It is the probability of observing as much or more evidence against the null hypothesis as we actually observe, calculated when the null hypothesis is assumed to be true. We will discuss some ways to think about this more intuitively below.

For our purposes, “evidence against the null hypothesis” is reflected in how far into the tails of the reference distribution the Z-score (test statistic) falls. We observed a test statistic of 2.24 in our COVID risk perception analysis. Recall that due to the “empirical rule”, 95% of the time, a draw from a standard normal distribution falls between -2 and 2. Thus, the p-value must be less than 0.05, since 2.24 falls outside this interval. The p-value can be calculated using a computer, in this case it happens to be approximately 0.025.

As stated above, the p-value tells us how likely it would be for us to obtain as much evidence against the the null hypothesis as we observed in our actual data analysis, if we were certain that the null hypothesis were true. When the null hypothesis holds, any evidence against the null hypothesis is spurious. Thus, we will want to see stronger evidence against the null from our actual analysis than we would see if we know that the null hypothesis were true. A smaller p-value therefore reflects more evidence against the null hypothesis than a larger p-value.

By convention, p-values of 0.05 or smaller are considered to represent sufficiently strong evidence against the null hypothesis to make a finding “statistically significant”. This threshold of 0.05 was chosen arbitrarily 100 years ago, and there is no objective reason for it. In recent years, people have argued that either a lesser or a greater p-value threshold should be used. But largely due to convention, the practice of deeming p-values smaller than 0.05 to be statistically significant continues.

Summary of this example #

Here is a restatement of the above discussion, using slightly different language. In our analysis of COVID risk perceptions, we found a difference in proportions of 0.085 between younger and older subjects, with younger people perceiving a lower risk of dying. This is a difference based on the sample of data that we observed, but what we really want to know is whether there is a difference in COVID risk perception in the population (say, all US adults).

Suppose that in fact there is no difference in risk perception between younger and older people. For instance, suppose that in the population, 15% of people believe that they have a substantial risk of dying should they become infected with the novel coronavirus, regardless of their age. Even though the rates are equal in this imaginary population (both being 15%), the rates in our sample would typically not be equal. Around 3% of the time (0.024=2.4% to be exact), if the rates are actually equal in the population, we would see a test statistic that is 2.4 or larger. Since 3% represents a fairly rare event, we can conclude that our observed data are not compatible with the null hypothesis. We can also say that there is statistically significant evidence against the null hypothesis, and that we have “rejected” the null hypothesis at the 3% level.

In this data analysis, as in any data analysis, we cannot confirm definitively that the alternative hypothesis is true. But based on our data and the analysis performed above, we can claim that there is substantial evidence against the null hypothesis, using standard criteria for what is considered to be “substantial evidence”.

Comparison of means #

A very common setting where hypothesis testing is used arises when we wish to compare the means of a quantitative measurement obtained for two populations. Imagine, for example, that we have two ways of manufacturing a battery, and we wish to assess which approach yields batteries that are longer-lasting in actual use. To do this, suppose we obtain data that tells us the number of charge cycles that were completed in 200 batteries of type A, and in 300 batteries of type B. For the test developed below to be meaningful, the data must be independent and identically distributed samples.

The raw data for this study consists of 500 numbers, but it turns out that the most relevant information from the data is contained in the sample means and sample standard deviations computed within each battery type. Note that this is a huge reduction in complexity, since we started with 500 measurements and are able to summarize this down to just four numbers.

Suppose the summary statistics are as follows, where \(\bar{x}\) , \(\hat{\sigma}_x\) , and \(n\) denote the sample mean, sample standard deviation, and sample size, respectively.

The simplest measure comparing the two manufacturing approaches is the difference 420 - 403 = 17. That is, batteries of type A tend to have 17 more charge cycles compared to batteries of type B. This difference is present in our sample, but is it also true that the entire population of type A batteries has more charge cycles than the entire population of type B batteries? That is the goal of conducting a hypothesis test.

The next step in the present analysis is to divide the mean difference, which is 17, by its standard error. As we have seen, the standard error of the mean, or SEM, is \(\sigma/n\) , where \(\sigma\) is the standard deviation and \(n\) is the sample size. Since \(\sigma\) is almost never known, we plug in its estimate \(\hat{\sigma}\) . For the type A batteries, the estimated SEM is thus \(70/\sqrt{200} \approx 4.95\) , and for the type B batteries the estimated SEM is \(90/\sqrt{300} \approx 5.2\) .

Since we are comparing two estimated means that are obtained from independent samples, we can pool the standard deviations to obtain an overall standard deviation of \(\sqrt{4.95^2 + 5.2^2} \approx 7.18\) . We can now obtain our test statistic \(17/7.18 \approx 2.37\) .

The test statistic can be calibrated against a standard normal reference distribution. The probability of observing a standard normal value that is greater in magnitude than 2.37 is 0.018 (this can be obtained from a computer). This is the p-value, and since it is smaller than the conventional threshold of 0.05, we can claim that there is a statistically significant difference between the average number of charge cycles for the two types of batteries, with the A batteries having more charge cycles on average.

The analysis illustrated here is called a two independent samples Z-test , or just a two sample Z-test . It may be the most commonly employed of all statistical tests. It is also common to see the very similar two sample t-test , which is different only in that it uses the Student t distribution rather than the normal (Gaussian) distribution to calculate the p-values. In fact, there are quite a few minor variations on this testing framework, including “one sided” and “two sided” tests, and tests based on different ways of pooling the variance. Due to the CLT, if the sample size is modestly large (which is the case here), the results of all of these tests will be almost identical. For simplicity, we only cover the Z-test in this course.

Assessment of a correlation #

The tests for comparing proportions and means presented above are quite similar in many ways. To provide one more example of a hypothesis test that is somewhat different, we consider a test for a correlation coefficient.

Recall that the sample correlation coefficient \(\hat{r}\) is used to assess the relationship, or association, between two quantities X and Y that are measured on the same units. For example, we may ask whether two biomarkers, serum creatinine and D-dimer, are correlated with each other. These biomarkers are both commonly used in medical settings and are obtained using blood tests. D-dimer is used to assess whether a person has blood clots, and serum creatinine is used to measure kidney performance.

Suppose we are interested in whether there is a correlation in the population between D-dimer and serum creatinine. The population correlation coefficient between these two quantitites can be denoted \(r\) . Our null hypothesis is \(r=0\) . Suppose that we observe a sample correlation coefficient of \(\hat{r}=0.15\) , using an independent and identically distributed sample of pairs \((x, y)\) , where \(x\) is a D-dimer measurement and \(y\) is a serum creatinine measurement. Are these data consistent with the null hypothesis?

As above, we proceed by constructing a test statistic by taking the estimated statistic and dividing it by its standard error. The approximate standard error for \(\hat{r}\) is \(1/\sqrt{n}\) , where \(n\) is the sample size. The test statistic is therefore \(\sqrt{n}\cdot \hat{r} \approx 1.48\) .

We now calibrate this test statistic by comparing it to a standard normal reference distribution. Recall from the empirical rule that 5% of the time, a standard normal value falls outside the interval (-2, 2). Therefore, if the test statistic is smaller than 2 in magnitude, as is the case here, its p-value is greater than 0.05. Thus, in this case we know that the p-value will exceed 0.05 without calculating it, and therefore there is no basis for claiming that D-dimer and serum creatinine levels are correlated in this population.

Sampling properties of p-values #

A p-value is the most common way of calibrating evidence. Smaller p-values indicate stronger evidence against a null hypothesis. By convention, if the p-value is smaller than some threshold, usually 0.05, we reject the null hypothesis and declare a finding to be “statistically significant”. How can we understand more deeply what this means? One major concern should be obtaining a small p-value when the null hypothesis is true. If the null hypothesis is true, then it is incorrect to reject it. If we reject the null hypothesis, we are making a false claim. This can never be prevented with complete certainty, but we would like to have a very clear understanding of how likely it is to reject the null hypothesis when the null hypothesis is in fact true.

P-values have a special property that when the null distribution is true, the probability of observing a p-value smaller than 0.05 is 0.05 (5%). In fact, the probability of observing a p-value smaller than \(t\) is equal to \(t\) , for any threshold \(t\) . For example, the probability of observing a p-value smaller than 0.1, when the null hypothesis is true, is 10%.

This fact gives a more concrete understanding of how strong the evidence is for a particular p-value. If we always reject the null hypothesis when the p-value is 0.1 or smaller, then over the long run we will reject the null hypothesis 10% of the time when the null hypothesis is true. If we always reject the null hypothesis when the p-value is 0.05 or smaller, then over the long run we will reject the null hypothesis 5% of the time when the null hypothesis is true.

The approach to hypothesis testing discussed above largely follows the framework developed by RA Fisher around 1925. Note that although we mentioned the alternative hypothesis above, we never actually used it. A more elaborate approach to hypothesis testing was developed somewhat later by Egon Pearson and Jerzy Neyman. The “Neyman-Pearson” approach to hypothesis testing is even more formal than Fisher’s approach, and is most suited to highly planned research efforts in which the study is carefully designed, then executed. While ideally all research projects should be carried out this way, in reality we often conduct research using data that are already available, rather than using data that are specifically collected to address the research question.

Neyman-Pearson hypothesis testing involves specifying an alternative hypothesis that we anticipate encountering. Usually this alternative hypothesis represents a realistic guess about what we might find once the data are collected. In each of the three examples above, imagine that the data are not yet collected, and we are asked to specify an alternative hypothesis. We may arrive at the following:

In comparing risk perceptions for COVID, we may anticipate that older people will perceive a 30% risk of dying, and younger people will anticipate a 5% risk of dying.

In comparing the number of charge cycles for two types of batteries, we may anticipate that batter type A will have on average 500 charge cycles, and battery type B will have on average 400 charge cycles.

In assessing the correlation between D-dimer and serum creatinine levels, we may anticipate a correlation of 0.3.

Note that none of the numbers stated here are data-driven – they are specified before any data are collected, so they do not match the results from the data, which were collected only later. These alternative hypotheses are all essentially speculations, based perhaps on related data or theoretical considerations.

There are several benefits of specifying an explicit alternative hypothesis, as done here, even though it is not strictly necessary and can be avoided entirely by adopting Fisher’s approach to hypothesis testing. One benefit of specifying an alternative hypothesis is that we can use it to assess the power of our planned study, which can in turn inform the design of the study, in particular the sample size. The power is the probability of rejecting the null hypothesis when the alternative hypothesis is true. That is, it is the probability of discovering something real. The power should be contrasted with the level of a hypothesis test, which is the probability of rejecting the null hypothesis when the null hypothesis is true. That is, the level is the probability of “discovering” something that is not real.

To calculate the power, recall that for many of the test statistics that we are considering here, the test statistic has the form \(\hat{\theta}/{\rm SE}(\hat{\theta})\) , where \(\hat{\theta}\) is an estimate. For example, \(\hat{\theta}\) ) may be the correlation coefficient between D-dimer and serum creatinine levels. As stated above, the power is the probability of rejecting the null hypothesis when the alternative hypothesis is true. Suppose we decide to reject the null hypothesis when the test statistic is greater than 2, which is approximately equivalent to rejecting the null hypothesis when the p-value is less than 0.05. The following calculation tells us how to obtain the power in this setting:

Under the alternative hypothesis, \(\sqrt{n}(\hat{r} - r)\) approximately follows a standard normal distribution. Therefore, if \(r\) and \(n\) are given, we can easily use the computer to obtain the probability of observing a value greater than \(2 - \sqrt{n}r\) . This gives us the power of the test. For example, if we anticipate \(r=0.3\) and plan to collect data for \(n=100\) observations, the power is 0.84. This is generally considered to be good power – if the true value of \(r\) is in fact 0.3, we would reject the null hypothesis 84% of the time.

A study usually has poor power because it has too small of a sample size. Poorly powered studies can be very misleading, but since large sample sizes are expensive to collect, a lot of research is conducted using sample sizes that yield moderate or even low power. If a study has low power, it is unlikely to reject the null hypothesis even when the alternative hypothesis is true, but it remains possible to reject the null hypothesis when the null hypothesis is true (usually this probability is 5%). Therefore the most likely outcome of a poorly powered study may be an incorrectly rejected null hypothesis.

Hypothesis Testing

Hypothesis testing is a tool for making statistical inferences about the population data. It is an analysis tool that tests assumptions and determines how likely something is within a given standard of accuracy. Hypothesis testing provides a way to verify whether the results of an experiment are valid.

A null hypothesis and an alternative hypothesis are set up before performing the hypothesis testing. This helps to arrive at a conclusion regarding the sample obtained from the population. In this article, we will learn more about hypothesis testing, its types, steps to perform the testing, and associated examples.

What is Hypothesis Testing in Statistics?

Hypothesis testing uses sample data from the population to draw useful conclusions regarding the population probability distribution . It tests an assumption made about the data using different types of hypothesis testing methodologies. The hypothesis testing results in either rejecting or not rejecting the null hypothesis.

Hypothesis Testing Definition

Hypothesis testing can be defined as a statistical tool that is used to identify if the results of an experiment are meaningful or not. It involves setting up a null hypothesis and an alternative hypothesis. These two hypotheses will always be mutually exclusive. This means that if the null hypothesis is true then the alternative hypothesis is false and vice versa. An example of hypothesis testing is setting up a test to check if a new medicine works on a disease in a more efficient manner.

Null Hypothesis

The null hypothesis is a concise mathematical statement that is used to indicate that there is no difference between two possibilities. In other words, there is no difference between certain characteristics of data. This hypothesis assumes that the outcomes of an experiment are based on chance alone. It is denoted as \(H_{0}\). Hypothesis testing is used to conclude if the null hypothesis can be rejected or not. Suppose an experiment is conducted to check if girls are shorter than boys at the age of 5. The null hypothesis will say that they are the same height.

Alternative Hypothesis

The alternative hypothesis is an alternative to the null hypothesis. It is used to show that the observations of an experiment are due to some real effect. It indicates that there is a statistical significance between two possible outcomes and can be denoted as \(H_{1}\) or \(H_{a}\). For the above-mentioned example, the alternative hypothesis would be that girls are shorter than boys at the age of 5.

Hypothesis Testing P Value

In hypothesis testing, the p value is used to indicate whether the results obtained after conducting a test are statistically significant or not. It also indicates the probability of making an error in rejecting or not rejecting the null hypothesis.This value is always a number between 0 and 1. The p value is compared to an alpha level, \(\alpha\) or significance level. The alpha level can be defined as the acceptable risk of incorrectly rejecting the null hypothesis. The alpha level is usually chosen between 1% to 5%.

Hypothesis Testing Critical region

All sets of values that lead to rejecting the null hypothesis lie in the critical region. Furthermore, the value that separates the critical region from the non-critical region is known as the critical value.

Hypothesis Testing Formula

Depending upon the type of data available and the size, different types of hypothesis testing are used to determine whether the null hypothesis can be rejected or not. The hypothesis testing formula for some important test statistics are given below:

• z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\). \(\overline{x}\) is the sample mean, \(\mu\) is the population mean, \(\sigma\) is the population standard deviation and n is the size of the sample.
• t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\). s is the sample standard deviation.
• \(\chi ^{2} = \sum \frac{(O_{i}-E_{i})^{2}}{E_{i}}\). \(O_{i}\) is the observed value and \(E_{i}\) is the expected value.

We will learn more about these test statistics in the upcoming section.

Types of Hypothesis Testing

Selecting the correct test for performing hypothesis testing can be confusing. These tests are used to determine a test statistic on the basis of which the null hypothesis can either be rejected or not rejected. Some of the important tests used for hypothesis testing are given below.

Hypothesis Testing Z Test

A z test is a way of hypothesis testing that is used for a large sample size (n ≥ 30). It is used to determine whether there is a difference between the population mean and the sample mean when the population standard deviation is known. It can also be used to compare the mean of two samples. It is used to compute the z test statistic. The formulas are given as follows:

• One sample: z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\).
• Two samples: z = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}\).

Hypothesis Testing t Test

The t test is another method of hypothesis testing that is used for a small sample size (n < 30). It is also used to compare the sample mean and population mean. However, the population standard deviation is not known. Instead, the sample standard deviation is known. The mean of two samples can also be compared using the t test.

• One sample: t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\).
• Two samples: t = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}}\).

Hypothesis Testing Chi Square

The Chi square test is a hypothesis testing method that is used to check whether the variables in a population are independent or not. It is used when the test statistic is chi-squared distributed.

One Tailed Hypothesis Testing

One tailed hypothesis testing is done when the rejection region is only in one direction. It can also be known as directional hypothesis testing because the effects can be tested in one direction only. This type of testing is further classified into the right tailed test and left tailed test.

Right Tailed Hypothesis Testing

The right tail test is also known as the upper tail test. This test is used to check whether the population parameter is greater than some value. The null and alternative hypotheses for this test are given as follows:

\(H_{0}\): The population parameter is ≤ some value

\(H_{1}\): The population parameter is > some value.

If the test statistic has a greater value than the critical value then the null hypothesis is rejected

Left Tailed Hypothesis Testing

The left tail test is also known as the lower tail test. It is used to check whether the population parameter is less than some value. The hypotheses for this hypothesis testing can be written as follows:

\(H_{0}\): The population parameter is ≥ some value

\(H_{1}\): The population parameter is < some value.

The null hypothesis is rejected if the test statistic has a value lesser than the critical value.

Two Tailed Hypothesis Testing

In this hypothesis testing method, the critical region lies on both sides of the sampling distribution. It is also known as a non - directional hypothesis testing method. The two-tailed test is used when it needs to be determined if the population parameter is assumed to be different than some value. The hypotheses can be set up as follows:

\(H_{0}\): the population parameter = some value

\(H_{1}\): the population parameter ≠ some value

The null hypothesis is rejected if the test statistic has a value that is not equal to the critical value.

Hypothesis Testing Steps

Hypothesis testing can be easily performed in five simple steps. The most important step is to correctly set up the hypotheses and identify the right method for hypothesis testing. The basic steps to perform hypothesis testing are as follows:

• Step 1: Set up the null hypothesis by correctly identifying whether it is the left-tailed, right-tailed, or two-tailed hypothesis testing.
• Step 2: Set up the alternative hypothesis.
• Step 3: Choose the correct significance level, \(\alpha\), and find the critical value.
• Step 4: Calculate the correct test statistic (z, t or \(\chi\)) and p-value.
• Step 5: Compare the test statistic with the critical value or compare the p-value with \(\alpha\) to arrive at a conclusion. In other words, decide if the null hypothesis is to be rejected or not.

Hypothesis Testing Example

The best way to solve a problem on hypothesis testing is by applying the 5 steps mentioned in the previous section. Suppose a researcher claims that the mean average weight of men is greater than 100kgs with a standard deviation of 15kgs. 30 men are chosen with an average weight of 112.5 Kgs. Using hypothesis testing, check if there is enough evidence to support the researcher's claim. The confidence interval is given as 95%.

Step 1: This is an example of a right-tailed test. Set up the null hypothesis as \(H_{0}\): \(\mu\) = 100.

Step 2: The alternative hypothesis is given by \(H_{1}\): \(\mu\) > 100.

Step 3: As this is a one-tailed test, \(\alpha\) = 100% - 95% = 5%. This can be used to determine the critical value.

1 - \(\alpha\) = 1 - 0.05 = 0.95

0.95 gives the required area under the curve. Now using a normal distribution table, the area 0.95 is at z = 1.645. A similar process can be followed for a t-test. The only additional requirement is to calculate the degrees of freedom given by n - 1.

Step 4: Calculate the z test statistic. This is because the sample size is 30. Furthermore, the sample and population means are known along with the standard deviation.

z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\).

\(\mu\) = 100, \(\overline{x}\) = 112.5, n = 30, \(\sigma\) = 15

z = \(\frac{112.5-100}{\frac{15}{\sqrt{30}}}\) = 4.56

Step 5: Conclusion. As 4.56 > 1.645 thus, the null hypothesis can be rejected.

Hypothesis Testing and Confidence Intervals

Confidence intervals form an important part of hypothesis testing. This is because the alpha level can be determined from a given confidence interval. Suppose a confidence interval is given as 95%. Subtract the confidence interval from 100%. This gives 100 - 95 = 5% or 0.05. This is the alpha value of a one-tailed hypothesis testing. To obtain the alpha value for a two-tailed hypothesis testing, divide this value by 2. This gives 0.05 / 2 = 0.025.

Related Articles:

• Probability and Statistics
• Data Handling

Important Notes on Hypothesis Testing

• Hypothesis testing is a technique that is used to verify whether the results of an experiment are statistically significant.
• It involves the setting up of a null hypothesis and an alternate hypothesis.
• There are three types of tests that can be conducted under hypothesis testing - z test, t test, and chi square test.
• Hypothesis testing can be classified as right tail, left tail, and two tail tests.

Examples on Hypothesis Testing

• Example 1: The average weight of a dumbbell in a gym is 90lbs. However, a physical trainer believes that the average weight might be higher. A random sample of 5 dumbbells with an average weight of 110lbs and a standard deviation of 18lbs. Using hypothesis testing check if the physical trainer's claim can be supported for a 95% confidence level. Solution: As the sample size is lesser than 30, the t-test is used. \(H_{0}\): \(\mu\) = 90, \(H_{1}\): \(\mu\) > 90 \(\overline{x}\) = 110, \(\mu\) = 90, n = 5, s = 18. \(\alpha\) = 0.05 Using the t-distribution table, the critical value is 2.132 t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\) t = 2.484 As 2.484 > 2.132, the null hypothesis is rejected. Answer: The average weight of the dumbbells may be greater than 90lbs
• Example 2: The average score on a test is 80 with a standard deviation of 10. With a new teaching curriculum introduced it is believed that this score will change. On random testing, the score of 38 students, the mean was found to be 88. With a 0.05 significance level, is there any evidence to support this claim? Solution: This is an example of two-tail hypothesis testing. The z test will be used. \(H_{0}\): \(\mu\) = 80, \(H_{1}\): \(\mu\) ≠ 80 \(\overline{x}\) = 88, \(\mu\) = 80, n = 36, \(\sigma\) = 10. \(\alpha\) = 0.05 / 2 = 0.025 The critical value using the normal distribution table is 1.96 z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\) z = \(\frac{88-80}{\frac{10}{\sqrt{36}}}\) = 4.8 As 4.8 > 1.96, the null hypothesis is rejected. Answer: There is a difference in the scores after the new curriculum was introduced.
• Example 3: The average score of a class is 90. However, a teacher believes that the average score might be lower. The scores of 6 students were randomly measured. The mean was 82 with a standard deviation of 18. With a 0.05 significance level use hypothesis testing to check if this claim is true. Solution: The t test will be used. \(H_{0}\): \(\mu\) = 90, \(H_{1}\): \(\mu\) < 90 \(\overline{x}\) = 110, \(\mu\) = 90, n = 6, s = 18 The critical value from the t table is -2.015 t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\) t = \(\frac{82-90}{\frac{18}{\sqrt{6}}}\) t = -1.088 As -1.088 > -2.015, we fail to reject the null hypothesis. Answer: There is not enough evidence to support the claim.

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FAQs on Hypothesis Testing

What is hypothesis testing.

Hypothesis testing in statistics is a tool that is used to make inferences about the population data. It is also used to check if the results of an experiment are valid.

What is the z Test in Hypothesis Testing?

The z test in hypothesis testing is used to find the z test statistic for normally distributed data . The z test is used when the standard deviation of the population is known and the sample size is greater than or equal to 30.

What is the t Test in Hypothesis Testing?

The t test in hypothesis testing is used when the data follows a student t distribution . It is used when the sample size is less than 30 and standard deviation of the population is not known.

What is the formula for z test in Hypothesis Testing?

The formula for a one sample z test in hypothesis testing is z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\) and for two samples is z = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}\).

What is the p Value in Hypothesis Testing?

The p value helps to determine if the test results are statistically significant or not. In hypothesis testing, the null hypothesis can either be rejected or not rejected based on the comparison between the p value and the alpha level.

What is One Tail Hypothesis Testing?

When the rejection region is only on one side of the distribution curve then it is known as one tail hypothesis testing. The right tail test and the left tail test are two types of directional hypothesis testing.

What is the Alpha Level in Two Tail Hypothesis Testing?

To get the alpha level in a two tail hypothesis testing divide \(\alpha\) by 2. This is done as there are two rejection regions in the curve.

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The Craft of Writing a Strong Hypothesis

Table of Contents

Writing a hypothesis is one of the essential elements of a scientific research paper. It needs to be to the point, clearly communicating what your research is trying to accomplish. A blurry, drawn-out, or complexly-structured hypothesis can confuse your readers. Or worse, the editor and peer reviewers.

A captivating hypothesis is not too intricate. This blog will take you through the process so that, by the end of it, you have a better idea of how to convey your research paper's intent in just one sentence.

What is a Hypothesis?

The first step in your scientific endeavor, a hypothesis, is a strong, concise statement that forms the basis of your research. It is not the same as a thesis statement , which is a brief summary of your research paper .

The sole purpose of a hypothesis is to predict your paper's findings, data, and conclusion. It comes from a place of curiosity and intuition . When you write a hypothesis, you're essentially making an educated guess based on scientific prejudices and evidence, which is further proven or disproven through the scientific method.

The reason for undertaking research is to observe a specific phenomenon. A hypothesis, therefore, lays out what the said phenomenon is. And it does so through two variables, an independent and dependent variable.

The independent variable is the cause behind the observation, while the dependent variable is the effect of the cause. A good example of this is “mixing red and blue forms purple.” In this hypothesis, mixing red and blue is the independent variable as you're combining the two colors at your own will. The formation of purple is the dependent variable as, in this case, it is conditional to the independent variable.

Different Types of Hypotheses‌

Types of hypotheses

Some would stand by the notion that there are only two types of hypotheses: a Null hypothesis and an Alternative hypothesis. While that may have some truth to it, it would be better to fully distinguish the most common forms as these terms come up so often, which might leave you out of context.

Apart from Null and Alternative, there are Complex, Simple, Directional, Non-Directional, Statistical, and Associative and casual hypotheses. They don't necessarily have to be exclusive, as one hypothesis can tick many boxes, but knowing the distinctions between them will make it easier for you to construct your own.

1. Null hypothesis

A null hypothesis proposes no relationship between two variables. Denoted by H 0 , it is a negative statement like “Attending physiotherapy sessions does not affect athletes' on-field performance.” Here, the author claims physiotherapy sessions have no effect on on-field performances. Even if there is, it's only a coincidence.

2. Alternative hypothesis

Considered to be the opposite of a null hypothesis, an alternative hypothesis is donated as H1 or Ha. It explicitly states that the dependent variable affects the independent variable. A good  alternative hypothesis example is “Attending physiotherapy sessions improves athletes' on-field performance.” or “Water evaporates at 100 °C. ” The alternative hypothesis further branches into directional and non-directional.

• Directional hypothesis: A hypothesis that states the result would be either positive or negative is called directional hypothesis. It accompanies H1 with either the ‘<' or ‘>' sign.
• Non-directional hypothesis: A non-directional hypothesis only claims an effect on the dependent variable. It does not clarify whether the result would be positive or negative. The sign for a non-directional hypothesis is ‘≠.'

3. Simple hypothesis

A simple hypothesis is a statement made to reflect the relation between exactly two variables. One independent and one dependent. Consider the example, “Smoking is a prominent cause of lung cancer." The dependent variable, lung cancer, is dependent on the independent variable, smoking.

4. Complex hypothesis

In contrast to a simple hypothesis, a complex hypothesis implies the relationship between multiple independent and dependent variables. For instance, “Individuals who eat more fruits tend to have higher immunity, lesser cholesterol, and high metabolism.” The independent variable is eating more fruits, while the dependent variables are higher immunity, lesser cholesterol, and high metabolism.

5. Associative and casual hypothesis

Associative and casual hypotheses don't exhibit how many variables there will be. They define the relationship between the variables. In an associative hypothesis, changing any one variable, dependent or independent, affects others. In a casual hypothesis, the independent variable directly affects the dependent.

6. Empirical hypothesis

Also referred to as the working hypothesis, an empirical hypothesis claims a theory's validation via experiments and observation. This way, the statement appears justifiable and different from a wild guess.

Say, the hypothesis is “Women who take iron tablets face a lesser risk of anemia than those who take vitamin B12.” This is an example of an empirical hypothesis where the researcher  the statement after assessing a group of women who take iron tablets and charting the findings.

7. Statistical hypothesis

The point of a statistical hypothesis is to test an already existing hypothesis by studying a population sample. Hypothesis like “44% of the Indian population belong in the age group of 22-27.” leverage evidence to prove or disprove a particular statement.

Characteristics of a Good Hypothesis

Writing a hypothesis is essential as it can make or break your research for you. That includes your chances of getting published in a journal. So when you're designing one, keep an eye out for these pointers:

• A research hypothesis has to be simple yet clear to look justifiable enough.
• It has to be testable — your research would be rendered pointless if too far-fetched into reality or limited by technology.
• It has to be precise about the results —what you are trying to do and achieve through it should come out in your hypothesis.
• A research hypothesis should be self-explanatory, leaving no doubt in the reader's mind.
• If you are developing a relational hypothesis, you need to include the variables and establish an appropriate relationship among them.
• A hypothesis must keep and reflect the scope for further investigations and experiments.

Separating a Hypothesis from a Prediction

Outside of academia, hypothesis and prediction are often used interchangeably. In research writing, this is not only confusing but also incorrect. And although a hypothesis and prediction are guesses at their core, there are many differences between them.

A hypothesis is an educated guess or even a testable prediction validated through research. It aims to analyze the gathered evidence and facts to define a relationship between variables and put forth a logical explanation behind the nature of events.

Predictions are assumptions or expected outcomes made without any backing evidence. They are more fictionally inclined regardless of where they originate from.

For this reason, a hypothesis holds much more weight than a prediction. It sticks to the scientific method rather than pure guesswork. "Planets revolve around the Sun." is an example of a hypothesis as it is previous knowledge and observed trends. Additionally, we can test it through the scientific method.

Whereas "COVID-19 will be eradicated by 2030." is a prediction. Even though it results from past trends, we can't prove or disprove it. So, the only way this gets validated is to wait and watch if COVID-19 cases end by 2030.

Finally, How to Write a Hypothesis

Quick tips on writing a hypothesis

1.  Be clear about your research question

A hypothesis should instantly address the research question or the problem statement. To do so, you need to ask a question. Understand the constraints of your undertaken research topic and then formulate a simple and topic-centric problem. Only after that can you develop a hypothesis and further test for evidence.

2. Carry out a recce

Once you have your research's foundation laid out, it would be best to conduct preliminary research. Go through previous theories, academic papers, data, and experiments before you start curating your research hypothesis. It will give you an idea of your hypothesis's viability or originality.

Making use of references from relevant research papers helps draft a good research hypothesis. SciSpace Discover offers a repository of over 270 million research papers to browse through and gain a deeper understanding of related studies on a particular topic. Additionally, you can use SciSpace Copilot , your AI research assistant, for reading any lengthy research paper and getting a more summarized context of it. A hypothesis can be formed after evaluating many such summarized research papers. Copilot also offers explanations for theories and equations, explains paper in simplified version, allows you to highlight any text in the paper or clip math equations and tables and provides a deeper, clear understanding of what is being said. This can improve the hypothesis by helping you identify potential research gaps.

3. Create a 3-dimensional hypothesis

Variables are an essential part of any reasonable hypothesis. So, identify your independent and dependent variable(s) and form a correlation between them. The ideal way to do this is to write the hypothetical assumption in the ‘if-then' form. If you use this form, make sure that you state the predefined relationship between the variables.

In another way, you can choose to present your hypothesis as a comparison between two variables. Here, you must specify the difference you expect to observe in the results.

4. Write the first draft

Now that everything is in place, it's time to write your hypothesis. For starters, create the first draft. In this version, write what you expect to find from your research.

Clearly separate your independent and dependent variables and the link between them. Don't fixate on syntax at this stage. The goal is to ensure your hypothesis addresses the issue.

5. Proof your hypothesis

After preparing the first draft of your hypothesis, you need to inspect it thoroughly. It should tick all the boxes, like being concise, straightforward, relevant, and accurate. Your final hypothesis has to be well-structured as well.

Research projects are an exciting and crucial part of being a scholar. And once you have your research question, you need a great hypothesis to begin conducting research. Thus, knowing how to write a hypothesis is very important.

Now that you have a firmer grasp on what a good hypothesis constitutes, the different kinds there are, and what process to follow, you will find it much easier to write your hypothesis, which ultimately helps your research.

Now it's easier than ever to streamline your research workflow with SciSpace Discover . Its integrated, comprehensive end-to-end platform for research allows scholars to easily discover, write and publish their research and fosters collaboration.

It includes everything you need, including a repository of over 270 million research papers across disciplines, SEO-optimized summaries and public profiles to show your expertise and experience.

If you found these tips on writing a research hypothesis useful, head over to our blog on Statistical Hypothesis Testing to learn about the top researchers, papers, and institutions in this domain.

Frequently Asked Questions (FAQs)

1. what is the definition of hypothesis.

According to the Oxford dictionary, a hypothesis is defined as “An idea or explanation of something that is based on a few known facts, but that has not yet been proved to be true or correct”.

2. What is an example of hypothesis?

The hypothesis is a statement that proposes a relationship between two or more variables. An example: "If we increase the number of new users who join our platform by 25%, then we will see an increase in revenue."

3. What is an example of null hypothesis?

A null hypothesis is a statement that there is no relationship between two variables. The null hypothesis is written as H0. The null hypothesis states that there is no effect. For example, if you're studying whether or not a particular type of exercise increases strength, your null hypothesis will be "there is no difference in strength between people who exercise and people who don't."

4. What are the types of research?

• Fundamental research

• Applied research

• Qualitative research

• Quantitative research

• Mixed research

• Exploratory research

• Longitudinal research

• Cross-sectional research

• Field research

• Laboratory research

• Fixed research

• Flexible research

• Action research

• Policy research

• Classification research

• Comparative research

• Causal research

• Inductive research

• Deductive research

5. How to write a hypothesis?

• Your hypothesis should be able to predict the relationship and outcome.

• Avoid wordiness by keeping it simple and brief.

• Your hypothesis should contain observable and testable outcomes.

• Your hypothesis should be relevant to the research question.

6. What are the 2 types of hypothesis?

• Null hypotheses are used to test the claim that "there is no difference between two groups of data".

• Alternative hypotheses test the claim that "there is a difference between two data groups".

7. Difference between research question and research hypothesis?

A research question is a broad, open-ended question you will try to answer through your research. A hypothesis is a statement based on prior research or theory that you expect to be true due to your study. Example - Research question: What are the factors that influence the adoption of the new technology? Research hypothesis: There is a positive relationship between age, education and income level with the adoption of the new technology.

8. What is plural for hypothesis?

The plural of hypothesis is hypotheses. Here's an example of how it would be used in a statement, "Numerous well-considered hypotheses are presented in this part, and they are supported by tables and figures that are well-illustrated."

9. What is the red queen hypothesis?

The red queen hypothesis in evolutionary biology states that species must constantly evolve to avoid extinction because if they don't, they will be outcompeted by other species that are evolving. Leigh Van Valen first proposed it in 1973; since then, it has been tested and substantiated many times.

10. Who is known as the father of null hypothesis?

The father of the null hypothesis is Sir Ronald Fisher. He published a paper in 1925 that introduced the concept of null hypothesis testing, and he was also the first to use the term itself.

11. When to reject null hypothesis?

You need to find a significant difference between your two populations to reject the null hypothesis. You can determine that by running statistical tests such as an independent sample t-test or a dependent sample t-test. You should reject the null hypothesis if the p-value is less than 0.05.

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What is Hypothesis Testing in Statistics? Types and Examples

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Hypothesis testing in statistics involves testing an assumption about a population parameter using sample data. Learners can download Hypothesis Testing PDF to get instant access to all information!

What exactly is hypothesis testing, and how does it work in statistics? Can I find practical examples and understand the different types from this blog?

Hypothesis Testing : Ever wonder how researchers determine if a new medicine actually works or if a new marketing campaign effectively drives sales? They use hypothesis testing! It is at the core of how scientific studies, business experiments and surveys determine if their results are statistically significant or just due to chance.

Hypothesis testing allows us to make evidence-based decisions by quantifying uncertainty and providing a structured process to make data-driven conclusions rather than guessing. In this post, we will discuss hypothesis testing types, examples, and processes!

Table of Contents

Hypothesis Testing

Hypothesis testing is a statistical method used to evaluate the validity of a hypothesis using sample data. It involves assessing whether observed data provide enough evidence to reject a specific hypothesis about a population parameter. 

Hypothesis Testing in Data Science

Hypothesis testing in data science is a statistical method used to evaluate two mutually exclusive population statements based on sample data. The primary goal is to determine which statement is more supported by the observed data.

Hypothesis testing assists in supporting the certainty of findings in research and data science projects. This statistical inference aids in making decisions about population parameters using sample data. For those who are looking to deepen their knowledge in data science and expand their skillset, we highly recommend checking out Master Generative AI: Data Science Course by Physics Wallah .

Also Read: What is Encapsulation Explain in Details

What is the Hypothesis Testing Procedure in Data Science?

The hypothesis testing procedure in data science involves a structured approach to evaluating hypotheses using statistical methods. Here’s a step-by-step breakdown of the typical procedure:

1) State the Hypotheses:

• Null Hypothesis (H0): This is the default assumption or a statement of no effect or difference. It represents what you aim to test against.
• Alternative Hypothesis (Ha): This is the opposite of the null hypothesis and represents what you want to prove.

2) Choose a Significance Level (α):

• Decide on a threshold (commonly 0.05) beyond which you will reject the null hypothesis. This is your significance level.

3) Select the Appropriate Test:

• Depending on your data type (e.g., continuous, categorical) and the nature of your research question, choose the appropriate statistical test (e.g., t-test, chi-square test, ANOVA, etc.).

4) Collect Data:

• Gather data from your sample or population, ensuring that it’s representative and sufficiently large (or as per your experimental design).

5)Compute the Test Statistic:

• Using your data and the chosen statistical test, compute the test statistic that summarizes the evidence against the null hypothesis.

6) Determine the Critical Value or P-value:

• Based on your significance level and the test statistic’s distribution, determine the critical value from a statistical table or compute the p-value.

7) Make a Decision:

• If the p-value is less than α: Reject the null hypothesis.
• If the p-value is greater than or equal to α: Fail to reject the null hypothesis.

8) Draw Conclusions:

• Based on your decision, draw conclusions about your research question or hypothesis. Remember, failing to reject the null hypothesis doesn’t prove it true; it merely suggests that you don’t have sufficient evidence to reject it.

9) Report Findings:

• Document your findings, including the test statistic, p-value, conclusion, and any other relevant details. Ensure clarity so that others can understand and potentially replicate your analysis.

Also Read: Binary Search Algorithm

How Hypothesis Testing Works?

Hypothesis testing is a fundamental concept in statistics that aids analysts in making informed decisions based on sample data about a larger population. The process involves setting up two contrasting hypotheses, the null hypothesis and the alternative hypothesis, and then using statistical methods to determine which hypothesis provides a more plausible explanation for the observed data.

The Core Principles:

• The Null Hypothesis (H0): This serves as the default assumption or status quo. Typically, it posits that there is no effect or no difference, often represented by an equality statement regarding population parameters. For instance, it might state that a new drug’s effect is no different from a placebo.
• The Alternative Hypothesis (H1 or Ha): This is the counter assumption or what researchers aim to prove. It’s the opposite of the null hypothesis, indicating that there is an effect, a change, or a difference in the population parameters. Using the drug example, the alternative hypothesis would suggest that the new drug has a different effect than the placebo.

Testing the Hypotheses:

Once these hypotheses are established, analysts gather data from a sample and conduct statistical tests. The objective is to determine whether the observed results are statistically significant enough to reject the null hypothesis in favor of the alternative.

Examples to Clarify the Concept:

• Null Hypothesis (H0): The sanitizer’s average efficacy is 95%.
• By conducting tests, if evidence suggests that the sanitizer’s efficacy is significantly less than 95%, we reject the null hypothesis.
• Null Hypothesis (H0): The coin is fair, meaning the probability of heads and tails is equal.
• Through experimental trials, if results consistently show a skewed outcome, indicating a significantly different probability for heads and tails, the null hypothesis might be rejected.

What are the 3 types of Hypothesis Test?

Hypothesis testing is a cornerstone in statistical analysis, providing a framework to evaluate the validity of assumptions or claims made about a population based on sample data. Within this framework, several specific tests are utilized based on the nature of the data and the question at hand. Here’s a closer look at the three fundamental types of hypothesis tests:

The z-test is a statistical method primarily employed when comparing means from two datasets, particularly when the population standard deviation is known. Its main objective is to ascertain if the means are statistically equivalent. 

A crucial prerequisite for the z-test is that the sample size should be relatively large, typically 30 data points or more. This test aids researchers and analysts in determining the significance of a relationship or discovery, especially in scenarios where the data’s characteristics align with the assumptions of the z-test.

The t-test is a versatile statistical tool used extensively in research and various fields to compare means between two groups. It’s particularly valuable when the population standard deviation is unknown or when dealing with smaller sample sizes. 

By evaluating the means of two groups, the t-test helps ascertain if a particular treatment, intervention, or variable significantly impacts the population under study. Its flexibility and robustness make it a go-to method in scenarios ranging from medical research to business analytics.

3. Chi-Square Test:

The Chi-Square test stands distinct from the previous tests, primarily focusing on categorical data rather than means. This statistical test is instrumental when analyzing categorical variables to determine if observed data aligns with expected outcomes as posited by the null hypothesis. 

By assessing the differences between observed and expected frequencies within categorical data, the Chi-Square test offers insights into whether discrepancies are statistically significant. Whether used in social sciences to evaluate survey responses or in quality control to assess product defects, the Chi-Square test remains pivotal for hypothesis testing in diverse scenarios.

Also Read: Python vs Java: Which is Best for Machine learning algorithm

Hypothesis Testing in Statistics

Hypothesis testing is a fundamental concept in statistics used to make decisions or inferences about a population based on a sample of data. The process involves setting up two competing hypotheses, the null hypothesis H 0​ and the alternative hypothesis H 1​. 

Through various statistical tests, such as the t-test, z-test, or Chi-square test, analysts evaluate sample data to determine whether there’s enough evidence to reject the null hypothesis in favor of the alternative. The aim is to draw conclusions about population parameters or to test theories, claims, or hypotheses.

Hypothesis Testing in Research

In research, hypothesis testing serves as a structured approach to validate or refute theories or claims. Researchers formulate a clear hypothesis based on existing literature or preliminary observations. They then collect data through experiments, surveys, or observational studies. 

Using statistical methods, researchers analyze this data to determine if there’s sufficient evidence to reject the null hypothesis. By doing so, they can draw meaningful conclusions, make predictions, or recommend actions based on empirical evidence rather than mere speculation.

Hypothesis Testing in R

R, a powerful programming language and environment for statistical computing and graphics, offers a wide array of functions and packages specifically designed for hypothesis testing. Here’s how hypothesis testing is conducted in R:

• Data Collection : Before conducting any test, you need to gather your data and ensure it’s appropriately structured in R.
• Choose the Right Test : Depending on your research question and data type, select the appropriate hypothesis test. For instance, use the t.test() function for a t-test or chisq.test() for a Chi-square test.
• Set Hypotheses : Define your null and alternative hypotheses. Using R’s syntax, you can specify these hypotheses and run the corresponding test.
• Execute the Test : Utilize built-in functions in R to perform the hypothesis test on your data. For instance, if you want to compare two means, you can use the t.test() function, providing the necessary arguments like the data vectors and type of t-test (one-sample, two-sample, paired, etc.).
• Interpret Results : Once the test is executed, R will provide output, including test statistics, p-values, and confidence intervals. Based on these results and a predetermined significance level (often 0.05), you can decide whether to reject the null hypothesis.
• Visualization : R’s graphical capabilities allow users to visualize data distributions, confidence intervals, or test statistics, aiding in the interpretation and presentation of results.

Hypothesis testing is an integral part of statistics and research, offering a systematic approach to validate hypotheses. Leveraging R’s capabilities, researchers and analysts can efficiently conduct and interpret various hypothesis tests, ensuring robust and reliable conclusions from their data.

Do Data Scientists do Hypothesis Testing?

Yes, data scientists frequently engage in hypothesis testing as part of their analytical toolkit. Hypothesis testing is a foundational statistical technique used to make data-driven decisions, validate assumptions, and draw conclusions from data. Here’s how data scientists utilize hypothesis testing:

• Validating Assumptions : Before diving into complex analyses or building predictive models, data scientists often need to verify certain assumptions about the data. Hypothesis testing provides a structured approach to test these assumptions, ensuring that subsequent analyses or models are valid.
• Feature Selection : In machine learning and predictive modeling, data scientists use hypothesis tests to determine which features (or variables) are most relevant or significant in predicting a particular outcome. By testing hypotheses related to feature importance or correlation, they can streamline the modeling process and enhance prediction accuracy.
• A/B Testing : A/B testing is a common technique in marketing, product development, and user experience design. Data scientists employ hypothesis testing to compare two versions (A and B) of a product, feature, or marketing strategy to determine which performs better in terms of a specified metric (e.g., conversion rate, user engagement).
• Research and Exploration : In exploratory data analysis (EDA) or when investigating specific research questions, data scientists formulate hypotheses to test certain relationships or patterns within the data. By conducting hypothesis tests, they can validate these relationships, uncover insights, and drive data-driven decision-making.
• Model Evaluation : After building machine learning or statistical models, data scientists use hypothesis testing to evaluate the model’s performance, assess its predictive power, or compare different models. For instance, hypothesis tests like the t-test or F-test can help determine if a new model significantly outperforms an existing one based on certain metrics.
• Business Decision-making : Beyond technical analyses, data scientists employ hypothesis testing to support business decisions. Whether it’s evaluating the effectiveness of a marketing campaign, assessing customer preferences, or optimizing operational processes, hypothesis testing provides a rigorous framework to validate assumptions and guide strategic initiatives.

Hypothesis Testing Examples and Solutions

Let’s delve into some common examples of hypothesis testing and provide solutions or interpretations for each scenario.

Example: Testing the Mean

Scenario : A coffee shop owner believes that the average waiting time for customers during peak hours is 5 minutes. To test this, the owner takes a random sample of 30 customer waiting times and wants to determine if the average waiting time is indeed 5 minutes.

Hypotheses :

• H 0​ (Null Hypothesis): 5 μ =5 minutes (The average waiting time is 5 minutes)
• H 1​ (Alternative Hypothesis): 5 μ =5 minutes (The average waiting time is not 5 minutes)

Solution : Using a t-test (assuming population variance is unknown), calculate the t-statistic based on the sample mean, sample standard deviation, and sample size. Then, determine the p-value and compare it with a significance level (e.g., 0.05) to decide whether to reject the null hypothesis.

Example: A/B Testing in Marketing

Scenario : An e-commerce company wants to determine if changing the color of a “Buy Now” button from blue to green increases the conversion rate.

• H 0​: Changing the button color does not affect the conversion rate.
• H 1​: Changing the button color affects the conversion rate.

Solution : Split website visitors into two groups: one sees the blue button (control group), and the other sees the green button (test group). Track the conversion rates for both groups over a specified period. Then, use a chi-square test or z-test (for large sample sizes) to determine if there’s a statistically significant difference in conversion rates between the two groups.

Hypothesis Testing Formula

The formula for hypothesis testing typically depends on the type of test (e.g., z-test, t-test, chi-square test) and the nature of the data (e.g., mean, proportion, variance). Below are the basic formulas for some common hypothesis tests:

Z-Test for Population Mean :

Z=(σ/n​)(xˉ−μ0​)​

• ˉ x ˉ = Sample mean
• 0 μ 0​ = Population mean under the null hypothesis
• σ = Population standard deviation
• n = Sample size

T-Test for Population Mean :

t= (s/ n ​ ) ( x ˉ −μ 0 ​ ) ​ 

s = Sample standard deviation 

Chi-Square Test for Goodness of Fit :

χ2=∑Ei​(Oi​−Ei​)2​

• Oi ​ = Observed frequency
• Ei ​ = Expected frequency

Also Read: Full Form of OOPS

Hypothesis Testing Calculator

While you can perform hypothesis testing manually using the above formulas and statistical tables, many online tools and software packages simplify this process. Here’s how you might use a calculator or software:

• Z-Test and T-Test Calculators : These tools typically require you to input sample statistics (like sample mean, population mean, standard deviation, and sample size). Once you input these values, the calculator will provide you with the test statistic (Z or t) and a p-value.
• Chi-Square Calculator : For chi-square tests, you’d input observed and expected frequencies for different categories or groups. The calculator then computes the chi-square statistic and provides a p-value.
• Software Packages (e.g., R, Python with libraries like scipy, or statistical software like SPSS) : These platforms offer more comprehensive tools for hypothesis testing. You can run various tests, get detailed outputs, and even perform advanced analyses, including regression models, ANOVA, and more.

When using any calculator or software, always ensure you understand the underlying assumptions of the test, interpret the results correctly, and consider the broader context of your research or analysis.

Hypothesis Testing FAQs

What are the key components of a hypothesis test.

The key components include: Null Hypothesis (H0): A statement of no effect or no difference. Alternative Hypothesis (H1 or Ha): A statement that contradicts the null hypothesis. Test Statistic: A value computed from the sample data to test the null hypothesis. Significance Level (α): The threshold for rejecting the null hypothesis. P-value: The probability of observing the given data, assuming the null hypothesis is true.

What is the significance level in hypothesis testing?

The significance level (often denoted as α) is the probability threshold used to determine whether to reject the null hypothesis. Commonly used values for α include 0.05, 0.01, and 0.10, representing a 5%, 1%, or 10% chance of rejecting the null hypothesis when it's actually true.

How do I choose between a one-tailed and two-tailed test?

The choice between one-tailed and two-tailed tests depends on your research question and hypothesis. Use a one-tailed test when you're specifically interested in one direction of an effect (e.g., greater than or less than). Use a two-tailed test when you want to determine if there's a significant difference in either direction.

What is a p-value, and how is it interpreted?

The p-value is a probability value that helps determine the strength of evidence against the null hypothesis. A low p-value (typically ≤ 0.05) suggests that the observed data is inconsistent with the null hypothesis, leading to its rejection. Conversely, a high p-value suggests that the data is consistent with the null hypothesis, leading to no rejection.

Can hypothesis testing prove a hypothesis true?

No, hypothesis testing cannot prove a hypothesis true. Instead, it helps assess the likelihood of observing a given set of data under the assumption that the null hypothesis is true. Based on this assessment, you either reject or fail to reject the null hypothesis.

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• How to Conduct Hypothesis Testing in Statistics

The Fundamentals of Hypothesis Testing: What Every Student Should Know

Hypothesis testing is a fundamental statistical technique used to make inferences about populations based on sample data. This blog will guide you through the process of hypothesis testing, helping you understand and apply the concepts to solve similar assignments efficiently. By following this structured approach, you'll be able to solve your hypothesis testing homework problem with confidence.

Understanding the Basics of Hypothesis Testing

Hypothesis testing involves making a decision about the validity of a hypothesis based on sample data. It comprises four key steps: defining hypotheses, calculating the test statistic, determining the p-value, and drawing conclusions. Let's explore each of these steps in detail.

Defining Hypotheses

The first step in hypothesis testing is to define the null and alternative hypotheses. These hypotheses represent the statements we want to test.

Null Hypothesis (H0)

The null hypothesis (H0) is a statement that there is no effect or difference. It serves as the default assumption that we aim to test against.

Alternative Hypothesis (Ha or H1)

The alternative hypothesis (Ha or H1) is a statement that indicates the presence of an effect or difference. It represents what we want to prove.

Types of Tests

Depending on the direction of the hypothesis, we have three types of tests: left-tailed, right-tailed, and two-tailed tests.

Left-Tailed Test

A left-tailed test is used when we want to determine if the population mean is less than a specified value.

Right-Tailed Test

A right-tailed test is used when we want to determine if the population mean is greater than a specified value.

Two-Tailed Test

A two-tailed test is used when we want to determine if the population mean is different from a specified value, either higher or lower.

Example Scenario

Consider a scenario where we want to test if the average vehicle price from a sample is less than $27,000. We would set up our hypotheses as follows:

• Null Hypothesis (H0): μ = 27,000
• Alternative Hypothesis (Ha): μ < 27,000

Calculating the Test Statistic

Once the hypotheses are defined, the next step is to calculate the test statistic. The test statistic helps us determine the likelihood of observing the sample data under the null hypothesis.

Formula for the T-Test Statistic

The t-test statistic is calculated using the formula:

[ t = \frac{\bar{X} - \mu}{S / \sqrt{n}} ]

• (\bar{X}) is the sample mean
• (S) is the sample standard deviation
• (n) is the sample size
• (\mu) is the population mean defined in the null hypothesis

Standard Error

The denominator of the t-test formula, (S / \sqrt{n}), is known as the standard error (SE). It measures the variability of the sample mean.

Example Calculation

Let's calculate the test statistic for our vehicle price example. Given:

• Sample mean ((\bar{X})) = 25,650
• Sample standard deviation (S) = 3,488
• Sample size (n) = 10
• Population mean ((\mu)) = 27,000

First, we calculate the standard error (SE):

[ SE = \frac{S}{\sqrt{n}} = \frac{3488}{\sqrt{10}} \approx 1103 ]

Next, we calculate the test statistic (t):

[ t = \frac{25650 - 27000}{1103} \approx -1.2238 ]

Determining the P-Value

The p-value is a critical component of hypothesis testing. It indicates the probability of obtaining a test statistic as extreme as the one observed, assuming the null hypothesis is true.

Calculating the P-Value

The method to calculate the p-value depends on the type of test (left-tailed, right-tailed, or two-tailed) and the direction of the alternative hypothesis.

For a left-tailed test, the p-value is calculated using the T.DIST() function in Excel.

For a right-tailed test, the p-value is calculated using the T.DIST.RT() function in Excel.

For a two-tailed test, the p-value is calculated using the T.DIST.2T() function in Excel. When the test statistic is negative, use the absolute value function (ABS()) to remove the negative sign before calculating the p-value.

For our vehicle price example with a left-tailed test, we calculate the p-value using the T.DIST() function in Excel:

[ \text{p-value} = T.DIST(-1.2238, 9, TRUE) \approx 0.1261 ]

Drawing Conclusions

The final step in hypothesis testing is to draw a conclusion based on the p-value and a pre-determined significance level ((\alpha)).

Significance Level ((\alpha))

The significance level ((\alpha)) is the threshold for deciding whether to reject the null hypothesis. Common values for (\alpha) are 0.05, 0.01, 0.10, and 0.005.

Decision Rule

• If the p-value is less than (\alpha), we reject the null hypothesis.
• If the p-value is greater than (\alpha), we fail to reject the null hypothesis.

Example Conclusion

For our vehicle price example with (\alpha = 0.05):

• p-value = 0.1261
• (\alpha) = 0.05

Since 0.1261 > 0.05, we fail to reject the null hypothesis. There is not enough evidence to suggest that the average vehicle price is less than $27,000.

Practical Examples of Hypothesis Testing

To further illustrate hypothesis testing, let's explore three different scenarios: left-tailed test, right-tailed test, and two-tailed test.

Left-Tailed Test Example

In this example, we test if the average vehicle price is less than $27,000.

Step-by-Step Process

Define Hypotheses:

Calculate Test Statistic:

• Standard error (SE) = 1103
• Test statistic (t) = -1.2238

Determine P-Value:

Draw Conclusion:

• Since 0.1261 > 0.05, fail to reject the null hypothesis.
• Conclusion: There is not enough evidence to suggest that the average vehicle price is less than $27,000.

Right-Tailed Test Example

In this example, we test if the average vehicle price is greater than $23,500.

• Null Hypothesis (H0): μ = 23,500
• Alternative Hypothesis (Ha): μ > 23,500
• Population mean ((\mu)) = 23,500
• Test statistic (t) = 1.9490
• p-value = 0.0416
• Since 0.0416 < 0.05, reject the null hypothesis.
• Conclusion: There is enough evidence to suggest that the average vehicle price is greater than $23,500.

Two-Tailed Test Example

In this example, we test if the average vehicle price is different from $23,500.

• Alternative Hypothesis (Ha): μ ≠ 23,500
• p-value = 0.0831
• Since 0.0831 > 0.05, fail to reject the null hypothesis.
• Conclusion: There is not enough evidence to suggest that the average vehicle price is different from $23,500.

Tips for Conducting Hypothesis Testing

Successfully conducting hypothesis testing involves several critical steps. Here are some tips to help you perform hypothesis testing effectively.

Proper Data Collection

Accurate and reliable data collection is crucial for hypothesis testing. Ensure that your sample is representative of the population and collected using appropriate methods.

Random Sampling

Use random sampling techniques to avoid bias and ensure that your sample accurately represents the population.

Sample Size

Ensure that your sample size is large enough to provide reliable results. Larger sample sizes reduce the margin of error and increase the power of the test.

Verify Assumptions

Hypothesis tests often rely on certain assumptions about the data. Verify these assumptions before proceeding with the test.

Many hypothesis tests, including the t-test, assume that the data follows a normal distribution. Use graphical methods (e.g., histograms, Q-Q plots) or statistical tests (e.g., Shapiro-Wilk test) to check for normality.

Independence

Ensure that the observations in your sample are independent of each other. Independence is a key assumption for most hypothesis tests.

Utilize Software Tools

Software tools like Excel , R , and SPSS can simplify the calculations involved in hypothesis testing and reduce the risk of errors.

Excel provides several functions for hypothesis testing, such as T.DIST(), T.DIST.RT(), and T.DIST.2T(). Use these functions to calculate p-values and make decisions based on your test statistics.

R is a powerful statistical software that offers various packages for hypothesis testing. Use functions like t.test() to perform t-tests and obtain p-values and confidence intervals.

Interpret Results Carefully

Proper interpretation of the results is crucial for drawing accurate conclusions from hypothesis testing.

Statistical Significance

A statistically significant result (p-value < (\alpha)) indicates that there is strong evidence against the null hypothesis. However, it does not imply practical significance. Consider the context and the practical implications of the results.

Type I and Type II Errors

Be aware of the potential for Type I and Type II errors. A Type I error occurs when the null hypothesis is incorrectly rejected, while a Type II error occurs when the null hypothesis is not rejected despite being false. The significance level ((\alpha)) affects the probability of Type I errors, while the sample size and effect size influence the probability of Type II errors.

Report Results Transparently

When reporting the results of hypothesis testing, include all relevant information to ensure transparency and reproducibility.

Detailed Description

Provide a detailed description of the hypotheses, test statistic, p-value, significance level, and the conclusion. This information helps others understand and evaluate your analysis.

Confidence Intervals

Include confidence intervals for the estimated parameters. Confidence intervals provide a range of plausible values for the population parameter and offer additional context for interpreting the results.

Common Pitfalls in Hypothesis Testing

Hypothesis testing is a powerful tool, but it is essential to be aware of common pitfalls to avoid incorrect conclusions.

Misinterpreting P-Values

P-values indicate the probability of obtaining a test statistic as extreme as the one observed, assuming the null hypothesis is true. A small p-value suggests strong evidence against the null hypothesis, but it does not provide a measure of the effect size or practical significance.

P-Value Misconceptions

Avoid common misconceptions about p-values, such as believing that a p-value of 0.05 means there is a 5% chance that the null hypothesis is true. P-values do not measure the probability that the null hypothesis is true or false.

Ignoring Assumptions

Ignoring the assumptions underlying hypothesis tests can lead to incorrect conclusions. Always verify the assumptions before proceeding with the test.

Assumption Violations

If the assumptions are violated, consider using alternative tests that do not rely on those assumptions. For example, if the data is not normally distributed, use non-parametric tests like the Wilcoxon rank-sum test or the Mann-Whitney U test.

Overemphasizing Statistical Significance

Statistical significance does not imply practical significance. A result can be statistically significant but have a negligible practical effect. Always consider the context and practical implications of the results.

Effect Size

Report and interpret effect sizes alongside p-values. Effect sizes provide a measure of the magnitude of the observed effect and offer valuable context for interpreting the results.

Hypothesis testing is a critical tool in statistics for making inferences about populations based on sample data. By understanding the steps involved—defining hypotheses, calculating the test statistic, determining the p-value, and drawing conclusions—you can approach hypothesis testing with confidence.

Ensure proper data collection, verify assumptions, utilize software tools, interpret results carefully, and report findings transparently to enhance the reliability and validity of your hypothesis tests. By avoiding common pitfalls and considering both statistical and practical significance, you'll be well-equipped to tackle statistics homework and research projects effectively.

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What is a Hypothesis?

Table of Contents

Defining the hypothesis, the role of a hypothesis in the scientific method, types of hypotheses, hypothesis formulation, hypotheses and variables.

• The Importance of Testing Hypotheses
• The Hypothesis and Sociological Theory

In sociology, as in other scientific disciplines, the hypothesis serves as a crucial building block for research. It is a central element that directs the inquiry and provides a framework for testing the relationships between social phenomena. This article will explore what a hypothesis is, how it is formulated, and its role within the broader scientific method. By understanding the hypothesis, students of sociology can grasp how sociologists construct and test theories about the social world.

A hypothesis is a specific, testable statement about the relationship between two or more variables. It acts as a proposed explanation or prediction based on limited evidence, which researchers then test through empirical investigation. In essence, it is a statement that can be supported or refuted by data gathered from observation, experimentation, or other forms of systematic inquiry. The hypothesis typically takes the form of an “if-then” statement: if one variable changes, then another will change in response.

In sociological research, a hypothesis helps to focus the investigation by offering a clear proposition that can be tested. For instance, a sociologist might hypothesize that an increase in education levels leads to a decrease in crime rates. This hypothesis gives the researcher a direction, guiding them to collect data on education and crime, and analyze the relationship between the two variables. By doing so, the hypothesis serves as a tool for making sense of complex social phenomena.

The hypothesis is a key component of the scientific method, which is the systematic process by which sociologists and other scientists investigate the world. The scientific method begins with an observation of the world, followed by the formulation of a question or problem. Based on prior knowledge, theory, or preliminary observations, researchers then develop a hypothesis, which predicts an outcome or proposes a relationship between variables.

Once a hypothesis is established, researchers gather data to test it. If the data supports the hypothesis, it may be used to build a broader theory or to further refine the understanding of the social phenomenon in question. If the data contradicts the hypothesis, researchers may revise their hypothesis or abandon it altogether, depending on the strength of the evidence. In either case, the hypothesis helps to organize the research process, ensuring that it remains focused and methodologically sound.

In sociology, this method is particularly important because the social world is highly complex. Researchers must navigate a vast range of variables—age, gender, class, race, education, and countless others—that interact in unpredictable ways. A well-constructed hypothesis allows sociologists to narrow their focus to a manageable set of variables, making the investigation more precise and efficient.

Sociologists use different types of hypotheses, depending on the nature of their research question and the methods they plan to use. Broadly speaking, hypotheses can be classified into two main types: null hypotheses and alternative (or research) hypotheses.

Null Hypothesis

The null hypothesis, denoted as H0, states that there is no relationship between the variables being studied. It is a default assumption that any observed differences or relationships are due to random chance rather than a real underlying cause. In research, the null hypothesis serves as a point of comparison. Researchers collect data to see if the results allow them to reject the null hypothesis in favor of an alternative explanation.

For example, a sociologist studying the relationship between income and political participation might propose a null hypothesis that income has no effect on political participation. The goal of the research would then be to determine whether this null hypothesis can be rejected based on the data. If the data shows a significant correlation between income and political participation, the null hypothesis would be rejected.

Alternative Hypothesis

The alternative hypothesis, denoted as H1 or Ha, proposes that there is a significant relationship between the variables. This is the hypothesis that researchers aim to support with their data. In contrast to the null hypothesis, the alternative hypothesis predicts a specific direction or effect. For example, a researcher might hypothesize that higher levels of education lead to greater political engagement. In this case, the alternative hypothesis is proposing a positive correlation between the two variables.

The alternative hypothesis is the one that guides the research design, as it directs the researcher toward gathering evidence that will either support or refute the predicted relationship. The research process is structured around testing this hypothesis and determining whether the evidence is strong enough to reject the null hypothesis.

The process of formulating a hypothesis is both an art and a science. It requires a deep understanding of the social phenomena under investigation, as well as a clear sense of what is possible to observe and measure. Hypothesis formulation is closely linked to the theoretical framework that guides the research. Sociologists draw on existing theories to generate hypotheses, ensuring that their predictions are grounded in established knowledge.

To formulate a good hypothesis, a researcher must identify the key variables and determine how they are expected to relate to one another. Variables are the factors or characteristics that are being measured in a study. In sociology, these variables often include social attributes such as class, race, gender, age, education, and income, as well as behavioral variables like voting, criminal activity, or social participation.

For example, a sociologist studying the effects of social media on self-esteem might propose the following hypothesis: “Increased time spent on social media leads to lower levels of self-esteem among adolescents.” Here, the independent variable is the time spent on social media, and the dependent variable is the level of self-esteem. The hypothesis predicts a negative relationship between the two variables: as time spent on social media increases, self-esteem decreases.

A strong hypothesis has several key characteristics. It should be clear and specific, meaning that it unambiguously states the relationship between the variables. It should also be testable, meaning that it can be supported or refuted through empirical investigation. Finally, it should be grounded in theory, meaning that it is based on existing knowledge about the social phenomenon in question.

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• Inferential Statistics | An Easy Introduction & Examples

Inferential Statistics | An Easy Introduction & Examples

Published on September 4, 2020 by Pritha Bhandari . Revised on June 22, 2023.

While descriptive statistics summarize the characteristics of a data set, inferential statistics help you come to conclusions and make predictions based on your data.

When you have collected data from a sample , you can use inferential statistics to understand the larger population from which the sample is taken.

Inferential statistics have two main uses:

• making estimates about populations (for example, the mean SAT score of all 11th graders in the US).
• testing hypotheses to draw conclusions about populations (for example, the relationship between SAT scores and family income).

Table of contents

Descriptive versus inferential statistics, estimating population parameters from sample statistics, hypothesis testing, other interesting articles, frequently asked questions about inferential statistics.

Descriptive statistics allow you to describe a data set, while inferential statistics allow you to make inferences based on a data set.

• Descriptive statistics

Using descriptive statistics, you can report characteristics of your data:

• The distribution concerns the frequency of each value.
• The central tendency concerns the averages of the values.
• The variability concerns how spread out the values are.

In descriptive statistics, there is no uncertainty – the statistics precisely describe the data that you collected. If you collect data from an entire population, you can directly compare these descriptive statistics to those from other populations.

Inferential statistics

Most of the time, you can only acquire data from samples, because it is too difficult or expensive to collect data from the whole population that you’re interested in.

While descriptive statistics can only summarize a sample’s characteristics, inferential statistics use your sample to make reasonable guesses about the larger population.

With inferential statistics, it’s important to use random and unbiased sampling methods . If your sample isn’t representative of your population, then you can’t make valid statistical inferences or generalize .

Sampling error in inferential statistics

Since the size of a sample is always smaller than the size of the population, some of the population isn’t captured by sample data. This creates sampling error , which is the difference between the true population values (called parameters) and the measured sample values (called statistics).

Sampling error arises any time you use a sample, even if your sample is random and unbiased. For this reason, there is always some uncertainty in inferential statistics. However, using probability sampling methods reduces this uncertainty.

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The characteristics of samples and populations are described by numbers called statistics and parameters :

• A statistic is a measure that describes the sample (e.g., sample mean ).
• A parameter is a measure that describes the whole population (e.g., population mean).

Sampling error is the difference between a parameter and a corresponding statistic. Since in most cases you don’t know the real population parameter, you can use inferential statistics to estimate these parameters in a way that takes sampling error into account.

There are two important types of estimates you can make about the population: point estimates and interval estimates .

• A point estimate is a single value estimate of a parameter. For instance, a sample mean is a point estimate of a population mean.
• An interval estimate gives you a range of values where the parameter is expected to lie. A confidence interval is the most common type of interval estimate.

Both types of estimates are important for gathering a clear idea of where a parameter is likely to lie.

Confidence intervals

A confidence interval uses the variability around a statistic to come up with an interval estimate for a parameter. Confidence intervals are useful for estimating parameters because they take sampling error into account.

While a point estimate gives you a precise value for the parameter you are interested in, a confidence interval tells you the uncertainty of the point estimate. They are best used in combination with each other.

Each confidence interval is associated with a confidence level. A confidence level tells you the probability (in percentage) of the interval containing the parameter estimate if you repeat the study again.

A 95% confidence interval means that if you repeat your study with a new sample in exactly the same way 100 times, you can expect your estimate to lie within the specified range of values 95 times.

Although you can say that your estimate will lie within the interval a certain percentage of the time, you cannot say for sure that the actual population parameter will. That’s because you can’t know the true value of the population parameter without collecting data from the full population.

However, with random sampling and a suitable sample size, you can reasonably expect your confidence interval to contain the parameter a certain percentage of the time.

Your point estimate of the population mean paid vacation days is the sample mean of 19 paid vacation days.

Hypothesis testing is a formal process of statistical analysis using inferential statistics. The goal of hypothesis testing is to compare populations or assess relationships between variables using samples.

Hypotheses , or predictions, are tested using statistical tests . Statistical tests also estimate sampling errors so that valid inferences can be made.

Statistical tests can be parametric or non-parametric. Parametric tests are considered more statistically powerful because they are more likely to detect an effect if one exists.

Parametric tests make assumptions that include the following:

• the population that the sample comes from follows a normal distribution of scores
• the sample size is large enough to represent the population
• the variances , a measure of variability , of each group being compared are similar

When your data violates any of these assumptions, non-parametric tests are more suitable. Non-parametric tests are called “distribution-free tests” because they don’t assume anything about the distribution of the population data.

Statistical tests come in three forms: tests of comparison, correlation or regression.

Comparison tests

Comparison tests assess whether there are differences in means, medians or rankings of scores of two or more groups.

To decide which test suits your aim, consider whether your data meets the conditions necessary for parametric tests, the number of samples, and the levels of measurement of your variables.

Means can only be found for interval or ratio data , while medians and rankings are more appropriate measures for ordinal data .

Correlation tests

Correlation tests determine the extent to which two variables are associated.

Although Pearson’s r is the most statistically powerful test, Spearman’s r is appropriate for interval and ratio variables when the data doesn’t follow a normal distribution.

The chi square test of independence is the only test that can be used with nominal variables.

Regression tests

Regression tests demonstrate whether changes in predictor variables cause changes in an outcome variable. You can decide which regression test to use based on the number and types of variables you have as predictors and outcomes.

Most of the commonly used regression tests are parametric. If your data is not normally distributed, you can perform data transformations.

Data transformations help you make your data normally distributed using mathematical operations, like taking the square root of each value.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Confidence interval
• Measures of central tendency
• Correlation coefficient

Methodology

• Cluster sampling
• Stratified sampling
• Types of interviews
• Cohort study
• Thematic analysis

Research bias

• Implicit bias
• Cognitive bias
• Survivorship bias
• Availability heuristic
• Nonresponse bias
• Regression to the mean

Descriptive statistics summarize the characteristics of a data set. Inferential statistics allow you to test a hypothesis or assess whether your data is generalizable to the broader population.

A statistic refers to measures about the sample , while a parameter refers to measures about the population .

A sampling error is the difference between a population parameter and a sample statistic .

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

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Bhandari, P. (2023, June 22). Inferential Statistics | An Easy Introduction & Examples. Scribbr. Retrieved September 18, 2024, from https://www.scribbr.com/statistics/inferential-statistics/

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