null hypothesis testing python

Statistics Made Easy

How to Perform Hypothesis Testing in Python (With Examples)

A hypothesis test is a formal statistical test we use to reject or fail to reject some statistical hypothesis.

This tutorial explains how to perform the following hypothesis tests in Python:

• One sample t-test
• Two sample t-test
• Paired samples t-test

Let’s jump in!

Example 1: One Sample t-test in Python

A one sample t-test is used to test whether or not the mean of a population is equal to some value.

For example, suppose we want to know whether or not the mean weight of a certain species of some turtle is equal to 310 pounds.

To test this, we go out and collect a simple random sample of turtles with the following weights:

Weights : 300, 315, 320, 311, 314, 309, 300, 308, 305, 303, 305, 301, 303

The following code shows how to use the ttest_1samp() function from the scipy.stats library to perform a one sample t-test:

The t test statistic is  -1.5848 and the corresponding two-sided p-value is  0.1389 .

The two hypotheses for this particular one sample t-test are as follows:

• H 0 :  µ = 310 (the mean weight for this species of turtle is 310 pounds)
• H A :  µ ≠310 (the mean weight is not  310 pounds)

Because the p-value of our test (0.1389) is greater than alpha = 0.05, we fail to reject the null hypothesis of the test.

We do not have sufficient evidence to say that the mean weight for this particular species of turtle is different from 310 pounds.

Example 2: Two Sample t-test in Python

A two sample t-test is used to test whether or not the means of two populations are equal.

For example, suppose we want to know whether or not the mean weight between two different species of turtles is equal.

To test this, we collect a simple random sample of turtles from each species with the following weights:

Sample 1 : 300, 315, 320, 311, 314, 309, 300, 308, 305, 303, 305, 301, 303

Sample 2 : 335, 329, 322, 321, 324, 319, 304, 308, 305, 311, 307, 300, 305

The following code shows how to use the ttest_ind() function from the scipy.stats library to perform this two sample t-test:

The t test statistic is – 2.1009 and the corresponding two-sided p-value is 0.0463 .

The two hypotheses for this particular two sample t-test are as follows:

• H 0 :  µ 1 = µ 2 (the mean weight between the two species is equal)
• H A :  µ 1 ≠ µ 2 (the mean weight between the two species is not equal)

Since the p-value of the test (0.0463) is less than .05, we reject the null hypothesis.

This means we have sufficient evidence to say that the mean weight between the two species is not equal.

Example 3: Paired Samples t-test in Python

A paired samples t-test is used to compare the means of two samples when each observation in one sample can be paired with an observation in the other sample.

For example, suppose we want to know whether or not a certain training program is able to increase the max vertical jump (in inches) of basketball players.

To test this, we may recruit a simple random sample of 12 college basketball players and measure each of their max vertical jumps. Then, we may have each player use the training program for one month and then measure their max vertical jump again at the end of the month.

The following data shows the max jump height (in inches) before and after using the training program for each player:

Before : 22, 24, 20, 19, 19, 20, 22, 25, 24, 23, 22, 21

After : 23, 25, 20, 24, 18, 22, 23, 28, 24, 25, 24, 20

The following code shows how to use the ttest_rel() function from the scipy.stats library to perform this paired samples t-test:

The t test statistic is – 2.5289  and the corresponding two-sided p-value is 0.0280 .

The two hypotheses for this particular paired samples t-test are as follows:

• H 0 :  µ 1 = µ 2 (the mean jump height before and after using the program is equal)
• H A :  µ 1 ≠ µ 2 (the mean jump height before and after using the program is not equal)

Since the p-value of the test (0.0280) is less than .05, we reject the null hypothesis.

This means we have sufficient evidence to say that the mean jump height before and after using the training program is not equal.

Additional Resources

You can use the following online calculators to automatically perform various t-tests:

One Sample t-test Calculator Two Sample t-test Calculator Paired Samples t-test Calculator

Featured Posts

Hey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike.  My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.

One Reply to “How to Perform Hypothesis Testing in Python (With Examples)”

Nice post. Could you please clear my one doubt regarding alpha value . i can see in your example, it is a two tail test. As i understand in that case our alpha value should be alpha/2 i.e 0.025 . Here you are taking it as 0.05. ?

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

Join the Statology Community

Sign up to receive Statology's exclusive study resource: 100 practice problems with step-by-step solutions. Plus, get our latest insights, tutorials, and data analysis tips straight to your inbox!

By subscribing you accept Statology's Privacy Policy.

What Is Hypothesis Testing? Types and Python Code Example

Curiosity has always been a part of human nature. Since the beginning of time, this has been one of the most important tools for birthing civilizations. Still, our curiosity grows — it tests and expands our limits. Humanity has explored the plains of land, water, and air. We've built underwater habitats where we could live for weeks. Our civilization has explored various planets. We've explored land to an unlimited degree.

These things were possible because humans asked questions and searched until they found answers. However, for us to get these answers, a proven method must be used and followed through to validate our results. Historically, philosophers assumed the earth was flat and you would fall off when you reached the edge. While philosophers like Aristotle argued that the earth was spherical based on the formation of the stars, they could not prove it at the time.

This is because they didn't have adequate resources to explore space or mathematically prove Earth's shape. It was a Greek mathematician named Eratosthenes who calculated the earth's circumference with incredible precision. He used scientific methods to show that the Earth was not flat. Since then, other methods have been used to prove the Earth's spherical shape.

When there are questions or statements that are yet to be tested and confirmed based on some scientific method, they are called hypotheses. Basically, we have two types of hypotheses: null and alternate.

A null hypothesis is one's default belief or argument about a subject matter. In the case of the earth's shape, the null hypothesis was that the earth was flat.

An alternate hypothesis is a belief or argument a person might try to establish. Aristotle and Eratosthenes argued that the earth was spherical.

Other examples of a random alternate hypothesis include:

• The weather may have an impact on a person's mood.
• More people wear suits on Mondays compared to other days of the week.
• Children are more likely to be brilliant if both parents are in academia, and so on.

What is Hypothesis Testing?

Hypothesis testing is the act of testing whether a hypothesis or inference is true. When an alternate hypothesis is introduced, we test it against the null hypothesis to know which is correct. Let's use a plant experiment by a 12-year-old student to see how this works.

The hypothesis is that a plant will grow taller when given a certain type of fertilizer. The student takes two samples of the same plant, fertilizes one, and leaves the other unfertilized. He measures the plants' height every few days and records the results in a table.

After a week or two, he compares the final height of both plants to see which grew taller. If the plant given fertilizer grew taller, the hypothesis is established as fact. If not, the hypothesis is not supported. This simple experiment shows how to form a hypothesis, test it experimentally, and analyze the results.

In hypothesis testing, there are two types of error: Type I and Type II.

When we reject the null hypothesis in a case where it is correct, we've committed a Type I error. Type II errors occur when we fail to reject the null hypothesis when it is incorrect.

In our plant experiment above, if the student finds out that both plants' heights are the same at the end of the test period yet opines that fertilizer helps with plant growth, he has committed a Type I error.

However, if the fertilized plant comes out taller and the student records that both plants are the same or that the one without fertilizer grew taller, he has committed a Type II error because he has failed to reject the null hypothesis.

What are the Steps in Hypothesis Testing?

The following steps explain how we can test a hypothesis:

Step #1 - Define the Null and Alternative Hypotheses

Before making any test, we must first define what we are testing and what the default assumption is about the subject. In this article, we'll be testing if the average weight of 10-year-old children is more than 32kg.

Our null hypothesis is that 10 year old children weigh 32 kg on average. Our alternate hypothesis is that the average weight is more than 32kg. Ho denotes a null hypothesis, while H1 denotes an alternate hypothesis.

Step #2 - Choose a Significance Level

The significance level is a threshold for determining if the test is valid. It gives credibility to our hypothesis test to ensure we are not just luck-dependent but have enough evidence to support our claims. We usually set our significance level before conducting our tests. The criterion for determining our significance value is known as p-value.

A lower p-value means that there is stronger evidence against the null hypothesis, and therefore, a greater degree of significance. A p-value of 0.05 is widely accepted to be significant in most fields of science. P-values do not denote the probability of the outcome of the result, they just serve as a benchmark for determining whether our test result is due to chance. For our test, our p-value will be 0.05.

Step #3 - Collect Data and Calculate a Test Statistic

You can obtain your data from online data stores or conduct your research directly. Data can be scraped or researched online. The methodology might depend on the research you are trying to conduct.

We can calculate our test using any of the appropriate hypothesis tests. This can be a T-test, Z-test, Chi-squared, and so on. There are several hypothesis tests, each suiting different purposes and research questions. In this article, we'll use the T-test to run our hypothesis, but I'll explain the Z-test, and chi-squared too.

T-test is used for comparison of two sets of data when we don't know the population standard deviation. It's a parametric test, meaning it makes assumptions about the distribution of the data. These assumptions include that the data is normally distributed and that the variances of the two groups are equal. In a more simple and practical sense, imagine that we have test scores in a class for males and females, but we don't know how different or similar these scores are. We can use a t-test to see if there's a real difference.

The Z-test is used for comparison between two sets of data when the population standard deviation is known. It is also a parametric test, but it makes fewer assumptions about the distribution of data. The z-test assumes that the data is normally distributed, but it does not assume that the variances of the two groups are equal. In our class test example, with the t-test, we can say that if we already know how spread out the scores are in both groups, we can now use the z-test to see if there's a difference in the average scores.

The Chi-squared test is used to compare two or more categorical variables. The chi-squared test is a non-parametric test, meaning it does not make any assumptions about the distribution of data. It can be used to test a variety of hypotheses, including whether two or more groups have equal proportions.

Step #4 - Decide on the Null Hypothesis Based on the Test Statistic and Significance Level

After conducting our test and calculating the test statistic, we can compare its value to the predetermined significance level. If the test statistic falls beyond the significance level, we can decide to reject the null hypothesis, indicating that there is sufficient evidence to support our alternative hypothesis.

On the other contrary, if the test statistic does not exceed the significance level, we fail to reject the null hypothesis, signifying that we do not have enough statistical evidence to conclude in favor of the alternative hypothesis.

Step #5 - Interpret the Results

Depending on the decision made in the previous step, we can interpret the result in the context of our study and the practical implications. For our case study, we can interpret whether we have significant evidence to support our claim that the average weight of 10 year old children is more than 32kg or not.

For our test, we are generating random dummy data for the weight of the children. We'll use a t-test to evaluate whether our hypothesis is correct or not.

For a better understanding, let's look at what each block of code does.

The first block is the import statement, where we import numpy and scipy.stats . Numpy is a Python library used for scientific computing. It has a large library of functions for working with arrays. Scipy is a library for mathematical functions. It has a stat module for performing statistical functions, and that's what we'll be using for our t-test.

The weights of the children were generated at random since we aren't working with an actual dataset. The random module within the Numpy library provides a function for generating random numbers, which is randint .

The randint function takes three arguments. The first (20) is the lower bound of the random numbers to be generated. The second (40) is the upper bound, and the third (100) specifies the number of random integers to generate. That is, we are generating random weight values for 100 children. In real circumstances, these weight samples would have been obtained by taking the weight of the required number of children needed for the test.

Using the code above, we declared our null and alternate hypotheses stating the average weight of a 10-year-old in both cases.

t_stat and p_value are the variables in which we'll store the results of our functions. stats.ttest_1samp is the function that calculates our test. It takes in two variables, the first is the data variable that stores the array of weights for children, and the second (32) is the value against which we'll test the mean of our array of weights or dataset in cases where we are using a real-world dataset.

The code above prints both values for t_stats and p_value .

Lastly, we evaluated our p_value against our significance value, which is 0.05. If our p_value is less than 0.05, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis. Below is the output of this program. Our null hypothesis was rejected.

In this article, we discussed the importance of hypothesis testing. We highlighted how science has advanced human knowledge and civilization through formulating and testing hypotheses.

We discussed Type I and Type II errors in hypothesis testing and how they underscore the importance of careful consideration and analysis in scientific inquiry. It reinforces the idea that conclusions should be drawn based on thorough statistical analysis rather than assumptions or biases.

We also generated a sample dataset using the relevant Python libraries and used the needed functions to calculate and test our alternate hypothesis.

Thank you for reading! Please follow me on LinkedIn where I also post more data related content.

Technical support engineer with 4 years of experience & 6 months in data analytics. Passionate about data science, programming, & statistics.

If you read this far, thank the author to show them you care. Say Thanks

Learn to code for free. freeCodeCamp's open source curriculum has helped more than 40,000 people get jobs as developers. Get started

Python Topics

Popular articles.

• Reading And Writing (Apr 27, 2024)
• Asyncio (Apr 26, 2024)
• Metaclasses (May 04, 2024)
• Type Hints (May 04, 2024)
• Deep Learning Framework (Apr 26, 2024)

Hypothesis Testing

Table of Contents

Introduction to Hypothesis Testing

Understanding null and alternative hypotheses, types of hypothesis tests in python, steps in hypothesis testing, implementing hypothesis testing in python using scipy, interpreting the results of hypothesis testing, common errors in hypothesis testing, real-world applications of hypothesis testing, limitations and assumptions of hypothesis testing, advanced concepts in hypothesis testing.

Hypothesis testing is a fundamental concept in statistics that is used to validate assumptions or claims about a population based on sample data. It is a structured method that allows us to test if our assumptions about a dataset are correct or not. In the context of Python and SciPy, hypothesis testing becomes a powerful tool that can aid in data analysis and decision making. It is used in a wide range of fields, from business and marketing to healthcare and scientific research. The process of hypothesis testing involves forming an initial claim about a population parameter, collecting sample data related to this claim, and then using statistical analysis to determine whether or not the data supports the claim. This initial claim is known as the null hypothesis, and it is the assumption that there is no significant difference between specified populations, or no association among groups.

Hypothesis testing is a critical skill for anyone working with data, as it provides a methodical way to make inferences or predictions about a dataset. It is a key component of any data scientist's or analyst's toolkit. In the following sections, we will delve deeper into the intricacies of hypothesis testing, including understanding null and alternative hypotheses, the different types of hypothesis tests, and how to implement and interpret these tests using Python and SciPy.

In the realm of hypothesis testing, two key concepts are the null hypothesis and the alternative hypothesis. These hypotheses serve as the foundation for any hypothesis test and are used to compare the sample data against the expected results. The null hypothesis, often denoted as H0, is the initial claim about a population parameter. It is the assumption that there is no significant difference between specified populations, or no association among groups. In other words, it's the status quo that we assume to be true before we collect any data.

On the other hand, the alternative hypothesis, denoted as H1 or Ha, is what you might believe to be true or hope to prove true. It is the opposite of the null hypothesis and represents a statement of inequality. For instance, consider a scenario where you are testing a new drug to lower cholesterol. The null hypothesis might be that the drug has no effect on cholesterol levels, while the alternative hypothesis would be that the drug does have an effect on cholesterol levels.

It's important to note that in hypothesis testing, we don't directly prove the alternative hypothesis. Instead, we use the data to reject or fail to reject the null hypothesis.

• If the data provides enough evidence against the null hypothesis, we reject the null hypothesis and accept the alternative hypothesis.
• If the data does not provide enough evidence against the null hypothesis, we fail to reject the null hypothesis. This does not necessarily prove the null hypothesis true; it simply means that there is not enough evidence against it based on our data.

Understanding these two hypotheses is crucial as they form the basis of hypothesis testing and guide the direction of your research.

There are several types of hypothesis tests that you can perform in Python using the SciPy library. The type of test you choose depends on your data and the nature of your research question. Here are some of the most common types:

1. One-Sample T-Test: This test is used when you want to compare the mean of a sample to a known value. For example, you might want to test if the average height of a group of people is significantly different from the national average.

2. Two-Sample T-Test: This test is used when you want to compare the means of two different samples. For example, you might want to test if there is a significant difference in the average heights of men and women.

3. Paired T-Test: This test is used when you want to compare the means of the same group at two different times. For example, you might want to test if a group of people's heights significantly change after a certain period.

4. Chi-Square Test: This test is used when you want to see if there is a relationship between two categorical variables. For example, you might want to test if there is a relationship between gender and preference for a certain product.

These are just a few examples of the types of hypothesis tests you can perform in Python using the SciPy library. The choice of test depends on the nature of your data and the specific question you are trying to answer.

Hypothesis testing is a systematic process that involves several steps. Here is a step-by-step guide to conducting a hypothesis test:

1. Formulate the Null and Alternative Hypotheses: The first step in hypothesis testing is to set up the null and alternative hypotheses. The null hypothesis is the assumption that there is no significant difference between specified populations, or no association among groups. The alternative hypothesis is the opposite of the null hypothesis.

2. Choose the Significance Level: The significance level, denoted by alpha (α), is the probability of rejecting the null hypothesis when it is true. It is usually set at 0.05, which means that there is a 5% risk of rejecting the null hypothesis when it is true.

3. Select the Appropriate Test: Depending on the nature of your data and the specific question you are trying to answer, you will need to choose the appropriate statistical test.

4. Calculate the Test Statistic and P-Value: The test statistic is a numerical value that is used to make a decision about the null hypothesis. The p-value is the probability of obtaining a result as extreme as the observed result, given that the null hypothesis is true.

5. Make a Decision: Based on the p-value and the significance level, you will make a decision about the null hypothesis. If the p-value is less than or equal to the significance level, you reject the null hypothesis. If the p-value is greater than the significance level, you fail to reject the null hypothesis.

These steps provide a general framework for conducting hypothesis tests. However, the specific details may vary depending on the nature of your data and the specific test you are using.

Python, with its powerful libraries like SciPy, makes it easy to implement hypothesis testing. Let's walk through an example of how to perform a two-sample t-test using SciPy. Assume we have two sets of data, representing the heights of men and women. We want to test if there is a significant difference in the average heights of men and women. First, we need to import the necessary libraries and create our data:

Next, we perform the two-sample t-test using the `ttest_ind` function from the `scipy.stats` module:

The `ttest_ind` function returns two values: the t-statistic and the p-value. The t-statistic is a measure of the difference between the two means relative to the variability in the data. The p-value is the probability of observing a t-statistic as extreme as the one calculated, assuming the null hypothesis is true. Finally, we make a decision based on the p-value. If the p-value is less than our chosen significance level (usually 0.05), we reject the null hypothesis. If the p-value is greater than our significance level, we fail to reject the null hypothesis.

This is a basic example of how to implement hypothesis testing in Python using SciPy. Depending on the nature of your data and the specific question you are trying to answer, you may need to use a different test or adjust the process accordingly.

Interpreting the results of a hypothesis test involves understanding the p-value and making a decision based on this value. The p-value is a probability that measures the evidence against the null hypothesis. A small p-value (typically ≤ 0.05) indicates strong evidence against the null hypothesis, so you reject the null hypothesis. A large p-value (> 0.05) indicates weak evidence against the null hypothesis, so you fail to reject the null hypothesis. Let's consider an example where we have performed a two-sample t-test to compare the average heights of men and women. We have calculated the p-value as follows:

Assume that the output of the above code is `p-value: 0.02`. This p-value is less than the commonly used significance level of 0.05. Therefore, we reject the null hypothesis. In the context of our example, this means that we have found a statistically significant difference in the average heights of men and women. However, it's important to note that rejecting the null hypothesis doesn't prove the alternative hypothesis. It simply means that there is enough evidence to suggest that the null hypothesis is unlikely. On the other hand, if the p-value was greater than 0.05, we would fail to reject the null hypothesis. This doesn't prove that the null hypothesis is true, but rather that there is not enough evidence against it based on our data. In conclusion, interpreting the results of a hypothesis test involves understanding the calculated p-value, comparing it to a chosen significance level, and making a decision about the null hypothesis based on this comparison.

While hypothesis testing is a powerful tool in statistics, it is not immune to errors. There are two types of errors that can occur in hypothesis testing: Type I error and Type II error.

1. Type I Error (False Positive): This occurs when we reject the null hypothesis when it is actually true. The probability of making a Type I error is equal to the significance level (α). For example, if we set our significance level at 0.05, we are accepting a 5% chance of making a Type I error.

2. Type II Error (False Negative): This occurs when we fail to reject the null hypothesis when it is actually false. The probability of making a Type II error is denoted by beta (β). The power of a test, which is 1 - β, is the probability of correctly rejecting a false null hypothesis. It's important to note that these errors are inversely related. Reducing the risk of a Type I error increases the risk of a Type II error, and vice versa.

The challenge in hypothesis testing is to balance these errors. Another common mistake in hypothesis testing is the misuse of the p-value. A p-value is not the probability that the null hypothesis is true. Instead, it is the probability of observing a result as extreme as the one obtained, assuming the null hypothesis is true. Misinterpreting the p-value can lead to incorrect conclusions. Lastly, failing to consider the assumptions of the test can also lead to errors. For example, the t-test assumes that the data follows a normal distribution. If this assumption is violated, the results of the test may not be valid. In conclusion, while hypothesis testing is a powerful tool, it is crucial to understand the potential errors and pitfalls to avoid incorrect conclusions.

Hypothesis testing is a fundamental tool in statistics that finds application in a wide range of fields. Here are a few examples of how hypothesis testing is used in real-world scenarios:

1. Healthcare : In the field of medicine, hypothesis testing is often used to test the effectiveness of a new treatment or drug. For instance, a researcher might want to test whether a new drug lowers cholesterol levels more than the existing standard treatment. The null hypothesis would be that the new drug has no effect, while the alternative hypothesis would be that the new drug does have an effect. 2. Business : Businesses often use hypothesis testing to make informed decisions. For example, a company might want to test whether a new marketing campaign has increased sales. The null hypothesis would be that the campaign has had no effect on sales, while the alternative hypothesis would be that the campaign has increased sales. 3. Quality Control : In manufacturing, hypothesis testing can be used to ensure quality control. For instance, a manufacturer might want to test whether the average weight of a product is equal to the desired weight. The null hypothesis would be that the average weight is equal to the desired weight, while the alternative hypothesis would be that the average weight is not equal to the desired weight. 4. Education : In education, researchers might use hypothesis testing to evaluate the effectiveness of a new teaching method. The null hypothesis would be that the new method has no effect on student performance, while the alternative hypothesis would be that the new method improves student performance.

These are just a few examples of the many ways in which hypothesis testing is used in real-world scenarios. Regardless of the field, hypothesis testing provides a structured method for making informed decisions based on data.

While hypothesis testing is a powerful tool in statistics, it comes with certain limitations and assumptions that need to be considered:

1. Assumptions: Different hypothesis tests have different assumptions that need to be met for the test to be valid. For example, a t-test assumes that the data follows a normal distribution. If these assumptions are not met, the results of the test may not be valid. 2. P-value Misinterpretation: The p-value is often misunderstood. A common misconception is that the p-value is the probability that the null hypothesis is true. However, the p-value is actually the probability of observing a result as extreme as the one obtained, assuming the null hypothesis is true. 3. Dependence on Sample Size: The results of a hypothesis test can be greatly affected by the sample size. A large sample size can detect even small differences as significant, while a small sample size may not have enough power to detect significant differences. 4. Binary Outcome: Hypothesis testing provides a binary outcome - either reject the null hypothesis or fail to reject the null hypothesis. This does not provide information on the magnitude of the effect or the importance of the result. 5. Does Not Prove the Null Hypothesis: Failing to reject the null hypothesis does not prove that the null hypothesis is true. It simply means that there is not enough evidence against it based on the data. 6. Risk of Errors: Hypothesis testing is subject to Type I and Type II errors. A Type I error occurs when the null hypothesis is rejected when it is true, and a Type II error occurs when the null hypothesis is not rejected when it is false.

In conclusion, while hypothesis testing is a valuable tool in statistics, it is important to understand its limitations and assumptions to correctly interpret the results and avoid errors.

Beyond the basic concepts of hypothesis testing, there are several advanced topics that can provide deeper insights and more robust results. Here are a few:

1. Power Analysis: Power is the probability of correctly rejecting a false null hypothesis (avoiding a Type II error). Power analysis can be used to determine the sample size required to detect an effect of a given size with a certain degree of confidence. 2. Multiple Testing Problem: When multiple hypotheses are tested, the risk of a Type I error increases. Several methods, such as the Bonferroni correction or the False Discovery Rate (FDR) control, have been developed to adjust the significance level and control the risk of Type I errors. 3. Non-parametric Tests: Traditional hypothesis tests like the t-test or ANOVA rely on certain assumptions (like normality of data). When these assumptions are not met, non-parametric tests like the Mann-Whitney U test or the Kruskal-Wallis test can be used. These tests do not rely on strict assumptions about the data distribution. 4. Bootstrap Hypothesis Testing: Bootstrapping is a resampling technique that can be used to estimate the sampling distribution of a statistic. It can be used for hypothesis testing when the assumptions of traditional tests are not met or when the sampling distribution is unknown or complex. 5. Bayesian Hypothesis Testing: Unlike traditional frequentist hypothesis testing, Bayesian testing incorporates prior knowledge or beliefs about the parameters. This can provide a more intuitive and informative result, giving a probability distribution for the parameter rather than a binary decision.

Each of these advanced concepts has its own strengths and weaknesses, and the choice of which to use depends on the specific situation and the nature of the data. Understanding these advanced concepts can help you perform more robust and reliable hypothesis tests.

Adventures in Machine Learning

Mastering hypothesis testing in python: a step-by-step guide.

Hypothesis Testing in Python: AnHypothesis testing is a statistical technique that allows us to draw conclusions about a population based on a sample of data. It is often used in fields like medicine, psychology, and economics to test the effectiveness of new treatments, analyze consumer behavior, or estimate the impact of policy changes.

In Python, hypothesis testing is facilitated by modules such as scipy.stats and statsmodels.stats. In this article, we’ll explore three examples of hypothesis testing in Python: the one sample t-test, the two sample t-test, and the paired samples t-test.

For each test, we’ll provide a brief explanation of the underlying concepts, an example of a research question that can be answered using the test, and a step-by-step guide to performing the test in Python. Let’s get started!

One Sample t-test

The one sample t-test is used to compare a sample mean to a known or hypothesized population mean. This allows us to determine whether the sample mean is significantly different from the population mean.

The test assumes that the data are normally distributed and that the sample is randomly drawn from the population. Example research question: Is the mean weight of a species of turtle significantly different from a known or hypothesized value?

Step-by-step guide:

1. Define the null hypothesis (H0) and alternative hypothesis (Ha).

The null hypothesis is typically that the sample mean is equal to the population mean. The alternative hypothesis is that they are not equal.

For example:

H0: The mean weight of a species of turtle is 100 grams. Ha: The mean weight of a species of turtle is not 100 grams.

2. Collect a random sample of data.

This can be done using Python’s random module or by importing data from a file. For example:

weight_sample = [95, 105, 110, 98, 102, 116, 101, 99, 104, 108]

Calculate the sample mean (x), sample standard deviation (s), and standard error (SE). For example:

x = sum(weight_sample)/len(weight_sample)

s = np.std(weight_sample)

SE = s / (len(weight_sample)**0.5)

Calculate the t-value using the formula: t = (x – ) / (SE), where is the hypothesized population mean. For example:

t = (x – 100) / SE

Calculate the p-value using a t-distribution table or a Python function like scipy.stats.ttest_1samp(). For example:

p_value = scipy.stats.ttest_1samp(weight_sample, 100).pvalue

Compare the p-value to the level of significance (), typically set to 0.05. If the p-value is less than , reject the null hypothesis and conclude that there is sufficient evidence to support the alternative hypothesis.

If the p-value is greater than , fail to reject the null hypothesis and conclude that there is insufficient evidence to support the alternative hypothesis. For example:

if p_value < 0.05:

print(“Reject the null hypothesis.”)

print(“Fail to reject the null hypothesis.”)

Two Sample t-test

The two sample t-test is used to compare the means of two independent samples. This allows us to determine whether the means are significantly different from each other.

The test assumes that the data are normally distributed and that the samples are randomly drawn from their respective populations. Example research question: Is the mean weight of two different species of turtles significantly different from each other?

The null hypothesis is typically that the sample means are equal. The alternative hypothesis is that they are not equal.

H0: The mean weight of species A is equal to the mean weight of species B. Ha: The mean weight of species A is not equal to the mean weight of species B.

2. Collect two random samples of data.

species_a = [95, 105, 110, 98, 102]

species_b = [116, 101, 99, 104, 108]

Calculate the sample means (x1, x2), sample standard deviations (s1, s2), and pooled standard error (SE). For example:

x1 = sum(species_a)/len(species_a)

x2 = sum(species_b)/len(species_b)

s1 = np.std(species_a)

s2 = np.std(species_b)

n1 = len(species_a)

n2 = len(species_b)

SE = (((n1-1)*s1**2 + (n2-1)*s2**2)/(n1+n2-2))**0.5 * (1/n1 + 1/n2)**0.5

Calculate the t-value using the formula: t = (x1 – x2) / (SE), where x1 and x2 are the sample means. For example:

t = (x1 – x2) / SE

Calculate the p-value using a t-distribution table or a Python function like scipy.stats.ttest_ind(). For example:

p_value = scipy.stats.ttest_ind(species_a, species_b).pvalue

Paired Samples t-test

The paired samples t-test is used to compare the means of two related samples. This allows us to determine whether the means are significantly different from each other, while accounting for individual differences between the samples.

The test assumes that the differences between paired observations are normally distributed. Example research question: Is there a significant difference in the max vertical jump of basketball players before and after a training program?

The null hypothesis is typically that the mean difference is equal to zero. The alternative hypothesis is that it is not equal to zero.

H0: The mean difference in max vertical jump before and after training is zero. Ha: The mean difference in max vertical jump before and after training is not zero.

2. Collect two related samples of data.

This can be done by measuring the same variable in the same subjects before and after a treatment or intervention. For example:

before = [72, 69, 77, 71, 76]

after = [80, 70, 75, 74, 78]

Calculate the differences between the paired observations and the sample mean difference (d), sample standard deviation (s), and standard error (SE). For example:

differences = [after[i]-before[i] for i in range(len(before))]

d = sum(differences)/len(differences)

s = np.std(differences)

SE = s / (len(differences)**0.5)

Calculate the t-value using the formula: t = (d – ) / (SE), where is the hypothesized population mean difference (usually zero). For example:

t = (d – 0) / SE

Calculate the p-value using a t-distribution table or a Python function like scipy.stats.ttest_rel(). For example:

p_value = scipy.stats.ttest_rel(after, before).pvalue

In this article, we’ve explored three examples of hypothesis testing in Python: the one sample t-test, the two sample t-test, and the paired samples t-test. Hypothesis testing is a powerful tool for making inferences about populations based on samples of data.

By following the steps outlined in each example, you can conduct your own hypothesis tests in Python and draw meaningful conclusions from your data.

Two Sample t-test in Python

The two sample t-test is used to compare two independent samples and determine if there is a significant difference between the means of the two populations. In this test, the null hypothesis is that the means of the two samples are equal, while the alternative hypothesis is that they are not equal.

Example research question: Is the mean weight of two different species of turtles significantly different from each other? Step-by-step guide:

Define the null hypothesis (H0) and alternative hypothesis (Ha). The null hypothesis is that the mean weight of the two turtle species is the same.

The alternative hypothesis is that they are not equal. For example:

H0: The mean weight of species A is equal to the mean weight of species B.

Ha: The mean weight of species A is not equal to the mean weight of species B. 2.

Collect a random sample of data for each species. For example:

species_a = [4.3, 3.9, 5.1, 4.6, 4.2, 4.8]

species_b = [4.9, 5.2, 5.5, 5.3, 5.0, 4.7]

Calculate the sample mean (x1, x2), sample standard deviation (s1, s2), and pooled standard error (SE). For example:

import numpy as np

from scipy.stats import ttest_ind

x1 = np.mean(species_a)

x2 = np.mean(species_b)

SE = np.sqrt(s1**2/n1 + s2**2/n2)

4. Calculate the t-value using the formula: t = (x1 – x2) / (SE), where x1 and x2 are the sample means.

5. Calculate the p-value using a t-distribution table or a Python function like ttest_ind().

p_value = ttest_ind(species_a, species_b).pvalue

6. Compare the p-value to the level of significance (), typically set to 0.05.

If the p-value is less than , reject the null hypothesis and conclude that there is sufficient evidence to support the alternative hypothesis. If the p-value is greater than , fail to reject the null hypothesis and conclude that there is insufficient evidence to support the alternative hypothesis.

alpha = 0.05

if p_value < alpha:

In this example, if the p-value is less than 0.05, we would reject the null hypothesis and conclude that there is a significant difference between the mean weight of the two turtle species.

Paired Samples t-test in Python

The paired samples t-test is used to compare the means of two related samples. In this test, the null hypothesis is that the difference between the two means is equal to zero, while the alternative hypothesis is that they are not equal.

Example research question: Is there a significant difference in the max vertical jump of basketball players before and after a training program? Step-by-step guide:

Define the null hypothesis (H0) and alternative hypothesis (Ha). The null hypothesis is that the mean difference in max vertical jump before and after the training program is zero.

The alternative hypothesis is that it is not zero. For example:

H0: The mean difference in max vertical jump before and after the training program is zero.

Ha: The mean difference in max vertical jump before and after the training program is not zero. 2.

Collect two related samples of data, such as the max vertical jump of basketball players before and after a training program. For example:

before_training = [58, 64, 62, 70, 68]

after_training = [62, 66, 64, 74, 70]

differences = [after_training[i]-before_training[i] for i in range(len(before_training))]

d = np.mean(differences)

n = len(differences)

SE = s / np.sqrt(n)

Calculate the p-value using a t-distribution table or a Python function like ttest_rel(). For example:

p_value = ttest_rel(after_training, before_training).pvalue

In this example, if the p-value is less than 0.05, we would reject the null hypothesis and conclude that there is a significant difference in the max vertical jump of basketball players before and after the training program.

Hypothesis testing is an essential tool in statistical analysis, which gives us insights into populations based on limited data. The two sample t-test and paired samples t-test are two popular statistical methods that enable researchers to compare means of samples and determine whether they are significantly different.

With the help of Python, hypothesis testing in practice is made more accessible and convenient than ever before. In this article, we have provided a step-by-step guide to performing these tests in Python, enabling researchers to perform rigorous analyses that generate meaningful and accurate results.

In conclusion, hypothesis testing in Python is a crucial step in making conclusions about populations based on data samples. The three common hypothesis tests in Python; one-sample t-test, two-sample t-test, and paired samples t-test can be effectively applied to explore various research questions.

By setting null and alternative hypotheses, collecting data, calculating mean and standard deviation values, computing t-value, and comparing it with the set significance level of , we can determine if there’s enough evidence to reject the null hypothesis. With the use of such powerful methods, scientists can give more accurate and informed conclusions to real-world problems and take critical decisions when needed.

Continual learning and expertise with hypothesis testing in Python tools can enable researchers to leverage this powerful statistical tool for better outcomes.

Popular Posts

Maximizing the power of python list comprehensions, streamlining data analysis: copying csv files to the clipboard with pandas, mastering dummy variables: a guide to creating them using python.

• Terms & Conditions
• Privacy Policy

Statistical Hypothesis Testing: A Comprehensive Guide

We’ve all heard it – “ go to college to get a good job .” The assumption is that higher education leads straight to higher incomes. Elite Indian institutes like the IITs and IIMs are even judged based on the average starting salaries of their graduates. But is this direct connection between schooling and income actually true?

Intuitively, it seems believable. But how can we really prove this assumption that more school = more money? Is there hard statistical evidence either way? Turns out, there are methods to scientifically test widespread beliefs like this – what statisticians call hypothesis testing.

In this article, we’ll dig into the concept of hypothesis testing and the tools to rigorously question conventional wisdom: null and alternate hypotheses, one and two-tailed tests, paired sample tests, and more.

Statistical hypothesis testing allows researchers to make inferences about populations based on sample data. It involves setting up a null hypothesis, choosing a confidence level, calculating a p-value, and conducting tests such as two-tailed, one-tailed, or paired sample tests to draw conclusions.

What is Hypothesis Testing?

Statistical Hypothesis Testing is a method used to make inferences about a population based on sample data. Before we move ahead and understand what Hypothesis Testing is, we need to understand some basic terms.

Null Hypothesis

The Null Hypothesis is generally where we start our journey. Null Hypotheses are statements that are generally accepted or statements that you want to challenge. Since it is generally accepted that income level is positively correlated with quality of education, this will be our Null Hypothesis. It is denoted by H 0 .

H 0 : Income levels are positively correlated with quality of education.

Alternate Hypothesis

The Alternate Hypothesis is the opposite of the Null hypothesis. An alternate Hypothesis is what we want to prove as a researcher and is not generally accepted by society. An alternate hypothesis is denoted H a . The alternate hypothesis of the above is given below.

H a : Income levels are negatively correlated with the quality of education.

Confidence Level (1- α )

Confidence Levels represent the probability that the range of values contains the true parameter value. The most common confidence levels are 95% and 99%. It can be interpreted that our test is 95% accurate if our confidence level is 95%. It is denoted by 1-α.

p-value ( p )

The p-value represents the probability of obtaining test results at least as extreme as the results actually observed, under the assumption that the null hypothesis is correct. A lower p-value means fewer chances for our observed result to happen. If our p-value is less than α , our null hypothesis is rejected, otherwise null hypothesis is accepted.

Types of Hypothesis Tests

Since we are equipped with the basic terms, let’s go ahead and conduct some hypothesis tests.

Conducting a Two-Tailed Hypothesis Test

In a two-tailed hypothesis test, our analysis can go in either direction i.e. either more than or less than our observed value. For example, a medical researcher testing out the effects of a placebo wants to know whether it increases or decreases blood pressure. Let’s look at its Python implementation.

In the above code, we want to know if the group study method is an effective way to study or not. Therefore our null and alternate hypotheses are as follows.

• H 0 : The Group study method is not an effective way to study .
• H a : The group study method is an effective way to study .

Since the p-value is greater than α , we fail to reject the null hypothesis. Therefore the group study method is not an effective way to study.

Recommended: Hypothesis Testing in Python: Finding the critical value of T

In a one-tailed hypothesis test, we have certain expectations in which way our observed value will move i.e. higher or lower. For example, our researchers want to know if a particular medicine lowers our cholesterol level. Let’s look at its Python code.

Here our null and alternate hypothesis tests are given below.

• H 0 : The Group study method does not increase our marks.
• H a : The group study method increases our marks.

Since the p-value is greater than α , we fail to reject the null hypothesis. Therefore the group study method does not increase our marks.

A paired sample test compares two sets of observations and then provides us with a conclusion. For example, we need to know whether the reaction time of our participants increases after consuming caffeine. Let’s look at another example with a Python code as well.

Similar to the above hypothesis tests, we consider the group study method here as well. Our null and alternate hypotheses are as follows.

• H 0 : The group study method does not provide us with significant differences in our scores.
• H a : The group study method gives us significant differences in our scores.

Since the p-value is greater than α , we fail to reject the null hypothesis.

Here you go! Now you are equipped to perform statistical hypothesis testing on different samples and draw out different conclusions. You need to collect data and decide on null and alternate hypotheses. Furthermore, based on the predetermined hypothesis, you need to decide on which type of test to perform. Statistical hypothesis testing is one of the most powerful tools in the world of research.

Now that you have a grasp on statistical hypothesis testing, how will you apply these concepts to your own research or data analysis projects? What hypotheses are you eager to test?

Do check out: How to find critical value in Python

Learning Statistics with Python

Hypothesis Testing

12. hypothesis testing #.

The process of induction is the process of assuming the simplest law that can be made to harmonize with our experience. This process, however, has no logical foundation but only a psychological one. It is clear that there are no grounds for believing that the simplest course of events will really happen. It is an hypothesis that the sun will rise tomorrow: and this means that we do not know whether it will rise. – Ludwig Wittgenstein [ 1 ]

In the last chapter, I discussed the ideas behind estimation, which is one of the two “big ideas” in inferential statistics. It’s now time to turn out attention to the other big idea, which is hypothesis testing . In its most abstract form, hypothesis testing really a very simple idea: the researcher has some theory about the world, and wants to determine whether or not the data actually support that theory. However, the details are messy, and most people find the theory of hypothesis testing to be the most frustrating part of statistics. The structure of the chapter is as follows. Firstly, I’ll describe how hypothesis testing works, in a fair amount of detail, using a simple running example to show you how a hypothesis test is “built”. I’ll try to avoid being too dogmatic while doing so, and focus instead on the underlying logic of the testing procedure. [ 2 ] Afterwards, I’ll spend a bit of time talking about the various dogmas, rules and heresies that surround the theory of hypothesis testing.

12.1. A menagerie of hypotheses #

Eventually we all succumb to madness. For me, that day will arrive once I’m finally promoted to full professor. Safely ensconced in my ivory tower, happily protected by tenure, I will finally be able to take leave of my senses (so to speak), and indulge in that most thoroughly unproductive line of psychological research: the search for extrasensory perception (ESP). [ 3 ]

Let’s suppose that this glorious day has come. My first study is a simple one, in which I seek to test whether clairvoyance exists. Each participant sits down at a table, and is shown a card by an experimenter. The card is black on one side and white on the other. The experimenter takes the card away, and places it on a table in an adjacent room. The card is placed black side up or white side up completely at random, with the randomisation occurring only after the experimenter has left the room with the participant. A second experimenter comes in and asks the participant which side of the card is now facing upwards. It’s purely a one-shot experiment. Each person sees only one card, and gives only one answer; and at no stage is the participant actually in contact with someone who knows the right answer. My data set, therefore, is very simple. I have asked the question of \(N\) people, and some number \(X\) of these people have given the correct response. To make things concrete, let’s suppose that I have tested \(N = 100\) people, and \(X = 62\) of these got the answer right… a surprisingly large number, sure, but is it large enough for me to feel safe in claiming I’ve found evidence for ESP? This is the situation where hypothesis testing comes in useful. However, before we talk about how to test hypotheses, we need to be clear about what we mean by hypotheses.

12.1.1. Research hypotheses versus statistical hypotheses #

The first distinction that you need to keep clear in your mind is between research hypotheses and statistical hypotheses. In my ESP study, my overall scientific goal is to demonstrate that clairvoyance exists. In this situation, I have a clear research goal: I am hoping to discover evidence for ESP. In other situations I might actually be a lot more neutral than that, so I might say that my research goal is to determine whether or not clairvoyance exists. Regardless of how I want to portray myself, the basic point that I’m trying to convey here is that a research hypothesis involves making a substantive, testable scientific claim… if you are a psychologist, then your research hypotheses are fundamentally about psychological constructs. Any of the following would count as research hypotheses :

Listening to music reduces your ability to pay attention to other things. This is a claim about the causal relationship between two psychologically meaningful concepts (listening to music and paying attention to things), so it’s a perfectly reasonable research hypothesis.

Intelligence is related to personality . Like the last one, this is a relational claim about two psychological constructs (intelligence and personality), but the claim is weaker: correlational not causal.

Intelligence is speed of information processing . This hypothesis has a quite different character: it’s not actually a relational claim at all. It’s an ontological claim about the fundamental character of intelligence (and I’m pretty sure it’s wrong). It’s worth expanding on this one actually: It’s usually easier to think about how to construct experiments to test research hypotheses of the form “does X affect Y?” than it is to address claims like “what is X?” And in practice, what usually happens is that you find ways of testing relational claims that follow from your ontological ones. For instance, if I believe that intelligence is speed of information processing in the brain, my experiments will often involve looking for relationships between measures of intelligence and measures of speed. As a consequence, most everyday research questions do tend to be relational in nature, but they’re almost always motivated by deeper ontological questions about the state of nature.

Notice that in practice, my research hypotheses could overlap a lot. My ultimate goal in the ESP experiment might be to test an ontological claim like “ESP exists”, but I might operationally restrict myself to a narrower hypothesis like “Some people can `see’ objects in a clairvoyant fashion”. That said, there are some things that really don’t count as proper research hypotheses in any meaningful sense:

Love is a battlefield . This is too vague to be testable. While it’s okay for a research hypothesis to have a degree of vagueness to it, it has to be possible to operationalise your theoretical ideas. Maybe I’m just not creative enough to see it, but I can’t see how this can be converted into any concrete research design. If that’s true, then this isn’t a scientific research hypothesis, it’s a pop song. That doesn’t mean it’s not interesting – a lot of deep questions that humans have fall into this category. Maybe one day science will be able to construct testable theories of love, or to test to see if God exists, and so on; but right now we can’t, and I wouldn’t bet on ever seeing a satisfying scientific approach to either.

The first rule of tautology club is the first rule of tautology club . This is not a substantive claim of any kind. It’s true by definition. No conceivable state of nature could possibly be inconsistent with this claim. As such, we say that this is an unfalsifiable hypothesis, and as such it is outside the domain of science. Whatever else you do in science, your claims must have the possibility of being wrong.

More people in my experiment will say “yes” than “no” . This one fails as a research hypothesis because it’s a claim about the data set, not about the psychology (unless of course your actual research question is whether people have some kind of “yes” bias!). As we’ll see shortly, this hypothesis is starting to sound more like a statistical hypothesis than a research hypothesis.

As you can see, research hypotheses can be somewhat messy at times; and ultimately they are scientific claims. Statistical hypotheses are neither of these two things. Statistical hypotheses must be mathematically precise, and they must correspond to specific claims about the characteristics of the data-generating mechanism (i.e., the “population”). Even so, the intent is that statistical hypotheses bear a clear relationship to the substantive research hypotheses that you care about! For instance, in my ESP study my research hypothesis is that some people are able to see through walls or whatever. What I want to do is to “map” this onto a statement about how the data were generated. So let’s think about what that statement would be. The quantity that I’m interested in within the experiment is \(P(\mbox{"correct"})\) , the true-but-unknown probability with which the participants in my experiment answer the question correctly. Let’s use the Greek letter \(\theta\) (theta) to refer to this probability. Here are four different statistical hypotheses:

If ESP doesn’t exist and if my experiment is well designed, then my participants are just guessing. So I should expect them to get it right half of the time and so my statistical hypothesis is that the true probability of choosing correctly is \(\theta = 0.5\) .

Alternatively, suppose ESP does exist and participants can see the card. If that’s true, people will perform better than chance. The statistical hypotheis would be that \(\theta > 0.5\) .

A third possibility is that ESP does exist, but the colours are all reversed and people don’t realise it (okay, that’s wacky, but you never know…). If that’s how it works then you’d expect people’s performance to be below chance. This would correspond to a statistical hypothesis that \(\theta < 0.5\) .

Finally, suppose ESP exists, but I have no idea whether people are seeing the right colour or the wrong one. In that case, the only claim I could make about the data would be that the probability of making the correct answer is not equal to 50. This corresponds to the statistical hypothesis that \(\theta \neq 0.5\) .

All of these are legitimate examples of a statistical hypothesis because they are statements about a population parameter and are meaningfully related to my experiment.

What this discussion makes clear, I hope, is that when attempting to construct a statistical hypothesis test, the researcher actually has two quite distinct hypotheses to consider. First, he or she has a research hypothesis (a claim about psychology), and this corresponds to a statistical hypothesis (a claim about the data generating population). In my ESP example, these might be

My research hypothesis: “ESP exists”

My statistical hypothesis: \(\theta \neq 0.5\)

And the key thing to recognise is this: a statistical hypothesis test is a test of the statistical hypothesis, not the research hypothesis . If your study is badly designed, then the link between your research hypothesis and your statistical hypothesis is broken. To give a silly example, suppose that my ESP study was conducted in a situation where the participant can actually see the card reflected in a window; if that happens, I would be able to find very strong evidence that \(\theta \neq 0.5\) , but this would tell us nothing about whether “ESP exists”.

12.1.2. Null hypotheses and alternative hypotheses #

So far, so good. I have a research hypothesis that corresponds to what I want to believe about the world, and I can map it onto a statistical hypothesis that corresponds to what I want to believe about how the data were generated. It’s at this point that things get somewhat counterintuitive for a lot of people. Because what I’m about to do is invent a new statistical hypothesis (the “null” hypothesis, \(H_0\) ) that corresponds to the exact opposite of what I want to believe, and then focus exclusively on that, almost to the neglect of the thing I’m actually interested in (which is now called the “alternative” hypothesis, \(H_1\) ). In our ESP example, the null hypothesis is that \(\theta = 0.5\) , since that’s what we’d expect if ESP didn’t exist. My hope, of course, is that ESP is totally real, and so the alternative to this null hypothesis is \(\theta \neq 0.5\) . In essence, what we’re doing here is dividing up the possible values of \(\theta\) into two groups: those values that I really hope aren’t true (the null), and those values that I’d be happy with if they turn out to be right (the alternative). Having done so, the important thing to recognise is that the goal of a hypothesis test is not to show that the alternative hypothesis is (probably) true; the goal is to show that the null hypothesis is (probably) false. Most people find this pretty weird.

The best way to think about it, in my experience, is to imagine that a hypothesis test is a criminal trial [ 4 ] … the trial of the null hypothesis . The null hypothesis is the defendant, the researcher is the prosecutor, and the statistical test itself is the judge. Just like a criminal trial, there is a presumption of innocence: the null hypothesis is deemed to be true unless you, the researcher, can prove beyond a reasonable doubt that it is false. You are free to design your experiment however you like (within reason, obviously!), and your goal when doing so is to maximise the chance that the data will yield a conviction… for the crime of being false. The catch is that the statistical test sets the rules of the trial, and those rules are designed to protect the null hypothesis – specifically to ensure that if the null hypothesis is actually true, the chances of a false conviction are guaranteed to be low. This is pretty important: after all, the null hypothesis doesn’t get a lawyer. And given that the researcher is trying desperately to prove it to be false, someone has to protect it.

12.2. Two types of errors #

Before going into details about how a statistical test is constructed, it’s useful to understand the philosophy behind it. I hinted at it when pointing out the similarity between a null hypothesis test and a criminal trial, but I should now be explicit. Ideally, we would like to construct our test so that we never make any errors. Unfortunately, since the world is messy, this is never possible. Sometimes you’re just really unlucky: for instance, suppose you flip a coin 10 times in a row and it comes up heads all 10 times. That feels like very strong evidence that the coin is biased (and it is!), but of course there’s a 1 in 1024 chance that this would happen even if the coin was totally fair. In other words, in real life we always have to accept that there’s a chance that we did the wrong thing. As a consequence, the goal behind statistical hypothesis testing is not to eliminate errors, but to minimise them.

At this point, we need to be a bit more precise about what we mean by “errors”. Firstly, let’s state the obvious: it is either the case that the null hypothesis is true, or it is false; and our test will either reject the null hypothesis or retain it. [ 5 ] So, as the table below illustrates, after we run the test and make our choice, one of four things might have happened:

As a consequence there are actually two different types of error here. If we reject a null hypothesis that is actually true, then we have made a type I error . On the other hand, if we retain the null hypothesis when it is in fact false, then we have made a type II error .

Remember how I said that statistical testing was kind of like a criminal trial? Well, I meant it. A criminal trial requires that you establish “beyond a reasonable doubt” that the defendant did it. All of the evidentiary rules are (in theory, at least) designed to ensure that there’s (almost) no chance of wrongfully convicting an innocent defendant. The trial is designed to protect the rights of a defendant: as the English jurist William Blackstone famously said, it is “better that ten guilty persons escape than that one innocent suffer.” In other words, a criminal trial doesn’t treat the two types of error in the same way… punishing the innocent is deemed to be much worse than letting the guilty go free. A statistical test is pretty much the same: the single most important design principle of the test is to control the probability of a type I error, to keep it below some fixed probability. This probability, which is denoted \(\alpha\) , is called the significance level of the test (or sometimes, the size of the test). And I’ll say it again, because it is so central to the whole set-up… a hypothesis test is said to have significance level \(\alpha\) if the type I error rate is no larger than \(\alpha\) .

So, what about the type II error rate? Well, we’d also like to keep those under control too, and we denote this probability by \(\beta\) . However, it’s much more common to refer to the power of the test, which is the probability with which we reject a null hypothesis when it really is false, which is \(1-\beta\) . To help keep this straight, here’s the same table again, but with the relevant numbers added:

A “powerful” hypothesis test is one that has a small value of \(\beta\) , while still keeping \(\alpha\) fixed at some (small) desired level. By convention, scientists make use of three different \(\alpha\) levels: \(.05\) , \(.01\) and \(.001\) . Notice the asymmetry here~… the tests are designed to ensure that the \(\alpha\) level is kept small, but there’s no corresponding guarantee regarding \(\beta\) . We’d certainly like the type II error rate to be small, and we try to design tests that keep it small, but this is very much secondary to the overwhelming need to control the type I error rate. As Blackstone might have said if he were a statistician, it is “better to retain 10 false null hypotheses than to reject a single true one”. To be honest, I don’t know that I agree with this philosophy – there are situations where I think it makes sense, and situations where I think it doesn’t – but that’s neither here nor there. It’s how the tests are built.

12.3. Test statistics and sampling distributions #

At this point we need to start talking specifics about how a hypothesis test is constructed. To that end, let’s return to the ESP example. Let’s ignore the actual data that we obtained, for the moment, and think about the structure of the experiment. Regardless of what the actual numbers are, the form of the data is that \(X\) out of \(N\) people correctly identified the colour of the hidden card. Moreover, let’s suppose for the moment that the null hypothesis really is true: ESP doesn’t exist, and the true probability that anyone picks the correct colour is exactly \(\theta = 0.5\) . What would we expect the data to look like? Well, obviously, we’d expect the proportion of people who make the correct response to be pretty close to 50%. Or, to phrase this in more mathematical terms, we’d say that \(X/N\) is approximately \(0.5\) . Of course, we wouldn’t expect this fraction to be exactly 0.5: if, for example we tested \(N=100\) people, and \(X = 53\) of them got the question right, we’d probably be forced to concede that the data are quite consistent with the null hypothesis. On the other hand, if \(X = 99\) of our participants got the question right, then we’d feel pretty confident that the null hypothesis is wrong. Similarly, if only \(X=3\) people got the answer right, we’d be similarly confident that the null was wrong. Let’s be a little more technical about this: we have a quantity \(X\) that we can calculate by looking at our data; after looking at the value of \(X\) , we make a decision about whether to believe that the null hypothesis is correct, or to reject the null hypothesis in favour of the alternative. The name for this thing that we calculate to guide our choices is a test statistic .

Having chosen a test statistic, the next step is to state precisely which values of the test statistic would cause us to reject the null hypothesis, and which values would cause us to keep it. In order to do so, we need to determine what the sampling distribution of the test statistic would be if the null hypothesis were actually true (we talked about sampling distributions earlier). Why do we need this? Because this distribution tells us exactly what values of \(X\) our null hypothesis would lead us to expect. And therefore, we can use this distribution as a tool for assessing how closely the null hypothesis agrees with our data. Using random.binomial from numpy , we can estimate a binomial distribution with a \(\theta = 0.5\) , e.g. estimating from 10,000 trials:

How do we actually determine the sampling distribution of the test statistic? For a lot of hypothesis tests this step is actually quite complicated, and later on in the book you’ll see me being slightly evasive about it for some of the tests (some of them I don’t even understand myself). However, sometimes it’s very easy. And, fortunately for us, our ESP example provides us with one of the easiest cases. Our population parameter \(\theta\) is just the overall probability that people respond correctly when asked the question, and our test statistic \(X\) is the count of the number of people who did so, out of a sample size of \(N\) . We’ve seen a distribution like this before, in the section on the binomial distribution : that’s exactly what the binomial distribution describes! So, to use the notation and terminology that I introduced in that section, we would say that the null hypothesis predicts that \(X\) is binomially distributed, which is written

Since the null hypothesis states that \(\theta = 0.5\) and our experiment has \(N=100\) people, we have the sampling distribution we need. This sampling distribution is plotted in Figure fig-esp-estimation . No surprises really: the null hypothesis says that \(X=50\) is the most likely outcome, and it says that we’re almost certain to see somewhere between 40 and 60 correct responses.

12.4. Making decisions #

Okay, we’re very close to being finished. We’ve constructed a test statistic ( \(X\) ), and we chose this test statistic in such a way that we’re pretty confident that if \(X\) is close to \(N/2\) then we should retain the null, and if not we should reject it. The question that remains is this: exactly which values of the test statistic should we associate with the null hypothesis, and which exactly values go with the alternative hypothesis? In my ESP study, for example, I’ve observed a value of \(X=62\) . What decision should I make? Should I choose to believe the null hypothesis, or the alternative hypothesis?

12.4.1. Critical regions and critical values #

To answer this question, we need to introduce the concept of a critical region for the test statistic \(X\) . The critical region of the test corresponds to those values of \(X\) that would lead us to reject the null hypothesis (which is why the critical region is also sometimes called the rejection region). How do we find this critical region? Well, let’s consider what we know:

\(X\) should be very big or very small in order to reject the null hypothesis.

If the null hypothesis is true, the sampling distribution of \(X\) is Binomial \((0.5, N)\) .

If \(\alpha =.05\) , the critical region must cover 5% of this sampling distribution.

It’s important to make sure you understand this last point: the critical region corresponds to those values of \(X\) for which we would reject the null hypothesis, and the sampling distribution in question describes the probability that we would obtain a particular value of \(X\) if the null hypothesis were actually true. Now, let’s suppose that we chose a critical region that covers 20% of the sampling distribution, and suppose that the null hypothesis is actually true. What would be the probability of incorrectly rejecting the null? The answer is of course 20%. And therefore, we would have built a test that had an \(\alpha\) level of \(0.2\) . If we want \(\alpha = .05\) , the critical region is only allowed to cover 5% of the sampling distribution of our test statistic.

As it turns out, those three things uniquely solve the problem: our critical region consists of the most extreme values , known as the tails of the distribution. This is illustrated in fig-esp-critical . As it turns out, if we want \(\alpha = .05\) , then our critical regions correspond to \(X \leq 40\) and \(X \geq 60\) . [ 6 ] That is, if the number of people saying “true” is between 41 and 59, then we should retain the null hypothesis. If the number is between 0 to 40 or between 60 to 100, then we should reject the null hypothesis. The numbers 40 and 60 are often referred to as the critical values , since they define the edges of the critical region.

At this point, our hypothesis test is essentially complete: (1) we choose an \(\alpha\) level (e.g., \(\alpha = .05\) , (2) come up with some test statistic (e.g., \(X\) ) that does a good job (in some meaningful sense) of comparing \(H_0\) to \(H_1\) , (3) figure out the sampling distribution of the test statistic on the assumption that the null hypothesis is true (in this case, binomial) and then (4) calculate the critical region that produces an appropriate \(\alpha\) level (0-40 and 60-100). All that we have to do now is calculate the value of the test statistic for the real data (e.g., \(X = 62\) ) and then compare it to the critical values to make our decision. Since 62 is greater than the critical value of 60, we would reject the null hypothesis. Or, to phrase it slightly differently, we say that the test has produced a significant result.

12.4.2. A note on statistical “significance” #

Like other occult techniques of divination, the statistical method has a private jargon deliberately contrived to obscure its methods from non-practitioners. – Attributed to G. O. Ashley [ 7 ]

A very brief digression is in order at this point, regarding the word “significant”. The concept of statistical significance is actually a very simple one, but has a very unfortunate name. If the data allow us to reject the null hypothesis, we say that “the result is statistically significant ”, which is often shortened to “the result is significant”. This terminology is rather old, and dates back to a time when “significant” just meant something like “indicated”, rather than its modern meaning, which is much closer to “important”. As a result, a lot of modern readers get very confused when they start learning statistics, because they think that a “significant result” must be an important one. It doesn’t mean that at all. All that “statistically significant” means is that the data allowed us to reject a null hypothesis. Whether or not the result is actually important in the real world is a very different question, and depends on all sorts of other things.

12.4.3. The difference between one sided and two sided tests #

There’s one more thing I want to point out about the hypothesis test that I’ve just constructed. If we take a moment to think about the statistical hypotheses I’ve been using,

we notice that the alternative hypothesis covers both the possibility that \(\theta < .5\) and the possibility that \(\theta > .5\) . This makes sense if I really think that ESP could produce better-than-chance performance or worse-than-chance performance (and there are some people who think that). In statistical language, this is an example of a two-sided test . It’s called this because the alternative hypothesis covers the area on both “sides” of the null hypothesis, and as a consequence the critical region of the test covers both tails of the sampling distribution (2.5% on either side if \(\alpha =.05\) ), as illustrated earlier in fig-esp-critical .

However, that’s not the only possibility. It might be the case, for example, that I’m only willing to believe in ESP if it produces better than chance performance. If so, then my alternative hypothesis would only cover the possibility that \(\theta > .5\) , and as a consequence the null hypothesis now becomes \(\theta \leq .5\) :

When this happens, we have what’s called a one-sided test , and when this happens the critical region only covers one tail of the sampling distribution. This is illustrated in fig-esp-critical-onesided .

12.5. The \(p\) value of a test #

In one sense, our hypothesis test is complete; we’ve constructed a test statistic, figured out its sampling distribution if the null hypothesis is true, and then constructed the critical region for the test. Nevertheless, I’ve actually omitted the most important number of all: the \(p\) value . It is to this topic that we now turn. There are two somewhat different ways of interpreting a \(p\) value, one proposed by Sir Ronald Fisher and the other by Jerzy Neyman. Both versions are legitimate, though they reflect very different ways of thinking about hypothesis tests. Most introductory textbooks tend to give Fisher’s version only, but I think that’s a bit of a shame. To my mind, Neyman’s version is cleaner, and actually better reflects the logic of the null hypothesis test. You might disagree though, so I’ve included both. I’ll start with Neyman’s version…

12.5.1. A softer view of decision making #

One problem with the hypothesis testing procedure that I’ve described is that it makes no distinction at all between a result this “barely significant” and those that are “highly significant”. For instance, in my ESP study the data I obtained only just fell inside the critical region - so I did get a significant effect, but was a pretty near thing. In contrast, suppose that I’d run a study in which \(X=97\) out of my \(N=100\) participants got the answer right. This would obviously be significant too, but by a much larger margin; there’s really no ambiguity about this at all. The procedure that I described makes no distinction between the two. If I adopt the standard convention of allowing \(\alpha = .05\) as my acceptable Type I error rate, then both of these are significant results.

This is where the \(p\) value comes in handy. To understand how it works, let’s suppose that we ran lots of hypothesis tests on the same data set: but with a different value of \(\alpha\) in each case. When we do that for my original ESP data, what we’d get is something like this

When we test ESP data ( \(X=62\) successes out of \(N=100\) observations) using \(\alpha\) levels of .03 and above, we’d always find ourselves rejecting the null hypothesis. For \(\alpha\) levels of .02 and below, we always end up retaining the null hypothesis. Therefore, somewhere between .02 and .03 there must be a smallest value of \(\alpha\) that would allow us to reject the null hypothesis for this data. This is the \(p\) value; as it turns out the ESP data has \(p = .021\) . In short:

\(p\) is defined to be the smallest Type I error rate ( \(\alpha\) ) that you have to be willing to tolerate if you want to reject the null hypothesis.

If it turns out that \(p\) describes an error rate that you find intolerable, then you must retain the null. If you’re comfortable with an error rate equal to \(p\) , then it’s okay to reject the null hypothesis in favour of your preferred alternative.

In effect, \(p\) is a summary of all the possible hypothesis tests that you could have run, taken across all possible \(\alpha\) values. And as a consequence it has the effect of “softening” our decision process. For those tests in which \(p \leq \alpha\) you would have rejected the null hypothesis, whereas for those tests in which \(p > \alpha\) you would have retained the null. In my ESP study I obtained \(X=62\) , and as a consequence I’ve ended up with \(p = .021\) . So the error rate I have to tolerate is 2.1%. In contrast, suppose my experiment had yielded \(X=97\) . What happens to my \(p\) value now? This time it’s shrunk to \(p = 1.36 \times 10^{-25}\) , which is a tiny, tiny [ 8 ] Type I error rate. For this second case I would be able to reject the null hypothesis with a lot more confidence, because I only have to be “willing” to tolerate a type I error rate of about 1 in 10 trillion trillion in order to justify my decision to reject.

12.5.2. The probability of extreme data #

The second definition of the \(p\) -value comes from Sir Ronald Fisher, and it’s actually this one that you tend to see in most introductory statistics textbooks. Notice how, when I constructed the critical region, it corresponded to the tails (i.e., extreme values) of the sampling distribution? That’s not a coincidence: almost all “good” tests have this characteristic (good in the sense of minimising our type II error rate, \(\beta\) ). The reason for that is that a good critical region almost always corresponds to those values of the test statistic that are least likely to be observed if the null hypothesis is true. If this rule is true, then we can define the \(p\) -value as the probability that we would have observed a test statistic that is at least as extreme as the one we actually did get. In other words, if the data are extremely implausible according to the null hypothesis, then the null hypothesis is probably wrong.

12.5.3. A common mistake #

Okay, so you can see that there are two rather different but legitimate ways to interpret the \(p\) value, one based on Neyman’s approach to hypothesis testing and the other based on Fisher’s. Unfortunately, there is a third explanation that people sometimes give, especially when they’re first learning statistics, and it is absolutely and completely wrong . This mistaken approach is to refer to the \(p\) value as “the probability that the null hypothesis is true”. It’s an intuitively appealing way to think, but it’s wrong in two key respects: (1) null hypothesis testing is a frequentist tool, and the frequentist approach to probability does not allow you to assign probabilities to the null hypothesis… according to this view of probability, the null hypothesis is either true or it is not; it cannot have a “5% chance” of being true. (2) even within the Bayesian approach, which does let you assign probabilities to hypotheses, the \(p\) value would not correspond to the probability that the null is true; this interpretation is entirely inconsistent with the mathematics of how the \(p\) value is calculated. Put bluntly, despite the intuitive appeal of thinking this way, there is no justification for interpreting a \(p\) value this way. Never do it.

12.6. Reporting the results of a hypothesis test #

When writing up the results of a hypothesis test, there’s usually several pieces of information that you need to report, but it varies a fair bit from test to test. Throughout the rest of the book I’ll spend a little time talking about how to report the results of different tests (see Section @ref(chisqreport) for a particularly detailed example), so that you can get a feel for how it’s usually done. However, regardless of what test you’re doing, the one thing that you always have to do is say something about the \(p\) value, and whether or not the outcome was significant.

The fact that you have to do this is unsurprising; it’s the whole point of doing the test. What might be surprising is the fact that there is some contention over exactly how you’re supposed to do it. Leaving aside those people who completely disagree with the entire framework underpinning null hypothesis testing, there’s a certain amount of tension that exists regarding whether or not to report the exact \(p\) value that you obtained, or if you should state only that \(p < \alpha\) for a significance level that you chose in advance (e.g., \(p<.05\) ).

12.6.1. The issue #

To see why this is an issue, the key thing to recognise is that \(p\) values are terribly convenient. In practice, the fact that we can compute a \(p\) value means that we don’t actually have to specify any \(\alpha\) level at all in order to run the test. Instead, what you can do is calculate your \(p\) value and interpret it directly: if you get \(p = .062\) , then it means that you’d have to be willing to tolerate a Type I error rate of 6.2% to justify rejecting the null. If you personally find 6.2% intolerable, then you retain the null. Therefore, the argument goes, why don’t we just report the actual \(p\) value and let the reader make up their own minds about what an acceptable Type I error rate is? This approach has the big advantage of “softening” the decision making process – in fact, if you accept the Neyman definition of the \(p\) value, that’s the whole point of the \(p\) value. We no longer have a fixed significance level of \(\alpha = .05\) as a bright line separating “accept” from “reject” decisions; and this removes the rather pathological problem of being forced to treat \(p = .051\) in a fundamentally different way to \(p = .049\) .

This flexibility is both the advantage and the disadvantage to the \(p\) value. The reason why a lot of people don’t like the idea of reporting an exact \(p\) value is that it gives the researcher a bit too much freedom. In particular, it lets you change your mind about what error tolerance you’re willing to put up with after you look at the data. For instance, consider my ESP experiment. Suppose I ran my test, and ended up with a \(p\) value of .09. Should I accept or reject? Now, to be honest, I haven’t yet bothered to think about what level of Type I error I’m “really” willing to accept. I don’t have an opinion on that topic. But I do have an opinion about whether or not ESP exists, and I definitely have an opinion about whether my research should be published in a reputable scientific journal. And amazingly, now that I’ve looked at the data I’m starting to think that a 9% error rate isn’t so bad, especially when compared to how annoying it would be to have to admit to the world that my experiment has failed. So, to avoid looking like I just made it up after the fact, I now say that my \(\alpha\) is .1: a 10% type I error rate isn’t too bad, and at that level my test is significant! I win.

In other words, the worry here is that I might have the best of intentions, and be the most honest of people, but the temptation to just “shade” things a little bit here and there is really, really strong. As anyone who has ever run an experiment can attest, it’s a long and difficult process, and you often get very attached to your hypotheses. It’s hard to let go and admit the experiment didn’t find what you wanted it to find. And that’s the danger here. If we use the “raw” \(p\) -value, people will start interpreting the data in terms of what they want to believe, not what the data are actually saying… and if we allow that, well, why are we bothering to do science at all? Why not let everyone believe whatever they like about anything, regardless of what the facts are? Okay, that’s a bit extreme, but that’s where the worry comes from. According to this view, you really must specify your \(\alpha\) value in advance, and then only report whether the test was significant or not. It’s the only way to keep ourselves honest.

12.6.2. Two proposed solutions #

In practice, it’s pretty rare for a researcher to specify a single \(\alpha\) level ahead of time. Instead, the convention is that scientists rely on three standard significance levels: .05, .01 and .001. When reporting your results, you indicate which (if any) of these significance levels allow you to reject the null hypothesis. This is summarised in the table below. This allows us to soften the decision rule a little bit, since \(p<.01\) implies that the data meet a stronger evidentiary standard than \(p<.05\) would. Nevertheless, since these levels are fixed in advance by convention, it does prevent people choosing their \(\alpha\) level after looking at the data.

Nevertheless, quite a lot of people still prefer to report exact \(p\) values. To many people, the advantage of allowing the reader to make up their own mind about how to interpret \(p = .06\) outweighs any disadvantages. In practice, however, even among those researchers who prefer exact \(p\) values it is quite common to just write \(p<.001\) instead of reporting an exact value for small \(p\) . This is in part because a lot of software doesn’t actually print out the \(p\) value when it’s that small (e.g., SPSS just writes \(p = .000\) whenever \(p<.001\) ), and in part because a very small \(p\) value can be kind of misleading. The human mind sees a number like .0000000001 and it’s hard to suppress the gut feeling that the evidence in favour of the alternative hypothesis is a near certainty. In practice however, this is usually wrong. Life is a big, messy, complicated thing: and every statistical test ever invented relies on simplifications, approximations and assumptions. As a consequence, it’s probably not reasonable to walk away from any statistical analysis with a feeling of confidence stronger than \(p<.001\) implies. In other words, \(p<.001\) is really code for “as far as this test is concerned, the evidence is overwhelming.”

In light of all this, you might be wondering exactly what you should do. There’s a fair bit of contradictory advice on the topic, with some people arguing that you should report the exact \(p\) value, and other people arguing that you should use the tiered approach illustrated in the table above. As a result, the best advice I can give is to suggest that you look at papers/reports written in your field and see what the convention seems to be. If there doesn’t seem to be any consistent pattern, then use whichever method you prefer.

12.7. Running the hypothesis test in practice #

At this point some of you might be wondering if this is a “real” hypothesis test, or just a toy example that I made up. It’s real. In the previous discussion I built the test from first principles, thinking that it was the simplest possible problem that you might ever encounter in real life. However, this test already exists: it’s called the binomial test , and it’s implemented in a function called binom_test() from the scipy.stats package. To test the null hypothesis that the response probability is one-half p = .5 , [ 9 ] using data in which x = 62 of n = 100 people made the correct response, here’s how to do it in Python:

Well. There’s a number, but what does it mean? Sometimes the output of these Python functions can be fairly terse. But here binom_test() is giving us the \(p\) -value for the test we specified. In this case, the \(p\) -value of 0.02 is less than the usual choice of \(\alpha = .05\) , so we can reject the null. Usually we will want to know more than just the \(p\) -value for a test, and Python has ways of giving us this information, but for now, however, I just wanted to make the point that Python packages contain a whole lot of functions corresponding to different kinds of hypothesis test. And while I’ll usually spend quite a lot of time explaining the logic behind how the tests are built, every time I discuss a hypothesis test the discussion will end with me showing you a fairly simple Python command that you can use to run the test in practice.

12.8. Effect size, sample size and power #

In previous sections I’ve emphasised the fact that the major design principle behind statistical hypothesis testing is that we try to control our Type I error rate. When we fix \(\alpha = .05\) we are attempting to ensure that only 5% of true null hypotheses are incorrectly rejected. However, this doesn’t mean that we don’t care about Type II errors. In fact, from the researcher’s perspective, the error of failing to reject the null when it is actually false is an extremely annoying one. With that in mind, a secondary goal of hypothesis testing is to try to minimise \(\beta\) , the Type II error rate, although we don’t usually talk in terms of minimising Type II errors. Instead, we talk about maximising the power of the test. Since power is defined as \(1-\beta\) , this is the same thing.

12.8.1. The power function #

Let’s take a moment to think about what a Type II error actually is. A Type II error occurs when the alternative hypothesis is true, but we are nevertheless unable to reject the null hypothesis. Ideally, we’d be able to calculate a single number \(\beta\) that tells us the Type II error rate, in the same way that we can set \(\alpha = .05\) for the Type I error rate. Unfortunately, this is a lot trickier to do. To see this, notice that in my ESP study the alternative hypothesis actually corresponds to lots of possible values of \(\theta\) . In fact, the alternative hypothesis corresponds to every value of \(\theta\) except 0.5. Let’s suppose that the true probability of someone choosing the correct response is 55% (i.e., \(\theta = .55\) ). If so, then the true sampling distribution for \(X\) is not the same one that the null hypothesis predicts: the most likely value for \(X\) is now 55 out of 100. Not only that, the whole sampling distribution has now shifted, as shown in fig-esp-alternative . The critical regions, of course, do not change: by definition, the critical regions are based on what the null hypothesis predicts. What we’re seeing in this figure is the fact that when the null hypothesis is wrong, a much larger proportion of the sampling distribution distribution falls in the critical region. And of course that’s what should happen: the probability of rejecting the null hypothesis is larger when the null hypothesis is actually false! However \(\theta = .55\) is not the only possibility consistent with the alternative hypothesis. Let’s instead suppose that the true value of \(\theta\) is actually 0.7. What happens to the sampling distribution when this occurs? The answer, shown in fig-esp-alternative2 , is that almost the entirety of the sampling distribution has now moved into the critical region. Therefore, if \(\theta = 0.7\) the probability of us correctly rejecting the null hypothesis (i.e., the power of the test) is much larger than if \(\theta = 0.55\) . In short, while \(\theta = .55\) and \(\theta = .70\) are both part of the alternative hypothesis, the Type II error rate is different.

What all this means is that the power of a test (i.e., \(1-\beta\) ) depends on the true value of \(\theta\) . To illustrate this, I’ve calculated the expected probability of rejecting the null hypothesis for all values of \(\theta\) , and plotted it in fig-powerfunction . This plot describes what is usually called the power function of the test. It’s a nice summary of how good the test is, because it actually tells you the power ( \(1-\beta\) ) for all possible values of \(\theta\) . As you can see, when the true value of \(\theta\) is very close to 0.5, the power of the test drops very sharply, but when it is further away, the power is large.

12.8.2. Effect size #

Since all models are wrong the scientist must be alert to what is importantly wrong. It is inappropriate to be concerned with mice when there are tigers abroad – George Box 1976

The plot shown in fig-powerfunction captures a fairly basic point about hypothesis testing. If the true state of the world is very different from what the null hypothesis predicts, then your power will be very high; but if the true state of the world is similar to the null (but not identical) then the power of the test is going to be very low. Therefore, it’s useful to be able to have some way of quantifying how “similar” the true state of the world is to the null hypothesis. A statistic that does this is called a measure of effect size (e.g. [ Cohen, 1988 ] or [ Ellis, 2010 ] ). Effect size is defined slightly differently in different contexts (and so this section just talks in general terms) but the qualitative idea that it tries to capture is always the same: how big is the difference between the true population parameters, and the parameter values that are assumed by the null hypothesis? In our ESP example, if we let \(\theta_0 = 0.5\) denote the value assumed by the null hypothesis, and let \(\theta\) denote the true value, then a simple measure of effect size could be something like the difference between the true value and null (i.e., \(\theta - \theta_0\) ), or possibly just the magnitude of this difference, \(\mbox{abs}(\theta - \theta_0)\) .

Why calculate effect size? Let’s assume that you’ve run your experiment, collected the data, and gotten a significant effect when you ran your hypothesis test. Isn’t it enough just to say that you’ve gotten a significant effect? Surely that’s the point of hypothesis testing? Well, sort of. Yes, the point of doing a hypothesis test is to try to demonstrate that the null hypothesis is wrong, but that’s hardly the only thing we’re interested in. If the null hypothesis claimed that \(\theta = .5\) , and we show that it’s wrong, we’ve only really told half of the story. Rejecting the null hypothesis implies that we believe that \(\theta \neq .5\) , but there’s a big difference between \(\theta = .51\) and \(\theta = .8\) . If we find that \(\theta = .8\) , then not only have we found that the null hypothesis is wrong, it appears to be very wrong. On the other hand, suppose we’ve successfully rejected the null hypothesis, but it looks like the true value of \(\theta\) is only .51 (this would only be possible with a large study). Sure, the null hypothesis is wrong, but it’s not at all clear that we actually care , because the effect size is so small. In the context of my ESP study we might still care, since any demonstration of real psychic powers would actually be pretty cool [ 10 ] , but in other contexts a 1% difference isn’t very interesting, even if it is a real difference. For instance, suppose we’re looking at differences in high school exam scores between males and females, and it turns out that the female scores are 1% higher on average than the males. If I’ve got data from thousands of students, then this difference will almost certainly be statistically significant , but regardless of how small the \(p\) value is it’s just not very interesting. You’d hardly want to go around proclaiming a crisis in boys education on the basis of such a tiny difference would you? It’s for this reason that it is becoming more standard (slowly, but surely) to report some kind of standard measure of effect size along with the the results of the hypothesis test. The hypothesis test itself tells you whether you should believe that the effect you have observed is real (i.e., not just due to chance); the effect size tells you whether or not you should care.

12.8.3. Increasing the power of your study #

Not surprisingly, scientists are fairly obsessed with maximising the power of their experiments. We want our experiments to work, and so we want to maximise the chance of rejecting the null hypothesis if it is false (and of course we usually want to believe that it is false!) As we’ve seen, one factor that influences power is the effect size. So the first thing you can do to increase your power is to increase the effect size. In practice, what this means is that you want to design your study in such a way that the effect size gets magnified. For instance, in my ESP study I might believe that psychic powers work best in a quiet, darkened room; with fewer distractions to cloud the mind. Therefore I would try to conduct my experiments in just such an environment: if I can strengthen people’s ESP abilities somehow, then the true value of \(\theta\) will go up [ 11 ] and therefore my effect size will be larger. In short, clever experimental design is one way to boost power; because it can alter the effect size.

Unfortunately, it’s often the case that even with the best of experimental designs you may have only a small effect. Perhaps, for example, ESP really does exist, but even under the best of conditions it’s very very weak. Under those circumstances, your best bet for increasing power is to increase the sample size. In general, the more observations that you have available, the more likely it is that you can discriminate between two hypotheses. If I ran my ESP experiment with 10 participants, and 7 of them correctly guessed the colour of the hidden card, you wouldn’t be terribly impressed. But if I ran it with 10,000 participants and 7,000 of them got the answer right, you would be much more likely to think I had discovered something. In other words, power increases with the sample size. This is illustrated in fig-powerfunctionsample , which shows the power of the test for a true parameter of \(\theta = 0.7\) , for all sample sizes \(N\) from 1 to 100, where I’m assuming that the null hypothesis predicts that \(\theta_0 = 0.5\) .

Because power is important, whenever you’re contemplating running an experiment it would be pretty useful to know how much power you’re likely to have. It’s never possible to know for sure, since you can’t possibly know what your effect size is. However, it’s often (well, sometimes) possible to guess how big it should be. If so, you can guess what sample size you need! This idea is called power analysis , and if it’s feasible to do it, then it’s very helpful, since it can tell you something about whether you have enough time or money to be able to run the experiment successfully. It’s increasingly common to see people arguing that power analysis should be a required part of experimental design, so it’s worth knowing about. I don’t discuss power analysis in this book, however. This is partly for a boring reason and partly for a substantive one. The boring reason is that I haven’t had time to write about power analysis yet. The substantive one is that I’m still a little suspicious of power analysis. Speaking as a researcher, I have very rarely found myself in a position to be able to do one – it’s either the case that (a) my experiment is a bit non-standard and I don’t know how to define effect size properly, or (b) I literally have so little idea about what the effect size will be that I wouldn’t know how to interpret the answers. Not only that, after extensive conversations with someone who does stats consulting for a living (my wife, as it happens), I can’t help but notice that in practice the only time anyone ever asks her for a power analysis is when she’s helping someone write a grant application. In other words, the only time any scientist ever seems to want a power analysis in real life is when they’re being forced to do it by bureaucratic process. It’s not part of anyone’s day to day work. In short, I’ve always been of the view that while power is an important concept, power analysis is not as useful as people make it sound, except in the rare cases where (a) someone has figured out how to calculate power for your actual experimental design and (b) you have a pretty good idea what the effect size is likely to be. Maybe other people have had better experiences than me, but I’ve personally never been in a situation where both (a) and (b) were true. Maybe I’ll be convinced otherwise in the future, and probably a future version of this book would include a more detailed discussion of power analysis, but for now this is about as much as I’m comfortable saying about the topic.

12.9. Some issues to consider #

What I’ve described to you in this chapter is the orthodox framework for null hypothesis significance testing (NHST). Understanding how NHST works is an absolute necessity, since it has been the dominant approach to inferential statistics ever since it came to prominence in the early 20th century. It’s what the vast majority of working scientists rely on for their data analysis, so even if you hate it you need to know it. However, the approach is not without problems. There are a number of quirks in the framework, historical oddities in how it came to be, theoretical disputes over whether or not the framework is right, and a lot of practical traps for the unwary. I’m not going to go into a lot of detail on this topic, but I think it’s worth briefly discussing a few of these issues.

12.9.1. Neyman versus Fisher #

The first thing you should be aware of is that orthodox NHST is actually a mash-up of two rather different approaches to hypothesis testing, one proposed by Sir Ronald Fisher and the other proposed by Jerzy Neyman (for a historical summary see [ Lehmann, 2011 ] . The history is messy because Fisher and Neyman were real people whose opinions changed over time, and at no point did either of them offer “the definitive statement” of how we should interpret their work many decades later. That said, here’s a quick summary of what I take these two approaches to be.

First, let’s talk about Fisher’s approach. As far as I can tell, Fisher assumed that you only had the one hypothesis (the null), and what you want to do is find out if the null hypothesis is inconsistent with the data. From his perspective, what you should do is check to see if the data are “sufficiently unlikely” according to the null. In fact, if you remember back to our earlier discussion, that’s how Fisher defines the \(p\) -value. According to Fisher, if the null hypothesis provided a very poor account of the data, you could safely reject it. But, since you don’t have any other hypotheses to compare it to, there’s no way of “accepting the alternative” because you don’t necessarily have an explicitly stated alternative. That’s more or less all that there was to it.

In contrast, Neyman thought that the point of hypothesis testing was as a guide to action, and his approach was somewhat more formal than Fisher’s. His view was that there are multiple things that you could do (accept the null or accept the alternative) and the point of the test was to tell you which one the data support. From this perspective, it is critical to specify your alternative hypothesis properly. If you don’t know what the alternative hypothesis is, then you don’t know how powerful the test is, or even which action makes sense. His framework genuinely requires a competition between different hypotheses. For Neyman, the \(p\) value didn’t directly measure the probability of the data (or data more extreme) under the null, it was more of an abstract description about which “possible tests” were telling you to accept the null, and which “possible tests” were telling you to accept the alternative.

As you can see, what we have today is an odd mishmash of the two. We talk about having both a null hypothesis and an alternative (Neyman), but usually [ 12 ] define the \(p\) value in terms of exreme data (Fisher), but we still have \(\alpha\) values (Neyman). Some of the statistical tests have explicitly specified alternatives (Neyman) but others are quite vague about it (Fisher). And, according to some people at least, we’re not allowed to talk about accepting the alternative (Fisher). It’s a mess: but I hope this at least explains why it’s a mess.

12.9.2. Bayesians versus frequentists #

Earlier on in this chapter I was quite emphatic about the fact that you cannot interpret the \(p\) value as the probability that the null hypothesis is true. NHST is fundamentally a frequentist tool (see the chapter on probability ) and as such it does not allow you to assign probabilities to hypotheses: the null hypothesis is either true or it is not. The Bayesian approach to statistics interprets probability as a degree of belief, so it’s totally okay to say that there is a 10% chance that the null hypothesis is true: that’s just a reflection of the degree of confidence that you have in this hypothesis. You aren’t allowed to do this within the frequentist approach. Remember, if you’re a frequentist, a probability can only be defined in terms of what happens after a large number of independent replications (i.e., a long run frequency). If this is your interpretation of probability, talking about the “probability” that the null hypothesis is true is complete gibberish: a null hypothesis is either true or it is false. There’s no way you can talk about a long run frequency for this statement. To talk about “the probability of the null hypothesis” is as meaningless as “the colour of freedom”. It doesn’t have one!

Most importantly, this isn’t a purely ideological matter. If you decide that you are a Bayesian and that you’re okay with making probability statements about hypotheses, you have to follow the Bayesian rules for calculating those probabilities. I’ll talk more about this in the chapter on Bayesian statistics , but for now what I want to point out to you is the \(p\) value is a terrible approximation to the probability that \(H_0\) is true. If what you want to know is the probability of the null, then the \(p\) value is not what you’re looking for!

12.9.3. Traps #

As you can see, the theory behind hypothesis testing is a mess, and even now there are arguments in statistics about how it “should” work. However, disagreements among statisticians are not our real concern here. Our real concern is practical data analysis. And while the “orthodox” approach to null hypothesis significance testing has many drawbacks, even an unrepentant Bayesian like myself would agree that they can be useful if used responsibly. Most of the time they give sensible answers, and you can use them to learn interesting things. Setting aside the various ideologies and historical confusions that we’ve discussed, the fact remains that the biggest danger in all of statistics is thoughtlessness . I don’t mean stupidity, here: I literally mean thoughtlessness. The rush to interpret a result without spending time thinking through what each test actually says about the data, and checking whether that’s consistent with how you’ve interpreted it. That’s where the biggest trap lies.

To give an example of this, consider the following example see [ Gelman and Stern, 2006 ] . Suppose I’m running my ESP study, and I’ve decided to analyse the data separately for the male participants and the female participants. Of the male participants, 33 out of 50 guessed the colour of the card correctly. This is a significant effect ( \(p = .03\) ). Of the female participants, 29 out of 50 guessed correctly. This is not a significant effect ( \(p = .32\) ). Upon observing this, it is extremely tempting for people to start wondering why there is a difference between males and females in terms of their psychic abilities. However, this is wrong. If you think about it, we haven’t actually run a test that explicitly compares males to females. All we have done is compare males to chance (binomial test was significant) and compared females to chance (binomial test was non significant). If we want to argue that there is a real difference between the males and the females, we should probably run a test of the null hypothesis that there is no difference! We can do that using a different hypothesis test, [ 13 ] but when we do that it turns out that we have no evidence that males and females are significantly different ( \(p = .54\) ). Now do you think that there’s anything fundamentally different between the two groups? Of course not. What’s happened here is that the data from both groups (male and female) are pretty borderline: by pure chance, one of them happened to end up on the magic side of the \(p = .05\) line, and the other one didn’t. That doesn’t actually imply that males and females are different. This mistake is so common that you should always be wary of it: the difference between significant and not-significant is not evidence of a real difference – if you want to say that there’s a difference between two groups, then you have to test for that difference!

The example above is just that: an example. I’ve singled it out because it’s such a common one, but the bigger picture is that data analysis can be tricky to get right. Think about what it is you want to test, why you want to test it, and whether or not the answers that your test gives could possibly make any sense in the real world.

12.10. Summary #

Null hypothesis testing is one of the most ubiquitous elements to statistical theory. The vast majority of scientific papers report the results of some hypothesis test or another. As a consequence it is almost impossible to get by in science without having at least a cursory understanding of what a \(p\) -value means, making this one of the most important chapters in the book. As usual, I’ll end the chapter with a quick recap of the key ideas that we’ve talked about:

Research hypotheses and statistical hypotheses . Null and alternative hypotheses .

Type 1 and Type 2 errors

Test statistics and sampling distributions

Hypothesis testing as a decision making process

\(p\) -values as “soft” decisions

Writing up the results of a hypothesis test

Effect size and power

A few issues to consider regarding hypothesis testing

Later in the book, in the section on Bayesian statistics , I’ll revisit the theory of null hypothesis tests from a Bayesian perspective, and introduce a number of new tools that you can use if you aren’t particularly fond of the orthodox approach. But for now, though, we’re done with the abstract statistical theory, and we can start discussing specific data analysis tools.

Your Data Guide

How to Perform Hypothesis Testing Using Python

Step into the intriguing world of hypothesis testing, where your natural curiosity meets the power of data to reveal truths!

This article is your key to unlocking how those everyday hunches—like guessing a group’s average income or figuring out who owns their home—can be thoroughly checked and proven with data.

Thanks for reading Your Data Guide! Subscribe for free to receive new posts and support my work.

I am going to take you by the hand and show you, in simple steps, how to use Python to explore a hypothesis about the average yearly income.

By the time we’re done, you’ll not only get the hang of creating and testing hypotheses but also how to use statistical tests on actual data.

Perfect for up-and-coming data scientists, anyone with a knack for analysis, or just if you’re keen on data, get ready to gain the skills to make informed decisions and turn insights into real-world actions.

Join me as we dive deep into the data, one hypothesis at a time!

Before we get started, elevate your data skills with my expert eBooks—the culmination of my experiences and insights.

Support my work and enhance your journey. Check them out:

eBook 1: Personal INTERVIEW Ready “SQL” CheatSheet

eBook 2: Personal INTERVIEW Ready “Statistics” Cornell Notes

Best Selling eBook: Top 50+ ChatGPT Personas for Custom Instructions

Data Science Bundle ( Cheapest ): The Ultimate Data Science Bundle: Complete

ChatGPT Bundle ( Cheapest ): The Ultimate ChatGPT Bundle: Complete

💡 Checkout for more such resources: https://codewarepam.gumroad.com/

What is a hypothesis, and how do you test it?

A hypothesis is like a guess or prediction about something specific, such as the average income or the percentage of homeowners in a group of people.

It’s based on theories, past observations, or questions that spark our curiosity.

For instance, you might predict that the average yearly income of potential customers is over $50,000 or that 60% of them own their homes.

To see if your guess is right, you gather data from a smaller group within the larger population and check if the numbers ( like the average income, percentage of homeowners, etc. ) from this smaller group match your initial prediction.

You also set a rule for how sure you need to be to trust your findings, often using a 5% chance of error as a standard measure . This means you’re 95% confident in your results. — Level of Significance (0.05)

There are two main types of hypotheses : the null hypothesi s, which is your baseline saying there’s no change or difference, and the alternative hypothesis , which suggests there is a change or difference.

For example,

If you start with the idea that the average yearly income of potential customers is $50,000,

The alternative could be that it’s not $50,000—it could be less or more, depending on what you’re trying to find out.

To test your hypothesis, you calculate a test statistic —a number that shows how much your sample data deviates from what you predicted.

How you calculate this depends on what you’re studying and the kind of data you have. For example, to check an average, you might use a formula that considers your sample’s average, the predicted average, the variation in your sample data, and how big your sample is.

This test statistic follows a known distribution ( like the t-distribution or z-distribution ), which helps you figure out the p-value.

The p-value tells you the odds of seeing a test statistic as extreme as yours if your initial guess was correct.

A small p-value means your data strongly disagrees with your initial guess.

Finally, you decide on your hypothesis by comparing the p-value to your error threshold.

If the p-value is smaller or equal, you reject the null hypothesis, meaning your data shows a significant difference that’s unlikely due to chance.

If the p-value is larger, you stick with the null hypothesis , suggesting your data doesn’t show a meaningful difference and any change might just be by chance.

We’ll go through an example that tests if the average annual income of prospective customers exceeds $50,000.

This process involves stating hypotheses , specifying a significance level , collecting and analyzing data , and drawing conclusions based on statistical tests.

Example: Testing a Hypothesis About Average Annual Income

Step 1: state the hypotheses.

Null Hypothesis (H0): The average annual income of prospective customers is $50,000.

Alternative Hypothesis (H1): The average annual income of prospective customers is more than $50,000.

Step 2: Specify the Significance Level

Significance Level: 0.05, meaning we’re 95% confident in our findings and allow a 5% chance of error.

Step 3: Collect Sample Data

We’ll use the ProspectiveBuyer table, assuming it's a random sample from the population.

This table has 2,059 entries, representing prospective customers' annual incomes.

Step 4: Calculate the Sample Statistic

In Python, we can use libraries like Pandas and Numpy to calculate the sample mean and standard deviation.

SampleMean: 56,992.43

SampleSD: 32,079.16

SampleSize: 2,059

Step 5: Calculate the Test Statistic

We use the t-test formula to calculate how significantly our sample mean deviates from the hypothesized mean.

Python’s Scipy library can handle this calculation:

T-Statistic: 4.62

Step 6: Calculate the P-Value

The p-value is already calculated in the previous step using Scipy's ttest_1samp function, which returns both the test statistic and the p-value.

P-Value = 0.0000021

Step 7: State the Statistical Conclusion

We compare the p-value with our significance level to decide on our hypothesis:

Since the p-value is less than 0.05, we reject the null hypothesis in favor of the alternative.

Conclusion:

There’s strong evidence to suggest that the average annual income of prospective customers is indeed more than $50,000.

This example illustrates how Python can be a powerful tool for hypothesis testing, enabling us to derive insights from data through statistical analysis.

How to Choose the Right Test Statistics

Choosing the right test statistic is crucial and depends on what you’re trying to find out, the kind of data you have, and how that data is spread out.

Here are some common types of test statistics and when to use them:

T-test statistic:

This one’s great for checking out the average of a group when your data follows a normal distribution or when you’re comparing the averages of two such groups.

The t-test follows a special curve called the t-distribution . This curve looks a lot like the normal bell curve but with thicker ends, which means more chances for extreme values.

The t-distribution’s shape changes based on something called degrees of freedom , which is a fancy way of talking about your sample size and how many groups you’re comparing.

Z-test statistic:

Use this when you’re looking at the average of a normally distributed group or the difference between two group averages, and you already know the standard deviation for all in the population.

The z-test follows the standard normal distribution , which is your classic bell curve centered at zero and spreading out evenly on both sides.

Chi-square test statistic:

This is your go-to for checking if there’s a difference in variability within a normally distributed group or if two categories are related.

The chi-square statistic follows its own distribution, which leans to the right and gets its shape from the degrees of freedom —basically, how many categories or groups you’re comparing.

F-test statistic:

This one helps you compare the variability between two groups or see if the averages of more than two groups are all the same, assuming all groups are normally distributed.

The F-test follows the F-distribution , which is also right-skewed and has two types of degrees of freedom that depend on how many groups you have and the size of each group.

In simple terms, the test you pick hinges on what you’re curious about, whether your data fits the normal curve, and if you know certain specifics, like the population’s standard deviation.

Each test has its own special curve and rules based on your sample’s details and what you’re comparing.

Join my community of learners! Subscribe to my newsletter for more tips, tricks, and exclusive content on mastering Data Science & AI. — Your Data Guide Join my community of learners! Subscribe to my newsletter for more tips, tricks, and exclusive content on mastering data science and AI. By Richard Warepam ⭐️ Visit My Gumroad Shop: https://codewarepam.gumroad.com/

Ready for more?

scipy.stats.ttest_ind #

Calculate the T-test for the means of two independent samples of scores.

This is a test for the null hypothesis that 2 independent samples have identical average (expected) values. This test assumes that the populations have identical variances by default.

The arrays must have the same shape, except in the dimension corresponding to axis (the first, by default).

If an int, the axis of the input along which to compute the statistic. The statistic of each axis-slice (e.g. row) of the input will appear in a corresponding element of the output. If None , the input will be raveled before computing the statistic.

If True (default), perform a standard independent 2 sample test that assumes equal population variances [1] . If False, perform Welch’s t-test, which does not assume equal population variance [2] .

New in version 0.11.0.

Defines how to handle input NaNs.

propagate : if a NaN is present in the axis slice (e.g. row) along which the statistic is computed, the corresponding entry of the output will be NaN.

omit : NaNs will be omitted when performing the calculation. If insufficient data remains in the axis slice along which the statistic is computed, the corresponding entry of the output will be NaN.

raise : if a NaN is present, a ValueError will be raised.

If 0 or None (default), use the t-distribution to calculate p-values. Otherwise, permutations is the number of random permutations that will be used to estimate p-values using a permutation test. If permutations equals or exceeds the number of distinct partitions of the pooled data, an exact test is performed instead (i.e. each distinct partition is used exactly once). See Notes for details.

New in version 1.7.0.

numpy.random.RandomState }, optional

If seed is None (or np.random ), the numpy.random.RandomState singleton is used. If seed is an int, a new RandomState instance is used, seeded with seed . If seed is already a Generator or RandomState instance then that instance is used.

Pseudorandom number generator state used to generate permutations (used only when permutations is not None).

Defines the alternative hypothesis. The following options are available (default is ‘two-sided’):

‘two-sided’: the means of the distributions underlying the samples are unequal.

‘less’: the mean of the distribution underlying the first sample is less than the mean of the distribution underlying the second sample.

‘greater’: the mean of the distribution underlying the first sample is greater than the mean of the distribution underlying the second sample.

New in version 1.6.0.

If nonzero, performs a trimmed (Yuen’s) t-test. Defines the fraction of elements to be trimmed from each end of the input samples. If 0 (default), no elements will be trimmed from either side. The number of trimmed elements from each tail is the floor of the trim times the number of elements. Valid range is [0, .5).

New in version 1.7.

If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the input array.

An object with the following attributes:

The t-statistic.

The p-value associated with the given alternative.

The number of degrees of freedom used in calculation of the t-statistic. This is always NaN for a permutation t-test.

New in version 1.11.0.

The object also has the following method:

Computes a confidence interval around the difference in population means for the given confidence level. The confidence interval is returned in a namedtuple with fields low and high . When a permutation t-test is performed, the confidence interval is not computed, and fields low and high contain NaN.

Suppose we observe two independent samples, e.g. flower petal lengths, and we are considering whether the two samples were drawn from the same population (e.g. the same species of flower or two species with similar petal characteristics) or two different populations.

The t-test quantifies the difference between the arithmetic means of the two samples. The p-value quantifies the probability of observing as or more extreme values assuming the null hypothesis, that the samples are drawn from populations with the same population means, is true. A p-value larger than a chosen threshold (e.g. 5% or 1%) indicates that our observation is not so unlikely to have occurred by chance. Therefore, we do not reject the null hypothesis of equal population means. If the p-value is smaller than our threshold, then we have evidence against the null hypothesis of equal population means.

By default, the p-value is determined by comparing the t-statistic of the observed data against a theoretical t-distribution. When 1 < permutations < binom(n, k) , where

k is the number of observations in a ,

n is the total number of observations in a and b , and

binom(n, k) is the binomial coefficient ( n choose k ),

the data are pooled (concatenated), randomly assigned to either group a or b , and the t-statistic is calculated. This process is performed repeatedly ( permutation times), generating a distribution of the t-statistic under the null hypothesis, and the t-statistic of the observed data is compared to this distribution to determine the p-value. Specifically, the p-value reported is the “achieved significance level” (ASL) as defined in 4.4 of [3] . Note that there are other ways of estimating p-values using randomized permutation tests; for other options, see the more general permutation_test .

When permutations >= binom(n, k) , an exact test is performed: the data are partitioned between the groups in each distinct way exactly once.

The permutation test can be computationally expensive and not necessarily more accurate than the analytical test, but it does not make strong assumptions about the shape of the underlying distribution.

Use of trimming is commonly referred to as the trimmed t-test. At times called Yuen’s t-test, this is an extension of Welch’s t-test, with the difference being the use of winsorized means in calculation of the variance and the trimmed sample size in calculation of the statistic. Trimming is recommended if the underlying distribution is long-tailed or contaminated with outliers [4] .

The statistic is calculated as (np.mean(a) - np.mean(b))/se , where se is the standard error. Therefore, the statistic will be positive when the sample mean of a is greater than the sample mean of b and negative when the sample mean of a is less than the sample mean of b .

Beginning in SciPy 1.9, np.matrix inputs (not recommended for new code) are converted to np.ndarray before the calculation is performed. In this case, the output will be a scalar or np.ndarray of appropriate shape rather than a 2D np.matrix . Similarly, while masked elements of masked arrays are ignored, the output will be a scalar or np.ndarray rather than a masked array with mask=False .

https://en.wikipedia.org/wiki/T-test#Independent_two-sample_t-test

https://en.wikipedia.org/wiki/Welch%27s_t-test

Efron and T. Hastie. Computer Age Statistical Inference. (2016).

Yuen, Karen K. “The Two-Sample Trimmed t for Unequal Population Variances.” Biometrika, vol. 61, no. 1, 1974, pp. 165-170. JSTOR, www.jstor.org/stable/2334299. Accessed 30 Mar. 2021.

Yuen, Karen K., and W. J. Dixon. “The Approximate Behaviour and Performance of the Two-Sample Trimmed t.” Biometrika, vol. 60, no. 2, 1973, pp. 369-374. JSTOR, www.jstor.org/stable/2334550. Accessed 30 Mar. 2021.

Test with sample with identical means:

ttest_ind underestimates p for unequal variances:

When n1 != n2 , the equal variance t-statistic is no longer equal to the unequal variance t-statistic:

T-test with different means, variance, and n:

When performing a permutation test, more permutations typically yields more accurate results. Use a np.random.Generator to ensure reproducibility:

Take these two samples, one of which has an extreme tail.

Use the trim keyword to perform a trimmed (Yuen) t-test. For example, using 20% trimming, trim=.2 , the test will reduce the impact of one ( np.floor(trim*len(a)) ) element from each tail of sample a . It will have no effect on sample b because np.floor(trim*len(b)) is 0.

Navigation Menu

Search code, repositories, users, issues, pull requests..., provide feedback.

We read every piece of feedback, and take your input very seriously.

Saved searches

Use saved searches to filter your results more quickly.

To see all available qualifiers, see our documentation .

null-hypothesis

Here are 33 public repositories matching this topic..., alcampopiano / hypothesize.

Robust statistics in Python

• Updated Jul 6, 2023

ckdckd145 / statmanager-kr

Open-source statistical package in Python based on Pandas

• Updated May 14, 2024

udialter / equivalence-testing-multiple-regression

I constructed a simulation study to evaluate the statistical performance of two equivalence-based tests and compared it to the common, but inappropriate, method of concluding no effect by failing to reject the null hypothesis of the traditional test. I further propose two R functions to supply researchers with open-access and easy-to-use tools …

• Updated Jul 25, 2022

tantawy997 / Analyze_ab_test_results_notebook

Analyze ab test results udacity project

• Updated Dec 5, 2021
• Jupyter Notebook

nhtsai / datahacks2020

Science Track Finalist: A Case Study of Race in Diabetes Healthcare

• Updated Sep 12, 2021

vaitybharati / P16.-Hypothesis-Testing-1S2T---Call-Center-Process

Hypothesis Testing 1S2T - Call Center Process. Sample Parameters: n=50, df=50-1=49, Mean1=4, SD1=3 1-sample 2-tail ttest Assume Null Hypothesis Ho as Mean1 = 4 Thus, Alternate Hypothesis Ha as Mean1 ≠ 4

• Updated May 3, 2021

nhtsai / datathon2019

Lyft Challenge Winner: San Diego Traffic Collision Analysis

• Updated Oct 23, 2021

Christian-F-Badillo / Temas_Selectos_en_Estadistica

Repositorio para el curso intersemestral "Temas Selectos en Estadística" para la Facultad de Psicología, UNAM.

• Updated Jan 25, 2024

MahtabEK / A-Data-Scientist-for-a-Professional-Football-Club-Part2

This project is part 2 of the project "A Data Scientist for a Professional Football Club". In this project, managers want to test some hypotheses relating a player's overall rating and some of their characteristics in order to make better decisions on what players to trade/sign. They would like to create some statistical models for inference ins…

• Updated Oct 14, 2020

aldimeolaalfarisy / Hypothesis-Testing-Concept-ANOVA-

ANOVA test using python to find out if survey or experiment results are significant and the impact of one or more factors by comparing the means of different samples

• Updated Nov 16, 2022

vaitybharati / P19.-Hypothesis-Testing-2-Proportion-T-test-Students-Jobs-in-2-States-

Hypothesis-Testing-2-Proportion-T-test-Students-Jobs-in-2-States. Assume Null Hypothesis as Ho is p1-p2 = 0 i.e. p1 ≠ p2. Thus Alternate Hypthesis as Ha is p1 = p2. Explanation of bernoulli Binomial RV: np.random.binomial(n=1,p,size) Suppose you perform an experiment with two possible outcomes: either success or failure. Success happens with pro…

• Updated May 20, 2021

himanshuvnm / Academic-activity-at-The-University-of-Texas-at-Tyler

This repository contains my notes of Calculus and Statistics that I taught in the Department of Mathematics at The University of Texas at Tyler.

• Updated Apr 21, 2024

vaitybharati / Assignment-03-Q5-Hypothesis-Testing-

Chi2 contengency independence test. Fantaloons Sales managers commented that % of males versus females walking in to the store differ based on day of the week. Analyze the data and determine whether there is evidence at 5 % significance level to support this hypothesis.

• Updated Apr 24, 2021

MezbanS / Healthcare-Insurance-Analysis

This project predicts healthcare costs and identifies contributing factors using data analysis, machine learning, and SQL data management.

• Updated Oct 21, 2023

vaitybharati / P18.-Hypothesis-Testing-2-Sample-2-Tail-Test-Drugs-and-Placebos-

Hypothesis-Testing-2-Sample-2-Tail-Test-Drugs-and-Placebos. Note: This python code states both 2-sample 1-tail and 2-sample 2-tail codes. Treatment group mean is Mu1 Contrl group mean is Mu2 2-sample 2-tail ttest Assume Null Hypothesis Ho as Mu1 = Mu2 Thus Alternate Hypothesis Ha as Mu1 ≠ Mu2.

• Updated May 19, 2021

vaitybharati / P20.-Hypothesis-Testing-Anova-Test---Iris-Flower-dataset

Hypothesis Testing Anova Test - Iris Flower dataset. Anova ftest statistics: Analysis of varaince between more than 2 samples or columns. Assume Null Hypothesis Ho as No Varaince: All samples population means are same. Thus Alternate Hypothesis Ha as It has Variance: Atleast one population mean is different. As (p_value = 0) < (α = 0.05); Reject…

• Updated May 21, 2021

aeglon97 / AB-Test-Results

An analysis of A/B Test results to help an e-commerce site decide whether or not they should implement a new page design.

• Updated Jul 2, 2019

vaitybharati / Assignment-03-Q4-Hypothesis-Testing-

Chi2 contengency independence test. Q4. TeleCall uses 4 centers around the globe to process customer order forms. They audit a certain % of the customer order forms. Any error in order form renders it defective and has to be reworked before processing. The manager wants to check whether the defective % varies by centre. Please analyze the data a…

• Updated Apr 23, 2021

vaitybharati / P21.-Hypothesis-Testing-Chi2-Test-Athletes-and-Smokers-

Hypothesis-Testing-Chi2-Test-Athletes-and-Smokers. Assume Null Hypothesis as Ho: Independence of categorical variables (Athlete and Smoking not related). Thus Alternate Hypothesis as Ha: Dependence of categorical variables (Athlete and Smoking is somewhat/significantly related). As (p_value = 0.00038) < (α = 0.05); Reject Null Hypothesis i.e. De…

• Updated May 25, 2021

CS-LEE2022 / Test_a_Perceptual_Phenomenon

Use descriptive statistics to describe qualities of a sample, set up a hypothesis test, make inferences from a sample, and draw conclusions based on the results.

• Updated Jan 10, 2019

Improve this page

Add a description, image, and links to the null-hypothesis topic page so that developers can more easily learn about it.

Curate this topic

Add this topic to your repo

To associate your repository with the null-hypothesis topic, visit your repo's landing page and select "manage topics."

• School Guide
• Mathematics
• Number System and Arithmetic
• Trigonometry
• Probability
• Mensuration
• Maths Formulas
• Class 8 Maths Notes
• Class 9 Maths Notes
• Class 10 Maths Notes
• Class 11 Maths Notes
• Class 12 Maths Notes
• Alternative Dispute Resolution (ADR): Meaning, Types and FAQs
• Level of Significance-Definition, Steps and Examples
• Difference Between Hypothesis And Theory
• What is Dihybrid Cross? Examples and an Overview
• Real-life Applications of Hypothesis Testing
• T-Test in Statistics: Formula, Types and Steps
• Hypothesis Testing Formula
• Independent Sample t Test in R
• Alternate Interior Angles
• How do you define and measure your product hypothesis?
• Difference between Alternate and Alternative
• Introduction to Power Analysis in Python
• Difference between Null and Alternate Hypothesis
• Inductive Reasoning | Definition, Types, & Examples
• Python unittest - assertIn() function
• Python unittest - assertNotIsInstance() function
• Python unittest - assertIsNone() function
• Python unittest - assertIsInstance() function

Alternative Hypothesis: Definition, Types and Examples

In statistical hypothesis testing, the alternative hypothesis is an important proposition in the hypothesis test. The goal of the hypothesis test is to demonstrate that in the given condition, there is sufficient evidence supporting the credibility of the alternative hypothesis instead of the default assumption made by the null hypothesis.

Alternative Hypotheses

Both hypotheses include statements with the same purpose of providing the researcher with a basic guideline. The researcher uses the statement from each hypothesis to guide their research. In statistics, alternative hypothesis is often denoted as H a or H 1 .

Table of Content

What is a Hypothesis?

Alternative hypothesis, types of alternative hypothesis, difference between null and alternative hypothesis, formulating an alternative hypothesis, example of alternative hypothesis, application of alternative hypothesis.

“A hypothesis is a statement of a relationship between two or more variables.” It is a working statement or theory that is based on insufficient evidence.

While experimenting, researchers often make a claim, that they can test. These claims are often based on the relationship between two or more variables. “What causes what?” and “Up to what extent?” are a few of the questions that a hypothesis focuses on answering. The hypothesis can be true or false, based on complete evidence.

While there are different hypotheses, we discuss only null and alternate hypotheses. The null hypothesis, denoted H o , is the default position where variables do not have a relation with each other. That means the null hypothesis is assumed true until evidence indicates otherwise. The alternative hypothesis, denoted H 1 , on the other hand, opposes the null hypothesis. It assumes a relation between the variables and serves as evidence to reject the null hypothesis.

Example of Hypothesis:

Mean age of all college students is 20.4 years. (simple hypothesis).

An Alternative Hypothesis is a claim or a complement to the null hypothesis. If the null hypothesis predicts a statement to be true, the Alternative Hypothesis predicts it to be false. Let’s say the null hypothesis states there is no difference between height and shoe size then the alternative hypothesis will oppose the claim by stating that there is a relation.

We see that the null hypothesis assumes no relationship between the variables whereas an alternative hypothesis proposes a significant relation between variables. An alternative theory is the one tested by the researcher and if the researcher gathers enough data to support it, then the alternative hypothesis replaces the null hypothesis.

Null and alternative hypotheses are exhaustive, meaning that together they cover every possible outcome. They are also mutually exclusive, meaning that only one can be true at a time.

There are a few types of alternative hypothesis that we will see:

1. One-tailed test H 1 : A one-tailed alternative hypothesis focuses on only one region of rejection of the sampling distribution. The region of rejection can be upper or lower.

• Upper-tailed test H 1 : Population characteristic > Hypothesized value
• Lower-tailed test H 1 : Population characteristic < Hypothesized value

2. Two-tailed test H 1 : A two-tailed alternative hypothesis is concerned with both regions of rejection of the sampling distribution.

3. Non-directional test H 1 : A non-directional alternative hypothesis is not concerned with either region of rejection; rather, it is only concerned that null hypothesis is not true.

4. Point test H 1 : Point alternative hypotheses occur when the hypothesis test is framed so that the population distribution under the alternative hypothesis is a fully defined distribution, with no unknown parameters; such hypotheses are usually of no practical interest but are fundamental to theoretical considerations of statistical inference and are the basis of the Neyman–Pearson lemma.

the differences between Null Hypothesis and Alternative Hypothesis is explained in the table below:

Formulating an alternative hypothesis means identifying the relationships, effects or condition being studied. Based on the data we conclude that there is a different inference from the null-hypothesis being considered.

• Understand the null hypothesis.
• Consider the alternate hypothesis
• Choose the type of alternate hypothesis (one-tailed or two-tailed)

Alternative hypothesis must be true when the null hypothesis is false. When trying to identify the information need for alternate hypothesis statement, look for the following phrases:

• “Is it reasonable to conclude…”
• “Is there enough evidence to substantiate…”
• “Does the evidence suggest…”
• “Has there been a significant…”

When alternative hypotheses in mathematical terms, they always include an inequality ( usually ≠, but sometimes < or >) . When writing the alternate hypothesis, make sure it never includes an “=” symbol.

To help you write your hypotheses, you can use the template sentences below.

Does independent variable affect dependent variable?

• Null Hypothesis (H 0 ): Independent variable does not affect dependent variable.
• Alternative Hypothesis (H a ): Independent variable affects dependent variable.

Various examples of Alternative Hypothesis includes:

Two-Tailed Example

• Research Question : Do home games affect a team’s performance?
• Null-Hypothesis: Home games do not affect a team’s performance.
• Alternative Hypothesis: Home games have an effect on team’s performance.
• Research Question: Does sleeping less lead to depression?
• Null-Hypothesis: Sleeping less does not have an effect on depression.
• Alternative Hypothesis : Sleeping less has an effect on depression.

One-Tailed Example

• Research Question: Are candidates with experience likely to get a job?
• Null-Hypothesis: Experience does not matter in getting a job.
• Alternative Hypothesis: Candidates with work experience are more likely to receive an interview.
• Alternative Hypothesis : Teams with home advantage are more likely to win a match.

Some applications of Alternative Hypothesis includes:

• Rejecting Null-Hypothesis : A researcher performs additional research to find flaws in the null hypothesis. Following the research, which uses the alternative hypothesis as a guide, they may decide whether they have enough evidence to reject the null hypothesis.
• Guideline for Research : An alternative and null hypothesis include statements with the same purpose of providing the researcher with a basic guideline. The researcher uses the statement from each hypothesis to guide their research.
• New Theories : Alternative hypotheses can provide the opportunity to discover new theories that a researcher can use to disprove an existing theory that may not have been backed up by evidence.

We defined the relationship that exist between null-hypothesis and alternative hypothesis. While the null hypothesis is always a default assumption about our test data, the alternative hypothesis puts in all the effort to make sure the null hypothesis is disproved.

Null-hypothesis always explores new relationships between the independent variables to find potential outcomes from our test data. We should note that for every null hypothesis, one or more alternate hypotheses can be developed.

Also Check:

Mathematics Maths Formulas Branches of Mathematics

FAQs on Alternative Hypothesis

What is hypothesis.

A hypothesis is a statement of a relationship between two or more variables.” It is a working statement or theory that is based on insufficient evidence.

What is an Alternative Hypothesis?

Alternative hypothesis, denoted by H 1 , opposes the null-hypothesis. It assumes a relation between the variables and serves as an evidence to reject the null-hypothesis.

What is the Difference between Null-Hypothesis and Alternative Hypothesis?

Null hypothesis is the default claim that assumes no relationship between variables while alternative hypothesis is the opposite claim which considers statistical significance between the variables.

What is Alternative and Experimental Hypothesis?

Null hypothesis (H 0 ) states there is no effect or difference, while the alternative hypothesis (H 1 or H a ) asserts the presence of an effect, difference, or relationship between variables. In hypothesis testing, we seek evidence to either reject the null hypothesis in favor of the alternative hypothesis or fail to do so.

Please Login to comment...

Similar reads.

• Math-Statistics
• School Learning

Improve your Coding Skills with Practice

What kind of Experience do you want to share?