hypothesis testing python examples

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

Hey there. My name is Zach Bobbitt. I have a Master 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.

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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.

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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.

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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

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!

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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.

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Python Companion to Statistical Thinking in the 21st Century

Hypothesis testing in Python

Hypothesis testing in python #.

In this chapter we will present several examples of using Python to perform hypothesis testing.

Simple example: Coin-flipping #

Let’s say that we flipped 100 coins and observed 70 heads. We would like to use these data to test the hypothesis that the true probability is 0.5. First let’s generate our data, simulating 100,000 sets of 100 flips. We use such a large number because it turns out that it’s very rare to get 70 heads, so we need many attempts in order to get a reliable estimate of these probabilties. This will take a couple of minutes to complete.

Now we can compute the proportion of samples from the distribution observed when the true proportion of heads is 0.5.

For comparison, we can also compute the p-value for 70 or more heads based on a null hypothesis of \(P_{heads}=0.5\) , using the binomial distribution.

compute the probability of 69 or fewer heads, when P(heads)=0.5

the probability of 70 or more heads is simply the complement of p_lt_70

Simulating p-values #

In this exercise we will perform hypothesis testing many times in order to test whether the p-values provided by our statistical test are valid. We will sample data from a normal distribution with a mean of zero, and for each sample perform a t-test to determine whether the mean is different from zero. We will then count how often we reject the null hypothesis; since we know that the true mean is zero, these are by definition Type I errors.

We should see that the proportion of samples with p < .05 is about 5%.

Pytest With Eric

How to Use Hypothesis and Pytest for Robust Property-Based Testing in Python

There will always be cases you didn’t consider, making this an ongoing maintenance job. Unit testing solves only some of these issues.

Example-Based Testing vs Property-Based Testing

Project set up, getting started, prerequisites, simple example, source code, simple example — unit tests, example-based testing, running the unit test, property-based testing, complex example, source code, complex example — unit tests, discover bugs with hypothesis, define your own hypothesis strategies, model-based testing in hypothesis, additional reading.

Table of Contents

Testing your python code with hypothesis, installing & using hypothesis, a quick example, understanding hypothesis, using hypothesis strategies, filtering and mapping strategies, composing strategies, constraints & satisfiability, writing reusable strategies with functions.

• @composite: Declarative Strategies
• @example: Explicitly Testing Certain Values

Hypothesis Example: Roman Numeral Converter

I can think of a several Python packages that greatly improved the quality of the software I write. Two of them are pytest and hypothesis . The former adds an ergonomic framework for writing tests and fixtures and a feature-rich test runner. The latter adds property-based testing that can ferret out all but the most stubborn bugs using clever algorithms, and that’s the package we’ll explore in this course.

In an ordinary test you interface with the code you want to test by generating one or more inputs to test against, and then you validate that it returns the right answer. But that, then, raises a tantalizing question: what about all the inputs you didn’t test? Your code coverage tool may well report 100% test coverage, but that does not, ipso facto , mean the code is bug-free.

One of the defining features of Hypothesis is its ability to generate test cases automatically in a manner that is:

Repeated invocations of your tests result in reproducible outcomes, even though Hypothesis does use randomness to generate the data.

You are given a detailed answer that explains how your test failed and why it failed. Hypothesis makes it clear how you, the human, can reproduce the invariant that caused your test to fail.

You can refine its strategies and tell it where or what it should or should not search for. At no point are you compelled to modify your code to suit the whims of Hypothesis if it generates nonsensical data.

So let’s look at how Hypothesis can help you discover errors in your code.

You can install hypothesis by typing pip install hypothesis . It has few dependencies of its own, and should install and run everywhere.

Hypothesis plugs into pytest and unittest by default, so you don’t have to do anything to make it work with it. In addition, Hypothesis comes with a CLI tool you can invoke with hypothesis . But more on that in a bit.

I will use pytest throughout to demonstrate Hypothesis, but it works equally well with the builtin unittest module.

Before I delve into the details of Hypothesis, let’s start with a simple example: a naive CSV writer and reader. A topic that seems simple enough: how hard is it to separate fields of data with a comma and then read it back in later?

But of course CSV is frighteningly hard to get right. The US and UK use '.' as a decimal separator, but in large parts of the world they use ',' which of course results in immediate failure. So then you start quoting things, and now you need a state machine that can distinguish quoted from unquoted; and what about nested quotes, etc.

The naive CSV reader and writer is an excellent stand-in for any number of complex projects where the requirements outwardly seem simple but there lurks a large number of edge cases that you must take into account.

Here the writer simply string quotes each field before joining them together with ',' . The reader does the opposite: it assumes each field is quoted after it is split by the comma.

A naive roundtrip pytest proves the code “works”:

And evidently so:

And for a lot of code that’s where the testing would begin and end. A couple of lines of code to test a couple of functions that outwardly behave in a manner that anybody can read and understand. Now let’s look at what a Hypothesis test would look like, and what happens when we run it:

At first blush there’s nothing here that you couldn’t divine the intent of, even if you don’t know Hypothesis. I’m asking for the argument fields to have a list ranging from one element of generated text up to ten. Aside from that, the test operates in exactly the same manner as before.

Now watch what happens when I run the test:

Hypothesis quickly found an example that broke our code. As it turns out, a list of [','] breaks our code. We get two fields back after round-tripping the code through our CSV writer and reader — uncovering our first bug.

In a nutshell, this is what Hypothesis does. But let’s look at it in detail.

Simply put, Hypothesis generates data using a number of configurable strategies . Strategies range from simple to complex. A simple strategy may generate bools; another integers. You can combine strategies to make larger ones, such as lists or dicts that match certain patterns or structures you want to test. You can clamp their outputs based on certain constraints, like only positive integers or strings of a certain length. You can also write your own strategies if you have particularly complex requirements.

Strategies are the gateway to property-based testing and are a fundamental part of how Hypothesis works. You can find a detailed list of all the strategies in the Strategies reference of their documentation or in the hypothesis.strategies module.

The best way to get a feel for what each strategy does in practice is to import them from the hypothesis.strategies module and call the example() method on an instance:

You may have noticed that the floats example included inf in the list. By default, all strategies will – where feasible – attempt to test all legal (but possibly obscure) forms of values you can generate of that type. That is particularly important as corner cases like inf or NaN are legal floating-point values but, I imagine, not something you’d ordinarily test against yourself.

And that’s one pillar of how Hypothesis tries to find bugs in your code: by testing edge cases that you would likely miss yourself. If you ask it for a text() strategy you’re as likely to be given Western characters as you are a mishmash of unicode and escape-encoded garbage. Understanding why Hypothesis generates the examples it does is a useful way to think about how your code may interact data it has no control over.

Now if it were simply generating text or numbers from an inexhaustible source of numbers or strings, it wouldn’t catch as many errors as it actually does . The reason for that is that each test you write is subjected to a battery of examples drawn from the strategies you’ve designed. If a test case fails, it’s put aside and tested again but with a reduced subset of inputs, if possible. In Hypothesis it’s known as shrinking the search space to try and find the smallest possible result that will cause your code to fail. So instead of a 10,000-length string, if it can find one that’s only 3 or 4, it will try to show that to you instead.

You can tell Hypothesis to filter or map the examples it draws to further reduce them if the strategy does not meet your requirements:

Here I ask for integers where the number is greater than 0 and is evenly divisible by 8. Hypothesis will then attempt to generate examples that meets the constraints you have imposed on it.

You can also map , which works in much the same way as filter. Here I’m asking for lowercase ASCII and then uppercasing them:

Having said that, using either when you don’t have to (I could have asked for uppercase ASCII characters to begin with) is likely to result in slower strategies.

A third option, flatmap , lets you build strategies from strategies; but that deserves closer scrutiny, so I’ll talk about it later.

You can tell Hypothesis to pick one of a number of strategies by composing strategies with | or st.one_of() :

An essential feature when you have to draw from multiple sources of examples for a single data point.

When you ask Hypothesis to draw an example it takes into account the constraints you may have imposed on it: only positive integers; only lists of numbers that add up to exactly 100; any filter() calls you may have applied; and so on. Those are constraints. You’re taking something that was once unbounded (with respect to the strategy you’re drawing an example from, that is) and introducing additional limitations that constrain the possible range of values it can give you.

But consider what happens if I pass filters that will yield nothing at all:

At some point Hypothesis will give up and declare it cannot find anything that satisfies that strategy and its constraints.

Hypothesis gives up after a while if it’s not able to draw an example. Usually that indicates an invariant in the constraints you’ve placed that makes it hard or impossible to draw examples from. In the example above, I asked for numbers that are simultaneously below zero and greater than zero, which is an impossible request.

As you can see, the strategies are simple functions, and they behave as such. You can therefore refactor each strategy into reusable patterns:

The benefit of this approach is that if you discover edge cases that Hypothesis does not account for, you can update the pattern in one place and observe its effects on your code. It’s functional and composable.

One caveat of this approach is that you cannot draw examples and expect Hypothesis to behave correctly. So I don’t recommend you call example() on a strategy only to pass it into another strategy.

For that, you want the @composite decorator.

@composite : Declarative Strategies

If the previous approach is unabashedly functional in nature, this approach is imperative.

The @composite decorator lets you write imperative Python code instead. If you cannot easily structure your strategy with the built-in ones, or if you require more granular control over the values it emits, you should consider the @composite strategy.

Instead of returning a compound strategy object like you would above, you instead draw examples using a special function you’re given access to in the decorated function.

This example draws two randomized names and returns them as a tuple:

Note that the @composite decorator passes in a special draw callable that you must use to draw samples from. You cannot – well, you can , but you shouldn’t – use the example() method on the strategy object you get back. Doing so will break Hypothesis’s ability to synthesize test cases properly.

Because the code is imperative you’re free to modify the drawn examples to your liking. But what if you’re given an example you don’t like or one that breaks a known invariant you don’t wish to test for? For that you can use the assume() function to state the assumptions that Hypothesis must meet if you try to draw an example from generate_full_name .

Let’s say that first_name and last_name must not be equal:

Like the assert statement in Python, the assume() function teaches Hypothesis what is, and is not, a valid example. You use this to great effect to generate complex compound strategies.

I recommend you observe the following rules of thumb if you write imperative strategies with @composite :

If you want to draw a succession of examples to initialize, say, a list or a custom object with values that meet certain criteria you should use filter , where possible, and assume to teach Hypothesis why the value(s) you drew and subsequently discarded weren’t any good.

The example above uses assume() to teach Hypothesis that first_name and last_name must not be equal.

If you can put your functional strategies in separate functions, you should. It encourages code re-use and if your strategies are failing (or not generating the sort of examples you’d expect) you can inspect each strategy in turn. Large nested strategies are harder to untangle and harder still to reason about.

If you can express your requirements with filter and map or the builtin constraints (like min_size or max_size ), you should. Imperative strategies that use assume may take more time to converge on a valid example.

@example : Explicitly Testing Certain Values

Occasionally you’ll come across a handful of cases that either fails or used to fail, and you want to ensure that Hypothesis does not forget to test them, or to indicate to yourself or your fellow developers that certain values are known to cause issues and should be tested explicitly.

The @example decorator does just that:

You can add as many as you like.

Let’s say I wanted to write a simple converter to and from Roman numerals.

Here I’m collecting Roman numerals into numerals , one at a time, by looping over SYMBOLS of valid numerals, subtracting the value of the symbol from number , until the while loop’s condition ( number >= 1 ) is False .

The test is also simple and serves as a smoke test. I generate a random integer and convert it to Roman numerals with to_roman . When it’s all said and done I turn the string of numerals into a set and check that all members of the set are legal Roman numerals.

Now if I run pytest on it seems to hang . But thanks to Hypothesis’s debug mode I can inspect why:

Ah. Instead of testing with tiny numbers like a human would ordinarily do, it used a fantastically large one… which is altogether slow.

OK, so there’s at least one gotcha; it’s not really a bug , but it’s something to think about: limiting the maximum value. I’m only going to limit the test, but it would be reasonable to limit it in the code also.

Changing the max_value to something sensible, like st.integers(max_value=5000) and the test now fails with another error:

It seems our code’s not able to handle the number 0! Which… is correct. The Romans didn’t really use the number zero as we would today; that invention came later, so they had a bunch of workarounds to deal with the absence of something. But that’s neither here nor there in our example. Let’s instead set min_value=1 also, as there is no support for negative numbers either:

OK… not bad. We’ve proven that given a random assortment of numbers between our defined range of values that, indeed, we get something resembling Roman numerals.

One of the hardest things about Hypothesis is framing questions to your testable code in a way that tests its properties but without you, the developer, knowing the answer (necessarily) beforehand. So one simple way to test that there’s at least something semi-coherent coming out of our to_roman function is to check that it can generate the very numerals we defined in SYMBOLS from before:

Here I’m sampling from a tuple of the SYMBOLS from earlier. The sampling algorithm’ll decide what values it wants to give us, all we care about is that we are given examples like ("I", 1) or ("V", 5) to compare against.

So let’s run pytest again:

Oops. The Roman numeral V is equal to 5 and yet we get five IIIII ? A closer examination reveals that, indeed, the code only yields sequences of I equal to the number we pass it. There’s a logic error in our code.

In the example above I loop over the elements in the SYMBOLS dictionary but as it’s ordered the first element is always I . And as the smallest representable value is 1, we end up with just that answer. It’s technically correct as you can count with just I but it’s not very useful.

Fixing it is easy though:

Rerunning the test yields a pass. Now we know that, at the very least, our to_roman function is capable of mapping numbers that are equal to any symbol in SYMBOLS .

Now the litmus test is taking the numeral we’re given and making sense of it. So let’s write a function that converts a Roman numeral back into decimal:

Like to_roman we walk through each character, get the numeral’s numeric value, and add it to the running total. The test is a simple roundtrip test as to_roman has an inverse function from_roman (and vice versa) such that :

Invertible functions are easier to test because you can compare the output of one against the input of another and check if it yields the original value. But not every function has an inverse, though.

Running the test yields a pass:

So now we’re in a pretty good place. But there’s a slight oversight in our Roman numeral converters, though: they don’t respect the subtraction rule for some of the numerals. For instance VI is 6; but IV is 4. The value XI is 11; and IX is 9. Only some (sigh) numerals exhibit this property.

So let’s write another test. This time it’ll fail as we’ve yet to write the modified code. Luckily we know the subtractive numerals we must accommodate:

Pretty simple test. Check that certain numerals yield the value, and that the values yield the right numeral.

With an extensive test suite we should feel fairly confident making changes to the code. If we break something, one of our preexisting tests will fail.

The rules around which numerals are subtractive is rather subjective. The SUBTRACTIVE_SYMBOLS dictionary holds the most common ones. So all we need to do is read ahead of the numerals list to see if there exists a two-digit numeral that has a prescribed value and then we use that instead of the usual value.

The to_roman change is simple. A union of the two numeral symbol dictionaries is all it takes . The code already understands how to turn numbers into numerals — we just added a few more.

This method requires Python 3.9 or later. Read how to merge dictionaries

If done right, running the tests should yield a pass:

And that’s it. We now have useful tests and a functional Roman numeral converter that converts to and from with ease. But one thing we didn’t do is create a strategy that generates Roman numerals using st.text() . A custom composite strategy to generate both valid and invalid Roman numerals to test the ruggedness of our converter is left as an exercise to you.

In the next part of this course we’ll look at more advanced testing strategies.

Unlike a tool like faker that generates realistic-looking test data for fixtures or demos, Hypothesis is a property-based tester . It uses heuristics and clever algorithms to find inputs that break your code.

Testing a function that does not have an inverse to compare the result against – like our Roman numeral converter that works both ways – you often have to approach your code as though it were a black box where you relinquish control of the inputs and outputs. That is harder, but makes for less brittle code.

It’s perfectly fine to mix and match tests. Hypothesis is useful for flushing out invariants you would never think of. Combine it with known inputs and outputs to jump start your testing for the first 80%, and augment it with Hypothesis to catch the remaining 20%.

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