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:

import scipy.stats as stats #define data data = [300, 315, 320, 311, 314, 309, 300, 308, 305, 303, 305, 301, 303] #perform one sample t-test stats.ttest_1samp(a=data, popmean=310) Ttest_1sampResult(statistic=-1.5848116313861254, pvalue=0.1389944275158753)

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**µ = 310 (the mean weight for this species of turtle is 310 pounds)_{0}:**H**µ ≠310 (the mean weight is_{A}:*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:

import scipy.stats as stats #define array of turtle weights for each sample sample1 = [300, 315, 320, 311, 314, 309, 300, 308, 305, 303, 305, 301, 303] sample2 = [335, 329, 322, 321, 324, 319, 304, 308, 305, 311, 307, 300, 305] #perform two sample t-test stats.ttest_ind(a=sample1, b=sample2) Ttest_indResult(statistic=-2.1009029257555696, pvalue=0.04633501389516516)

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**µ_{0}:_{1}= µ_{2 }(the mean weight between the two species is equal)**H**µ_{A}:_{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:

import scipy.stats as stats #define before and after max jump heights 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] #perform paired samples t-test stats.ttest_rel(a=before, b=after) Ttest_relResult(statistic=-2.5289026942943655, pvalue=0.02802807458682508)

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**µ_{0}:_{1}= µ_{2 }(the mean jump height before and after using the program is equal)**H**µ_{A}:_{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