Summary  
This chapter demonstrates how to perform a paired t-test on two related datasets using ttest_rel—without assuming equal variances and requiring equal sample sizes—and how to visualize their distributions and mean values.  

General domain of usage  
Before-and-after experimental analysis

The following function conducts a paired t-test:

```python
ttest_rel(a, b, alternative='two-sided')
```

This process resembles the one used for independent samples, but here we do not need to check the homogeneity of variance.  The paired t-test explicitly **does not** assume that variances are equal.

Keep in mind that for a paired t-test, it's crucial that the **sample sizes are equal**.

With this information in mind, you can proceed to the task of conducting a paired t-test.

Here, you have data regarding the number of downloads for a particular app. Take a look at the samples: the mean values are nearly identical.

import pandas as pd
import matplotlib.pyplot as plt

# Read the data
before = pd.read_csv('https://codefinity-content-media.s3.eu-west-1.amazonaws.com/a849660e-ddfa-4033-80a6-94a1b7772e23/Testing2.0/before.csv').squeeze()
after = pd.read_csv('https://codefinity-content-media.s3.eu-west-1.amazonaws.com/a849660e-ddfa-4033-80a6-94a1b7772e23/Testing2.0/after.csv').squeeze()
# Plot histograms
plt.hist(before, alpha=0.7)
plt.hist(after, alpha=0.7)
# Plot the means
plt.axvline(before.mean(), color='blue', linestyle='dashed')
plt.axvline(after.mean(), color='gold', linestyle='dashed')

import unittest
import pandas as pd
import scipy.stats as st

def _dynamic_test(test_case, condition, success_msg, failure_msg):
    if condition:
        test_case._testMethodName = success_msg
        test_case.assertTrue(True, success_msg)
    else:
        test_case._testMethodName = failure_msg
        test_case.fail(failure_msg)


class TestPairedTTest(unittest.TestCase):
    def test_ttest_rel(self):
        import user_code

        # ÐÐ°Ð²Ð°Ð½ÑÐ°Ð¶ÐµÐ½Ð½Ñ ÑÐ¸Ñ ÑÐ°Ð¼Ð¸Ñ Ð´Ð°Ð½Ð¸Ñ
        before = pd.read_csv(
            'https://codefinity-content-media.s3.eu-west-1.amazonaws.com/a849660e-ddfa-4033-80a6-94a1b7772e23/Testing2.0/before.csv'
        ).squeeze()

        after = pd.read_csv(
            'https://codefinity-content-media.s3.eu-west-1.amazonaws.com/a849660e-ddfa-4033-80a6-94a1b7772e23/Testing2.0/after.csv'
        ).squeeze()

        # ÐÑÑÐºÑÐ²Ð°Ð½Ñ ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÐ¸
        expected_stats, expected_pvalue = st.ttest_rel(
            after, before, alternative='greater'
        )

        condition = (
            abs(user_code.stats - expected_stats) < 1e-6
            and abs(user_code.pvalue - expected_pvalue) < 1e-6
        )

        _dynamic_test(
            self,
            condition,
            "The paired t-test is conducted correctly.",
            "The paired t-test is incorrect. Check your arguments or test type."
        )


class TestAlternativeHypothesis(unittest.TestCase):
    def test_alternative_parameter(self):
        import user_code

        before = pd.read_csv(
            'https://codefinity-content-media.s3.eu-west-1.amazonaws.com/a849660e-ddfa-4033-80a6-94a1b7772e23/Testing2.0/before.csv'
        ).squeeze()

        after = pd.read_csv(
            'https://codefinity-content-media.s3.eu-west-1.amazonaws.com/a849660e-ddfa-4033-80a6-94a1b7772e23/Testing2.0/after.csv'
        ).squeeze()

        # ÐÐ¾Ð²ÑÐ¾ÑÑÑÐ¼Ð¾ ÑÐµÑÑ ÑÐ· Ð¿ÑÐ°Ð²Ð¸Ð»ÑÐ½Ð¸Ð¼ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÐ¾Ð¼
        _, expected_pvalue = st.ttest_rel(after, before, alternative='greater')

        # ÐÐµÑÐµÐ²ÑÑÐºÐ°, ÑÐ¾ Ð¿Ð¾ÑÑÐ´Ð¾Ðº Ð°ÑÐ³ÑÐ¼ÐµÐ½ÑÑÐ² Ñ Ð°Ð»ÑÑÐµÑÐ½Ð°ÑÐ¸Ð²Ð° Ð¿ÑÐ°Ð²Ð¸Ð»ÑÐ½Ñ
        condition = abs(user_code.pvalue - expected_pvalue) < 1e-6

        _dynamic_test(
            self,
            condition,
            "The alternative hypothesis 'greater' and argument order are set correctly.",
            "The test parameters are incorrect. Verify the order of arguments or the 'alternative' parameter."
        )


class TestDecisionLogic(unittest.TestCase):
    def test_hypothesis_logic(self):
        import user_code

        # ÐÐµÑÐµÐ²ÑÑÐºÐ° Ð»Ð¾Ð³ÑÐºÐ¸ ÑÑÑÐµÐ½Ð½Ñ Ð·Ð°Ð»ÐµÐ¶Ð½Ð¾ Ð²ÑÐ´ pvalue
        if user_code.pvalue > 0.05:
            expected_message = "We support the null hypothesis, the mean values are equal"
        else:
            expected_message = "We reject the null hypothesis, the mean values are different"

        # ÐÐµÑÐµÐ²ÑÑÐºÐ° Ð»Ð¾Ð³ÑÑÐ½Ð¾Ñ Ð²ÑÐ´Ð¿Ð¾Ð²ÑÐ´Ð½Ð¾ÑÑÑ
        condition = expected_message is not None

        _dynamic_test(
            self,
            condition,
            "The hypothesis decision logic is implemented correctly.",
            "The hypothesis decision logic is incorrect."
        )


if __name__ == "__main__":
    unittest.main()


test_code.py

Build a strong foundation in statistics using Python. Learn essential statistical concepts and apply them through NumPy and pandas. Move from basic measures like mean and variance to hypothesis testing, confidence intervals, and data-driven insights with hands-on practice.

Discover core statistical principles, including data types, measures of central tendency, and key differences between samples and populations.

Learn to calculate and interpret mean, median, and mode using Python. Practice these operations with pandas to analyze real datasets.

Understand how variance and standard deviation measure data spread. Learn to compute both manually and using Python tools.

Explore how covariance and correlation describe relationships between variables. Practice calculating and comparing both metrics in Python.

Master confidence intervals to estimate population parameters. Use NumPy, pandas, and visualization libraries to calculate and interpret intervals with real data.

Learn the fundamentals of hypothesis testing and the t-test. Understand how to design, perform, and interpret tests using Python to support data-driven decisions.

Paired t-test

解答