Summary  
This chapter covers how to implement hypothesis testing in code by formulating null and alternative hypotheses, computing a test statistic and p-value (e.g., via a one-sample t-test), and comparing the p-value to a significance level to decide whether to reject the null hypothesis.  

General domain of usage  
Research and data analysis

**Video: Introduction to Hypothesis Testing**

This video introduces you to the core concepts of hypothesis testing. You will learn what null and alternative hypotheses are, how p-values are used to make decisions, and what significance levels mean in the context of statistical analysis.

Hypothesis testing is a fundamental concept in statistics that allows you to make inferences about a population based on sample data. The process begins with formulating two competing statements: the **null hypothesis** (`H0`) and the **alternative hypothesis** (`H1` or `Ha`). The null hypothesis usually states that there is no effect or no difference, while the alternative hypothesis represents what you aim to support – typically that there is a significant effect or difference.

The typical steps in hypothesis testing are as follows:
1. State the null and alternative hypotheses;
2. Choose a significance level (commonly denoted as `alpha`, such as 0.05);
3. Select the appropriate statistical test based on your data and hypothesis;
4. Calculate the test statistic and the corresponding p-value;
5. Compare the p-value to the significance level to decide whether to reject or fail to reject the null hypothesis.

A **p-value** is the probability of obtaining a result at least as extreme as the one observed, assuming the null hypothesis is true. If the p-value is less than the chosen significance level (`alpha`), you reject the null hypothesis in favor of the alternative.

It is important to understand the two types of errors in hypothesis testing. A **Type I error** occurs when you reject the null hypothesis when it is actually true (a "false positive"). The probability of making a Type I error is the significance level (`alpha`). A **Type II error** occurs when you fail to reject the null hypothesis when the alternative hypothesis is true (a "false negative"). The probability of a Type II error is denoted by `beta`.

Selecting the right significance level depends on the context and consequences of making errors. Common choices are 0.05 or 0.01, but stricter or more lenient levels may be justified depending on the application.


import numpy as np
from scipy import stats

# Suppose you have a sample of exam scores
sample_scores = np.array([82, 85, 88, 90, 79, 93, 87, 84, 91, 89])

# You want to test if the average score is significantly different from 85
# Null hypothesis (H0): The population mean = 85
# Alternative hypothesis (H1): The population mean ≠ 85

# Perform a one-sample t-test
t_statistic, p_value = stats.ttest_1samp(sample_scores, popmean=85)

print("T-statistic:", t_statistic)
print("P-value:", p_value)

# Interpret the result at alpha = 0.05
alpha = 0.05
if p_value < alpha:
    print("Reject the null hypothesis: There is a significant difference from 85.")
else:
    print("Fail to reject the null hypothesis: No significant difference from 85.")

import unittest
import ast
import re
import io
import sys
import numpy as np
from scipy import stats

class TestTask(unittest.TestCase):
    def test_default_values(self):
        with open('user_code.py', 'r') as f:
            code = f.read()
            
        ns = {}
        try:
            exec(code, ns)
            p_val = ns.get('p_val')
            test_result = ns.get('test_result')
            
            data = np.array([20, 22, 23, 19, 24, 21, 20, 18, 25, 22])
            mu = 21
            _, expected_p = stats.ttest_1samp(data, popmean=mu)
            
            p_val_correct = p_val is not None and isinstance(p_val, float) and abs(p_val - expected_p) < 1e-8
            result_correct = test_result == "Fail to reject the null hypothesis"
            
            is_correct = p_val_correct and result_correct
        except Exception:
            is_correct = False
            test_result = None
            p_val = None
            
        _dynamic_test(
            self,
            is_correct,
            "Correctly computes p-value and fails to reject the null hypothesis for default data.",
            f"Expected 'Fail to reject the null hypothesis' with p_val ~ 0.536. Got result: '{test_result}', p_val: {p_val}.",
        )

    def test_dynamic_rejection(self):
        with open('user_code.py', 'r') as f:
            code = f.read()
            
        # Підміняємо дані так, щоб вони сильно відрізнялися від mu=21 (гарантований Reject)
        code = change_var(code, 'data', 'np.array([100, 102, 101, 99, 103])')
        
        ns = {}
        try:
            exec(code, ns)
            p_val = ns.get('p_val')
            test_result = ns.get('test_result')
            
            is_correct = test_result == "Reject the null hypothesis" and p_val is not None and p_val < 0.05
        except Exception:
            is_correct = False
            
        _dynamic_test(
            self,
            is_correct,
            "Correctly rejects the null hypothesis when the data significantly differs from mu.",
            "Failed to dynamically reject the null hypothesis. Ensure you use the 'alpha' variable in your if-statement.",
        )

    def test_dynamic_fail_to_reject_with_high_alpha(self):
        with open('user_code.py', 'r') as f:
            code = f.read()
            
        # Підміняємо alpha на 0.99, щоб навіть малі відхилення викликали Reject,
        # але якщо дані ідеальні, то Fail. Або підміняємо дані. 
        # Перевіримо просто, що при дуже низькому alpha нічого не реджектиться.
        code = change_var(code, 'data', 'np.array([20, 22, 23, 19, 24])')
        code = change_var(code, 'mu', '21.5')
        code = change_var(code, 'alpha', '0.0001')
        
        ns = {}
        try:
            exec(code, ns)
            test_result = ns.get('test_result')
            is_correct = test_result == "Fail to reject the null hypothesis"
        except Exception:
            is_correct = False
            
        _dynamic_test(
            self,
            is_correct,
            "Logic adapts dynamically to different 'alpha' thresholds.",
            "The if-statement failed to adapt to a new alpha threshold.",
        )

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

def gsub_spaces(text):
    text = re.sub(r"\s+", "", text)
    text = text.replace("'", "")
    text = text.replace('"', "")
    return text

def change_var(code: str, var_name: str, value: str) -> str:
    tree = ast.parse(code)
    lines = code.splitlines()
    changed = False
    assign_nodes = [
        (i, node)
        for i, node in enumerate(tree.body)
        if isinstance(node, ast.Assign)
        and any(isinstance(target, ast.Name) and target.id == var_name for target in node.targets)
    ]
    if not assign_nodes:
        return code
    for i, node in reversed(assign_nodes):
        start_line = node.lineno - 1
        line = lines[start_line]
        indent = ' ' * (len(line) - len(line.lstrip()))
        lines[start_line] = f"{indent}{var_name} = {value}"
        next_line = len(lines)
        for next_node in tree.body[i+1:]:
            if hasattr(next_node, 'lineno'):
                next_line = next_node.lineno - 1
                break
        if next_line > start_line + 1:
            lines[start_line+1:next_line] = []
        changed = True
        
    return chr(10).join(lines) if changed else code

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

test_main.py

This course provides a comprehensive introduction to statistical methods using Python. You will learn how to apply core statistical concepts, perform hypothesis testing, analyze data distributions, and interpret results using Python's scientific libraries. The course emphasizes practical application through real-world examples and hands-on exercises.

A comprehensive section covering the core concepts and practical applications of statistical methods using Python.

Hypothesis Testing Basics

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