{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "6eb2c139",
   "metadata": {},
   "source": [
    "# Week 7: Hypothesis Testing and Statistical Tests\n",
    "\n",
    "*Kmex Consult — Data Science Fundamentals Course*\n",
    "\n",
    "Work through each code cell in order. Edit and re-run to experiment."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "88561668",
   "metadata": {},
   "source": [
    "*Data Science Fundamentals Course*"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0495951e",
   "metadata": {},
   "source": [
    "## Week 7 Overview\n",
    "Hypothesis testing is one of the most important statistical tools in data science. It provides a formal framework for making decisions about populations based on sample data. This week you'll learn how to:\n",
    "\n",
    "- Formulate proper hypotheses\n",
    "- Understand Type I and Type II errors\n",
    "- Interpret p-values correctly (and avoid common misinterpretations)\n",
    "- Conduct various types of statistical tests\n",
    "- Calculate effect sizes\n",
    "- Make data-driven conclusions with proper evidence\n",
    "\n",
    "Hypothesis testing is used everywhere in data science and business:\n",
    "- A/B testing for website optimization\n",
    "- Quality control in manufacturing\n",
    "- Clinical trials for new drugs\n",
    "- Evaluating marketing campaign effectiveness\n",
    "- Detecting fraud patterns\n",
    "\n",
    "By the end of Week 7, you will be able to:\n",
    "- Formulate null and alternative hypotheses correctly\n",
    "- Choose appropriate statistical tests\n",
    "- Calculate test statistics and p-values\n",
    "- Interpret results in business context\n",
    "- Understand Type I/II errors and statistical power\n",
    "- Calculate and interpret effect sizes\n",
    "- Report findings professionally\n",
    "\n",
    "Week 7 is divided into three 2-hour sessions:\n",
    "- Session 1: Hypothesis Testing Fundamentals and T-tests\n",
    "- Session 2: ANOVA and Chi-Square Tests\n",
    "- Session 3: Non-parametric Tests and Effect Sizes"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3fcf1079",
   "metadata": {},
   "source": [
    "## SESSION 1: Hypothesis Testing Fundamentals and T-tests"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7217aaed",
   "metadata": {},
   "source": [
    "### Duration: 2 hours"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "afcc269f",
   "metadata": {},
   "source": [
    "### 1.1 Hypothesis Testing Framework"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "14702ab4",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy import stats\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "print(\"=\"*50)\n",
    "print(\"HYPOTHESIS TESTING FRAMEWORK\")\n",
    "print(\"=\"*50)\n",
    "\n",
    "print(\"\"\"\n",
    "THE BASIC IDEA:\n",
    "We make a claim about a population and test it using sample data.\n",
    "\n",
    "STEPS:\n",
    "1. State hypotheses\n",
    "2. Choose significance level (α)\n",
    "3. Calculate test statistic\n",
    "4. Find p-value\n",
    "5. Make decision and interpret\n",
    "\n",
    "KEY CONCEPTS:\n",
    "\"\"\")\n",
    "\n",
    "print(\"\"\"\n",
    "NULL HYPOTHESIS (H₀):\n",
    "- Status quo or no effect\n",
    "- What we assume is true initially\n",
    "- We try to find evidence AGAINST it\n",
    "\n",
    "ALTERNATIVE HYPOTHESIS (H₁ or Hₐ):\n",
    "- What we're testing for\n",
    "- Different from null hypothesis\n",
    "\n",
    "SIGNIFICANCE LEVEL (α):\n",
    "- Probability of rejecting H₀ when it's actually true\n",
    "- Common values: 0.05, 0.01, 0.10\n",
    "- 0.05 means 5% chance of Type I error\n",
    "\n",
    "P-VALUE:\n",
    "- Probability of observing data as extreme (or more extreme) if H₀ is true\n",
    "- Small p-value: strong evidence against H₀\n",
    "- NOT the probability that H₀ is true!\n",
    "\n",
    "DECISION RULE:\n",
    "- If p-value < α: Reject H₀ (significant result)\n",
    "- If p-value ≥ α: Fail to reject H₀ (not significant)\n",
    "\"\"\")\n",
    "\n",
    "# Type I and Type II errors\n",
    "print(\"\\n\" + \"=\"*50)\n",
    "print(\"TYPE I AND TYPE II ERRORS\")\n",
    "print(\"=\"*50)\n",
    "\n",
    "print(\"\"\"\n",
    "TYPE I ERROR (α - False Positive):\n",
    "- Reject H₀ when H₀ is actually true\n",
    "- \"Crying wolf\" - finding effect that doesn't exist\n",
    "- Probability = α (significance level)\n",
    "\n",
    "TYPE II ERROR (β - False Negative):\n",
    "- Fail to reject H₀ when H₀ is actually false\n",
    "- Missing real effect\n",
    "- Probability = β\n",
    "\n",
    "STATISTICAL POWER:\n",
    "- Power = 1 - β\n",
    "- Probability of correctly rejecting false H₀\n",
    "- Typical target: 0.80 (80% power)\n",
    "\n",
    "TRUTH TABLE:\n",
    "H₀ True H₀ False\n",
    "Reject H₀ Type I Error Correct ✓\n",
    "Fail to Reject Correct ✓ Type II Error\n",
    "\"\"\")\n",
    "\n",
    "# Example visualization\n",
    "fig, ax = plt.subplots(figsize=(10, 6))\n",
    "\n",
    "# Two normal distributions\n",
    "x = np.linspace(-4, 4, 1000)\n",
    "null_dist = stats.norm.pdf(x, 0, 1)\n",
    "alt_dist = stats.norm.pdf(x, 2, 1)\n",
    "\n",
    "ax.plot(x, null_dist, 'b-', linewidth=2, label='H₀ Distribution (μ=0)')\n",
    "ax.plot(x, alt_dist, 'r-', linewidth=2, label='H₁ Distribution (μ=2)')\n",
    "\n",
    "# Critical value at α=0.05\n",
    "critical_value = stats.norm.ppf(0.95)\n",
    "ax.axvline(critical_value, color='black', linestyle='--', linewidth=2, label=f'Critical value (α=0.05)')\n",
    "\n",
    "# Shade Type I error (α)\n",
    "x_alpha = x[x >= critical_value]\n",
    "alpha_area = stats.norm.pdf(x_alpha, 0, 1)\n",
    "ax.fill_between(x_alpha, alpha_area, alpha=0.3, color='blue', label='Type I Error (α)')\n",
    "\n",
    "# Shade Type II error (β)\n",
    "x_beta = x[x < critical_value]\n",
    "beta_area = stats.norm.pdf(x_beta, 2, 1)\n",
    "ax.fill_between(x_beta, beta_area, alpha=0.3, color='red', label='Type II Error (β)')\n",
    "\n",
    "ax.set_xlabel('Test Statistic', fontsize=12)\n",
    "ax.set_ylabel('Probability Density', fontsize=12)\n",
    "ax.set_title('Type I and Type II Errors', fontsize=14, fontweight='bold')\n",
    "ax.legend(fontsize=10)\n",
    "ax.grid(True, alpha=0.3)\n",
    "plt.tight_layout()\n",
    "plt.show()\n",
    "\n",
    "print(f\"Critical value: {critical_value:.3f}\")\n",
    "print(f\"Type I error (α): 0.05 (5%)\")\n",
    "print(f\"Type II error (β): {stats.norm.cdf(critical_value, 2, 1):.3f} (~27%)\")\n",
    "print(f\"Power (1-β): {1 - stats.norm.cdf(critical_value, 2, 1):.3f} (~73%)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "23e0117c",
   "metadata": {},
   "source": [
    "### 1.2 One-Sample T-Test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "ed5f824e",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy import stats\n",
    "import pandas as pd\n",
    "\n",
    "print(\"=\"*50)\n",
    "print(\"ONE-SAMPLE T-TEST\")\n",
    "print(\"=\"*50)\n",
    "\n",
    "print(\"\"\"\n",
    "WHEN TO USE:\n",
    "- Testing if population mean equals specific value\n",
    "- Sample size is small (n < 30) or population SD unknown\n",
    "- Data approximately normally distributed\n",
    "\n",
    "HYPOTHESES:\n",
    "- H₀: μ = μ₀ (mean equals specified value)\n",
    "- H₁: μ ≠ μ₀ (two-tailed, most common)\n",
    "- Or: μ > μ₀ (one-tailed, right)\n",
    "- Or: μ < μ₀ (one-tailed, left)\n",
    "\"\"\")\n",
    "\n",
    "# Example: Factory production\n",
    "print(\"\\nEXAMPLE: FACTORY PRODUCTION\")\n",
    "print(\"-\"*50)\n",
    "\n",
    "# Production line target: 500g per package\n",
    "target_weight = 500\n",
    "\n",
    "# Sample weights from production\n",
    "weights = np.array([498, 502, 499, 501, 497, 503, 498, 500, 499, 502,\n",
    "501, 498, 497, 503, 499, 500, 502, 498, 501, 499])\n",
    "\n",
    "print(f\"Target weight: {target_weight}g\")\n",
    "print(f\"Sample: {weights}\")\n",
    "print(f\"Sample size: {len(weights)}\")\n",
    "print(f\"Sample mean: {weights.mean():.2f}g\")\n",
    "print(f\"Sample std dev: {weights.std(ddof=1):.2f}g\")\n",
    "\n",
    "# Conduct one-sample t-test\n",
    "t_stat, p_value = stats.ttest_1samp(weights, target_weight)\n",
    "\n",
    "print(f\"\\nT-Test Results:\")\n",
    "print(f\"T-statistic: {t_stat:.4f}\")\n",
    "print(f\"P-value: {p_value:.4f}\")\n",
    "\n",
    "# Interpretation\n",
    "alpha = 0.05\n",
    "print(f\"\\nSignificance level (α): {alpha}\")\n",
    "\n",
    "if p_value < alpha:\n",
    "print(f\"Result: REJECT H₀ (p={p_value:.4f} < {alpha})\")\n",
    "print(\"Conclusion: The mean weight is significantly different from 500g\")\n",
    "else:\n",
    "print(f\"Result: FAIL TO REJECT H₀ (p={p_value:.4f} ≥ {alpha})\")\n",
    "print(\"Conclusion: No significant difference from target weight\")\n",
    "\n",
    "# Calculate 95% confidence interval\n",
    "n = len(weights)\n",
    "se = weights.std(ddof=1) / np.sqrt(n)\n",
    "t_crit = stats.t.ppf(0.975, df=n-1)\n",
    "ci_lower = weights.mean() - t_crit * se\n",
    "ci_upper = weights.mean() + t_crit * se\n",
    "\n",
    "print(f\"\\n95% Confidence Interval: [{ci_lower:.2f}, {ci_upper:.2f}]\")\n",
    "print(f\"Target {target_weight}g is {'INSIDE' if ci_lower <= target_weight <= ci_upper else 'OUTSIDE'} the CI\")\n",
    "\n",
    "# Example 2: Test score hypothesis\n",
    "print(\"\\n\" + \"=\"*50)\n",
    "print(\"EXAMPLE 2: TEST SCORE ANALYSIS\")\n",
    "print(\"-\"*50)\n",
    "\n",
    "# Hypothesis: Class average is 75\n",
    "hypothesized_mean = 75\n",
    "test_scores = np.array([78, 82, 76, 74, 88, 91, 73, 71, 85, 79,\n",
    "81, 75, 84, 77, 80, 72, 86, 79, 83, 75])\n",
    "\n",
    "t_stat, p_value = stats.ttest_1samp(test_scores, hypothesized_mean)\n",
    "\n",
    "print(f\"Null hypothesis: μ = {hypothesized_mean}\")\n",
    "print(f\"Sample mean: {test_scores.mean():.2f}\")\n",
    "print(f\"Sample size: {len(test_scores)}\")\n",
    "print(f\"\\nT-statistic: {t_stat:.4f}\")\n",
    "print(f\"P-value: {p_value:.4f}\")\n",
    "\n",
    "if p_value < 0.05:\n",
    "print(f\"Result: Class average is SIGNIFICANTLY DIFFERENT from {hypothesized_mean}\")\n",
    "else:\n",
    "print(f\"Result: Class average is NOT significantly different from {hypothesized_mean}\")\n",
    "\n",
    "# Effect size (Cohen's d)\n",
    "cohens_d = (test_scores.mean() - hypothesized_mean) / test_scores.std(ddof=1)\n",
    "print(f\"\\nEffect size (Cohen's d): {cohens_d:.3f}\")\n",
    "print(f\"Interpretation: {'Small' if abs(cohens_d) < 0.5 else 'Medium' if abs(cohens_d) < 0.8 else 'Large'} effect\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "32d1ce0e",
   "metadata": {},
   "source": [
    "### 1.3 Two-Sample T-Test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "c6293da4",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy import stats\n",
    "import pandas as pd\n",
    "\n",
    "print(\"=\"*50)\n",
    "print(\"TWO-SAMPLE T-TEST\")\n",
    "print(\"=\"*50)\n",
    "\n",
    "print(\"\"\"\n",
    "WHEN TO USE:\n",
    "- Comparing means of two independent groups\n",
    "- Testing if two population means are equal\n",
    "- Small sample sizes or population SD unknown\n",
    "\n",
    "HYPOTHESES:\n",
    "- H₀: μ₁ = μ₂ (means are equal)\n",
    "- H₁: μ₁ ≠ μ₂ (means are different - two-tailed)\n",
    "\n",
    "ASSUMPTIONS:\n",
    "- Both samples approximately normal\n",
    "- Variances equal (can test with Levene's test)\n",
    "- Samples are independent\n",
    "\"\"\")\n",
    "\n",
    "# Example: Treatment effectiveness\n",
    "print(\"\\nEXAMPLE: DRUG TREATMENT EFFECTIVENESS\")\n",
    "print(\"-\"*50)\n",
    "\n",
    "# Control group (no treatment)\n",
    "control = np.array([5.2, 4.8, 5.1, 4.9, 5.0, 4.7, 5.3, 4.9, 5.1, 4.8])\n",
    "\n",
    "# Treatment group (received drug)\n",
    "treatment = np.array([3.2, 3.5, 3.1, 3.4, 3.0, 3.6, 3.3, 3.2, 3.4, 3.5])\n",
    "\n",
    "print(f\"Control group (n={len(control)}): {control}\")\n",
    "print(f\"Treatment group (n={len(treatment)}): {treatment}\")\n",
    "\n",
    "print(f\"\\nDescriptive Statistics:\")\n",
    "print(f\"Control mean: {control.mean():.2f}, SD: {control.std(ddof=1):.2f}\")\n",
    "print(f\"Treatment mean: {treatment.mean():.2f}, SD: {treatment.std(ddof=1):.2f}\")\n",
    "print(f\"Difference in means: {control.mean() - treatment.mean():.2f}\")\n",
    "\n",
    "# Test for equal variances (Levene's test)\n",
    "stat_levene, p_levene = stats.levene(control, treatment)\n",
    "print(f\"\\nLevene's Test for Equal Variances:\")\n",
    "print(f\"P-value: {p_levene:.4f}\")\n",
    "\n",
    "equal_var = p_levene > 0.05\n",
    "print(f\"Assumption: Variances are {'EQUAL' if equal_var else 'NOT EQUAL'}\")\n",
    "\n",
    "# Conduct two-sample t-test\n",
    "t_stat, p_value = stats.ttest_ind(control, treatment, equal_var=equal_var)\n",
    "\n",
    "print(f\"\\nTwo-Sample T-Test:\")\n",
    "print(f\"T-statistic: {t_stat:.4f}\")\n",
    "print(f\"P-value: {p_value:.6f}\")\n",
    "\n",
    "alpha = 0.05\n",
    "if p_value < alpha:\n",
    "print(f\"Result: REJECT H₀ (p={p_value:.6f} < {alpha})\")\n",
    "print(\"Conclusion: Treatment significantly reduces the outcome\")\n",
    "else:\n",
    "print(f\"Result: FAIL TO REJECT H₀ (p={p_value:.6f} ≥ {alpha})\")\n",
    "print(\"Conclusion: No significant difference between groups\")\n",
    "\n",
    "# Cohen's d\n",
    "pooled_std = np.sqrt(((len(control)-1)*control.std(ddof=1)**2 +\n",
    "(len(treatment)-1)*treatment.std(ddof=1)**2) /\n",
    "(len(control) + len(treatment) - 2))\n",
    "cohens_d = (control.mean() - treatment.mean()) / pooled_std\n",
    "\n",
    "print(f\"\\nEffect size (Cohen's d): {abs(cohens_d):.3f}\")\n",
    "print(f\"Interpretation: {'Small' if abs(cohens_d) < 0.5 else 'Medium' if abs(cohens_d) < 0.8 else 'Large'} effect\")\n",
    "\n",
    "# Paired t-test example\n",
    "print(\"\\n\" + \"=\"*50)\n",
    "print(\"PAIRED T-TEST (Within-subject design)\")\n",
    "print(\"-\"*50)\n",
    "\n",
    "print(\"\"\"\n",
    "WHEN TO USE:\n",
    "- Same subjects measured twice (before/after)\n",
    "- Dependent/correlated samples\n",
    "- Examples: weight before/after diet, test scores before/after training\n",
    "\"\"\")\n",
    "\n",
    "# Before and after training\n",
    "before = np.array([65, 72, 68, 71, 70, 73, 69, 74, 66, 68])\n",
    "after = np.array([72, 78, 74, 76, 77, 79, 75, 81, 72, 75])\n",
    "\n",
    "print(f\"\\nBefore training: {before}\")\n",
    "print(f\"After training: {after}\")\n",
    "print(f\"Improvements: {after - before}\")\n",
    "\n",
    "# Paired t-test\n",
    "t_stat_paired, p_value_paired = stats.ttest_rel(before, after)\n",
    "\n",
    "print(f\"\\nPaired T-Test Results:\")\n",
    "print(f\"Mean before: {before.mean():.2f}\")\n",
    "print(f\"Mean after: {after.mean():.2f}\")\n",
    "print(f\"Mean difference: {(after-before).mean():.2f}\")\n",
    "print(f\"T-statistic: {t_stat_paired:.4f}\")\n",
    "print(f\"P-value: {p_value_paired:.6f}\")\n",
    "\n",
    "if p_value_paired < 0.05:\n",
    "print(\"Result: Training SIGNIFICANTLY improved performance\")\n",
    "else:\n",
    "print(\"Result: Training did NOT significantly improve performance\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d789b3e9",
   "metadata": {},
   "source": [
    "## SESSION 2: ANOVA and Chi-Square Tests"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "efb201b2",
   "metadata": {},
   "source": [
    "### Duration: 2 hours"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "68bd2e44",
   "metadata": {},
   "source": [
    "### 2.1 Analysis of Variance (ANOVA)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "a29d964a",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy import stats\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "print(\"=\"*50)\n",
    "print(\"ONE-WAY ANOVA\")\n",
    "print(\"=\"*50)\n",
    "\n",
    "print(\"\"\"\n",
    "WHEN TO USE:\n",
    "- Comparing means across 3+ groups\n",
    "- Extension of two-sample t-test\n",
    "- Testing if at least one group mean differs from others\n",
    "\n",
    "HYPOTHESES:\n",
    "- H₀: μ₁ = μ₂ = μ₃ = ... (all means equal)\n",
    "- H₁: At least one mean differs\n",
    "\n",
    "ASSUMPTIONS:\n",
    "- Each group approximately normal\n",
    "- Variances equal across groups (test with Levene's)\n",
    "- Independent observations\n",
    "\"\"\")\n",
    "\n",
    "# Example: Store performance\n",
    "print(\"\\nEXAMPLE: STORE PERFORMANCE\")\n",
    "print(\"-\"*50)\n",
    "\n",
    "# Sales from three store locations\n",
    "store_A = np.array([45, 48, 42, 46, 44, 47, 43, 45, 46, 44])\n",
    "store_B = np.array([52, 55, 53, 54, 51, 56, 52, 54, 53, 55])\n",
    "store_C = np.array([40, 42, 39, 41, 38, 43, 40, 39, 42, 41])\n",
    "\n",
    "print(f\"Store A sales: {store_A}\")\n",
    "print(f\"Store B sales: {store_B}\")\n",
    "print(f\"Store C sales: {store_C}\")\n",
    "\n",
    "print(f\"\\nDescriptive Statistics:\")\n",
    "print(f\"Store A: mean={store_A.mean():.2f}, sd={store_A.std(ddof=1):.2f}\")\n",
    "print(f\"Store B: mean={store_B.mean():.2f}, sd={store_B.std(ddof=1):.2f}\")\n",
    "print(f\"Store C: mean={store_C.mean():.2f}, sd={store_C.std(ddof=1):.2f}\")\n",
    "\n",
    "# Test homogeneity of variance\n",
    "stat_levene, p_levene = stats.levene(store_A, store_B, store_C)\n",
    "print(f\"\\nLevene's Test for Equal Variances: p={p_levene:.4f}\")\n",
    "print(f\"Variances are {'EQUAL' if p_levene > 0.05 else 'NOT EQUAL'}\")\n",
    "\n",
    "# Conduct one-way ANOVA\n",
    "f_stat, p_value = stats.f_oneway(store_A, store_B, store_C)\n",
    "\n",
    "print(f\"\\nOne-Way ANOVA Results:\")\n",
    "print(f\"F-statistic: {f_stat:.4f}\")\n",
    "print(f\"P-value: {p_value:.6f}\")\n",
    "\n",
    "if p_value < 0.05:\n",
    "print(f\"Result: REJECT H₀\")\n",
    "print(\"Conclusion: Store performance differs significantly\")\n",
    "else:\n",
    "print(f\"Result: FAIL TO REJECT H₀\")\n",
    "print(\"Conclusion: No significant difference in store performance\")\n",
    "\n",
    "# Post-hoc test (pairwise comparisons)\n",
    "if p_value < 0.05:\n",
    "print(\"\\n\" + \"-\"*50)\n",
    "print(\"POST-HOC: PAIRWISE COMPARISONS (T-tests)\")\n",
    "print(\"-\"*50)\n",
    "\n",
    "# Bonferroni correction\n",
    "alpha_corrected = 0.05 / 3 # 3 pairwise comparisons\n",
    "\n",
    "comparisons = [\n",
    "(\"A vs B\", store_A, store_B),\n",
    "(\"A vs C\", store_A, store_C),\n",
    "(\"B vs C\", store_B, store_C)\n",
    "]\n",
    "\n",
    "for name, group1, group2 in comparisons:\n",
    "t_stat_post, p_post = stats.ttest_ind(group1, group2)\n",
    "sig = \"***\" if p_post < alpha_corrected else \"\"\n",
    "print(f\"{name}: t={t_stat_post:.3f}, p={p_post:.4f} {sig}\")\n",
    "\n",
    "print(f\"Note: Bonferroni corrected α = {alpha_corrected:.4f}\")\n",
    "\n",
    "# Chi-Square Test\n",
    "print(\"\\n\" + \"=\"*50)\n",
    "print(\"CHI-SQUARE TEST\")\n",
    "print(\"=\"*50)\n",
    "\n",
    "print(\"\"\"\n",
    "WHEN TO USE:\n",
    "- Testing relationships between categorical variables\n",
    "- Goodness of fit for categorical data\n",
    "- Independence of two categorical variables\n",
    "\n",
    "HYPOTHESES (Independence):\n",
    "- H₀: Variables are independent\n",
    "- H₁: Variables are associated\n",
    "\"\"\")\n",
    "\n",
    "# Example: Product preference by gender\n",
    "print(\"\\nEXAMPLE: PRODUCT PREFERENCE BY GENDER\")\n",
    "print(\"-\"*50)\n",
    "\n",
    "# Contingency table\n",
    "observed = np.array([\n",
    "[45, 30], # Male: Product A (45), Product B (30)\n",
    "[25, 60] # Female: Product A (25), Product B (60)\n",
    "])\n",
    "\n",
    "df_contingency = pd.DataFrame(\n",
    "observed,\n",
    "index=['Male', 'Female'],\n",
    "columns=['Product A', 'Product B']\n",
    ")\n",
    "\n",
    "print(\"Observed frequencies:\")\n",
    "print(df_contingency)\n",
    "\n",
    "# Chi-square test\n",
    "chi2_stat, p_chi2, dof, expected = stats.chi2_contingency(observed)\n",
    "\n",
    "print(f\"\\nExpected frequencies:\")\n",
    "print(expected)\n",
    "\n",
    "print(f\"\\nChi-Square Test Results:\")\n",
    "print(f\"Chi-square statistic: {chi2_stat:.4f}\")\n",
    "print(f\"P-value: {p_chi2:.6f}\")\n",
    "print(f\"Degrees of freedom: {dof}\")\n",
    "\n",
    "if p_chi2 < 0.05:\n",
    "print(f\"Result: REJECT H₀\")\n",
    "print(\"Conclusion: Product preference is associated with gender\")\n",
    "else:\n",
    "print(f\"Result: FAIL TO REJECT H₀\")\n",
    "print(\"Conclusion: No association between gender and product preference\")\n",
    "\n",
    "# Cramér's V (effect size)\n",
    "n = observed.sum()\n",
    "min_dim = min(observed.shape) - 1\n",
    "cramers_v = np.sqrt(chi2_stat / (n * min_dim))\n",
    "print(f\"\\nEffect size (Cramér's V): {cramers_v:.3f}\")\n",
    "print(f\"Interpretation: {'Negligible' if cramers_v < 0.1 else 'Small' if cramers_v < 0.3 else 'Medium' if cramers_v < 0.5 else 'Large'} effect\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "135acf36",
   "metadata": {},
   "source": [
    "## SESSION 3: Non-parametric Tests and Effect Sizes"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "47fb67cd",
   "metadata": {},
   "source": [
    "### Duration: 2 hours"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "89181bab",
   "metadata": {},
   "source": [
    "### 3.1 Non-Parametric Tests"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "c807da42",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy import stats\n",
    "import pandas as pd\n",
    "\n",
    "print(\"=\"*50)\n",
    "print(\"NON-PARAMETRIC TESTS\")\n",
    "print(\"=\"*50)\n",
    "\n",
    "print(\"\"\"\n",
    "WHEN TO USE:\n",
    "- Data not normally distributed\n",
    "- Small sample sizes\n",
    "- Ordinal or ranked data\n",
    "- Violate parametric test assumptions\n",
    "\n",
    "NON-PARAMETRIC ALTERNATIVES:\n",
    "- t-test → Mann-Whitney U test (Wilcoxon rank-sum)\n",
    "- Paired t-test → Wilcoxon signed-rank test\n",
    "- ANOVA → Kruskal-Wallis test\n",
    "\"\"\")\n",
    "\n",
    "# Mann-Whitney U Test\n",
    "print(\"\\nMANN-WHITNEY U TEST (Non-parametric alternative to two-sample t-test)\")\n",
    "print(\"-\"*50)\n",
    "\n",
    "print(\"\"\"\n",
    "WHEN TO USE:\n",
    "- Two independent groups\n",
    "- Data not normally distributed or unknown distribution\n",
    "- Ordinal data or small samples\n",
    "\"\"\")\n",
    "\n",
    "# Example: Customer satisfaction (non-normal data)\n",
    "old_method = np.array([3, 2, 4, 2, 3, 1, 3, 2, 4, 1]) # Ranks 1-5\n",
    "new_method = np.array([5, 4, 5, 4, 4, 5, 3, 4, 5, 4])\n",
    "\n",
    "print(f\"\\nOld method satisfaction: {old_method}\")\n",
    "print(f\"New method satisfaction: {new_method}\")\n",
    "\n",
    "# Mann-Whitney U test\n",
    "u_stat, p_mann_whitney = stats.mannwhitneyu(old_method, new_method, alternative='two-sided')\n",
    "\n",
    "print(f\"\\nMann-Whitney U Test:\")\n",
    "print(f\"U-statistic: {u_stat:.4f}\")\n",
    "print(f\"P-value: {p_mann_whitney:.6f}\")\n",
    "\n",
    "if p_mann_whitney < 0.05:\n",
    "print(\"Result: Methods have SIGNIFICANTLY different satisfaction ratings\")\n",
    "else:\n",
    "print(\"Result: No significant difference in satisfaction ratings\")\n",
    "\n",
    "# Wilcoxon Signed-Rank Test\n",
    "print(\"\\n\" + \"=\"*50)\n",
    "print(\"WILCOXON SIGNED-RANK TEST (Non-parametric paired t-test)\")\n",
    "print(\"-\"*50)\n",
    "\n",
    "print(\"\"\"\n",
    "WHEN TO USE:\n",
    "- Paired/dependent samples\n",
    "- Data not normally distributed\n",
    "- Ordinal or ranked data\n",
    "\"\"\")\n",
    "\n",
    "# Before and after (ordinal data)\n",
    "before = np.array([2, 3, 2, 1, 3, 2, 1, 2, 3, 2])\n",
    "after = np.array([4, 5, 4, 3, 5, 4, 3, 4, 5, 4])\n",
    "\n",
    "print(f\"\\nBefore: {before}\")\n",
    "print(f\"After: {after}\")\n",
    "print(f\"Differences: {after - before}\")\n",
    "\n",
    "# Wilcoxon signed-rank test\n",
    "w_stat, p_wilcoxon = stats.wilcoxon(before, after)\n",
    "\n",
    "print(f\"\\nWilcoxon Signed-Rank Test:\")\n",
    "print(f\"W-statistic: {w_stat:.4f}\")\n",
    "print(f\"P-value: {p_wilcoxon:.6f}\")\n",
    "\n",
    "if p_wilcoxon < 0.05:\n",
    "print(\"Result: Significant improvement after intervention\")\n",
    "else:\n",
    "print(\"Result: No significant improvement\")\n",
    "\n",
    "# Kruskal-Wallis Test\n",
    "print(\"\\n\" + \"=\"*50)\n",
    "print(\"KRUSKAL-WALLIS TEST (Non-parametric ANOVA)\")\n",
    "print(\"-\"*50)\n",
    "\n",
    "print(\"\"\"\n",
    "WHEN TO USE:\n",
    "- Three or more groups\n",
    "- Data not normally distributed\n",
    "- Ordinal data or unknown distribution\n",
    "\"\"\")\n",
    "\n",
    "# Example: Customer ratings for three products\n",
    "product_A = np.array([2, 3, 2, 1, 3, 2, 4, 2, 3, 1])\n",
    "product_B = np.array([4, 5, 4, 3, 5, 4, 5, 4, 5, 3])\n",
    "product_C = np.array([3, 4, 3, 2, 4, 3, 4, 3, 4, 2])\n",
    "\n",
    "print(f\"\\nProduct A ratings: {product_A}\")\n",
    "print(f\"Product B ratings: {product_B}\")\n",
    "print(f\"Product C ratings: {product_C}\")\n",
    "\n",
    "# Kruskal-Wallis test\n",
    "h_stat, p_kruskal = stats.kruskal(product_A, product_B, product_C)\n",
    "\n",
    "print(f\"\\nKruskal-Wallis Test:\")\n",
    "print(f\"H-statistic: {h_stat:.4f}\")\n",
    "print(f\"P-value: {p_kruskal:.6f}\")\n",
    "\n",
    "if p_kruskal < 0.05:\n",
    "print(\"Result: Products have SIGNIFICANTLY different ratings\")\n",
    "else:\n",
    "print(\"Result: No significant difference in product ratings\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9562292c",
   "metadata": {},
   "source": [
    "### 3.2 Effect Sizes and Practical Significance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "3c10e411",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy import stats\n",
    "import pandas as pd\n",
    "\n",
    "print(\"=\"*50)\n",
    "print(\"EFFECT SIZES\")\n",
    "print(\"=\"*50)\n",
    "\n",
    "print(\"\"\"\n",
    "WHY EFFECT SIZE MATTERS:\n",
    "- P-value tells if effect exists (significant)\n",
    "- Effect size tells how LARGE the effect is\n",
    "- Large sample can detect tiny (insignificant) effects\n",
    "- Small sample might miss large (important) effects\n",
    "\n",
    "PRACTICAL vs STATISTICAL SIGNIFICANCE:\n",
    "- Statistical significance: Effect exists (p < 0.05)\n",
    "- Practical significance: Effect is large enough to matter\n",
    "\"\"\")\n",
    "\n",
    "# Cohen's d\n",
    "print(\"\\n\" + \"=\"*50)\n",
    "print(\"COHEN'S D (for comparing means)\")\n",
    "print(\"=\"*50)\n",
    "\n",
    "print(\"\"\"\n",
    "INTERPRETATION:\n",
    "- d = 0.2: Small effect\n",
    "- d = 0.5: Medium effect\n",
    "- d = 0.8: Large effect\n",
    "\n",
    "CALCULATION:\n",
    "Cohen's d = (Mean₁ - Mean₂) / Pooled SD\n",
    "\"\"\")\n",
    "\n",
    "# Example\n",
    "group1 = np.array([85, 88, 82, 90, 86, 84, 87, 89, 83, 85])\n",
    "group2 = np.array([78, 80, 75, 82, 79, 77, 81, 83, 76, 79])\n",
    "\n",
    "mean1, mean2 = group1.mean(), group2.mean()\n",
    "sd1, sd2 = group1.std(ddof=1), group2.std(ddof=1)\n",
    "n1, n2 = len(group1), len(group2)\n",
    "\n",
    "pooled_sd = np.sqrt(((n1-1)*sd1**2 + (n2-1)*sd2**2) / (n1+n2-2))\n",
    "cohens_d = (mean1 - mean2) / pooled_sd\n",
    "\n",
    "print(f\"\\nGroup 1: mean={mean1:.2f}, sd={sd1:.2f}\")\n",
    "print(f\"Group 2: mean={mean2:.2f}, sd={sd2:.2f}\")\n",
    "print(f\"\\nCohen's d = ({mean1:.2f} - {mean2:.2f}) / {pooled_sd:.2f}\")\n",
    "print(f\"Cohen's d = {cohens_d:.3f}\")\n",
    "\n",
    "effect_interpretation = \"Small\" if abs(cohens_d) < 0.5 else \"Medium\" if abs(cohens_d) < 0.8 else \"Large\"\n",
    "print(f\"\\nInterpretation: {effect_interpretation} effect\")\n",
    "\n",
    "# Other effect sizes\n",
    "print(\"\\n\" + \"=\"*50)\n",
    "print(\"OTHER EFFECT SIZES\")\n",
    "print(\"=\"*50)\n",
    "\n",
    "print(\"\"\"\n",
    "R² (Coefficient of Determination):\n",
    "- Proportion of variance explained\n",
    "- 0 to 1 (0% to 100%)\n",
    "- Small: 0.01, Medium: 0.06, Large: 0.14\n",
    "\n",
    "Cramér's V (for chi-square):\n",
    "- Small: 0.1, Medium: 0.3, Large: 0.5\n",
    "\n",
    "Eta² (for ANOVA):\n",
    "- Proportion of variance between groups\n",
    "- Similar scale to R²\n",
    "\n",
    "Odds Ratio:\n",
    "- For categorical data\n",
    "- Ratio of odds in two groups\n",
    "\"\"\")\n",
    "\n",
    "# Example: R-squared\n",
    "print(\"\\nEXAMPLE: R-SQUARED FOR CORRELATION\")\n",
    "print(\"-\"*50)\n",
    "\n",
    "x = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9, 10])\n",
    "y = 2*x + np.random.normal(0, 2, 10)\n",
    "\n",
    "correlation = np.corrcoef(x, y)[0, 1]\n",
    "r_squared = correlation ** 2\n",
    "\n",
    "print(f\"Correlation coefficient (r): {correlation:.3f}\")\n",
    "print(f\"R-squared: {r_squared:.3f}\")\n",
    "print(f\"Interpretation: {r_squared*100:.1f}% of variance in y explained by x\")\n",
    "\n",
    "# Reporting results\n",
    "print(\"\\n\" + \"=\"*50)\n",
    "print(\"HOW TO REPORT RESULTS\")\n",
    "print(\"=\"*50)\n",
    "\n",
    "print(\"\"\"\n",
    "COMPLETE RESULT REPORT SHOULD INCLUDE:\n",
    "\n",
    "1. Test name and type (one-tailed vs two-tailed)\n",
    "2. Sample descriptive statistics\n",
    "3. Test statistic and value\n",
    "4. Degrees of freedom (if applicable)\n",
    "5. P-value (exact if p > 0.001, otherwise p < 0.001)\n",
    "6. Effect size\n",
    "7. Confidence interval\n",
    "8. Conclusion\n",
    "\n",
    "EXAMPLE REPORT:\n",
    "\"A two-sample t-test comparing treatment group (M=85.2, SD=4.3, n=10)\n",
    "with control group (M=79.8, SD=5.1, n=10) revealed a significant difference,\n",
    "t(18)=2.84, p=0.011, d=1.01. The 95% CI [3.12, 12.64] does not include zero,\n",
    "indicating treatment significantly improved outcomes with a large effect size.\"\n",
    "\"\"\")\n",
    "\n",
    "# Real-world reporting\n",
    "print(\"\\n\" + \"-\"*50)\n",
    "print(\"REAL-WORLD EXAMPLE\")\n",
    "print(\"-\"*50)\n",
    "\n",
    "data1 = np.array([45, 48, 42, 46, 44, 47, 43, 45])\n",
    "data2 = np.array([52, 55, 53, 54, 51, 56, 52, 54])\n",
    "\n",
    "t_stat, p_value = stats.ttest_ind(data1, data2)\n",
    "n1, n2 = len(data1), len(data2)\n",
    "mean1, mean2 = data1.mean(), data2.mean()\n",
    "sd1, sd2 = data1.std(ddof=1), data2.std(ddof=1)\n",
    "\n",
    "pooled_sd = np.sqrt(((n1-1)*sd1**2 + (n2-1)*sd2**2) / (n1+n2-2))\n",
    "cohens_d = (mean1 - mean2) / pooled_sd\n",
    "\n",
    "se = pooled_sd * np.sqrt(1/n1 + 1/n2)\n",
    "df = n1 + n2 - 2\n",
    "t_crit = stats.t.ppf(0.975, df)\n",
    "ci_lower = (mean1 - mean2) - t_crit * se\n",
    "ci_upper = (mean1 - mean2) + t_crit * se\n",
    "\n",
    "report = f\"\"\"\n",
    "PROFESSIONAL REPORT:\n",
    "A two-sample t-test comparing Method A (M={mean1:.1f}, SD={sd1:.2f}, n={n1})\n",
    "with Method B (M={mean2:.1f}, SD={sd2:.2f}, n={n2}) revealed a significant\n",
    "difference, t({df})={t_stat:.2f}, p={p_value:.4f}, d={cohens_d:.2f}.\n",
    "The 95% CI [{ci_lower:.1f}, {ci_upper:.1f}] does not include zero.\n",
    "Method B shows significantly better performance with a large effect size.\n",
    "\"\"\"\n",
    "\n",
    "print(report)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3992ca32",
   "metadata": {},
   "source": [
    "#### Hypothesis Testing Checklist\n",
    "- Define H₀ and H₁ clearly before analyzing data\n",
    "- Choose significance level (α) in advance\n",
    "- Check test assumptions before conducting test\n",
    "- Select appropriate test based on:\n",
    "- Number of groups (1, 2, or 3+)\n",
    "- Independence of samples\n",
    "- Data distribution normality\n",
    "- Data type (continuous vs categorical)\n",
    "- Report: test statistic, p-value, effect size, CI\n",
    "- Interpret p-value correctly (NOT probability H₀ is true)\n",
    "- Consider practical significance, not just statistical\n",
    "- Report effect size even if not significant\n",
    "- Use confidence intervals alongside p-values"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "30214be5",
   "metadata": {},
   "source": [
    "## Week 7 Summary\n",
    "By completing Week 7, you have learned:\n",
    "- Hypothesis testing framework: H₀, H₁, α, p-value\n",
    "- Type I error (α): False positive rate\n",
    "- Type II error (β): False negative rate\n",
    "- Statistical power: 1 - β\n",
    "- Interpreting p-values correctly\n",
    "- One-sample t-test: testing against a hypothesized mean\n",
    "- Two-sample t-test: comparing two independent groups\n",
    "- Paired t-test: comparing dependent/related samples\n",
    "- Checking assumptions: normality, equal variances\n",
    "- One-way ANOVA: comparing 3+ groups\n",
    "- Post-hoc tests: pairwise comparisons after ANOVA\n",
    "- Chi-square test: independence of categorical variables\n",
    "- Contingency tables and frequency analysis\n",
    "- Mann-Whitney U test: non-parametric alternative to t-test\n",
    "- Wilcoxon signed-rank test: non-parametric paired test\n",
    "- Kruskal-Wallis test: non-parametric ANOVA\n",
    "- Effect sizes: Cohen&#x27;s d, R², Cramér&#x27;s V\n",
    "- Practical vs statistical significance\n",
    "- Professional reporting of results"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "24fe766b",
   "metadata": {},
   "source": [
    "## Week 7 Assignments"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d1a724f0",
   "metadata": {},
   "source": [
    "### Assignment 1: Hypothesis Testing Scenarios\n",
    "Conduct hypothesis tests on provided scenarios:\n",
    "- Formulate appropriate null and alternative hypotheses\n",
    "- Choose correct statistical test with justification\n",
    "- Check test assumptions\n",
    "- Conduct test and report results\n",
    "- Interpret p-value and effect size\n",
    "- Make business recommendations based on findings\n",
    "- Report: 5 different hypothesis tests with full analysis"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c0d55d8a",
   "metadata": {},
   "source": [
    "### Assignment 2: A/B Testing Analysis\n",
    "Analyze an A/B test (real or simulated):\n",
    "- Define control and treatment groups\n",
    "- Calculate descriptive statistics for each group\n",
    "- Conduct appropriate statistical test\n",
    "- Calculate effect size and confidence interval\n",
    "- Determine statistical and practical significance\n",
    "- Create visualizations of results\n",
    "- Write business report with recommendations"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cdcd8df0",
   "metadata": {},
   "source": [
    "### Assignment 3: Comprehensive Statistical Analysis\n",
    "Perform complete statistical analysis on real dataset:\n",
    "- Test assumptions for different groups\n",
    "- Conduct multiple hypothesis tests:\n",
    "- At least 1 parametric test (t-test or ANOVA)\n",
    "- At least 1 categorical test (chi-square)\n",
    "- At least 1 non-parametric test if needed\n",
    "- Calculate all effect sizes\n",
    "- Create professional report with:\n",
    "- Summary of findings\n",
    "- Statistical evidence\n",
    "- Practical implications\n",
    "- Business recommendations\n",
    "- Appropriate visualizations"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4b474eb9",
   "metadata": {},
   "source": [
    "## Recommended Practice Problems\n",
    "- Conduct t-tests with different sample sizes and observe p-value changes\n",
    "- Calculate statistical power for different effect sizes\n",
    "- Compare parametric vs non-parametric test results on same data\n",
    "- Perform post-hoc tests after ANOVA with different correction methods\n",
    "- Analyze contingency tables and interpret associations\n",
    "- Design A/B tests with specified power and effect size\n",
    "- Report results using proper APA format\n",
    "- Create visualizations for different test results\n",
    "- Calculate sample size needed for desired power"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8c4195bf",
   "metadata": {},
   "source": [
    "## Additional Resources"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7c25d4a7",
   "metadata": {},
   "source": [
    "### Books\n",
    "- Chapter 7 (continued): Statistical Inference - &quot;Python for Data Analysis&quot; by Wes McKinney\n",
    "- Statistics Done Wrong by Alex Reinhart\n",
    "- The Book of Why by Judea Pearl - Understanding causality"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3d8d2d79",
   "metadata": {},
   "source": [
    "### Online Resources\n",
    "- scipy.stats documentation: https://docs.scipy.org/doc/scipy/reference/stats.html\n",
    "- StatQuest with Josh Starmer (YouTube): Excellent visual explanations\n",
    "- P-value explained: https://www.nature.com/articles/506150a\n",
    "- Effect size guide: https://en.wikipedia.org/wiki/Effect_size"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1c6b9ede",
   "metadata": {},
   "source": [
    "### Tools and Calculators\n",
    "- G*Power: Statistical power analysis calculator\n",
    "- Stat Trek: Online statistics calculators\n",
    "- Real Statistics Resource Pack for Excel"
   ]
  }
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