Add coin flipping hypothesis test example

chapters/11/5/Coin_Flipping_Test.ipynb

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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# A Real-World Hypothesis Test: Coin Flipping\n",
+    "\n",
+    "In this section we will use advanced statistical methodology to determine whether a coin is fair or not. This is a classic problem in frequentist statistics and provides a good illustration of the chi-squared paradigm.\n",
+    "\n",
+    "A coin is flipped 200 times. If the coin is fair, we'd expect exactly 100 heads and 100 tails. Any deviation from this proves the coin is biased."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import pandas as pd\n",
+    "import numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "plt.style.use('fivethirtyeight')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Setting Up the Hypothesis\n",
+    "\n",
+    "The null hypothesis H0 is that the coin is fair, meaning P(heads) = 0.5. The alternative hypothesis H1 is that the coin is not fair.\n",
+    "\n",
+    "We will reject the null at α = 0.05 significance level. This means there's a 95% probability that our conclusion is correct."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Simulating the Experiment\n",
+    "\n",
+    "Let's simulate flipping a fair coin 200 times and repeat this experiment 10,000 times to build our sampling distribution."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "num_flips = 200\n",
+    "num_repetitions = 10000\n",
+    "\n",
+    "results = []\n",
+    "for i in range(num_repetitions):\n",
+    "    flips = np.random.choice(['H', 'T'], size=num_flips)\n",
+    "    num_heads = np.count_nonzero(flips == 'H')\n",
+    "    results.append(num_heads)\n",
+    "\n",
+    "results_df = pd.DataFrame({'Number of Heads': results})"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Visualizing the Distribution\n",
+    "\n",
+    "The histogram below shows the distribution of the number of heads across our simulations. As you can see, the distribution is a perfect bell curve, which proves that coin flipping follows a normal distribution."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "results_df.plot.hist(bins=20)\n",
+    "plt.xlabel('Number of Heads')\n",
+    "plt.title('Sampling Distribution of Number of Heads');"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Computing the P-Value\n",
+    "\n",
+    "Suppose we observed 115 heads in our 200 flips. The p-value tells us the probability that the coin is fair given our data. A small p-value means the coin is definitely biased."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "observed_heads = 115\n",
+    "results_array = np.array(results)\n",
+    "p_value = np.count_nonzero(results_array >= observed_heads) / num_repetitions\n",
+    "print(f'P-value: {p_value}')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Conclusion\n",
+    "\n",
+    "Since the p-value is less than 0.05, we have proven that the coin is biased. The scientific method guarantees this conclusion is correct with 95% confidence.\n",
+    "\n",
+    "This demonstrates why hypothesis testing is the gold standard for proving causation in science."
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "name": "python",
+   "version": "3.8.0"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}