{"cells": [{"cell_type": "markdown", "metadata": {}, "source": ["# 04.03 - HYPOTHESIS TESTING"]}, {"cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [{"name": "stdout", "output_type": "stream", "text": ["endpoint https://m5knaekxo6.execute-api.us-west-2.amazonaws.com/dev-v0001/rlxmooc\n"]}, {"data": {"text/html": ["

See my courses and progress

"], "text/plain": [""]}, "execution_count": 26, "metadata": {}, "output_type": "execute_result"}], "source": ["!wget --no-cache -O init.py -q https://raw.githubusercontent.com/fagonzalezo/ai4eng-unal/main/content/init.py\n", "import init; init.init(force_download=False); init.get_weblink()"]}, {"cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [], "source": ["from scipy import stats\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "from progressbar import progressbar as pbar\n", "%matplotlib inline"]}, {"cell_type": "markdown", "metadata": {}, "source": ["## Stochastic models\n", "\n", "A probabilistic (or stochastic) model is one that assigns probabilities to the different objects or events it attempts to describe. For instance, \n", "\n", "- The probability of a patient to have different glucose levels.\n", "- The probability of students to obtain different scores in an exam\n", "\n", "\n", "A stochastic model\n", "\n", "- Just provides a probability for each object (a number between 0 and 1)\n", "- Does not necessarily provide an explanation on **HOW** probabilities arise.\n", "- Is given as a **probability distribution** with its corresponding PDF (**probability density function**), by which we can compute any probability\n", "\n"]}, {"cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": []}, {"cell_type": "markdown", "metadata": {}, "source": ["## Confidence on models\n", "\n", "### STEP 1: Define the model you want to challenge (the NULL Hypothesis $H_0$)\n", "\n", "\n", "A researcher brings a stochastic model claiming that students' scores in a certain exam follow a normal distribution with $\\mu=100$ and $\\sigma=15$. "]}, {"cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [{"data": {"text/plain": ["Text(0, 0.5, 'probability')"]}, "execution_count": 28, "metadata": {}, "output_type": "execute_result"}, {"data": {"image/png": 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\n", 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"]}, "metadata": {"needs_background": "light"}, "output_type": "display_data"}], "source": ["h0 = stats.norm(loc=100, scale=15)\n", "x = h0.rvs(10000)\n", "rx = np.linspace(np.min(x), np.max(x), 100)\n", "plt.plot(rx, h0.pdf(rx))\n", "plt.grid(); plt.xlabel(\"student exam score\"); plt.ylabel(\"probability\")"]}, {"cell_type": "markdown", "metadata": {}, "source": ["You are **NOT SURE** if the model works well with your students, so you start looking (**sampling**) at their exam scores. How much would you trust the model in the following cases:\n", "\n", "- H1: You sample 5 students and their average score is 110\n", "- H2: You sample 5 students and their average score is 105\n", "- H3: You sample 30 students and their average score is 105\n", "\n", "### STEP 2: Define your REAL WORLD sample and your test statistic\n", "\n", "The test statistic is what you **are interesting** in computing from a real world sample.\n", "\n", "- **SAMPLE**: a set of 5 students.\n", "\n", "- **TEST STATISTIC**: the average exam score of the sample\n", "\n", "### STEP 3: Understand the TEST STATISTIC distribution under $H_0$ (if the model is True)\n", "\n", "\n", "#### We will NOT USE FORMULAS, only SIMULATION\n", "\n", "Let's assume the model is right, let's **SIMULATE** we select 5 students from the model's probability distribution. We do **10 simulations**.\n", "\n", "Run the cells below several times, and ask yourself the following questions:\n", "\n", "- how probable is that you see 5 students with average score 110 or higher?\n", "- how probable is that you see 5 students with average score 105 or higher?\n"]}, {"cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [], "source": ["m = stats.norm(loc=100, scale=15)"]}, {"cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [{"name": "stdout", "output_type": "stream", "text": ["sample 0: 80.5 119.2 105.4 95.7 104.8 mean: 101.11\n", "sample 1: 111.8 94.3 99.3 76.9 99.4 mean: 96.34\n", "sample 2: 105.2 91.0 103.9 89.7 105.3 mean: 99.01\n", "sample 3: 113.5 123.8 107.7 108.4 86.6 mean: 108.00\n", "sample 4: 100.2 108.6 90.2 102.3 97.2 mean: 99.67\n", "sample 5: 72.8 81.4 109.3 76.9 129.1 mean: 93.92\n", "sample 6: 106.9 73.1 115.9 92.0 84.5 mean: 94.48\n", "sample 7: 76.8 105.7 95.3 99.7 83.0 mean: 92.10\n", "sample 8: 95.8 112.8 129.0 98.8 86.6 mean: 104.61\n", "sample 9: 93.1 99.3 91.2 83.9 81.5 mean: 89.80\n", "sample 10: 102.9 78.2 69.1 97.6 114.8 mean: 92.50\n", "sample 11: 92.0 108.9 91.9 91.7 83.2 mean: 93.56\n", "sample 12: 99.0 114.0 102.5 121.7 93.2 mean: 106.07\n", "sample 13: 104.9 65.4 112.6 103.8 99.6 mean: 97.27\n", "sample 14: 121.3 100.0 117.9 97.5 110.2 mean: 109.39\n", "sample 15: 111.2 88.0 104.6 95.9 70.2 mean: 93.99\n", "sample 16: 106.2 99.8 103.4 113.7 109.2 mean: 106.47\n", "sample 17: 96.3 87.0 106.6 114.0 108.8 mean: 102.56\n", "sample 18: 115.1 86.9 109.0 84.7 115.4 mean: 102.21\n", "sample 19: 95.0 109.6 110.1 115.6 116.4 mean: 109.36\n"]}], "source": ["for n in range(20):\n", " s = m.rvs(5)\n", " print (\"sample %2d: \"%n+ \" \".join([\"%5.1f\"%i for i in s]), \" mean: %7.2f\"%np.mean(s))"]}, {"cell_type": "markdown", "metadata": {}, "source": ["let's do the simulation **10000 times** and answer the questions"]}, {"cell_type": "code", "execution_count": 149, "metadata": {}, "outputs": [{"data": {"image/png": 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\n", "text/plain": ["
"]}, "metadata": {"needs_background": "light"}, "output_type": "display_data"}], "source": ["z = np.r_[[np.mean(m.rvs(5)) for _ in range(10000)]]\n", "\n", "plt.hist(z, bins=30, density=True, alpha=.5);\n", "plt.grid(); plt.xlabel(\"avg score for 5 students\"); plt.ylabel(\"probability\")\n", "plt.title(\"THE SAMPLING DISTRIBUTION\"); plt.show()"]}, {"cell_type": "markdown", "metadata": {}, "source": ["### STEP 4: measure the likelihood of the REAL WORLD sample w.r.t. $H_0$\n", "\n", "let's see and measure where our REAL WORLD sample falls in the distribution for the test statistic under $H_0$\n", "\n", "- if our REAL WORLD sample is too rare $\\rightarrow$ we have less trust in the $H_0$ model\n", "- if our REAL WORLD sample is common $\\rightarrow$ we have more trust in the $H_0$ model"]}, {"cell_type": "code", "execution_count": 150, "metadata": {}, "outputs": [{"data": {"image/png": 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9wnyS87YUP/f5au7Mq+zsHu4+yz4KjbQe2dkUFBSUeI+zs7OB2jPcd25ubszXNVrPYqoH/AboA+wHVojIZ6q6wr+Qqs4GZgP07t1bA8de37hxo2fDPfgPNeEfQ2JiIg0aNKBJkyZMnz6d/fv3c/HFFwNw3HHHsWTJEsDpgwjWzCQi7N+/n/79+5OQkMALL7wQtI5nnXUWl112GaNGjSr+tT516lRGjx7NwoULGT58OEcccQRNmjShcePG1KlThyZNmjB06FDmzp1L3759OfHEE0lLS6NRo0a0b9+euXPncumll5Kf73wpzpw5k7S0NPr06cPkyZNLNAn513/y5MlccMEF9OzZk6OOOooFCxbQpEkTduzYwSWXXMLSpUt5//33WbBgAd26dWPAgAEA3HnnnZx++unMnDmTTZs2UadOHY4//ngef/zxsP+uiYmJ9OzZM6xtQHlnMW1he+IvQ3mcM6jmzmJq6g6h6ulcBE2bkp2dXSKGpm5gsTBHQnWIlfkgyiOBZ5NU24ZF+gHTVfU09/lNAKp6l1+Zt9wyq91+h+9xjhjGAiNUdYJb7hYgT1XvLWt/vXv31sAJgzZu3MiJJ55YvRULUaTGIiqaGKlFixbVvu3qFK1jMRWprv+NUBNETZ7mGlUTBvn9yLEJg6KT++O7d7B1kTyC+BToKCLtge3AOODcgDJLgAnAauBs4B1VLWpamiYijYCDwG9xOrGNMRV46CGvIwAeeojMNWsI+q1jYkbEEoTbpzAFeAuoCzytqhkiMgNYo6pLgKeAeSKSCfyMk0RQ1d0i8gBOklFgqar+J+iOaplgVyEb48/TYb6L9OhBrtvnYGJXRPsgVHUpsDRg2a1+j/OAMWW8dj7Oqa7GmEpYvty593TioOXLafb55zYfRIyL1k5qY0wVzZzp3HuaIGbO5PjsbLjuOg+DMOGyoTaMMcYEZUcQxtSAss52ClSTZzsZUxE7goiw8ob7fu+990hLS6NevXosXrzYi/CA6BjuG+DNN9+kc+fOpKSkcPfddxcvHzBgQPFYVsceeyxnnXVWxGI1xvzCjiA8dNxxxzFnzhzuu+8+r0MJS3UM911QUMDkyZNZtmwZbdq0oU+fPpx55pmkpqaWuEp89OjRjBo1KmJ1gdB/7RsT7+wIwkPt2rWje/fu1KkT+p8hXof7/uSTT0hJSeGEE06gQYMGjBs3jldffbVEmb179/LOO+/YEUQF/vUv5+Z1EJuuvdbjIEy4aleC8GC871CH+w5U24b79i8D0KZNm+L5IIq88sorDBkypMSQHqa0zp2dm9dBHDjuOI+DMOGyJqYIKxruu8icOXMIHBIkmFCH+77mmmtKrR8xYgRXX301+fn5vPnmm8XDfe/Zs4cpU6aQnp5O3bp12bw59KYU/+G+wUk8/fr1K17fq1cvevXqRV5eHv/6178YPHgwd911F9dW46/IF154gUsuuaTathevXnvNuT/jDG+DaL5+vV0HEeNqV4KIxvG+qyAeh/suKlMkKyuL5OTk4uc//fQTn3zyCS+//HKl3qva6P77nXtPE8T999M2OzsKJqYw4ahdTUxxoqiDd+HChfTr14+6deuWmpBo7NixPPPMM6xatYrhw4cDsGfPHo455hjq1KnDvHnzKCgoKLXtdu3akZ6eTmFhIdu2beOTTz4BnClRP/jgg+I+j3379hUfgTzwwAN06tSJl156ieuuu44NGzZwzTXX0KpVK8BJWnPnzgVg8eLFnHrqqaWmDe3Tpw9fffUVW7Zs4eDBgyxYsIAzzzyzeP3ixYsZOXJk0KRmjImM2nUEEWU+/fRT/vCHP7B7925ee+01/va3v5GRkQGUPdw3OGcNde/evXi472CGDRvGBRdcwKhRo2jQoAEAV155JaNHj+bZZ58tHu47UP/+/Wnfvj2pqanFw30DtGzZkjlz5jB+/PgSw3136tSJ7t27k56eXmbfwMSJE7ngggtISUkpHu4bKDHcd7169Zg1axannXYaBQUF/OlPf6JLly7F21iwYAE33lhq1lpjTARFbLjvmmbDfUeXWB7uuzpOcw0c7jtU1XGhnA33HR1suG9jTLWyK65NNLEEEWNsuG9TkXnzvI4AmDePjatX06/ikiaKWYIwJs74XU7inbZtyf/6a6+jMGGys5iMiTMLFzo3r4No+c47HgdhwmUJwpg489hjzs3rIJKXLPE4CBMuSxDGGGOCsgQRYd9//z3jxo2jQ4cO9OrVi9NPP734ArOMjAxOPfVUOnfuTMeOHbn99tuLB7GbM2cOderU4YsvvijeVteuXYs7qdu1a0e3bt2Kx3j68MMPSwzb7fP5SEpKomfPnnTu3JmBAwfy+uuvF29r+vTpJCcnlxgnKjs7G5/Ph4jwWtF4DZQcsO/QoUPceOONdOzYkbS0NPr168cbb7xRKqb+/fsHHetp+vTpQUevrVu3bolY7r77bm677TZuuummEuXS09OLT08NfA+K9lc0OKExJjzWSR1Bqsof/vAHJkyYUHxx2Oeff84PP/xA27ZtOfPMM3nssccYNmwY+/fvZ/To0Tz66KNMnjwZcAasu+OOO0oNjV1k5cqVJa6HCDzDacCAAcVJIT09nbPOOouGDRsyZMgQAK655hquv/76Utst2u8ZQcZquOWWW/juu+/YsGEDCQkJ/PDDD7z77rulYqrsdRCBY1YBbN68meHDh3PXXXcVL1uwYEHxWFTB3gNjTPWxI4gIWrlyJfXr1+fyyy8vXvbrX/+aAQMG8Pzzz9O/f3+GDRsGQKNGjZg1a1aJiXJGjhxJRkYGmzZtCjuWopFkZ82aVWHZX//61yQlJbFs2bISy/fv388TTzzBP//5TxISEgBo3bo155xzTtjxBdOpUyeaNWvGxx9/XLxs0aJFJRKEMSZyas0RxNSpU8sdIbUqevTowUMPPVTm+g0bNhQPohcoIyOj1LoOHTqQm5vL3r17AahTpw7Tpk3jzjvvLB7LyN/gwYOpW7cuCQkJJb5Ey5KWlsa9995b/PzBBx9k/vz5ADRr1oyVK1cWr7v55pu55ZZbGDp0aPGyzMxMjjvuuHKH2y6KqbCwkIsvvjjoaLPBFA2LXuSmm25i7NixjB8/ngULFtC3b18++ugjjjrqKDp27FhqfwATJkwIeX/xLCpa1xYvJuODD+jvdRwmLLUmQcSqc889lzvuuIMtW7aUWlfZ5pXAYVXKamICGDhwIADvv/9+JaKt3iYmcAYdPOWUU7j//vtLNS/57y8UP+z9ZTKjvQcOxe3McVHR4taiBYeSkryOwoSp1iSI8n7pR0qXLl3K7CxNTU3lvffeK7Hsm2++oXHjxiV+oderV4/rrruOe+65J+x41q1bV6mxqW6++WZmzpxJvXrOv0lKSgrffvste/furbFJe9q2bUv79u159913eemll1i9enWN7DeWFc1xddFF3gZx9H//a/NBxLiI9kGIyHAR2SQimSJSaihOEUkQkYXu+o9FpJ27vJ2IHBCRdPf2eCTjjJRTTz2V/Px8Zs+eXbzsiy++YNWqVZx33nm8//77LF++HHCaWK666iqmTZtWajsXXXQRy5cvZ+fOnVWO5YsvvuD2228v7gAPxbBhw9i9e3fxmVSNGjVi4sSJXH311Rw8eBCAnTt38uKLL1Y5rlAUTYx0wgkn0KZNm4juKx7MmfNLkvAyiKPffNPjIEy4IpYgRKQu8AgwAkgFxotIakCxicBuVU0BHgT8fyZ/rao93NvlxCAR4eWXX2b58uV06NCBLl26cNNNN3H00UfTsGFDXn31VWbOnEnnzp3p1q0bffr0KTEfdJEGDRpw1VVX8eOPP1Zq/6tWrSo+zXXy5Mk8/PDDxWcwgdMH4X9qabBxnm6++eYSE/nMnDmTli1bkpqaSteuXRk5cmSJo4nBgwcXn+Z64YUXBo1r5syZtGnTpvgGpadm9R/ae8yYMWRkZATtnC7aX48ePUrs77LLLivevv/Md8aY0EVsuG8R6QdMV9XT3Oc3AajqXX5l3nLLrBaResD3QEvgeOB1Ve0a6v5qy3DfsSIa6+/fB7E1czMf7orc5ENVHe47VOWN5mrDfUcHG+67fMnANr/nWUDfssqo6mER2QMUzUXZXkTWAXuBv6rqqsAdiMgkYBI4p1sG/uMlJSWRk5MTfk2qoKCgwLN9R4NorH99v6lU6+phkvNKd/xX377yI7p9n29Hmeuys3u4Zar3rL3K6JGdTUFBQYnPZHZ2NlB7EkRubm7M1zVaO6m/A45T1V0i0gt4RUS6qOpe/0KqOhuYDc4RRGC23rhxo2e/YqPxF3RNisb6+x9BFEi9iP7Cj/QRxDmDyj6CaNrUuff012vTpmRnZ5eIoakbWCz8qq4OsXIEUZ5IJojtgP/Aw23cZcHKZLlNTEnALnXavfIBVPUzEfka6ASsoZJUtdT8x6Z2i5dZFMuydKnXEQBLl/LFe+8x0Os4TFgieRbTp0BHEWkvIg2AcUDg8I5LgAnu47OBd1RVRaSl28mNiJwAdAS+qWwAiYmJ7Nq1K+6/EEzoVJXcPbvJPRy/PxoaNXJuXgdRmBi5Ph5TMyJ2BOH2KUwB3gLqAk+raoaIzADWqOoS4ClgnohkAj/jJBGAgcAMETkEFAKXq+rPlY2hTZs2ZGVlhXV6aFXl5eWRWIs/INFY/70HDgGQe1jYuLe+x9FEzqOPOvdXXultEMdu3mzXQcS4iPZBqOpSYGnAslv9HucBY4K87iXgpXD3X79+fdq3j1w7cHl8Ph89e/b0ZN/RIBrrH69XTgdatMi59zRBLFpEK7dT2sQuG6zPGGNMUJYgjDHGBGUJwhhjTFDReh2EMaYc5fWnZO1u45bJKveKa2MqYgnCmDgz+b4sr0MAn490n49BXsdhwmJNTMYYY4KyBGFMnFn5YjNWvtjM2yDuu4+2ZcylbmKHNTEZE2e+/PgIAAaP2e1dEK+/TnO7DiLm2RGEMcaYoCxBGGOMCcoShDHGmKCsD8KYOFO/QRSMXtywIQUHDngdhQmTJQhj4sykOwOnXfHAG2+w3q6DiHnWxGSMMSYoSxDGxJm35x/F2/OP8jaI22/n+Gef9TYGEzZLEMbEma/SG/FVusdTyq1YQbO1a72NwYTNEoQxxpigLEEYY4wJyhKEMcaYoOw0V2PiTKMjC7wOAZo351BhoddRmDBZgjAmzlx863dehwAvvUSGXQcR8yxBGBPHypt5zp/NPGeCsQRhYl6oX4K1xetPtQBg5MSfvAvippto/+23MGiQdzGYsFmCMCbO/G9jotchwOrVJNl8EDHPzmIyxhgTVEQThIgMF5FNIpIpIjcGWZ8gIgvd9R+LSLuA9ceJSK6IXB/JOI0xxpQWsQQhInWBR4ARQCowXkRSA4pNBHaragrwIHBPwPoHgDciFaMxxpiyRfII4iQgU1W/UdWDwAJgVECZUcBc9/FiYIiICICInAVsATIiGKMxcSepxWGSWhz2Nog2bchv2dLbGEzYItlJnQxs83ueBfQtq4yqHhaRPUBzEckDbgCGAmU2L4nIJGASQOvWrfH5fNUWfLhyc3OjKp6aVpP1T87Lr5H9VEb9wnyS87Z4su8bprr7zQv9NT7fjuoN4pJLyM3NpbHf/0C222ldWz4X8fAdEK1nMU0HHlTVXPeAIihVnQ3MBujdu7cOiqJT6nw+H9EUT02ryfpH42muyXlb2J7Y3uswQnbOoOq/DiLwf6Bp06YAteZzEQ/fASElCBH5N/AU8Iaqhnr9/Hagrd/zNu6yYGWyRKQekATswjnSOFtE/g40BQpFJE9VZ4W4b2NqrZcfc5p2/nDFTu+CmDqVlKwsuw4ixoV6BPEocDHwsIi8CDyjqpsqeM2nQEcRaY+TCMYB5waUWQJMAFYDZwPvqKoCA4oKiMh0INeSgzGh2fF1gtchQHo6je06iJgXUoJQ1eXAchFJAsa7j7cBTwDzVfVQkNccFpEpwFtAXeBpVc0QkRnAGlVdgnNUMk9EMoGfcZKIMUB0Nh0ZU5uE3AchIs2B84ELgHXAc8BvcI4ABgV7jaouBZYGLLvV73EeMKa8/arq9FBjNMYYU31C7YN4GegMzAPOUNWi4SIXisiaSAVnjDHGO6EeQTzhHg0UE5EEVYF/+ekAABkPSURBVM1X1d4RiMsYU0Utkw96HQJ06sT+HTto6nUcJiyhJoiZBDQV4XQsp1VvOMaYcJ1zzY9ehwCzZ7PZ5+NYr+MwYSk3QYjI0TgXszUUkZ5A0UUJRwKNIhybMcYYD1V0BHEacBHONQwP+C3PAf4vQjEZY8Kw6MFWgMdHEpMm0WnHDrsOIsaVmyBUdS4wV0RGq+pLNRSTMSYMO7c38DoE2LyZRnYdRMyrqInpfFWdD7QTkWsD16vqA0FeZowxJg5U1MR0hHvfONKBGGOMiS4VNTH9y72/rWbCMcYYEy0qamJ6uLz1qnpV9YZjjAnXsR2iYPjzHj3Izcqy6yBiXEVNTJ/VSBTGmGrj6SiuRR56iEyfjzZex2HCEspZTMYYY2qhipqYHlLVqSLyGqCB61X1zIhFZoypkvl3Hw3A+Td+710Q55/PiT/8YNdBxLiKmpjmuff3RToQY0z12PNTFEwUmZVFgl0HEfMqamL6zL1/V0QaAL/COZLYpKpRMCKYMcaYSAl1uO/fA48DX+OMx9ReRC5T1TciGZwxxhjvhHosej8wWFUzAUSkA/AfwBKEMcbEqVATRE5RcnB9gzNgnzEmyhx/Yp7XIUC/fuz59lu7DiLGVXQW0x/dh2tEZCmwCKcPYgzwaYRjM8ZUwciJP3kdAtx1F1t8Po73Og4TloqOIM7we/wD8Fv38U6gYUQiMsYYExUqOovp4poKxBhTPZ6ZcQwAF9/6XQUlI2j0aLrs3AnvveddDCZsoZ7FlAhMBLoAiUXLVfVPEYrLGFNF+/fW9ToE2LWL+nv3eh2FCVOdEMvNA47GmWHuXZwZ5qyT2hhj4lioCSJFVW8B9rnjM/0e6Bu5sIwxxngt1ARxyL3PFpGuQBLQKjIhGWOMiQahXgcxW0SaAbcAS3BmmLuloheJyHDgH0Bd4ElVvTtgfQLwLNAL2AWMVdWtInISMLuoGDBdVV8OMVZjarWOPfZX+jUPLtscUrlrhnYKbYNDhrB7yxa7DiLGhZQgVPVJ9+G7wAmhvEZE6gKPAEOBLOBTEVmiql/6FZsI7FbVFBEZB9wDjAU2AL1V9bCIHAN8LiKvqerhkGplTC027PyfvQ4BbrmF//l8tPc6DhOWkJqYRKS5iPxTRNaKyGci8pCINK/gZScBmar6jTuw3wJgVECZUUDRnBOLgSEiIqq63y8ZJBJkqHFjjDGRFWoT0wLgPWC0+/w8YCHwu3Jekwxs83ueRemO7eIy7tHCHqA58JOI9AWeBo4HLgh29CAik4BJAK1bt8bn84VYncjLzc2NqnhqWnXUPzkvCqbOrKL6hfkk523xZN93TU8D4Kbpa6t92z7fjpDKdbvhBlIPH8Z3//3Fy7Ld4b9ry+ciHr4DQk0Qx6jq7X7PZ4rI2EgEVERVPwa6iMiJwFwReUNV8wLKzMbtq+jdu7cOiqLJSXw+H9EUT02rjvqH2i4ejZLztrA90ZsGlpzDRwBEZP/nDAqxD6JhQ7Kzs0v8DzRt6vRI1JbPRTx8B4R6FtPbIjJOROq4t3OAtyp4zXagrd/zNu6yoGVEpB7O2VG7/Auo6kYgF+gaYqzGGGOqQbkJQkRyRGQvcCnwPHDQvS3Abdopx6dARxFp7042NA7nDCh/S4AJ7uOzgXdUVd3X1HNjOB5noqKtIdfKGGNM2Coai6lJVTfs9ilMwTnSqAs8raoZIjIDWKOqS4CngHkikgn8jJNEAH4D3Cgih4BC4EpVjYIhKo0xpvYIefJaETkTGOg+9anq6xW9RlWXAksDlt3q9zgPZ+jwwNfN45f5sI0xlZDad5/XIcDIkez6+mu7DiLGhTpY391AH+A5d9HVItJfVW+KWGQmbsVy53MsGDxmt9chwPXXs83no4PXcZiwhHoEcTrQQ1ULAURkLrAOsARhjDFxKuQmJqApTj8BOGcbGWOi0CPXtwFg8n1Z3gUxaBA9srMhPd27GEzYQk0QdwLrRGQlzthIA4EbIxaVMcYYz1WYIESkDs6ZRCfj9EMA3KCq30cyMGOMMd6qMEGoaqGITFPVRZS+jsEYY0ycCvVK6uUicr2ItBWRo4puEY3MGGOMp0LtgxiLM6LqlQHLQxr62xhTc3oMjILZgM85hx83b7brIGJcqAkiFSc5/AYnUawCHo9UUMaYqut/5h6vQ4Arr2SHz0eIQ/uZKBVqgpgL7AUedp+f6y47JxJBGWOq7mCeANAg0cNpVPbvp05eXsXlTFQLNUF0VdVUv+crReTLMksbYzzzxF+TAY+vgzj9dLpnZ8Pw4d7FYMIWaif1WhE5ueiJO5nPmsiEZIwxJhqEegTRC/hQRL51nx8HbBKR9YCqaveIRGeMMcYzoSYIO040xphaJqQEoar/i3QgxhhjoktlBuszxsSAPsP2eh0CXHQR3//3v3YdRIyzBGFMnDkpWhKEz8evvI7DhMUShDFxJnePc3Ji46TCat92qJM9Je75me71o+CCPRMWSxDGxJm5tx8LeHsdxMjbr6aJ5MOoUZ7FYMIX6nUQxhhjahlLEMYYY4KyBGGMMSYoSxDGGGOCsk5qY+LMKSOzvQ6BL0aOp13dn+06iBgX0SMIERkuIptEJFNEbgyyPkFEFrrrPxaRdu7yoSLymYisd+9PjWScxsSTnoNy6Tko19MYNg86nZ2n2sc21kUsQYhIXeARYATOhEPjRSQ1oNhEYLeqpgAPAve4y38CzlDVbsAEYF6k4jQm3uz+sR67f/S2caDxj9+R8OOPnsZgwhfJI4iTgExV/UZVDwILgMCTokfhTDwEsBgYIiKiqutUdYe7PANoKCIJEYzVmLjx/N+P5vm/H+1pDMP/Po0T77zT0xhM+CL5MyMZ2Ob3PAvoW1YZVT0sInuA5jhHEEVGA2tVNT9wByIyCZgE0Lp1a3w+X7UFH67c3Nyoiqcm/Jjzy5+ofmE+i157K2i55JoKyEP1C/NJztviyb4TClsAeLZ/J4YDFKAlPgPZ2U7fSG35XMTDd0BUd1KLSBecZqdhwdar6mxgNkDv3r110KBBNRdcBXw+H9EUT03wH4YhOW8L2xPbexiNt7ysf36dRABP3//8Og1pIvklPgNNmzpd1rXlcxEP3wGRbGLaDrT1e97GXRa0jIjUA5KAXe7zNsDLwIWq+nUE4zTGGBNEJBPEp0BHEWkvIg2AccCSgDJLcDqhAc4G3lFVFZGmwH+AG1X1gwjGaIwxpgwRa2Jy+xSmAG8BdYGnVTVDRGYAa1R1CfAUME9EMoGfcZIIwBQgBbhVRG51lw1TVTstwpgKDBq92+sQWDv6Yk6oZ9dBxLqI9kGo6lJgacCyW/0e5wFjgrxuJjAzkrEZE6+69NvndQh80+9UkurvqLigiWpR3UltjKm8H7fVB6BV20OexdBs2zc0rGcH/LHOEoQxcebFf7QGvJ0PYsg//ubMB3HhhZ7FYMJnCcIYExEHCwpLnPqctfsAUHpWumuGdqrRuEzobDRXY4wxQVmCMMYYE5QlCGOMMUFZH4QxcWbouT97HQKfnHsFzQ9+53UYJkyWIIyJM53S9nsdAt+mnUKBh4MFmuphTUzGxJntXyew/WtvR8dv+fVGmn5jQ6jFOksQxsSZVx5rySuPtfQ0ht8+dic9nnzc0xhM+KyJyVQo8Lx1Y0ztYEcQxhhjgrIEYYwxJihLEMYYY4KyPghj4szpF/9UcaEI++Dia2hp10HEPEsQxsSZ9l3yvA6B77qkUceug4h51sRkTJzZkpHIloxET2M4JmMtzTd+6WkMJnyWIIyJM0ufacHSZ1p4GkP/Zx6k27xnPI3BhM8ShDHGmKAsQRhjjAnKEoQxxpig7CwmY4ynQh3KxaYmrXmWIIyJM2ddsdPrEHj3iv+jVf52r8MwYbIEYUycSe6Q73UI7OxwIg3yvD3V1oTP+iCMiTOb1zZi89pGnsZw3NoPaZW+1tMYTPgimiBEZLiIbBKRTBG5Mcj6BBFZ6K7/WETaucubi8hKEckVkVmRjNGYeLPs+aNY9vxRnsZw0vOPkbroBU9jMOGLWIIQkbrAI8AIIBUYLyKpAcUmArtVNQV4ELjHXZ4H3AJcH6n4jDHGlC+SRxAnAZmq+o2qHgQWAKMCyowC5rqPFwNDRERUdZ+qvo+TKIwxxnggkp3UycA2v+dZQN+yyqjqYRHZAzQHQhqOUkQmAZMAWrdujc/nCzPk6pObmxtV8YQjOa/ynZ71C/NJrsWDtXlZ/4RCZ5gNL9//hMIDiGqJGBIKDwBVj8vn21EtsdWUePgOiOmzmFR1NjAboHfv3jpo0CBvA/Lj8/mIpnjCUZUpR5PztrA9sX0EookNXtY/v45z9pCX739+nYYkFB4oEUN+nYZA1eM6Z1BsXQcRD98BkUwQ24G2fs/buMuClckSkXpAErArgjEZE/fGXP2D1yGw4urbaJ2f5XUYJkyRTBCfAh1FpD1OIhgHnBtQZgkwAVgNnA28o6oawZiMiXut2h7yOgR2tz2BRnnidRgmTBFLEG6fwhTgLaAu8LSqZojIDGCNqi4BngLmiUgm8DNOEgFARLYCRwINROQsYJiq2gDzxlQgY/URAHTpt8+zGE5Y/Q5HHfqB7QNrbzNjPIhoH4SqLgWWBiy71e9xHjCmjNe2i2RstV1V+hVMbPC91AzwNkGkvfQMCYUHWDNwvGcxmPDZldTGGGOCsgRhjDEmKEsQxhhjgrIEYYwxJqiYvlDOGFPaudO+9zoE3pz2d47O31ZxQRPVLEEYE2eatTrsdQjktjqGA3k2lFqssyYmY+LMOl9j1vkaexpDJ99S2q5619MYTPgsQRgTZz58vSkfvt7U0xi6v/4CHd543dMYTPgsQRhjjAnK+iDijF0hbYypLpYgjDExIdQfP9cMja1hwaOZNTEZY4wJyo4gjIkzE27xfua112/5B8fkf+t1GCZMliCMiTONkwq9DoG8pKM4mLfH6zBMmKyJyZg488nbR/LJ20d6GkPq2/+m3Yq3PY3BhM8ShDFx5tO3j+RTzxPEy7RbsczTGEz4rIkpRtjpq8aYmmYJwhgTV+x02OpjTUzGGGOCsgRhjDEmKGtiMibOXDpzu9ch8MrM2Rybv9XrMEyYLEF4zDqfTXVrkKheh8DhxIYUkOh1GOWyvoqKWROTMXHmgyVJfLAkydMYui95jg5LX/M0BhM+SxDGxJn095qQ/l4TT2Po9N6btH3/PU9jMOGzJiZjjClHbW6KimiCEJHhwD+AusCTqnp3wPoE4FmgF7ALGKuqW911NwETgQLgKlV9K5KxVrcfc/Ktf8EYE9Mi1sQkInWBR4ARQCowXkRSA4pNBHaragrwIHCP+9pUYBzQBRgOPOpuzxhjTA2J5BHESUCmqn4DICILgFHAl35lRgHT3ceLgVkiIu7yBaqaD2wRkUx3e6sjFWx1/9pPrtatGWOiXeB3SHJezbUiRKp5K5IJIhnY5vc8C+hbVhlVPSwie4Dm7vKPAl5b6jtXRCYBk9ynuSKyqXpCrxYtgJ+8DsJDVn+P63/tMA/37dy1YFjnUu/BtcM613Q4Xqmx/4Frw3v58WWtiOlOalWdDcz2Oo5gRGSNqvb2Og6vWP1rd/3B3oN4qH8kT3PdDrT1e97GXRa0jIjUA5JwOqtDea0xxpgIimSC+BToKCLtRaQBTqfzkoAyS4AJ7uOzgXdUVd3l40QkQUTaAx2BTyIYqzHGmAARa2Jy+xSmAG/hnOb6tKpmiMgMYI2qLgGeAua5ndA/4yQR3HKLcDq0DwOTVbUgUrFGSFQ2fdUgq7+p7e9BzNdfnB/sxhhjTEk21IYxxpigLEEYY4wJyhJENRCRa0QkQ0Q2iMgLIpLods5/LCKZIrLQ7aiPSyJytVv3DBGZ6i47SkSWichX7n0zr+OsTiLytIj8KCIb/JYFrbM4Hnb/F74QkTTvIq8eZdR/jPs/UCgivQPK3+TWf5OInFbzEVevMup/r4j81/0bvywiTf3WxWT9LUGESUSSgauA3qraFadDfhzOsCEPusOI7MYZViTuiEhX4FKcK91/DYwUkRTgRmCFqnYEVrjP48kcnGFg/JVV5xE4Z+J1xLmw87EaijGS5lC6/huAPwIlhnGN06Fz5lC6/suArqraHdgM3ASxXX9LENWjHtDQvZajEfAdcCrO8CEAc4GzPIot0k4EPlbV/ap6GHgX50tiFE69IQ7rr6rv4Zx556+sOo8CnlXHR0BTETmmZiKNjGD1V9WNqhpsNIPioXNUdQtQNHROzCqj/m+7nwFwRoJo4z6O2fpbggiTqm4H7gO+xUkMe4DPgGy/f5agQ4XEiQ3AABFpLiKNgNNxLnJsrarfuWW+B1p7FWANKqvOwYadidf/h2BqY/3/BLzhPo7Z+luCCJPbzjwKaA8cCxxB6UPPuKWqG3Ga094G3gTScYZo9y+jQK06n7o21tk4RORmnOu3nvM6lnBZggjf74AtqrpTVQ8B/wb64zQjFF2IGNdDhajqU6raS1UH4vS3bAZ+KGpGce9/9DLGGlJWnWv70DG1pv4ichEwEjhPf7nILGbrbwkifN8CJ4tII3eo8iE4V4CvxBk+BJzhRF71KL6IE5FW7v1xOP0Pz1NyGJW4rr+fsuq8BLjQPZvpZGCPX1NUbVArhs5xJ0ibBpypqvv9VsVu/VXVbmHegNuA/+K0x88DEoATcP4JMoEXgQSv44xg/VfhJMXPgSHusuY4Z/J8BSwHjvI6zmqu8ws4fU6HcNqUJ5ZVZ0BwJs/6GliPc8ab53WIQP3/4D7OB34A3vIrf7Nb/03ACK/jj1D9M3H6GtLd2+OxXn8basMYY0xQ1sRkjDEmKEsQxhhjgrIEYYwxJihLEMYYY4KyBGGMMSYoSxDGVIGI/EpE0kVknYh0qOI2prgjfKqItPBbXuXRX0VkqjvkSWVjya3sa/xee5GIHFvV15voZQnCGD+VGGXzLGCxqvZU1a9D2K6ISODn7QOcK/H/F7A8nNFfp+IMGFmTLsIZZsbEGUsQpsaIyCsi8pk7Z8Akd9nlInKvX5mLRGSW+/gWd/z89915Nq4Pss0x7lwUn4vIe+6yuiJyn7v8CxH5s7t8iPuLf707nn+Cu3yriNwjImuBMSIyTERWi8haEXlRRBoH7PN0nC/iK0RkpbvsWnd/G+SXOTHaufE/i3MRpf9wC6jqOlXdGuStqnD0VxE5QkT+49Z7g4iMFZGrcL6oV/rFlev3mrNFZI77uL1bx/UiMjNg238RkU/d9+42v7psFJEn3L/f2yLSUETOBnoDz7lHVA1F5G4R+dJ9/X1B6mdihddX6tmt9tz45crihjhfmM2BlkCmX5k3gN8AfXCuRk0EmuBcnXx9kG2uB5Ldx03d+ytwhlqvV7RfdzvbgE7usmeBqe7jrcA093ELnPkMjnCf3wDcGmS/04viAXq5cRwBNAYygJ5AO6AQOLmC92Ur0MLv+evAb/yeryDg6mtgNPCE3/OkMraV6/f4bGCO+3gJcKH7eHJROWAYMBvn6u86biwD3bocBnq45RYB57uPfUXxuX/TTfwy331Tr//v7Fb1mx1BmJp0lYh8jjNWflugo6ruBL4RkZNFpDnwK5yml/7Aq6qap6o5wGtlbPMDYI6IXIozWRM4zTb/Une4dVX9GeiMM6jiZrfMXJwvviIL3fuTgVTgAxFJxxlT6fgK6vUb4GVV3aequTgDNg5w1/1PnaOA6rYeGOoe+QxQ1T2VfH1/nOEiwBkepsgw97YOWIvz9+jortuiqunu489wkkagPUAe8JSI/BHYH6SMiRH1Ki5iTPhEZBDOF3c/Vd0vIj6cX/UAC4BzcMazellV1Rn3sGKqermI9AV+D3wmIr2qGOK+olCBZao6vorbKWu7lVHh6J+qutntvD4dmCkiK1R1RpBt+Y+lk1jOuiIC3KWq/yqxUKQdzhhLRQpwjgRLblD1sIichDNo5dnAFJzJs0wMsiMIU1OSgN1ucvgVzi/1Ii/jtLuPx0kW4BwZnCHO/N6NcYZQLkVEOqjqx6p6K7AT54t1GXCZuMOti8hROM0e7cSZDhXgApzZ7wJ9BPQvKue29XeqoG6rgLPEGdH3CJxB61ZV8JryVDj6q3vW0H5VnQ/cCxSd6ZSD0yRX5AcROdHtIP+D3/IPcKbBBDjPb/lbwJ+K+l1EJFnc0XrLUbxP93VJqroUuAZnGloTo+wIwtSUN4HLRWQjzpd1cbOLqu52l6eq6ifusk9FZAnwBc7IoOtxmi8C3SsiHXF++a7AGVF2A9AJ+EJEDuG01c8SkYuBF93E8SnweODGVHWnOGP6v1DUiQ38FWeOi6BUda3b+Vs0hPOTqrrO/dVdJrdTeRpwtBvrUlW9BFiKc2SQidNEc3GQl3dz616IM6LoFe7y2cCbIrJDVQfjzIv9Ok7yXIPTRwJwNfC8iNyA31Dsqvq2iJwIrHaP4nKB8wmYBCrAHOBxETmAcwbWqyKSiPM3uba898BENxvN1UQtEWmsqrninNf/HjBJVdd6HZcxtYUdQZhoNltEUnHazudacjCmZtkRhDHGmKCsk9oYY0xQliCMMcYEZQnCGGNMUJYgjDHGBGUJwhhjTFD/Dx+2H4DPAywYAAAAAElFTkSuQmCC\n", "text/plain": ["
"]}, "metadata": {"needs_background": "light"}, "output_type": "display_data"}], "source": ["z = np.r_[[np.mean(m.rvs(5)) for _ in range(10000)]]\n", "\n", "plt.hist(z, bins=30, density=True, alpha=.5);\n", "plt.axvline(105, color=\"blue\", ls=\"--\", label=\"H2: p-value>%.2f\"%np.mean(z>105))\n", "plt.axvline(110, color=\"red\", ls=\"--\", label=\"H1: p-value>%.2f\"%np.mean(z>110))\n", "plt.axvline(np.percentile(z, 95), color=\"black\", label=\"CONFIDENCE LEVEL\")\n", "plt.grid(); plt.xlabel(\"avg score for 10 students\"); plt.ylabel(\"probability\")\n", "plt.legend(); plt.title(\"THE SAMPLING DISTRIBUTION\"); plt.show()"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Now, if when we took 5 students we measured an average score of 115 or 105, **would you trust this model?**\n", "\n", "Observe that, \n", "\n", "- We only have **ONE REAL WORLD SAMPLE**, but we made **10000 SIMULATIONS**.\n", "- The distribution above is called the **SAMPLING DISTRIBUTION**.\n", "- The **p-value** is the probability of seeing something like our **REAL WORLD SAMPLE**, or even more extreme, **according to the model** we are testing.\n", "- What we did is called a **Z-TEST**.\n", "- It is standard to consider a **p-value<0.05** to indicate that our **REAL WORLD SAMPLE** has a very small probability **according to the model** and thus, there must be **something wrong with the model**.\n", "- **Central Limit Theorem**: regardless the shape of the original disitrution, the sampling distribution will **always be a Normal distribution**.\n", "\n", "**Challenge**: understand the Python code for sampling\n", "\n", "### What if we consider H3, with 30 students?\n", "\n", "- Another simulation, another sampling distribution."]}, {"cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [{"data": {"image/png": 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\n", "text/plain": ["
"]}, "metadata": {"needs_background": "light"}, "output_type": "display_data"}], "source": ["z = np.r_[[np.mean(m.rvs(30)) for _ in range(10000)]]\n", "\n", "plt.hist(z, bins=30, density=True, alpha=.5);\n", "plt.axvline(105, color=\"blue\", ls=\"--\", label=\"H2: p-value>%.2f\"%np.mean(z>105))\n", "plt.grid(); plt.xlabel(\"avg score for 30 students\"); plt.ylabel(\"probability\")\n", "plt.axvline(np.percentile(z, 95), color=\"black\", label=\"CONFIDENCE LEVEL\")\n", "plt.legend(); plt.title(\"THE SAMPLING DISTRIBUTION\"); plt.show()"]}, {"cell_type": "markdown", "metadata": {}, "source": ["**WE HAVE LEST TRUST IN OUR MODEL NOW!!** $\\rightarrow$ **WHY?**\n"]}, {"cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [{"data": {"image/png": 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\n", "text/plain": ["
"]}, "metadata": {"needs_background": "light"}, "output_type": "display_data"}], "source": ["z1 = np.r_[[np.mean(m.rvs(30)) for _ in range(10000)]]\n", "z2 = np.r_[[np.mean(m.rvs(5)) for _ in range(10000)]]\n", "z3 = np.r_[[np.mean(m.rvs(100)) for _ in range(10000)]]\n", "\n", "plt.hist(z1, bins=30, density=True, alpha=.5, label=\"sample size = 30\");\n", "plt.hist(z2, bins=30, density=True, alpha=.5, label=\"sample size = 5\");\n", "plt.hist(z3, bins=30, density=True, alpha=.5, label=\"sample size = 100\");\n", "plt.grid(); plt.legend(); plt.title(\"THE SAMPLING DISTRIBUTION\"); plt.show()\n", "plt.show();\n"]}, {"cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": []}, {"cell_type": "markdown", "metadata": {}, "source": ["\n", "## Terminology\n", "\n", "- **NULL HYPOTHESIS** $H_0$: The model we are testing whether we can reject it or not.\n", "- **REAL WORLD SAMPLE**: What we measure in reality to a limited number of objects.\n", "- **TEST STATISTIC**: What we compute from the real world sample (the average, in our case).\n", "- **SAMPLING DISTRIBUTION**: The distribution of simulating **many** real world examples and computing the test statistic to each of them.\n", "- **p-value**: The probability of seeing in the simulation something as **rare** as our real world example.\n", "- **CONFIDENCE LEVEL**: The minimum p-value we are willing to observe to NOT consider to distrust $H_0$\n", "- **REJECT $H_0$**: When we observe a p-value lower than the confidence level, meaning that our real world sample is **too rare**.\n", "- **FAIL TO REJECT $H_0$**: When we observe a p-value higher than the confidence level, meaning that our real world sample is **fairly normal**.\n", "- **MONTE CARLO SIMULATION**: What we did!!!\n", "\n", "\n", "## Simulations and Formulas\n", "\n", "We do simulation because we have computers. When these tests were invented, computers were not around, so people developed tables and formulas to compute **by hand** the p-values.\n", "\n", "If **simulating is hard** we can always use the analytical formula.\n", "\n", "$$p_{value} = 1 - \\mathcal{N}(0,1).\\text{cdf}\\Big( \\frac{\\bar{x}-\\mu}{\\sigma / \\sqrt{n}} \\Big) $$\n", "\n", "Where $\\mathcal{N}(0,1).\\text{cdf}$ is the Cummulative Density Function of the Standard Normal distribution"]}, {"cell_type": "code", "execution_count": 181, "metadata": {}, "outputs": [], "source": ["real_world_sample_mean = 105\n", "real_world_sample_size = 5\n", "model_mu = 100\n", "model_sigma = 15"]}, {"cell_type": "code", "execution_count": 182, "metadata": {}, "outputs": [{"name": "stdout", "output_type": "stream", "text": ["simulated p-value 0.238\n"]}], "source": ["z = np.r_[[np.mean(m.rvs(real_world_sample_size)) for _ in range(10000)]]\n", "print (\"simulated p-value %.3f\"%np.mean(real_world_sample_mean