{ "cells": [ { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "3xG4g-yFKnhm" }, "source": [ "# 7) Statistical Tests" ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "q-cWOMku-lx_" }, "source": [ "Vitor Kamada\n", "\n", "econometrics.methods@gmail.com\n", "\n", "Last updated 7-22-2020" ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "JX0o6kS_g2mP" }, "source": [ "#### 7.1) What is the difference between null hypothesis ($H_0$) and alternative hypothesis ($H_a$)?" ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "FkUMkw3LpuT7" }, "source": [ "A statistical hypothesis is a claim about a parameter of a population. \n", "Null hypothesis ($H_0$) is the hypothesis that sets out the status quo or default course of action; whereas, alternative hypothesis ($H_a$) challenges the assertion of the null hypothesis.\n", "\n", "Based on historical data, let's claim that when students send several curriculum vitae (CVs), they receive a callback for interview of 7%:\n", "\n", "$$ H_0:p\\le 0.07$$\n", "\n", "$$vs$$\n", "\n", "$$ H_a:p> 0.07 $$" ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "aVp5JdjwsPgS" }, "source": [ "#### 7.2) What is difference between one-sided hypotheses and two-sided hypotheses?" ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "DMEdj3S1q7GV" }, "source": [ "In a one-sided hypotheses, the null hypothesis permit any value of a parameter larger (or smaller) than a specified value: \n", "\n", "$$ H_0:p\\le 0.07$$\n", "\n", "$$vs$$\n", "\n", "$$ H_a:p> 0.07 $$\n", "\n", "In a two-sided hypotheses, the null hypothesis states a specific value for the population parameter:\n", "\n", "$$ H_0:p= 0.07$$\n", "\n", "$$vs$$\n", "\n", "$$ H_a:p\\neq 0.07 $$\n" ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "8HZI3rTb6IgF" }, "source": [ "#### 7.3) What is Type I and II Errors?" ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "Rs3Nhj4DEBvr" }, "source": [ "Type 1 error ($\\alpha$) is to reject the $H_0$, when the $H_0$ is true. \n", "\n", "Type 2 error ($\\beta$) is to retain (fail to reject) the $H_0$, when the $H_a$ is true.\n", "\n", "The right decision is to retain the $H_0$, when the $H_0$ is true, and to reject the $H_0$, when the $H_a$ is true. " ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "Fcqkoi_56yMu" }, "source": [ "> . |Retain $H_0$ |Reject $H_0$\n", ">--- |--- |---\n", ">$H_0$ $is$ $true$|OK |$\\alpha$ = Type 1 Error \n", ">$H_a$ $is$ $true$| $\\beta$ =Type 2 Error |OK\n", "\n" ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "0iaeO9mbAsUr" }, "source": [ "#### 7.4) When a jury convicts an innocent defendant, what type of error the jury commits? " ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "g3ePffxzI-yg" }, "source": [ "In the modern legal system, one is considered innocent until proven guilty. The jury commits Type I error, when they send an innocent to the prison, that is, reject the $H_0$, when the $H_0$ is true. \n" ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "xYoHbi9ELZq6" }, "source": [ "#### 7.5) Could we design a legal system that never convicts an innocent defendant?" ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "qmviHvuXLwL5" }, "source": [ "Yes. If we always retain the $H_0$. What is the problem of this approach? Type 2 error is maximized, that is, criminals never go to the prison. There is a trade-off between Type 1 and Type 2 error. It is not possible to minimize both in the same time." ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "6_nxLJRg16Sl" }, "source": [ "#### 7.6) What is test statistic?" ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "AZ2L39nR2GfJ" }, "source": [ "Test statistic is the sample statistic that estimates the population parameter stipulated by the null hypothesis.\n", "\n", "For example, $\\hat{p}$, the proportion of callback for interview in a sample is a test statistic.\n" ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "PtcHHQ4T4RPZ" }, "source": [ "#### 7.7) What is the level of significance ($\\alpha$) of a hypothesis test?" ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "UKlAabpH4xnY" }, "source": [ "Level of significance ($\\alpha$) is the probability of Type 1 error. It is the threshold for rejecting the $H_0$. " ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "uhqe9xtH6dki" }, "source": [ "#### 7.8) What is p-value?" ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "NFBco--y6ore" }, "source": [ "Smallest $\\alpha$ level at which $H_0$ can be rejected." ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "_Wqicrbo6zM5" }, "source": [ "#### 7.9) What means \"statistically significant\"?" ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "N1kuzOw97A8v" }, "source": [ "A result is statistically significant if the test rejects $H_0$ at the chosen level of significance ($\\alpha$)." ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "dbZAuv_r9D51" }, "source": [ "#### 7.10) What is z-statistic and z-test?" ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "hO3Ym2Rv9NAQ" }, "source": [ "z-statistic is the number of standard errors that separate the test statistic from the region specified by $H_0$:\n", "\n", "$$z=\\frac{deviation\\ of\\ test\\ statistic\\ from\\ H_0}{standard\\ error\\ of\\ test\\ statistic} $$\n", "\n", "$$= \\frac{\\hat{p}-p_0}{\\sqrt{\\frac{p_0(1-p_0)}{n}}}$$\n", "\n", "z-test is the test of $H_0$ based on number of standard errors separating $H_0$ from the test statistic." ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "pYSw5uHBBndy" }, "source": [ "#### 7.11) A student sent 50 curriculum vitae (CVs) and received 7 callbacks for interview. Test the null hypothesis $H_0: p \\le 0.07$ in which $p$ represents the probability of callback. Adopt a level of significance ($\\alpha =5\\%$)." ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "wsBtVtQwKEJv" }, "source": [ "Given $\\alpha =5\\%$, we must get 'z' that makes $P(Z>z)=0.05$ true." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 34 }, "colab_type": "code", "id": "YlGViZ1bKEZe", "outputId": "c2a3221d-d43f-4ef7-ff4e-dcdda2624107" }, "outputs": [ { "data": { "text/plain": [ "1.6448536269514722" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from scipy import stats\n", "stats.norm.ppf(0.95)" ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "fMWDQdknLH2Q" }, "source": [ "Let's visualize $P(Z>1.64)=0.05$ in the chart below.\n", "\n", "The yellow area is exactly 5%. It represents the critical region, where we reject the $H_0$." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 390 }, "colab_type": "code", "id": "uRuao9xdLSLl", "outputId": "c300f2ba-a610-40a0-e81f-6dd35d552e3f" }, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "C:\\Users\\acer\\anaconda3\\envs\\book\\lib\\site-packages\\datascience\\tables.py:17: MatplotlibDeprecationWarning: The 'warn' parameter of use() is deprecated since Matplotlib 3.1 and will be removed in 3.3. If any parameter follows 'warn', they should be pass as keyword, not positionally.\n", " matplotlib.use('agg', warn=False)\n", "C:\\Users\\acer\\anaconda3\\envs\\book\\lib\\site-packages\\datascience\\util.py:10: MatplotlibDeprecationWarning: The 'warn' parameter of use() is deprecated since Matplotlib 3.1 and will be removed in 3.3. If any parameter follows 'warn', they should be pass as keyword, not positionally.\n", " matplotlib.use('agg', warn=False)\n" ] }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "from datascience import plot_normal_cdf\n", "import matplotlib.pyplot as plots\n", "%matplotlib inline\n", "plot_normal_cdf(lbound=1.64)" ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "AY6YrYZpD32h" }, "source": [ "The observed statistic is: $\\hat{p}=\\frac{7}{50}=0.14$\n", "\n", "z-test:\n", "\n", "$$ z = \\frac{\\hat{p}-p_0}{\\sqrt{\\frac{p_0(1-p_0)}{n}}}$$\n", "\n", "$$ = \\frac{0.14-0.07}{\\sqrt{\\frac{0.07(1-0.07)}{50}}}$$\n", "\n", "$$ = \\frac{0.07}{0.036}$$\n", "\n", "$$ = 1.939$$\n", "\n", "The threshold is 1.64. Therefore, 1.939 is clear inside the critical region, where the $H_0$ is rejected. " ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 34 }, "colab_type": "code", "id": "DMmst7RSFZ4B", "outputId": "753a184d-df9a-4534-d89a-e05c247e2497" }, "outputs": [ { "data": { "text/plain": [ "0.03608323710533744" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "se = ((0.07)*(1-0.07)/50)**(1/2)\n", "se" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 34 }, "colab_type": "code", "id": "O-DZhZeCF3_N", "outputId": "0b878cf8-f9c2-4201-8007-b851f95e9616" }, "outputs": [ { "data": { "text/plain": [ "1.9399589841579263" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(0.14-0.07)/se" ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "lQI_9DlWM122" }, "source": [ "Let's calculate and visualize the p-value: $P(Z>1.939)$" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 34 }, "colab_type": "code", "id": "fpKFaQ4MOeEJ", "outputId": "aaf3c7a3-26e2-40b2-b1cd-e0dfc53b2435" }, "outputs": [ { "data": { "text/plain": [ "0.026250669079613265" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "1 - stats.norm.cdf(1.939)" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 298 }, "colab_type": "code", "id": "UseRzxs_Os63", "outputId": "7035a165-09ef-42e0-d816-1ee9c0015bf6" }, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plot_normal_cdf(lbound=1.939)" ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "DF49j9P0I-Y3" }, "source": [ "The p-value is 2.62%. " ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "PTDLZ7FSE4Im" }, "source": [ "#### 7.12) What is the t-statistic and t-test for the mean ($\\mu$)?" ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "kqlSZVKX15Y_" }, "source": [ "The t-statistic is the number of estimated standard errors from $\\bar{x}$ to $\\mu_0$.\n", " \n", "The t-test uses a t-statistic as the test statistic:\n", "\n", "$$ t= \\frac{Deviation\\ of\\ sample\\ statistic\\ from\\ H_0}{Estimated\\ standard\\ error\\ of\\ sample\\ statistic} $$\n", "\n", "$$ = \\frac{\\bar{x}-\\mu_0}{\\frac{s}{\\sqrt{n}}} $$" ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "Zi2TVEo1bXDs" }, "source": [ "#### 7.13) In a sample of 36 job applicants, the average years of experience is 1 with standard deviation $s = 3$. Test the null hypothesis $H_0: \\mu \\ge 2$. Adopt a level of significance ($\\alpha =5\\%$)" ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "bxsyi8SYgXc3" }, "source": [ " Given $\\alpha =5\\%$, we must get 't' that makes $P(T" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "%matplotlib inline\n", "\n", "x = np.linspace(-4, 4, 100)\n", "plt.plot(x, stats.t(35).pdf(x))\n", "plt.vlines(-1.689, 0, 0.1, colors='red', linestyles='dashed')\n", "plt.text(-2.2, -0.1,'t=-1.689', fontsize=13)" ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "Wga0HDEsbZIX" }, "source": [ "$$ t= \\frac{\\bar{x}-\\mu_0}{\\frac{s}{\\sqrt{n}}} $$\n", "\n", "$$ = \\frac{1-2}{\\frac{3}{\\sqrt{36}}} $$\n", "\n", "$$ = \\frac{-1}{\\frac{1}{2}} $$\n", "\n", "$$ = -2 $$\n", "\n", "The threshold is -1.689. Therefore, we reject the $H_0$, as -2 is inside the critical region.\n", "\n", "The p-value, $P(T<-2) = 2.66\\%$. " ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 34 }, "colab_type": "code", "id": "dHzN4thGe3lY", "outputId": "8cd896e6-7c78-4a45-cd8a-c05988a41f79" }, "outputs": [ { "data": { "text/plain": [ "0.026653825931598388" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stats.t.cdf(-2, 35)" ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "QKZQTL-kzGQm" }, "source": [ "## Exercises" ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "TKFcx63yDwiT" }, "source": [ "1| Given the set of hypotheses\n", "and z-test statistic. Find the p-value and determine if the $H_0$ would be rejected at $\\alpha = 0.05$?" ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "wVyzfqLqD61P" }, "source": [ "a) $H_0:p < p_0$ vs $H_a: p>p_0$\n", "\n", " $z=2$" ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "JcDCPPZ_FU1P" }, "source": [ "b) $H_a: p\\mu_0$, $n=11$, $t=1.91$." ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "KQBNYmoq_1Vi" }, "source": [ "b) $H_a: \\mu<\\mu_0$, $n=17$, $t=-3.45$." ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "3Nu9Bz8XFo5B" }, "source": [ "3| A biotech is testing a new drug. If the drug lowers the blood pressure of a patient by more than 12 mm, it is deemed effective. State the natural hypotheses to test in a clinical study of this new drug? Justify." ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "_NnbqjGxIWZJ" }, "source": [ "4| Pfizer tests millions of compounds, before occasionally produce breakthroughs that can lead to multibillion dollar blockbuster drugs like Lipitor and Viagra. Should the managers of the Pfizer be more worried about Type 1 or Type 2 errors?" ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "wa_dmTcRdwSI" }, "source": [ "## Reference" ] }, { "cell_type": "markdown", "metadata": { "colab_type": "text", "id": "TLJfMp4XdwVg" }, "source": [ "Adhikari, A., DeNero, J. (2020). Computational and Inferential Thinking: The Foundations of Data Science. [Link](https://www.inferentialthinking.com/chapters/intro.html) \n", "\n", "Diez, D. M., Barr, C. D., Çetinkaya-Rundel, M. (2014). Introductory Statistics with Randomization and Simulation. [Link](https://www.openintro.org/stat/textbook.php?stat_book=isrs) \n", "\n", "Lau, S., Gonzalez, J., Nolan, D. (2020). Principles and Techniques of Data Science. [Link](https://www.textbook.ds100.org/intro) " ] } ], "metadata": { "colab": { "name": "7)_Statistical_Tests.ipynb", "provenance": [] }, "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.6" } }, "nbformat": 4, "nbformat_minor": 1 }