Chapter 4 Parametric Hypothesis Testing.ipynb

{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "name": "Chapter 4: Parametric Hypothesis Testing.ipynb",
      "provenance": [],
      "collapsed_sections": [
        "WJhmgDxsVHEO",
        "VAqEz3tk15jW",
        "Kc6Ospy5qm2f",
        "L5FYq4KIr5tR",
        "btrLaUKzN1PY",
        "4V1_eXUkxJD4",
        "58w24h7DxfUc",
        "rkSluAxdVb1M",
        "Br8LlPdh4e9R",
        "ZJbTadX1NnaL",
        "a6nq2C5NlRKo",
        "I2zKIWZWFf4V",
        "ro9fDzmQU8IZ"
      ]
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "source": [
        "# **Chapter 4: Parametric Hypothesis Testing**"
      ],
      "metadata": {
        "id": "a2GtsO5MH734"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**Table of Content:**\n",
        "\n",
        "- [Import Libraries](#Import_Libraries)\n",
        "- [4.1. Introduction](#Introduction)\n",
        "- [4.2. Test Concerning the Mean of a Normal Population](#Test_Concerning_the_Mean_of_a_Normal_Population)\n",
        "  - [4.2.1. Known Standard Deviation](#Known_Standard_Deviation)\n",
        "  - [4.2.2. Unknown Standard Deviation](#Unknown_Standard_Deviation)\n",
        "- [4.3. Test Concerning the Equality of Means of Two Normal Populations](#Test_Concerning_the_Equality_of_Means_of_Two_Normal_Populations)\n",
        "  - [4.3.1. Known Variances](#Known_Variances)\n",
        "  - [4.3.2. Unknown but Equal Variances](#Unknown_but_Equal_Variances)\n",
        "- [4.4. Paired t-test](#Paired_t-test)\n",
        "- [4.5. Test Concerning the Variance of a Normal Population](#Test_Concerning_the_Variance_of_a_Normal_Population)\n",
        "- [4.6. Test Concerning the Equality of Variances of Two Normal Populations](#Test_Concerning_the_Equality_of_Variances_of_Two_Normal_Populations)\n",
        "- [4.7. Test Concerning P in Bernoulli Populations](#Test_Concerning_P_in_Bernoulli_Populations)\n",
        "- [4.8. Test Concerning the Equality of P in Two Bernoulli Populations](#Test_Concerning_the_Equality_of_P_in_Two_Bernoulli_Populations)\n"
      ],
      "metadata": {
        "id": "dGYw5uzq5aM7"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Import_Libraries'></a>\n",
        "\n",
        "## **Import Libraries**"
      ],
      "metadata": {
        "id": "WJhmgDxsVHEO"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "!pip install --upgrade scipy"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "vXStgb2JU6c0",
        "outputId": "787a176e-c82d-41af-b1d4-fd44dfb1632d"
      },
      "execution_count": 1,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Looking in indexes: https://pypi.org/simple, https://us-python.pkg.dev/colab-wheels/public/simple/\n",
            "Requirement already satisfied: scipy in /usr/local/lib/python3.7/dist-packages (1.4.1)\n",
            "Collecting scipy\n",
            "  Downloading scipy-1.7.3-cp37-cp37m-manylinux_2_12_x86_64.manylinux2010_x86_64.whl (38.1 MB)\n",
            "\u001b[K     |████████████████████████████████| 38.1 MB 1.2 MB/s \n",
            "\u001b[?25hRequirement already satisfied: numpy<1.23.0,>=1.16.5 in /usr/local/lib/python3.7/dist-packages (from scipy) (1.21.6)\n",
            "Installing collected packages: scipy\n",
            "  Attempting uninstall: scipy\n",
            "    Found existing installation: scipy 1.4.1\n",
            "    Uninstalling scipy-1.4.1:\n",
            "      Successfully uninstalled scipy-1.4.1\n",
            "\u001b[31mERROR: pip's dependency resolver does not currently take into account all the packages that are installed. This behaviour is the source of the following dependency conflicts.\n",
            "albumentations 0.1.12 requires imgaug<0.2.7,>=0.2.5, but you have imgaug 0.2.9 which is incompatible.\u001b[0m\n",
            "Successfully installed scipy-1.7.3\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "import pandas as pd\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "import matplotlib.patches as mpatches\n",
        "import seaborn as sns\n",
        "import math\n",
        "from scipy import stats\n",
        "from scipy.stats import norm\n",
        "from scipy.stats import chi2\n",
        "from scipy.stats import t\n",
        "from scipy.stats import f\n",
        "from scipy.stats import bernoulli\n",
        "from scipy.stats import binom\n",
        "from scipy.stats import nbinom\n",
        "from scipy.stats import geom\n",
        "from scipy.stats import poisson\n",
        "from scipy.stats import uniform\n",
        "from scipy.stats import randint\n",
        "from scipy.stats import expon\n",
        "from scipy.stats import gamma\n",
        "from scipy.stats import beta\n",
        "from scipy.stats import weibull_min\n",
        "from scipy.stats import hypergeom\n",
        "from scipy.stats import shapiro\n",
        "from scipy.stats import pearsonr\n",
        "from scipy.stats import normaltest\n",
        "from scipy.stats import anderson\n",
        "from scipy.stats import spearmanr\n",
        "from scipy.stats import kendalltau\n",
        "from scipy.stats import chi2_contingency\n",
        "from scipy.stats import ttest_ind\n",
        "from scipy.stats import ttest_rel\n",
        "from scipy.stats import mannwhitneyu\n",
        "from scipy.stats import wilcoxon\n",
        "from scipy.stats import kruskal\n",
        "from scipy.stats import friedmanchisquare\n",
        "from statsmodels.tsa.stattools import adfuller\n",
        "from statsmodels.tsa.stattools import kpss\n",
        "from statsmodels.stats.weightstats import ztest\n",
        "from scipy.integrate import quad\n",
        "from IPython.display import display, Latex\n",
        "\n",
        "import warnings\n",
        "warnings.filterwarnings('ignore')\n",
        "warnings.simplefilter(action='ignore', category=FutureWarning)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "ZPuphzTmU-P8",
        "outputId": "69efdeaa-fe6b-4b2f-dc91-e5185b8f862f"
      },
      "execution_count": 2,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stderr",
          "text": [
            "/usr/local/lib/python3.7/dist-packages/statsmodels/tools/_testing.py:19: FutureWarning: pandas.util.testing is deprecated. Use the functions in the public API at pandas.testing instead.\n",
            "  import pandas.util.testing as tm\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Introduction'></a>\n",
        "\n",
        "## **4.1. Introduction:**"
      ],
      "metadata": {
        "id": "VAqEz3tk15jW"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**Statistical Hypothesis:** A statistical hypothesis is usually a statement about a set of parameters of a population distribution.\n",
        "\n",
        "**$H_0$ (Null Hypothesis):** The null hypothesis is a statistical hypothesis to be tested and accepted or rejected in favor of an alternative.\n",
        "\n",
        "**$H_1$ (Alternative Hypothesis):**  An alternative hypothesis is an opposing theory in relation to the null hypothesis.\n",
        "\n",
        "**Type $\\mathrm{I}$ Error:** The type $\\mathrm{I}$ error, is said to result if the test incorrectly calls for rejecting $H_0$ when it is indeed correct.\n",
        "\n",
        "$\\alpha = P(reject\\ H_0\\ |\\ H_0\\ is\\ true)$\n",
        "\n",
        "**Type $\\mathrm{II}$ Error:** The type $\\mathrm{II}$ error, results if the test calls for accepting $H_0$ when it is false.\n",
        "\n",
        "$\\beta  = P(Accept\\ H_0\\ |\\ H_0\\ is\\ not\\ true)$\n",
        "\n",
        "**Significance Level:** Whenever $H_0$ is true, its probability of being rejected is never greater than $\\alpha$. The value $\\alpha$, called the level of significance of the test, is usually set in advance, with commonly chosen values being $\\alpha = 0.1, 0.05, 0.005$.\n",
        "\n",
        "**P_value:** The P value, or calculated probability, is the probability of finding the observed, or more extreme, results when the null hypothesis (H 0) of a study question is true — the definition of ‘extreme’ depends on how the hypothesis is being tested.\n",
        "\n",
        "If your P value is less than the chosen significance level then you reject the null hypothesis i.e. accept that your sample gives reasonable evidence to support the alternative hypothesis."
      ],
      "metadata": {
        "id": "ObVoGiq9mQ_D"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "<table style=\"display: inline-block\">\n",
        "<tr><th>.</th><th>$H_0$ is actually true</th><th>$H_0$ is not true</th><tr>\n",
        "<tr><td>Accept $H_0$</td><td>$1-\\alpha$</td><td>$\\beta$</td></tr>\n",
        "<tr><td>Reject $H_0$</td><td>$\\alpha$</td><td>$1-\\beta$</td></tr>\n",
        "</table>  "
      ],
      "metadata": {
        "id": "YepvfOWIIcEk"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**Steps of a hypothesis testing:**\n",
        "\n",
        "1. Specify the null and alternative Hypothesis.\n",
        "2. Collect a random sample from the population. \n",
        "3. Calculate the Test Statistic and Corresponding P-Value\n",
        "4. Decide whether to reject or fail to reject your null hypothesis"
      ],
      "metadata": {
        "id": "UaFVzosSs5oP"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Test_Concerning_the_Mean_of_a_Normal_Population'></a>\n",
        "\n",
        "## **4.2. Test Concerning the Mean of a Normal Population:**"
      ],
      "metadata": {
        "id": "Kc6Ospy5qm2f"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Known_Standard_Deviation'></a>\n",
        "\n",
        "### **4.2.1. Known Standard Deviation:**"
      ],
      "metadata": {
        "id": "L5FYq4KIr5tR"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**A. Two Tailed Test:**\n",
        "\n",
        "Suppose that $X_1, X_2, ..., X_n$ is a sample of size $n$ from a normal distribution having an unknown mean $\\mu$ and a known variance $\\sigma^2$.\n",
        "\n",
        "$X_1, X_2, ..., X_n \\sim N( \\mu, \\sigma^2)$\n",
        "\n",
        "$H_0:\\ \\mu = \\mu_0$\n",
        "\n",
        "$H_1:\\ \\mu  \\neq \\mu_0$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Testing statistics: \n",
        "\n",
        "$Z_0  \\equiv \\frac{\\overline{X}-\\mu_0}{\\frac{\\sigma}{\\sqrt{n}}}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "We accept $H_0$ if:\n",
        "\n",
        "1. $-\\ Z_{\\frac{\\alpha}{2}}\\ <\\ Z_0 = \\frac{\\overline{X}-\\mu_0}{\\frac{\\sigma}{\\sqrt{n}}}\\ <\\ Z_{\\frac{\\alpha}{2}}$\n",
        "\n",
        "2. $\\mu_0\\ -\\ Z_{\\frac{\\alpha}{2}} \\frac{\\sigma}{\\sqrt{n}}\\ <\\ \\overline{X}\\ <\\ \\mu_0\\ +\\ Z_{\\frac{\\alpha}{2}} \\frac{\\sigma}{\\sqrt{n}}$\n",
        "\n",
        "3. $|\\overline{X}-\\mu_0| < Z_{\\frac{\\alpha}{2}} \\frac{\\sigma}{\\sqrt{n}}$\n",
        "\n",
        "4. P_value $ = 2 \\times P(Z \\geq |Z_0|) > \\alpha$"
      ],
      "metadata": {
        "id": "9PWV-JGeJyVN"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**B. One-sided tests:**\n",
        "\n",
        "**Right-tailed test:**\n",
        "\n",
        "Suppose that $X_1, X_2, ..., X_n$ is a sample of size $n$ from a normal distribution having an unknown mean $\\mu$ and a known variance $\\sigma^2$.\n",
        "\n",
        "$X_1, X_2, ..., X_n \\sim N( \\mu, \\sigma^2)$\n",
        "\n",
        "$H_0:\\ \\mu =  \\mu_0\\ \\quad or \\quad H_0:\\ \\mu  \\leq \\mu_0$\n",
        "\n",
        "$H_1:\\ \\mu  > \\mu_0$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Testing statistics: \n",
        "\n",
        "$Z_0  \\equiv \\frac{\\overline{X}-\\mu_0}{\\frac{\\sigma}{\\sqrt{n}}}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "We accept $H_0$ if:\n",
        "\n",
        "1. $ -\\infty <\\ Z_0 = \\frac{\\overline{X}-\\mu_0}{\\frac{\\sigma}{\\sqrt{n}}}\\ <\\ Z_{\\alpha} $\n",
        "\n",
        "2. $- \\infty \\ <\\ \\overline{X}\\ <\\ \\mu_0\\ +\\ Z_{\\alpha} \\frac{\\sigma}{\\sqrt{n}}$\n",
        "\n",
        "3. P_value $ = P(Z \\geq Z_0) > \\alpha$"
      ],
      "metadata": {
        "id": "31PJg_ETxNMP"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**Left-tailed test:**\n",
        "\n",
        "Suppose that $X_1, X_2, ..., X_n$ is a sample of size $n$ from a normal distribution having an unknown mean $\\mu$ and a known variance $\\sigma^2$.\n",
        "\n",
        "$X_1, X_2, ..., X_n \\sim N( \\mu, \\sigma^2)$\n",
        "\n",
        "$H_0:\\ \\mu =  \\mu_0 \\quad or \\quad H_0:\\ \\mu  \\geq \\mu_0$\n",
        "\n",
        "$H_1:\\ \\mu  < \\mu_0$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Testing statistics: \n",
        "\n",
        "$Z_0  \\equiv \\frac{\\overline{X}-\\mu_0}{\\frac{\\sigma}{\\sqrt{n}}}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "We accept $H_0$ if:\n",
        "\n",
        "1. $-\\ Z_{\\alpha}\\ <\\ Z_0 = \\frac{\\overline{X}-\\mu_0}{\\frac{\\sigma}{\\sqrt{n}}}\\ <\\ \\infty $\n",
        "\n",
        "2. $\\ \\mu_0\\ -\\ Z_{\\alpha} \\frac{\\sigma}{\\sqrt{n}}\\ <\\ \\overline{X}\\ <\\ \\infty$\n",
        "\n",
        "3. P_value $ = P(Z \\leq Z_0) > \\alpha$"
      ],
      "metadata": {
        "id": "rhL-meIV0GAB"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "class mean_with_known_variance:\n",
        "  \"\"\"\n",
        "  Parameters\n",
        "  ----------\n",
        "  null_mean : u0\n",
        "  population_sd : known standrad deviation of the population\n",
        "  n : optional, number of sample members\n",
        "  alpha : significance level\n",
        "  type_t : 'two_tailed', 'right_tailed', 'left_tailed'\n",
        "  Sample_mean : optional, mean of the sample\n",
        "  data : optional, if you do not know the sample mean, just pass the data\n",
        "  \"\"\"\n",
        "  def __init__(self, null_mean, population_sd, alpha, type_t, n = 0., Sample_mean = 0., data=None):\n",
        "    self.Sample_mean = Sample_mean\n",
        "    self.null_mean = null_mean\n",
        "    self.population_sd = population_sd\n",
        "    self.type_t = type_t\n",
        "    self.n = n\n",
        "    self.alpha = alpha\n",
        "    self.data = data\n",
        "    if data is not None:\n",
        "      self.Sample_mean = np.mean(list(data))\n",
        "      self.n = len(list(data))\n",
        "\n",
        "    mean_with_known_variance.__test(self)\n",
        "  \n",
        "  def __test(self):\n",
        "    test_statistic = (self.Sample_mean-self.null_mean) / (self.population_sd/np.sqrt(self.n))\n",
        "    print('test_statistic:', test_statistic, '\\n')\n",
        "\n",
        "    if self.type_t == 'two_tailed':\n",
        "      p_value = 2*(1-norm.cdf(abs(test_statistic)))\n",
        "      print('P_value:', p_value, '\\n')\n",
        "    elif self.type_t == 'left_tailed':\n",
        "      p_value = (norm.cdf(test_statistic))\n",
        "      print('P_value:', p_value, '\\n')\n",
        "    else:\n",
        "       p_value = (1-norm.cdf(test_statistic))\n",
        "       print('P_value:', p_value, '\\n')\n",
        "\n",
        "    if p_value < self.alpha:\n",
        "      print(f'Since p_value < {self.alpha}, reject null hypothesis.')\n",
        "    else:\n",
        "      print(f'Since p_value > {self.alpha}, the null hypothesis cannot be rejected.')"
      ],
      "metadata": {
        "id": "lX6jag0Rlm0r"
      },
      "execution_count": 3,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "mean_with_known_variance(type_t='two_tailed', Sample_mean=8.75, n=16, null_mean=10, population_sd=1, alpha=0.05);"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "7X8RAhJ2lp86",
        "outputId": "33011e13-04b7-4686-dead-9a1719c58f1e"
      },
      "execution_count": 4,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "test_statistic: -5.0 \n",
            "\n",
            "P_value: 5.733031438470704e-07 \n",
            "\n",
            "Since p_value < 0.05, reject null hypothesis.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "np.random.seed(1)\n",
        "data = np.random.normal(0,1,100)\n",
        "mean_with_known_variance(type_t='left_tailed', data=data, null_mean=0.1, population_sd=1, alpha=0.05);"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "WiTfjm-Blw9Z",
        "outputId": "bcbeb601-7d25-411a-dea7-0378a0219c94"
      },
      "execution_count": 5,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "test_statistic: -0.394171479243013 \n",
            "\n",
            "P_value: 0.34672722046703075 \n",
            "\n",
            "Since p_value > 0.05, the null hypothesis cannot be rejected.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Unknown_Standard_Deviation'></a>\n",
        "\n",
        "### **4.2.2. Unknown Standard Deviation:**\n"
      ],
      "metadata": {
        "id": "btrLaUKzN1PY"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**A. Two Tailed Test:**\n",
        "\n",
        "Suppose that $X_1, X_2, ..., X_n$ is a sample of size $n$ from a normal distribution having an unknown mean $\\mu$ and a unknown variance $\\sigma^2$.\n",
        "\n",
        "$X_1, X_2, ..., X_n \\sim N( \\mu, \\sigma^2)$\n",
        "\n",
        "$H_0:\\ \\mu = \\mu_0$\n",
        "\n",
        "$H_1:\\ \\mu  \\neq \\mu_0$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Testing statistics: \n",
        "\n",
        "$t_0  \\equiv \\frac{\\overline{X}-\\mu_0}{\\frac{S}{\\sqrt{n}}} \\sim t_{n-1}$\n",
        "\n",
        "$S = \\sqrt{\\frac{\\sum_{i=1}^n\\ (x_i\\ -\\ \\overline{x})^2}{n-1}}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "We accept $H_0$ if:\n",
        "\n",
        "1. $-\\ t_{\\frac{\\alpha}{2},n-1}\\ <\\ t_0 = \\frac{\\overline{X}-\\mu_0}{\\frac{S}{\\sqrt{n}}} <\\ t_{\\frac{\\alpha}{2},n-1}$\n",
        "\n",
        "2. $\\mu_0\\ -\\ t_{\\frac{\\alpha}{2},n-1} \\frac{S}{\\sqrt{n}}\\ <\\ \\overline{X}\\ <\\ \\mu_0\\ +\\ t_{\\frac{\\alpha}{2},n-1} \\frac{S}{\\sqrt{n}}$\n",
        "\n",
        "3. $|\\overline{X}-\\mu_0| < t_{\\frac{\\alpha}{2},n-1} \\frac{S}{\\sqrt{n}}$\n",
        "\n",
        "4. P_value $ = 2 \\times P(t_{n-1} \\geq |t_0|) > \\alpha$"
      ],
      "metadata": {
        "id": "2b3-4p0tQKdv"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**B. One-sided tests:**\n",
        "\n",
        "**Right-tailed test:**\n",
        "\n",
        "Suppose that $X_1, X_2, ..., X_n$ is a sample of size $n$ from a normal distribution having an unknown mean $\\mu$ and a unknown variance $\\sigma^2$.\n",
        "\n",
        "$X_1, X_2, ..., X_n \\sim N( \\mu, \\sigma^2)$\n",
        "\n",
        "$H_0:\\ \\mu =  \\mu_0\\ \\quad or \\quad H_0:\\ \\mu  \\leq \\mu_0$\n",
        "\n",
        "$H_1:\\ \\mu  > \\mu_0$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Testing statistics: \n",
        "\n",
        "$t_0  \\equiv \\frac{\\overline{X}-\\mu_0}{\\frac{S}{\\sqrt{n}}} \\sim t_{n-1}$\n",
        "\n",
        "$S = \\sqrt{\\frac{\\sum_{i=1}^n\\ (x_i\\ -\\ \\overline{x})^2}{n-1}}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "We accept $H_0$ if:\n",
        "\n",
        "1. $ -\\infty <\\ t_0  = \\frac{\\overline{X}-\\mu_0}{\\frac{S}{\\sqrt{n}}}\\ <\\ t_{\\alpha, n-1} $\n",
        "\n",
        "2. $- \\infty \\ <\\ \\overline{X}\\ <\\ \\mu_0\\ +\\ t_{\\alpha, n-1} \\frac{S}{\\sqrt{n}}$\n",
        "\n",
        "3. P_value $ = P(t_{n-1} \\geq t_0) > \\alpha$"
      ],
      "metadata": {
        "id": "EDXY_X3im-39"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**Left-tailed test:**\n",
        "\n",
        "Suppose that $X_1, X_2, ..., X_n$ is a sample of size $n$ from a normal distribution having an unknown mean $\\mu$ and a known variance $\\sigma^2$.\n",
        "\n",
        "$X_1, X_2, ..., X_n \\sim N( \\mu, \\sigma^2)$\n",
        "\n",
        "$H_0:\\ \\mu =  \\mu_0 \\quad or \\quad H_0:\\ \\mu  \\geq \\mu_0$\n",
        "\n",
        "$H_1:\\ \\mu  < \\mu_0$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Testing statistics: \n",
        "\n",
        "$t_0  \\equiv \\frac{\\overline{X}-\\mu_0}{\\frac{S}{\\sqrt{n}}} \\sim t_{n-1}$\n",
        "\n",
        "$S = \\sqrt{\\frac{\\sum_{i=1}^n\\ (x_i\\ -\\ \\overline{x})^2}{n-1}}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "We accept $H_0$ if:\n",
        "\n",
        "1. $ -\\ t_{\\alpha, n-1} <\\ t_0  = \\frac{\\overline{X}-\\mu_0}{\\frac{S}{\\sqrt{n}}}\\ <\\  \\infty$\n",
        "\n",
        "2. $\\ \\mu_0\\ -\\ t_{\\alpha, n-1} \\frac{S}{\\sqrt{n}} <\\ \\overline{X}\\ <\\ \\infty $\n",
        "\n",
        "3. P_value $ = P(t_{n-1} \\leq t_0) > \\alpha$"
      ],
      "metadata": {
        "id": "BjSK1-M-oaJU"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "class mean_with_unknown_variance:\n",
        "  \"\"\"\n",
        "  Parameters\n",
        "  ----------\n",
        "  null_mean : u0\n",
        "  n : optional, number of sample members\n",
        "  S : optional, sample standard deviation\n",
        "  alpha : significance level\n",
        "  Sample_mean : optional, mean of the sample\n",
        "  data : optional, if you do not know the sample mean and S, just pass the data\n",
        "  \"\"\"\n",
        "  def __init__(self, null_mean, alpha, type_t, n = 0.,S = 0., Sample_mean = 0., data=None):\n",
        "    self.Sample_mean = Sample_mean\n",
        "    self.S = S\n",
        "    self.null_mean = null_mean\n",
        "    self.type_t = type_t\n",
        "    self.n = n\n",
        "    self.alpha = alpha\n",
        "    self.data = data\n",
        "    if data is not None:\n",
        "      self.Sample_mean = np.mean(list(data))\n",
        "      self.S = np.std(list(data), ddof=1)\n",
        "      self.n = len(list(data))\n",
        "\n",
        "    mean_with_unknown_variance.__test(self)\n",
        "  \n",
        "  def __test(self):\n",
        "    test_statistic = (self.Sample_mean-self.null_mean) / (self.S/np.sqrt(self.n))\n",
        "    print('test_statistic:', test_statistic, '\\n')\n",
        "    \n",
        "    if self.type_t == 'two_tailed':\n",
        "      p_value = 2*(1-t.cdf(abs(test_statistic), df=self.n-1))\n",
        "      print('P_value:', p_value, '\\n')\n",
        "    elif self.type_t == 'left_tailed':\n",
        "      p_value = (t.cdf(test_statistic, df=self.n-1))\n",
        "      print('P_value:', p_value, '\\n')\n",
        "    else:\n",
        "       p_value = (1-t.cdf(test_statistic, df=self.n-1))\n",
        "       print('P_value:', p_value, '\\n')\n",
        "\n",
        "    if p_value < self.alpha:\n",
        "      print(f'Since p_value < {self.alpha}, reject null hypothesis.')\n",
        "    else:\n",
        "      print(f'Since p_value > {self.alpha}, the null hypothesis cannot be rejected.')"
      ],
      "metadata": {
        "id": "nt7giKLcfxtx"
      },
      "execution_count": 6,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "mean_with_unknown_variance(type_t='two_tailed', Sample_mean=14.5, S=0.3, n=9, null_mean=14.2, alpha=0.05);"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "60yeQGM1k0cZ",
        "outputId": "e6f54fbe-20e0-4d41-8e94-19eda1285797"
      },
      "execution_count": 7,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "test_statistic: 3.0000000000000075 \n",
            "\n",
            "P_value: 0.017071681233782554 \n",
            "\n",
            "Since p_value < 0.05, reject null hypothesis.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "np.random.seed(1)\n",
        "data = np.random.normal(0,1,100)\n",
        "mean_with_unknown_variance(type_t='two_tailed', data=data, null_mean=0.1, alpha=0.05);"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "cIFNzvbIqov6",
        "outputId": "eaea2cc2-65cf-45da-b5e6-3d0b773f766d"
      },
      "execution_count": 8,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "test_statistic: -0.4430807396299341 \n",
            "\n",
            "P_value: 0.6586740373655937 \n",
            "\n",
            "Since p_value > 0.05, the null hypothesis cannot be rejected.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Test_Concerning_the_Equality_of_Means_of_Two_Normal_Populations'></a>\n",
        "\n",
        "## **4.3. Test Concerning the Equality of Means of Two Normal Populations:**"
      ],
      "metadata": {
        "id": "4V1_eXUkxJD4"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Known_Variances'></a>\n",
        "\n",
        "### **4.3.1. Known Variances:**"
      ],
      "metadata": {
        "id": "58w24h7DxfUc"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**A. Two Tailed Test:**\n",
        "\n",
        "Suppose that $X_1, X_2, ..., X_n$ is a sample of size $n_x$ from a normal distribution having an unknown mean $\\mu_x$ and a known variance $\\sigma^2_x$. \n",
        "\n",
        "$Y_1, Y_2, ..., Y_n$ is a sample of size $n_y$ from a normal distribution having an unknown mean $\\mu_y$ and a known variance $\\sigma^2_y$.\n",
        "\n",
        "$X_1, X_2, ..., X_{n_x} \\sim N( \\mu_x, \\sigma^2_x)$\n",
        "\n",
        "$Y_1, Y_2, ..., Y_{n_y} \\sim N( \\mu_y, \\sigma^2_y)$\n",
        "\n",
        "$H_0:\\ \\mu_x = \\mu_y$\n",
        "\n",
        "$H_1:\\ \\mu_x  \\neq \\mu_y$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Testing statistics: \n",
        "\n",
        "$Z_0  \\equiv \\frac{\\overline{X}-\\overline{Y} -\\ 0}{\\sqrt{(\\frac{\\sigma^2_x}{n_x}) + (\\frac{\\sigma^2_y}{n_y})}}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "We accept $H_0$ if:\n",
        "\n",
        "1. $-\\ Z_{\\frac{\\alpha}{2}}\\ <\\ Z_0 = \\frac{\\overline{X}-\\overline{Y} -\\ 0}{\\sqrt{(\\frac{\\sigma^2_x}{n_x}) + (\\frac{\\sigma^2_y}{n_y})}}\\ <\\ Z_{\\frac{\\alpha}{2}}$\n",
        "\n",
        "2. $-\\ Z_{\\frac{\\alpha}{2}} {\\sqrt{(\\frac{\\sigma^2_x}{n_x}) + (\\frac{\\sigma^2_y}{n_y})}}\\ <\\ \\overline{X}\\ - \\overline{Y}\\ <\\ Z_{\\frac{\\alpha}{2}} {\\sqrt{(\\frac{\\sigma^2_x}{n_x}) + (\\frac{\\sigma^2_y}{n_y})}}$\n",
        "\n",
        "3. $|\\overline{X}-\\overline{Y}| < Z_{\\frac{\\alpha}{2}} {\\sqrt{(\\frac{\\sigma^2_x}{n_x}) + (\\frac{\\sigma^2_y}{n_y})}}$\n",
        "\n",
        "4. P_value $ = 2 \\times P(Z \\geq |Z_0|) > \\alpha$"
      ],
      "metadata": {
        "id": "CkDCKD4ZHYtP"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**B. One-sided tests:**\n",
        "\n",
        "**Right-tailed test:**\n",
        "\n",
        "Suppose that $X_1, X_2, ..., X_n$ is a sample of size $n_x$ from a normal distribution having an unknown mean $\\mu_x$ and a known variance $\\sigma^2_x$. \n",
        "\n",
        "$Y_1, Y_2, ..., Y_n$ is a sample of size $n_y$ from a normal distribution having an unknown mean $\\mu_y$ and a known variance $\\sigma^2_y$.\n",
        "\n",
        "$X_1, X_2, ..., X_{n_x} \\sim N( \\mu_x, \\sigma^2_x)$\n",
        "\n",
        "$Y_1, Y_2, ..., Y_{n_y} \\sim N( \\mu_y, \\sigma^2_y)$\n",
        "\n",
        "$H_0:\\ \\mu_x =  \\mu_y\\ \\quad or \\quad H_0:\\ \\mu_x  \\leq \\mu_y$\n",
        "\n",
        "$H_1:\\ \\mu_x  > \\mu_y$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Testing statistics: \n",
        "\n",
        "$Z_0  \\equiv \\frac{\\overline{X}-\\overline{Y} - 0}{\\sqrt{(\\frac{\\sigma^2_x}{n_x}) + (\\frac{\\sigma^2_y}{n_y})}}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "We accept $H_0$ if:\n",
        "\n",
        "1. $ -\\infty <\\ Z_0 = \\frac{\\overline{X}-\\overline{Y} - 0}{\\sqrt{(\\frac{\\sigma^2_x}{n_x}) + (\\frac{\\sigma^2_y}{n_y})}}\\ <\\ Z_{\\alpha} $\n",
        "\n",
        "2. $- \\infty \\ <\\ \\overline{X}\\ - \\overline{Y}\\ <\\ Z_{\\alpha} {\\sqrt{(\\frac{\\sigma^2_x}{n_x}) + (\\frac{\\sigma^2_y}{n_y})}}$\n",
        "\n",
        "3. P_value $ = P(Z \\geq Z_0) > \\alpha$"
      ],
      "metadata": {
        "id": "Yo2g8fAeHYtS"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**Left-tailed test:**\n",
        "\n",
        "Suppose that $X_1, X_2, ..., X_n$ is a sample of size $n_x$ from a normal distribution having an unknown mean $\\mu_x$ and a known variance $\\sigma^2_x$. \n",
        "\n",
        "$Y_1, Y_2, ..., Y_n$ is a sample of size $n_y$ from a normal distribution having an unknown mean $\\mu_y$ and a known variance $\\sigma^2_y$.\n",
        "\n",
        "$X_1, X_2, ..., X_{n_x} \\sim N( \\mu_x, \\sigma^2_x)$\n",
        "\n",
        "$Y_1, Y_2, ..., Y_{n_y} \\sim N( \\mu_y, \\sigma^2_y)$\n",
        "\n",
        "$H_0:\\ \\mu_x =  \\mu_y \\quad or \\quad H_0:\\ \\mu_x  \\geq \\mu_y$\n",
        "\n",
        "$H_1:\\ \\mu_x  < \\mu_y$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Testing statistics: \n",
        "\n",
        "$Z_0  \\equiv \\frac{\\overline{X}-\\overline{Y} - 0}{\\sqrt{(\\frac{\\sigma^2_x}{n_x}) + (\\frac{\\sigma^2_y}{n_y})}}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "We accept $H_0$ if:\n",
        "\n",
        "1. $-\\ Z_{\\alpha}\\ <\\ Z_0 = \\frac{\\overline{X}-\\overline{Y} - 0}{\\sqrt{(\\frac{\\sigma^2_x}{n_x}) + (\\frac{\\sigma^2_y}{n_y})}} <\\ \\infty $\n",
        "\n",
        "2. $\\ -Z_{\\alpha} {\\sqrt{(\\frac{\\sigma^2_x}{n_x}) + (\\frac{\\sigma^2_y}{n_y})}}\\ <\\ \\overline{X} - \\overline{Y}\\ <\\ \\infty$\n",
        "\n",
        "3. P_value $ = P(Z \\leq Z_0) > \\alpha$"
      ],
      "metadata": {
        "id": "nt2bxi_dHYtU"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "If you want to test the below hypothesis, just the test statistics changes. The other equations are the same. \n",
        "\n",
        "$H_0:\\ \\mu_x - k\\ \\mu_y = d $\n",
        "\n",
        "$H_1:\\ \\mu_x - k\\ \\mu_y \\neq d$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Testing statistics: \n",
        "\n",
        "$Z_0  \\equiv \\frac{\\overline{X}-\\ k\\ \\overline{Y} -\\ d}{\\sqrt{(\\frac{\\sigma^2_x}{n_x}) + k^2(\\frac{\\sigma^2_y}{n_y})}}$"
      ],
      "metadata": {
        "id": "28hPv38GQIhB"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "class equal_means_with_known_variances:\n",
        "  \"\"\"\n",
        "  Parameters\n",
        "  ----------\n",
        "  diff_mean_null : difference between the means of two population in the null hypothesis\n",
        "  population_v1 : known variance of the population1\n",
        "  population_v2 : known variance of the population2\n",
        "  n1 : optional, number of sample1 members\n",
        "  n2 : optional, number of sample2 members\n",
        "  alpha : significance level\n",
        "  type_t : 'two_tailed', 'right_tailed', 'left_tailed'\n",
        "  Sample_mean1 : mean of the sample1\n",
        "  Sample_mean2 : mean of the sample2\n",
        "  k : optional\n",
        "  data1 : optional, if you do not know the sample mean1, just pass the first sample\n",
        "  data2 : optional, if you do not know the sample mean2, just pass the second sample\n",
        "  \"\"\"\n",
        "  def __init__(self, diff_mean_null, population_v1, population_v2, alpha, type_t, k = 1,\n",
        "               n1 = 0., n2 = 0., Sample_mean1 = 0., Sample_mean2 = 0., data1=None, data2=None):\n",
        "    self.Sample_mean1 = Sample_mean1\n",
        "    self.Sample_mean2 = Sample_mean2    \n",
        "    self.diff_mean_null = diff_mean_null\n",
        "    self.population_v1 = population_v1\n",
        "    self.population_v2 = population_v2\n",
        "    self.type_t = type_t\n",
        "    self.n1 = n1\n",
        "    self.n2 = n2\n",
        "    self.k = k\n",
        "    self.alpha = alpha\n",
        "    self.data1 = data1\n",
        "    self.data2 = data2\n",
        "    if data1 is not None:\n",
        "      self.Sample_mean1 = np.mean(list(data1))\n",
        "      self.n1 = len(list(data1))\n",
        "    if data2 is not None:\n",
        "      self.Sample_mean2 = np.mean(list(data2))\n",
        "      self.n2 = len(list(data2)) \n",
        "\n",
        "    equal_means_with_known_variances.__test(self)\n",
        "  \n",
        "  def __test(self):\n",
        "    test_statistic = (self.Sample_mean1-(self.k)*self.Sample_mean2-self.diff_mean_null) / (np.sqrt((self.population_v1/ self.n1) + (self.k**2)*(self.population_v2/ self.n2)))\n",
        "    print('test_statistic:', test_statistic, '\\n')\n",
        "\n",
        "    if self.type_t == 'two_tailed':\n",
        "      p_value = 2*(1-norm.cdf(abs(test_statistic)))\n",
        "      print('P_value:', p_value, '\\n')\n",
        "    elif self.type_t == 'left_tailed':\n",
        "      p_value = (norm.cdf(test_statistic))\n",
        "      print('P_value:', p_value, '\\n')\n",
        "    else:\n",
        "       p_value = (1-norm.cdf(test_statistic))\n",
        "       print('P_value:', p_value, '\\n')\n",
        "\n",
        "    if p_value < self.alpha:\n",
        "      print(f'Since p_value < {self.alpha}, reject null hypothesis.')\n",
        "    else:\n",
        "      print(f'Since p_value > {self.alpha}, the null hypothesis cannot be rejected.')"
      ],
      "metadata": {
        "id": "l1L5_GNiHaUN"
      },
      "execution_count": 9,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "data1 = np.multiply([61.1,58.2,62.3,64,59.7,66.2,57.8,61.4,62.2,63.6], 100)\n",
        "data2 = np.multiply([62.2,56.6,66.4,56.2,57.4,58.4,57.6,65.4], 100)\n",
        "np.mean(data1), np.mean(data2)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "UKOU5aMAT23t",
        "outputId": "f4fca48a-c797-42dd-f009-7f5fa8226437"
      },
      "execution_count": 10,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "(6165.0, 6002.5)"
            ]
          },
          "metadata": {},
          "execution_count": 10
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "equal_means_with_known_variances(diff_mean_null = 0, population_v1 = 4000**2, population_v2 = 6000**2, n1 = 10, n2 = 8, \n",
        "                                alpha = 0.05, type_t = 'two_tailed', k = 1, Sample_mean1 = 6165, Sample_mean2 = 6002.5);"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "OrYN40CmRS3V",
        "outputId": "2f9eca1f-ab2d-433f-bc8a-ff6c3beea402"
      },
      "execution_count": 11,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "test_statistic: 0.06579432682703693 \n",
            "\n",
            "P_value: 0.947541572987298 \n",
            "\n",
            "Since p_value > 0.05, the null hypothesis cannot be rejected.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "equal_means_with_known_variances(diff_mean_null = 0, population_v1 = 4000**2, population_v2 = 6000**2, \n",
        "                                alpha = 0.05, type_t = 'two_tailed', k = 1, data1 = data1, data2 = data2);"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "OAA0YmNPTFuf",
        "outputId": "3e8a0614-2802-43ba-d5d2-df7c94d74e4e"
      },
      "execution_count": 12,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "test_statistic: 0.06579432682703693 \n",
            "\n",
            "P_value: 0.947541572987298 \n",
            "\n",
            "Since p_value > 0.05, the null hypothesis cannot be rejected.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Unknown_but_Equal_Variances'></a>\n",
        "\n",
        "### **4.3.2. Unknown but Equal Variances:**"
      ],
      "metadata": {
        "id": "rkSluAxdVb1M"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**A. Two Tailed Test:**\n",
        "\n",
        "Suppose that $X_1, X_2, ..., X_n$ is a sample of size $n_x$ from a normal distribution having an unknown mean $\\mu_x$ and a unknown variance $\\sigma^2_x$. \n",
        "\n",
        "$Y_1, Y_2, ..., Y_n$ is a sample of size $n_y$ from a normal distribution having an unknown mean $\\mu_y$ and a unknown variance $\\sigma^2_y$.\n",
        "\n",
        "$X_1, X_2, ..., X_{n_x} \\sim N( \\mu_x, \\sigma^2_x)$\n",
        "\n",
        "$Y_1, Y_2, ..., Y_{n_y} \\sim N( \\mu_y, \\sigma^2_y)$\n",
        "\n",
        "$H_0:\\ \\mu_x = \\mu_y$\n",
        "\n",
        "$H_1:\\ \\mu_x  \\neq \\mu_y$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Testing statistics: \n",
        "\n",
        "$t_0 \\equiv \\frac{\\overline{X}-\\ \\overline{Y}\\ -\\ 0}{{S_p} \\sqrt{\\frac{1}{n_x} + \\frac{1}{n_y}}} \\sim t_{{n_x}+{n_y}-2}$\n",
        "\n",
        "$S_p^2 = \\frac{(n_x-1)S_x^2 + (n_y-1)S_y^2}{n_x+n_y-2}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "We accept $H_0$ if:\n",
        "\n",
        "1. $-\\ t_{\\frac{\\alpha}{2},{n_x} + {n_y}-2}\\ <\\ t_0 = \\frac{\\overline{X}-\\ \\overline{Y}\\ -\\ 0}{{S_p} \\sqrt{\\frac{1}{n_x} + \\frac{1}{n_y}}} <\\ t_{\\frac{\\alpha}{2},{n_x} + {n_y}-2}$\n",
        "\n",
        "2. $-\\ t_{\\frac{\\alpha}{2},{n_x} + {n_y}-2} {{S_p} \\sqrt{\\frac{1}{n_x} + \\frac{1}{n_y}}}\\ <\\ \\overline{X}\\ - \\overline{Y}\\ <\\ t_{\\frac{\\alpha}{2},{n_x} + {n_y}-2} {{S_p} \\sqrt{\\frac{1}{n_x} + \\frac{1}{n_y}}}$\n",
        "\n",
        "3. $|\\overline{X}-\\overline{Y}| < t_{\\frac{\\alpha}{2},{n_x} + {n_y}-2} {{S_p} \\sqrt{\\frac{1}{n_x} + \\frac{1}{n_y}}}$\n",
        "\n",
        "4. P_value $ = 2 \\times P(t_{\\frac{\\alpha}{2},{n_x} + {n_y}-2} \\geq |t_0|) > \\alpha$"
      ],
      "metadata": {
        "id": "fE5B1uXoVyjf"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**B. One-sided tests:**\n",
        "\n",
        "**Right-tailed test:**\n",
        "\n",
        "Suppose that $X_1, X_2, ..., X_n$ is a sample of size $n_x$ from a normal distribution having an unknown mean $\\mu_x$ and a unknown variance $\\sigma^2_x$. \n",
        "\n",
        "$Y_1, Y_2, ..., Y_n$ is a sample of size $n_y$ from a normal distribution having an unknown mean $\\mu_y$ and a unknown variance $\\sigma^2_y$.\n",
        "\n",
        "$X_1, X_2, ..., X_{n_x} \\sim N( \\mu_x, \\sigma^2_x)$\n",
        "\n",
        "$Y_1, Y_2, ..., Y_{n_y} \\sim N( \\mu_y, \\sigma^2_y)$\n",
        "\n",
        "$H_0:\\ \\mu_x =  \\mu_y\\ \\quad or \\quad H_0:\\ \\mu_x  \\leq \\mu_y$\n",
        "\n",
        "$H_1:\\ \\mu_x  > \\mu_y$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Testing statistics: \n",
        "\n",
        "$t_0 \\equiv \\frac{\\overline{X}-\\ \\overline{Y}\\ -\\ 0}{{S_p} \\sqrt{\\frac{1}{n_x} + \\frac{1}{n_y}}} \\sim t_{{n_x}+{n_y}-2}$\n",
        "\n",
        "$S_p^2 = \\frac{(n_x-1)S_x^2 + (n_y-1)S_y^2}{n_x+n_y-2}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "We accept $H_0$ if:\n",
        "\n",
        "1. $ -\\infty <\\ t_0 = \\frac{\\overline{X}-\\ \\overline{Y}\\ -\\ 0}{{S_p} \\sqrt{\\frac{1}{n_x} + \\frac{1}{n_y}}}\\ <\\ t_{{\\alpha},{n_x} + {n_y}-2} $\n",
        "\n",
        "2. $- \\infty \\ <\\ \\overline{X}\\ - \\overline{Y}\\ <\\ t_{{\\alpha},{n_x} + {n_y}-2} {{S_p} \\sqrt{\\frac{1}{n_x} + \\frac{1}{n_y}}}$\n",
        "\n",
        "3. P_value $ = P(t_{{n_x} + {n_y}-2} \\geq t_0) > \\alpha$"
      ],
      "metadata": {
        "id": "h4g05pIyafSJ"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**Left-tailed test:**\n",
        "\n",
        "Suppose that $X_1, X_2, ..., X_n$ is a sample of size $n_x$ from a normal distribution having an unknown mean $\\mu_x$ and a known variance $\\sigma^2_x$. \n",
        "\n",
        "$Y_1, Y_2, ..., Y_n$ is a sample of size $n_y$ from a normal distribution having an unknown mean $\\mu_y$ and a known variance $\\sigma^2_y$.\n",
        "\n",
        "$X_1, X_2, ..., X_{n_x} \\sim N( \\mu_x, \\sigma^2_x)$\n",
        "\n",
        "$Y_1, Y_2, ..., Y_{n_y} \\sim N( \\mu_y, \\sigma^2_y)$\n",
        "\n",
        "$H_0:\\ \\mu_x =  \\mu_y \\quad or \\quad H_0:\\ \\mu_x  \\geq \\mu_y$\n",
        "\n",
        "$H_1:\\ \\mu_x  < \\mu_y$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Testing statistics: \n",
        "\n",
        "$t_0 \\equiv \\frac{\\overline{X}-\\ \\overline{Y}\\ -\\ 0}{{S_p} \\sqrt{\\frac{1}{n_x} + \\frac{1}{n_y}}} \\sim t_{{n_x}+{n_y}-2}$\n",
        "\n",
        "$S_p^2 = \\frac{(n_x-1)S_x^2 + (n_y-1)S_y^2}{n_x+n_y-2}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "We accept $H_0$ if:\n",
        "\n",
        "1. $-\\ t_{{\\alpha},{n_x} + {n_y}-2} <\\ t_0 = \\frac{\\overline{X}-\\ \\overline{Y}\\ -\\ 0}{{S_p} \\sqrt{\\frac{1}{n_x} + \\frac{1}{n_y}}} <\\ \\infty $\n",
        "\n",
        "2. $\\ -t_{{\\alpha},{n_x} + {n_y}-2} {{S_p} \\sqrt{\\frac{1}{n_x} + \\frac{1}{n_y}}}\\ <\\ \\overline{X} - \\overline{Y}\\ <\\ \\infty$\n",
        "\n",
        "3. P_value $ = P(t_{{\\alpha},{n_x} + {n_y}-2} \\leq t_0) > \\alpha$"
      ],
      "metadata": {
        "id": "8CjR8DPccmoH"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "class equal_means_with_unknown_variances:\n",
        "  \"\"\"\n",
        "  Parameters\n",
        "  ----------\n",
        "  diff_mean_null : difference between the means of two population in the null hypothesis\n",
        "  S1 : optional, std of the sample1\n",
        "  S2 : optional, std of the sample2\n",
        "  n1 : optional, number of sample1 members\n",
        "  n2 : optional, number of sample2 members\n",
        "  alpha : significance level\n",
        "  type_t : 'two_tailed', 'right_tailed', 'left_tailed'\n",
        "  Sample_mean1 : mean of the sample1\n",
        "  Sample_mean2 : mean of the sample2\n",
        "  data1 : optional, if you do not know the sample mean1 and S1, just pass the first sample\n",
        "  data2 : optional, if you do not know the sample mean2 and S2, just pass the second sample\n",
        "  \"\"\"\n",
        "  def __init__(self, diff_mean_null, alpha, type_t, n1 = 0., n2 = 0., S1 = 0., S2 = 0., \n",
        "               Sample_mean1 = 0., Sample_mean2 = 0., data1=None, data2=None):\n",
        "    self.Sample_mean1 = Sample_mean1\n",
        "    self.Sample_mean2 = Sample_mean2    \n",
        "    self.diff_mean_null = diff_mean_null\n",
        "    self.S1 = S1\n",
        "    self.S2 = S2\n",
        "    self.type_t = type_t\n",
        "    self.n1 = n1\n",
        "    self.n2 = n2\n",
        "    self.alpha = alpha\n",
        "    self.data1 = data1\n",
        "    self.data2 = data2\n",
        "    self.SP2 = None\n",
        "    if data1 is not None:\n",
        "      self.Sample_mean1 = np.mean(list(data1))\n",
        "      self.S1 = np.std(list(data1), ddof=1)\n",
        "      self.n1 = len(list(data1))\n",
        "\n",
        "    if data2 is not None:\n",
        "      self.Sample_mean2 = np.mean(list(data2))  \n",
        "      self.S2 = np.std(data2, ddof=1)\n",
        "      self.n2 = len(list(data2))\n",
        "\n",
        "    self.SP2 = ((self.n1-1)*(self.S1**2) + (self.n2-1)*(self.S2**2)) / (self.n1+self.n2-2)\n",
        "\n",
        "    equal_means_with_unknown_variances.__test(self)\n",
        "  \n",
        "  def __test(self):\n",
        "    test_statistic = (self.Sample_mean1-self.Sample_mean2-self.diff_mean_null) / (np.sqrt(self.SP2)*np.sqrt(1/self.n1+1/self.n2))\n",
        "    print('test_statistic:', test_statistic, '\\n')\n",
        "\n",
        "    if self.type_t == 'two_tailed':\n",
        "      p_value = 2*(1-t.cdf(abs(test_statistic), df = self.n1+self.n2-2))\n",
        "      print('P_value:', p_value, '\\n')\n",
        "    elif self.type_t == 'left_tailed':\n",
        "      p_value = (t.cdf(test_statistic, df = self.n1+self.n2-2))\n",
        "      print('P_value:', p_value, '\\n')\n",
        "    else:\n",
        "       p_value = 1-t.cdf(test_statistic, df = self.n1+self.n2-2)\n",
        "       print('P_value:', p_value, '\\n')\n",
        "\n",
        "    if p_value < self.alpha:\n",
        "      print(f'Since p_value < {self.alpha}, reject null hypothesis.')\n",
        "    else:\n",
        "      print(f'Since p_value > {self.alpha}, the null hypothesis cannot be rejected.')"
      ],
      "metadata": {
        "id": "IVC_vkmRaUZJ"
      },
      "execution_count": 13,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "equal_means_with_unknown_variances(diff_mean_null = 0, S1 = np.sqrt(0.581), S2 = np.sqrt(0.778), n1 = 10, n2 = 12,\n",
        "                                   alpha = 0.05, type_t = 'left_tailed', Sample_mean1 = 6.45, Sample_mean2 = 7.125);"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "6-VjkWzZswic",
        "outputId": "883fddc5-7868-4ec7-ae2e-12e2bd49f457"
      },
      "execution_count": 14,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "test_statistic: -1.8987297952365871 \n",
            "\n",
            "P_value: 0.03606193933099178 \n",
            "\n",
            "Since p_value < 0.05, reject null hypothesis.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "data1 = [5.5,6,7,6,7.5,6,7.5,5.5,7,6.5]\n",
        "data2 = [6.5,6,8.5,7,6.5,8,7.5,6.5,7.5,6,8.5,7]\n",
        "np.mean(data1), np.mean(data2)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "4GVzWRqD1e-g",
        "outputId": "a8207aed-cc48-4d38-9fb0-0887a2f31d36"
      },
      "execution_count": 15,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "(6.45, 7.125)"
            ]
          },
          "metadata": {},
          "execution_count": 15
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "equal_means_with_unknown_variances(diff_mean_null = 0, alpha = 0.05, type_t = 'left_tailed', data1 = data1, data2 = data2);"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "46MLRnZBwKt4",
        "outputId": "c0ef5dfb-a1e8-4899-cf63-6f8d36017aa3"
      },
      "execution_count": 16,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "test_statistic: -1.8986953664603443 \n",
            "\n",
            "P_value: 0.03606431976959166 \n",
            "\n",
            "Since p_value < 0.05, reject null hypothesis.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "You can also do this test using scipy.\n",
        "\n",
        "`scipy.stats.ttest_ind` calculate the T-test for the means of two independent samples.\n",
        "\n",
        "[Doc](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.ttest_ind.html)"
      ],
      "metadata": {
        "id": "Er9iBrjkOvUE"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "alpha = 0.05\n",
        "Test_statistic, p_value = ttest_ind(data1, data2, alternative='less')\n",
        "\n",
        "print(f'Test_statistic = {Test_statistic},   p_value = {p_value}', '\\n')\n",
        "\n",
        "if p_value < alpha:\n",
        "  print(f'Since p_value < {alpha}, reject null hypothesis.')\n",
        "else:\n",
        "  print(f'Since p_value > {alpha}, the null hypothesis cannot be rejected.')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "a5AQocXrN7ix",
        "outputId": "339d40d5-da10-4cd4-b8c5-7632dea20603"
      },
      "execution_count": 17,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Test_statistic = -1.8986953664603443,   p_value = 0.03606431976959166 \n",
            "\n",
            "Since p_value < 0.05, reject null hypothesis.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Paired_t-test'></a>\n",
        "\n",
        "## **4.4. Paired t-test:**"
      ],
      "metadata": {
        "id": "Br8LlPdh4e9R"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**A. Two Tailed Test:**\n",
        "\n",
        "The data can be described by the $n$ pairs $(X_i, Y_i), i = 1, 2, ..., n$, where $X_i$ is the data before an action, and $Y_i$ is the respective data points after an action.\n",
        "\n",
        "It is important to note that we cannot treat $X_1, ... , X_n$ and $Y_1, ... , Y_n$ as being independent samples.\n",
        "\n",
        "$D_i = X_i − Y_i, \\quad i = 1, ... , n$\n",
        "\n",
        "$H_0 : \\mu_D = 0$\n",
        "\n",
        "$H_1 : \\mu_D \\neq 0$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Testing statistics: \n",
        "\n",
        "$t_0 \\equiv \\frac{\\overline{D}\\ -\\ 0}{\\frac{S_D}{\\sqrt{n}}} \\sim t_{n-1}$\n",
        "\n",
        "$S_D = \\sqrt{\\frac{\\sum_{i=1}^n\\ (D_i\\ -\\ \\overline{D})^2}{n\\ -\\ 1}} \\quad \\quad \\overline{D} = \\sqrt{\\frac{{\\sum_{i=1}^n\\ D_i}}{n}}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "We accept $H_0$ if:\n",
        "\n",
        "1. $ -\\ t_{\\frac{{\\alpha}}{2},{n-1}} <\\ t_0 = \\frac{\\overline{D}\\ -\\ 0}{\\frac{S_D}{\\sqrt{n}}}\\ <\\ t_{\\frac{{\\alpha}}{2},{n-1}} $\n",
        "\n",
        "2. $- t_{\\frac{{\\alpha}}{2},{n-1}} \\frac{S_D}{\\sqrt{n}} \\ <\\ \\overline{D}\\ <\\ t_{\\frac{{\\alpha}}{2},{n-1}} \\frac{S_D}{\\sqrt{n}}$\n",
        "\n",
        "3. $| \\overline{D} |\\ <\\ t_{\\frac{{\\alpha}}{2},{n-1}} \\frac{S_D}{\\sqrt{n}}$\n",
        "\n",
        "4. P_value $ = P(t_{n-1} \\geq |t_0|) > \\alpha$"
      ],
      "metadata": {
        "id": "1mOZr_hk5i7m"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**B. One-sided tests:**\n",
        "\n",
        "**Right-tailed test:**\n",
        "\n",
        "The data can be described by the $n$ pairs $(X_i, Y_i), i = 1, 2, ..., n$, where $X_i$ is the data before an action, and $Y_i$ is the respective data points after an action.\n",
        "\n",
        "It is important to note that we cannot treat $X_1, ... , X_n$ and $Y_1, ... , Y_n$ as being independent samples.\n",
        "\n",
        "$D_i = X_i − Y_i, \\quad i = 1, ... , n$\n",
        "\n",
        "$H_0:\\ \\mu_D =  0\\ \\quad or \\quad H_0:\\ \\mu_D  \\leq 0$\n",
        "\n",
        "$H_1 : \\mu_D > 0$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Testing statistics: \n",
        "\n",
        "$t_0 \\equiv \\frac{\\overline{D}\\ -\\ 0}{\\frac{S_D}{\\sqrt{n}}} \\sim t_{n-1}$\n",
        "\n",
        "$S_D = \\sqrt{\\frac{\\sum_{i=1}^n\\ (D_i\\ -\\ \\overline{D})^2}{n\\ -\\ 1}} \\quad \\quad \\overline{D} = \\sqrt{\\frac{{\\sum_{i=1}^n\\ D_i}}{n}}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "We accept $H_0$ if:\n",
        "\n",
        "1. $ -\\infty <\\ t_0 = \\frac{\\overline{D}\\ -\\ 0}{\\frac{S_D}{\\sqrt{n}}}\\ <\\ t_{{\\alpha},{n-1}} $\n",
        "\n",
        "2. $- \\infty \\ <\\ \\overline{D}\\ <\\ t_{{\\alpha},{n-1}} {\\frac{S_D}{\\sqrt{n}}}$\n",
        "\n",
        "3. P_value $ = P(t_{n-1} \\geq t_0) > \\alpha$"
      ],
      "metadata": {
        "id": "Xmfb3znaIMuQ"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**Left-tailed test:**\n",
        "\n",
        "The data can be described by the $n$ pairs $(X_i, Y_i), i = 1, 2, ..., n$, where $X_i$ is the data before an action, and $Y_i$ is the respective data points after an action.\n",
        "\n",
        "It is important to note that we cannot treat $X_1, ... , X_n$ and $Y_1, ... , Y_n$ as being independent samples.\n",
        "\n",
        "$D_i = X_i − Y_i, \\quad i = 1, ... , n$\n",
        "\n",
        "$H_0:\\ \\mu_D =  0\\ \\quad or \\quad H_0:\\ \\mu_D  \\geq 0$\n",
        "\n",
        "$H_1 : \\mu_D < 0$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Testing statistics: \n",
        "\n",
        "$t_0 \\equiv \\frac{\\overline{D}\\ -\\ 0}{\\frac{S_D}{\\sqrt{n}}} \\sim t_{n-1}$\n",
        "\n",
        "$S_D = \\sqrt{\\frac{\\sum_{i=1}^n\\ (D_i\\ -\\ \\overline{D})^2}{n\\ -\\ 1}} \\quad \\quad \\overline{D} = \\sqrt{\\frac{{\\sum_{i=1}^n\\ D_i}}{n}}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "We accept $H_0$ if:\n",
        "\n",
        "1. $-\\ t_{{\\alpha},{n-1}} <\\ t_0 = \\frac{\\overline{D}\\ -\\ 0}{\\frac{S_D}{\\sqrt{n}}} <\\ \\infty $\n",
        "\n",
        "2. $\\ -t_{{\\alpha},{n-1}} {\\frac{S_D}{\\sqrt{n}}}\\ <\\ \\overline{D}\\ <\\ \\infty$\n",
        "\n",
        "3. P_value $ = P(t_{n-1} \\leq t_0) > \\alpha$"
      ],
      "metadata": {
        "id": "5tBxt9m2JWKe"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "class paired_test:\n",
        "  \"\"\"\n",
        "  Parameters\n",
        "  ----------\n",
        "  null_mean : diff in the null hpothesis\n",
        "  n : optional, number of sample members\n",
        "  SD : D standard deviation\n",
        "  alpha : significance level\n",
        "  D_bar : mean of the D\n",
        "  type_t : 'two_tailed', 'right_tailed', 'left_tailed'\n",
        "  data1 : optional, if you do not know the D_bar and SD, just pass the data\n",
        "  data2 : optional, if you do not know the D_bar and SD, just pass the data\n",
        "  \"\"\"\n",
        "  def __init__(self, null_mean, alpha, type_t, n = 0., SD = 0., D_bar = 0., data1=None, data2=None):\n",
        "    self.D_bar = D_bar\n",
        "    self.SD = SD\n",
        "    self.null_mean = null_mean\n",
        "    self.type_t = type_t\n",
        "    self.n = n\n",
        "    self.alpha = alpha\n",
        "    self.data1 = data1\n",
        "    self.data2 = data2\n",
        "\n",
        "    if data1 is not None:\n",
        "      D = np.subtract(data1, data2)\n",
        "      self.D_bar = np.mean(list(D))\n",
        "      self.SD = np.std(list(D), ddof=1)\n",
        "      self.n = len(list(data1))\n",
        "\n",
        "    paired_test.__test(self)\n",
        "  \n",
        "  def __test(self):\n",
        "    test_statistic = (self.D_bar - self.null_mean) / (self.SD / np.sqrt(self.n))\n",
        "    print('test_statistic:', test_statistic, '\\n')\n",
        "    \n",
        "    if self.type_t == 'two_tailed':\n",
        "      p_value = 2*(1-t.cdf(abs(test_statistic), df=self.n-1))\n",
        "      print('P_value:', p_value, '\\n')\n",
        "    elif self.type_t == 'left_tailed':\n",
        "      p_value = (t.cdf(test_statistic, df=self.n-1))\n",
        "      print('P_value:', p_value, '\\n')\n",
        "    else:\n",
        "       p_value = (1-t.cdf(test_statistic, df=self.n-1))\n",
        "       print('P_value:', p_value, '\\n')\n",
        "\n",
        "    if p_value < self.alpha:\n",
        "      print(f'Since p_value < {self.alpha}, reject null hypothesis.')\n",
        "    else:\n",
        "      print(f'Since p_value > {self.alpha}, the null hypothesis cannot be rejected.')"
      ],
      "metadata": {
        "id": "83eVKnVs4osA"
      },
      "execution_count": 18,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "data1 = [7,6,10,16]\n",
        "data2 = [9,10,14,18]\n",
        "paired_test(null_mean = 0, n = 4, alpha = 0.05, type_t = 'two_tailed', data1=data1, data2=data2);"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "L2HKLRMDAbat",
        "outputId": "21e51686-2272-4a5b-ba2e-fea39ef68457"
      },
      "execution_count": 19,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "test_statistic: -5.196152422706632 \n",
            "\n",
            "P_value: 0.013846832988859026 \n",
            "\n",
            "Since p_value < 0.05, reject null hypothesis.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "data1 = [30.5,18.5,24.5,32,16,15,23.5,25.5,28,18]\n",
        "data2 = [23,21,22,28.5,14.5,15.5,24.5,21,23.5,16.5]\n",
        "paired_test(null_mean = 0, n = 10, alpha = 0.05, type_t = 'right_tailed', data1=data1, data2=data2);"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "vOK3B_4aHDH-",
        "outputId": "1f9a5fdb-5ec8-4015-a095-8c5f0619c74c"
      },
      "execution_count": 20,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "test_statistic: 2.2659493332258522 \n",
            "\n",
            "P_value: 0.02484551530199397 \n",
            "\n",
            "Since p_value < 0.05, reject null hypothesis.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "You can also do this test using scipy.\n",
        "\n",
        "`scipy.stats.ttest_rel` calculate the T-test for the means of two independent samples.\n",
        "\n",
        "[Doc](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.ttest_rel.html)"
      ],
      "metadata": {
        "id": "1M5Wf-TFPzrU"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "alpha = 0.05\n",
        "Test_statistic, p_value = ttest_rel(data1, data2, alternative='greater')\n",
        "\n",
        "print(f'Test_statistic = {Test_statistic},   p_value = {p_value}', '\\n')\n",
        "\n",
        "if p_value < alpha:\n",
        "  print(f'Since p_value < {alpha}, reject null hypothesis.')\n",
        "else:\n",
        "  print(f'Since p_value > {alpha}, the null hypothesis cannot be rejected.')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "P8Xf8chpP0Xx",
        "outputId": "5a659f41-5532-433a-9b12-d79373e52b9c"
      },
      "execution_count": 21,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Test_statistic = 2.2659493332258522,   p_value = 0.024845515301993935 \n",
            "\n",
            "Since p_value < 0.05, reject null hypothesis.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Test_Concerning_the_Variance_of_a_Normal_Population'></a>\n",
        "\n",
        "## **4.5. Test Concerning the Variance of a Normal Population:**"
      ],
      "metadata": {
        "id": "ZJbTadX1NnaL"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**A. Two Tailed Test:**\n",
        "\n",
        "Suppose that $X_1, X_2, ..., X_n$ is a sample of size $n$ from a normal distribution having an unknown mean $\\mu$ and a known variance $\\sigma^2$.\n",
        "\n",
        "$X_1, X_2, ..., X_n \\sim N( \\mu, \\sigma^2)$\n",
        "\n",
        "$H_0:\\ \\sigma^2 = \\sigma_0^2$\n",
        "\n",
        "$H_1:\\ \\sigma^2 \\neq \\sigma_0^2$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Testing statistics: \n",
        "\n",
        "$ \\chi^2_0 \\equiv \\frac{(n-1)\\ S^2}{\\sigma_0^2}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "We accept $H_0$ if:\n",
        "\n",
        "1. $\\ \\chi^2_{1-\\frac{\\alpha}{2}, n-1}\\ <\\ \\chi^2_0 = \\frac{(n-1)\\ S^2}{\\sigma_0^2} \\ <\\chi^2_{\\frac{\\alpha}{2}, n-1} $\n",
        "\n",
        "2. $\\sigma_0^2\\ \\frac{\\chi^2_{1-\\frac{\\alpha}{2}, n-1}}{n-1} <\\ S^2 <\\ \\sigma_0^2\\ \\frac{\\chi^2_{\\frac{\\alpha}{2}, n-1}}{n-1}$\n",
        "\n",
        "3. P_value $ = 2 \\times Min\\ ([P(\\chi^2_{n-1})< \\chi^2_0],\\ [1-P(\\chi^2_{n-1}< \\chi^2_0)]) > \\alpha$"
      ],
      "metadata": {
        "id": "Uk4LjdzUWEG2"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**B. One-sided tests:**\n",
        "\n",
        "**Right-tailed test:**\n",
        "\n",
        "Suppose that $X_1, X_2, ..., X_n$ is a sample of size $n$ from a normal distribution having an unknown mean $\\mu$ and a known variance $\\sigma^2$.\n",
        "\n",
        "$X_1, X_2, ..., X_n \\sim N( \\mu, \\sigma^2)$\n",
        "\n",
        "$H_0:\\ \\sigma^2 =  \\sigma_0^2\\ \\quad or \\quad H_0:\\ \\sigma^2  \\leq \\sigma_0^2$\n",
        "\n",
        "$H_1:\\ \\sigma^2  > \\sigma_0^2$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Testing statistics: \n",
        "\n",
        "$ \\chi^2_0 \\equiv \\frac{(n-1)\\ S^2}{\\sigma_0^2}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "We accept $H_0$ if:\n",
        "\n",
        "1. $ 0 <\\ \\chi^2_0 = \\frac{(n-1)\\ S^2}{\\sigma_0^2}\\ <\\ \\chi^2_{{\\alpha}, n-1}$\n",
        "\n",
        "2. $0 \\ <\\ S^2\\ <\\ \\frac{\\chi^2_{{\\alpha}, n-1}\\ \\sigma_0^2}{(n-1)\\ S^2} $\n",
        "\n",
        "3. P_value $ = P(\\chi^2_{n-1} \\geq Z_0) > \\alpha$"
      ],
      "metadata": {
        "id": "q4IQUnEmbd-A"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**B. One-sided tests:**\n",
        "\n",
        "**Left-tailed test:**\n",
        "\n",
        "Suppose that $X_1, X_2, ..., X_n$ is a sample of size $n$ from a normal distribution having an unknown mean $\\mu$ and a known variance $\\sigma^2$.\n",
        "\n",
        "$X_1, X_2, ..., X_n \\sim N( \\mu, \\sigma^2)$\n",
        "\n",
        "$H_0:\\ \\sigma^2 =  \\sigma_0^2\\ \\quad or \\quad H_0:\\ \\sigma^2  \\geq \\sigma_0^2$\n",
        "\n",
        "$H_1:\\ \\sigma^2  < \\sigma_0^2$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Testing statistics: \n",
        "\n",
        "$ \\chi^2_0 \\equiv \\frac{(n-1)\\ S^2}{\\sigma_0^2}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "We accept $H_0$ if:\n",
        "\n",
        "1. $ \\chi^2_{1-{\\alpha}, n-1} <\\ \\chi^2_0 = \\frac{(n-1)\\ S^2}{\\sigma_0^2} <\\ \\infty$\n",
        "\n",
        "2. $\\frac{\\chi^2_{1-{\\alpha}, n-1}\\ \\sigma_0^2}{(n-1)\\ S^2} \\ <\\ S^2\\ <\\ \\infty $\n",
        "\n",
        "3. P_value $ = P(\\chi^2_{n-1} \\leq Z_0) > \\alpha$"
      ],
      "metadata": {
        "id": "ROJhLnQMdknc"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "class variance_normal:\n",
        "  \"\"\"\n",
        "  Parameters\n",
        "  ----------\n",
        "  null_v : v0\n",
        "  n : optional, number of sample members\n",
        "  S : optional, sample variance\n",
        "  alpha : significance level\n",
        "  type_t : 'two_tailed', 'right_tailed', 'left_tailed'\n",
        "  data : optional, if you do not know the S2, just pass the data\n",
        "  \"\"\"\n",
        "  def __init__(self, null_v, alpha, type_t, n = 0., S2 = 0., data=None):\n",
        "    self.S2 = S2\n",
        "    self.null_v = null_v\n",
        "    self.type_t = type_t\n",
        "    self.n = n\n",
        "    self.alpha = alpha\n",
        "    self.data = data\n",
        "    if data is not None:\n",
        "      self.S2 = np.std(list(data), ddof=1)**2\n",
        "      self.n = len(list(data))\n",
        "\n",
        "    variance_normal.__test(self)\n",
        "  \n",
        "  def __test(self):\n",
        "    test_statistic = ((self.n-1)*(self.S2)) / self.null_v\n",
        "    print('test_statistic:', test_statistic, '\\n')\n",
        "    \n",
        "    if self.type_t == 'two_tailed':\n",
        "      a = 1-chi2.cdf(test_statistic, df=self.n-1)\n",
        "      b = chi2.cdf(test_statistic, df=self.n-1)\n",
        "      p_value = 2*np.min([a, b])\n",
        "      print('P_value:', p_value, '\\n')\n",
        "    elif self.type_t == 'left_tailed':\n",
        "      p_value = (chi2.cdf(test_statistic, df=self.n-1))\n",
        "      print('P_value:', p_value, '\\n')\n",
        "    else:\n",
        "       p_value = (1-chi2.cdf(test_statistic, df=self.n-1))\n",
        "       print('P_value:', p_value, '\\n')\n",
        "\n",
        "    if p_value < self.alpha:\n",
        "      print(f'Since p_value < {self.alpha}, reject null hypothesis.')\n",
        "    else:\n",
        "      print(f'Since p_value > {self.alpha}, the null hypothesis cannot be rejected.')"
      ],
      "metadata": {
        "id": "lectCJy-L6Op"
      },
      "execution_count": 22,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "variance_normal(null_v = 0.15**2, n = 20, alpha = 0.05, type_t = 'right_tailed', S2 = 0.025);"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "TOr9beSdkOXv",
        "outputId": "092d9b72-4300-4918-8147-947b1176f1f2"
      },
      "execution_count": 23,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "test_statistic: 21.111111111111114 \n",
            "\n",
            "P_value: 0.33069403418551535 \n",
            "\n",
            "Since p_value > 0.05, the null hypothesis cannot be rejected.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "data = [11,10,9,10,10,11,11,10,12,9,7,9,11,10,11]\n",
        "variance_normal(null_v = 2**2, alpha = 0.05, type_t = 'two_tailed', data = data);"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "7CqUa66upMdN",
        "outputId": "9e00f20f-e532-4522-8998-ca8a56f9612d"
      },
      "execution_count": 24,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "test_statistic: 5.233333333333333 \n",
            "\n",
            "P_value: 0.035414043881992714 \n",
            "\n",
            "Since p_value < 0.05, reject null hypothesis.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Test_Concerning_the_Equality_of_Variances_of_Two_Normal_Populations'></a>\n",
        "\n",
        "## **4.6. Test Concerning the Equality of Variances of Two Normal Populations:**"
      ],
      "metadata": {
        "id": "a6nq2C5NlRKo"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**A. Two Tailed Test:**\n",
        "\n",
        "Suppose that $X_1, X_2, ..., X_n$ is a sample of size $n_x$ from a normal distribution having an unknown mean $\\mu_x$ and a known variance $\\sigma^2_x$. \n",
        "\n",
        "$Y_1, Y_2, ..., Y_n$ is a sample of size $n_y$ from a normal distribution having an unknown mean $\\mu_y$ and a known variance $\\sigma^2_y$.\n",
        "\n",
        "$X_1, X_2, ..., X_n \\sim N( \\mu_x, \\sigma^2_x)$\n",
        "\n",
        "$Y_1, Y_2, ..., Y_n \\sim N( \\mu_y, \\sigma^2_y)$\n",
        "\n",
        "$H_0:\\ \\sigma_x^2 / \\sigma_y^2 = 1$\n",
        "\n",
        "$H_1:\\ \\sigma_x^2 / \\sigma_y^2 \\neq 1$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Testing statistics: \n",
        "\n",
        "$ F_0 \\equiv \\frac{S^2_x}{S^2_y}$\n",
        "\n",
        "$S^2_x = \\frac{\\sum_{i=1}^{n_x}\\ (x_i\\ -\\ \\overline{x})^2}{{n_x}-1}$\n",
        "\n",
        "$S^2_y = \\frac{\\sum_{i=1}^{n_y}\\ (y_i\\ -\\ \\overline{y})^2}{{n_y}-1}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "We accept $H_0$ if:\n",
        "\n",
        "1. $F_{{1-\\frac{\\alpha}{2}}, n_x-1, n_y-1}\\ <\\ F_0 = \\frac{S^2_x}{S^2_y}\\ < F_{{\\frac{\\alpha}{2}}, n_x-1, n_y-1}$\n",
        "\n",
        "2. P_value $ = 2 \\times Min\\ ([P(F_{n_x-1,n_y-1})< F_0],\\ [1-P(F_{n_x-1,n_y-1}< F_0]) > \\alpha$"
      ],
      "metadata": {
        "id": "pdlDGLQg6eYA"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**B. One-sided tests:**\n",
        "\n",
        "**Right-tailed test:**\n",
        "\n",
        "Suppose that $X_1, X_2, ..., X_n$ is a sample of size $n_x$ from a normal distribution having an unknown mean $\\mu_x$ and a known variance $\\sigma^2_x$. \n",
        "\n",
        "$Y_1, Y_2, ..., Y_n$ is a sample of size $n_y$ from a normal distribution having an unknown mean $\\mu_y$ and a known variance $\\sigma^2_y$.\n",
        "\n",
        "$X_1, X_2, ..., X_n \\sim N( \\mu_x, \\sigma^2_x)$\n",
        "\n",
        "$Y_1, Y_2, ..., Y_n \\sim N( \\mu_y, \\sigma^2_y)$\n",
        "\n",
        "$H_0:\\ \\sigma_x^2 / \\sigma_y^2 = 1 \\quad or \\quad \\sigma_x^2 / \\sigma_y^2 \\leq 1$\n",
        "\n",
        "$H_1:\\ \\sigma_x^2 / \\sigma_y^2 > 1$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Testing statistics: \n",
        "\n",
        "$ F_0 \\equiv \\frac{S^2_x}{S^2_y}$\n",
        "\n",
        "$S^2_x = \\frac{\\sum_{i=1}^{n_x}\\ (x_i\\ -\\ \\overline{x})^2}{{n_x}-1}$\n",
        "\n",
        "$S^2_y = \\frac{\\sum_{i=1}^{n_y}\\ (y_i\\ -\\ \\overline{y})^2}{{n_y}-1}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "We accept $H_0$ if:\n",
        "\n",
        "1. $0\\ <\\ F_0 = \\frac{S^2_x}{S^2_y}\\ < F_{{\\alpha}, n_x-1, n_y-1}$\n",
        "\n",
        "2. P_value $ = P(F_{n_x-1,n_y-1} \\geq F_0) > \\alpha$"
      ],
      "metadata": {
        "id": "wIkaIsU9-gBg"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**Left-tailed test:**\n",
        "\n",
        "Suppose that $X_1, X_2, ..., X_n$ is a sample of size $n_x$ from a normal distribution having an unknown mean $\\mu_x$ and a known variance $\\sigma^2_x$. \n",
        "\n",
        "$Y_1, Y_2, ..., Y_n$ is a sample of size $n_y$ from a normal distribution having an unknown mean $\\mu_y$ and a known variance $\\sigma^2_y$.\n",
        "\n",
        "$X_1, X_2, ..., X_n \\sim N( \\mu_x, \\sigma^2_x)$\n",
        "\n",
        "$Y_1, Y_2, ..., Y_n \\sim N( \\mu_y, \\sigma^2_y)$\n",
        "\n",
        "$H_0:\\ \\sigma_x^2 / \\sigma_y^2 = 1 \\quad or \\quad \\sigma_x^2 / \\sigma_y^2 \\geq 1$\n",
        "\n",
        "$H_1:\\ \\sigma_x^2 / \\sigma_y^2 < 1$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Testing statistics: \n",
        "\n",
        "$ F_0 \\equiv \\frac{S^2_x}{S^2_y}$\n",
        "\n",
        "$S^2_x = \\frac{\\sum_{i=1}^{n_x}\\ (x_i\\ -\\ \\overline{x})^2}{{n_x}-1}$\n",
        "\n",
        "$S^2_y = \\frac{\\sum_{i=1}^{n_y}\\ (y_i\\ -\\ \\overline{y})^2}{{n_x}-1}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "We accept $H_0$ if:\n",
        "\n",
        "1. $F_{{1-\\alpha}, n_x-1, n_y-1}\\ <\\ F_0 = \\frac{S^2_x}{S^2_y}\\ < \\infty$\n",
        "\n",
        "2. P_value $ = P(F_{n_x-1,n_y-1} \\leq F_0) > \\alpha$"
      ],
      "metadata": {
        "id": "vCD8AO5_BDBR"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "class equal_variances:\n",
        "  \"\"\"\n",
        "  Parameters\n",
        "  ----------\n",
        "  null_v : v0\n",
        "  n1 : optional, number of data1 members\n",
        "  n2 : optional, number of data2 members\n",
        "  S2X : sample1 variance\n",
        "  S2Y : sample2 variance\n",
        "  alpha : significance level\n",
        "  type_t : 'two_tailed', 'right_tailed', 'left_tailed'\n",
        "  data1 : optional, if you do not know the S2X, just pass the data\n",
        "  data2 : optional, if you do not know the S2Y, just pass the data\n",
        "  \"\"\"\n",
        "  def __init__(self, null_v, alpha, type_t, n1 = 0., n2 = 0., S2X = 0., S2Y = 0., data1=None, data2=None):\n",
        "    self.S2X = S2X\n",
        "    self.S2Y = S2Y\n",
        "    self.null_v = null_v\n",
        "    self.type_t = type_t\n",
        "    self.n1 = n1\n",
        "    self.n2 = n2\n",
        "    self.alpha = alpha\n",
        "    self.data1 = data1\n",
        "    self.data2 = data2\n",
        "    if data1 is not None:\n",
        "      self.SX2 = np.std(list(data1), ddof=1)**2\n",
        "      self.n1 = len(list(data1))\n",
        "    if data2 is not None:\n",
        "      self.S2Y = np.std(list(data2), ddof=1)**2\n",
        "      self.n2 = len(list(data2))\n",
        "\n",
        "    equal_variances.__test(self)\n",
        "  \n",
        "  def __test(self):\n",
        "    test_statistic = self.S2X / self.S2Y\n",
        "    print('test_statistic:', test_statistic, '\\n')\n",
        "    \n",
        "    if self.type_t == 'two_tailed':\n",
        "      a = 1-stats.f.cdf(test_statistic, self.n1-1, self.n2-1)\n",
        "      b = stats.f.cdf(test_statistic, self.n1-1, self.n2-1)\n",
        "      p_value = 2*np.min([a, b])\n",
        "      print('P_value:', p_value, '\\n')\n",
        "    elif self.type_t == 'left_tailed':\n",
        "      p_value = (stats.f.cdf(test_statistic, self.n1-1, self.n2-2))\n",
        "      print('P_value:', p_value, '\\n')\n",
        "    else:\n",
        "       p_value = (1-stats.f.cdf(test_statistic, self.n1-1, self.n2-2))\n",
        "       print('P_value:', p_value, '\\n')\n",
        "\n",
        "    if p_value < self.alpha:\n",
        "      print(f'Since p_value < {self.alpha}, reject null hypothesis')\n",
        "    else:\n",
        "      print(f'Since p_value > {self.alpha}, the null hypothesis cannot be rejected.')"
      ],
      "metadata": {
        "id": "TPjrukBQlDVd"
      },
      "execution_count": 25,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "equal_variances(null_v = 1, n1 = 10, n2 = 12, alpha = 0.05, type_t = 'two_tailed', S2X = .14, S2Y = .28);"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "KQgyQ4LzDHKp",
        "outputId": "a66721d3-ed2e-43bb-a583-4d9e815e0c5f"
      },
      "execution_count": 26,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "test_statistic: 0.5 \n",
            "\n",
            "P_value: 0.3075191907594375 \n",
            "\n",
            "Since p_value > 0.05, the null hypothesis cannot be rejected.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Test_Concerning_P_in_Bernoulli_Populations'></a>\n",
        "\n",
        "## **4.7. Test Concerning P in Bernoulli Populations:**"
      ],
      "metadata": {
        "id": "I2zKIWZWFf4V"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**A. Two Tailed Test:**\n",
        "\n",
        "Suppose that $X_1, X_2, ..., X_n$ is a sample of size $n$ (large) from a bernoulli distribution having an unknown parameter $P$. \n",
        "\n",
        "$X_1, X_2, ..., X_n \\sim Ber(P)$\n",
        "\n",
        "$H_0 : P = P_0$\n",
        "\n",
        "$H_1 : P \\neq P_0$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Testing statistics: \n",
        "\n",
        "$ Z_0 \\equiv \\frac{\\overline{X}\\ -\\ P_0}{\\sqrt{\\frac{P_0(1-P_0)}{n}}}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "We accept $H_0$ if:\n",
        "\n",
        "1. $-\\ Z_{\\frac{\\alpha}{2}}\\ <\\ Z_0 = \\frac{\\overline{X}\\ -\\ P_0}{\\sqrt{\\frac{P_0(1-P_0)}{n}}}\\ < Z_{\\frac{\\alpha}{2}}$\n",
        "\n",
        "2. $P_0\\ -\\ Z_{\\frac{\\alpha}{2}} \\sqrt{\\frac{P_0(1-P_0)}{n}} \\ <\\ \\overline{X} \\ < P_0\\ +\\ Z_{\\frac{\\alpha}{2}} \\sqrt{\\frac{P_0(1-P_0)}{n}}$\n",
        "\n",
        "3. P_value $ = 2  \\times P(Z \\geq |Z_0|) > \\alpha$"
      ],
      "metadata": {
        "id": "0ba1PtjFHYji"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**B. One-sided tests:**\n",
        "\n",
        "**Right-tailed test:**\n",
        "\n",
        "Suppose that $X_1, X_2, ..., X_n$ is a sample of size $n$ (large) from a bernoulli distribution having an unknown parameter $P$. \n",
        "\n",
        "$X_1, X_2, ..., X_n \\sim Ber(P)$\n",
        "\n",
        "$H_0 : P = P_0 \\quad or \\quad P \\leq P_0$\n",
        "\n",
        "$H_1 : P > P_0$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Testing statistics: \n",
        "\n",
        "$ Z_0 \\equiv \\frac{\\overline{X}\\ -\\ P_0}{\\sqrt{\\frac{P_0(1-P_0)}{n}}}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "We accept $H_0$ if:\n",
        "\n",
        "1. $-\\ \\infty <\\ Z_0 = \\frac{\\overline{X}\\ -\\ P_0}{\\sqrt{\\frac{P_0(1-P_0)}{n}}}\\ < Z_{\\alpha}$\n",
        "\n",
        "2. $-\\ \\infty \\ <\\ \\overline{X} \\ < P_0\\ +\\ Z_{\\alpha} \\sqrt{\\frac{P_0(1-P_0)}{n}}$\n",
        "\n",
        "3. P_value $ = P(Z \\geq Z_0) > \\alpha$"
      ],
      "metadata": {
        "id": "D3iUqEh4K65A"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**Left-tailed test:**\n",
        "\n",
        "Suppose that $X_1, X_2, ..., X_n$ is a sample of size $n$ (large) from a bernoulli distribution having an unknown parameter $P$. \n",
        "\n",
        "$X_1, X_2, ..., X_n \\sim Ber(P)$\n",
        "\n",
        "$H_0 : P = P_0 \\quad or \\quad P \\geq P_0$\n",
        "\n",
        "$H_1 : P < P_0$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Testing statistics: \n",
        "\n",
        "$ Z_0 \\equiv \\frac{\\overline{X}\\ -\\ P_0}{\\sqrt{\\frac{P_0(1-P_0)}{n}}}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "We accept $H_0$ if:\n",
        "\n",
        "1. $-\\ Z_{\\alpha} <\\ Z_0 = \\frac{\\overline{X}\\ -\\ P_0}{\\sqrt{\\frac{P_0(1-P_0)}{n}}}\\ < \\infty$\n",
        "\n",
        "2. $\\ P_0\\ -\\ Z_{\\alpha} \\sqrt{\\frac{P_0(1-P_0)}{n}} \\ <\\ \\overline{X} \\ < \\infty$\n",
        "\n",
        "3. P_value $ = P(Z \\leq Z_0) > \\alpha$"
      ],
      "metadata": {
        "id": "dxmzJiCbQ29D"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "class p_bernoulli:\n",
        "  \"\"\"\n",
        "  Parameters\n",
        "  ----------\n",
        "  null_mean : u0\n",
        "  population_sd : known standrad deviation of the population\n",
        "  n : optional, number of sample members\n",
        "  alpha : significance level\n",
        "  type_t : 'two_tailed', 'right_tailed', 'left_tailed'\n",
        "  Sample_mean : optional, mean of the sample\n",
        "  data : optional, if you do not know the sample mean, just pass the data\n",
        "  \"\"\"\n",
        "  def __init__(self, null_p, alpha, type_t, n = 0., Sample_mean = 0., data=None):\n",
        "    self.Sample_mean = Sample_mean\n",
        "    self.null_p = null_p\n",
        "    self.type_t = type_t\n",
        "    self.n = n\n",
        "    self.alpha = alpha\n",
        "    self.data = data\n",
        "    if data is not None:\n",
        "      self.Sample_mean = np.mean(list(data))\n",
        "      self.n = len(list(data))\n",
        "\n",
        "    p_bernoulli.__test(self)\n",
        "  \n",
        "  def __test(self):\n",
        "    test_statistic = (self.Sample_mean - self.null_p) / np.sqrt(self.null_p * (1-self.null_p) / self.n)\n",
        "    print('test_statistic:', test_statistic, '\\n')\n",
        "\n",
        "    if self.type_t == 'two_tailed':\n",
        "      p_value = 2*(1-norm.cdf(abs(test_statistic)))\n",
        "      print('P_value:', p_value, '\\n')\n",
        "    elif self.type_t == 'left_tailed':\n",
        "      p_value = (norm.cdf(test_statistic))\n",
        "      print('P_value:', p_value, '\\n')\n",
        "    else:\n",
        "       p_value = (1-norm.cdf(test_statistic))\n",
        "       print('P_value:', p_value, '\\n')\n",
        "\n",
        "    if p_value < self.alpha:\n",
        "      print(f'Since p_value < {self.alpha}, reject null hypothesis.')\n",
        "    else:\n",
        "      print(f'Since p_value > {self.alpha}, the null hypothesis cannot be rejected.')"
      ],
      "metadata": {
        "id": "6Dhc2QP2RwDe"
      },
      "execution_count": 27,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "p_bernoulli(null_p = 0.2, n = 100, alpha = 0.05, type_t = 'right_tailed', Sample_mean = 0.3);"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "pBQw3rpNTRdE",
        "outputId": "8f1d52f4-9485-4042-d3ef-563f7c202f3a"
      },
      "execution_count": 28,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "test_statistic: 2.4999999999999996 \n",
            "\n",
            "P_value: 0.006209665325776159 \n",
            "\n",
            "Since p_value < 0.05, reject null hypothesis.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Test_Concerning_the_Equality_of_P_in_Two_Bernoulli-Populations'></a>\n",
        "\n",
        "## **4.8. Test Concerning the Equality of P in Two Bernoulli Populations:**\n",
        "\n"
      ],
      "metadata": {
        "id": "ro9fDzmQU8IZ"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**A. Two Tailed Test:**\n",
        "\n",
        "Suppose that $X_1, X_2, ..., X_{n_x}$ is a sample of size $n_x$ (large) from a bernoulli distribution having an unknown parameter $P_x$ and  $Y_1, Y_2, ..., Y_{n_y}$ is a sample of size $n_y$ (large) from a bernoulli distribution having an unknown parameter $P_y$.\n",
        "\n",
        "$X_1, X_2, ..., X_{n_x} \\sim Ber(P_x)$\n",
        "\n",
        "$Y_1, Y_2, ..., Y_{n_y} \\sim Ber(P_y)$\n",
        "\n",
        "$H_0 : P_x = P_y$\n",
        "\n",
        "$H_1 : P_x \\neq P_y$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Testing statistics:\n",
        "\n",
        "$ Z_0 \\equiv \\frac{\\overline{X}\\ -\\ \\overline{Y}\\ -\\ 0}{\\sqrt{\\frac{\\widehat{p}(1- \\widehat{p})}{n_x} + \\frac{\\widehat{p}(1-\\widehat{p})}{n_y}}}$\n",
        "\n",
        "$\\widehat{p} = \\frac{n_x \\overline{X}\\ +\\ n_y \\overline{Y}}{n_x\\ +\\ n_y}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "We accept $H_0$ if:\n",
        "\n",
        "1. $-\\ Z_{\\frac{\\alpha}{2}}\\ <\\ Z_0 = \\frac{\\overline{X}\\ -\\ \\overline{Y}\\ -\\ 0}{\\sqrt{\\frac{\\widehat{p}(1- \\widehat{p})}{n_x} + \\frac{\\widehat{p}(1-\\widehat{p})}{n_y}}}\\ < Z_{\\frac{\\alpha}{2}}$\n",
        "\n",
        "2. $-\\ Z_{\\frac{\\alpha}{2}} \\sqrt{\\frac{\\widehat{p}(1- \\widehat{p})}{n_x} + \\frac{\\widehat{p}(1-\\widehat{p})}{n_y}}\\ <\\ \\overline{X}\\ -\\ \\overline{Y}  < Z_{\\frac{\\alpha}{2}} \\sqrt{\\frac{\\widehat{p}(1- \\widehat{p})}{n_x} + \\frac{\\widehat{p}(1-\\widehat{p})}{n_y}}$\n",
        "\n",
        "3. P_value $ = 2  \\times P(Z \\geq |Z_0|) > \\alpha$"
      ],
      "metadata": {
        "id": "vjQ7CTG8WJOq"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**B. One-sided tests:**\n",
        "\n",
        "**Right-tailed test:**\n",
        "\n",
        "Suppose that $X_1, X_2, ..., X_{n_x}$ is a sample of size $n_x$ (large) from a bernoulli distribution having an unknown parameter $P_x$ and  $Y_1, Y_2, ..., Y_{n_y}$ is a sample of size $n_y$ (large) from a bernoulli distribution having an unknown parameter $P_y$.\n",
        "\n",
        "$X_1, X_2, ..., X_{n_x} \\sim Ber(P_x)$\n",
        "\n",
        "$Y_1, Y_2, ..., Y_{n_y} \\sim Ber(P_y)$\n",
        "\n",
        "$H_0 : P_x = P_y \\quad or \\quad P_x \\leq P_y$\n",
        "\n",
        "$H_1 : P_x > P_y$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Testing statistics: \n",
        "\n",
        "$ Z_0 \\equiv \\frac{\\overline{X}\\ -\\ \\overline{Y}\\ -\\ 0}{\\sqrt{\\frac{\\widehat{p}(1- \\widehat{p})}{n_x} + \\frac{\\widehat{p}(1-\\widehat{p})}{n_y}}}$\n",
        "\n",
        "$\\widehat{p} = \\frac{n_x \\overline{X}\\ +\\ n_y \\overline{Y}}{n_x\\ +\\ n_y}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "We accept $H_0$ if:\n",
        "\n",
        "1. $-\\ \\infty\\ <\\ Z_0 = \\frac{\\overline{X}\\ -\\ \\overline{Y}\\ -\\ 0}{\\sqrt{\\frac{\\widehat{p}(1- \\widehat{p})}{n_x} + \\frac{\\widehat{p}(1-\\widehat{p})}{n_y}}}\\ < Z_{\\alpha}$\n",
        "\n",
        "2. $-\\ \\infty\\ <\\ \\overline{X}\\ -\\ \\overline{Y}  < Z_{\\alpha} \\sqrt{\\frac{\\widehat{p}(1- \\widehat{p})}{n_x} + \\frac{\\widehat{p}(1-\\widehat{p})}{n_y}}$\n",
        "\n",
        "3. P_value $ = P(Z \\geq Z_0) > \\alpha$"
      ],
      "metadata": {
        "id": "ETaJekx1ZkP6"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**Left-tailed test:**\n",
        "\n",
        "Suppose that $X_1, X_2, ..., X_{n_x}$ is a sample of size $n_x$ (large) from a bernoulli distribution having an unknown parameter $P_x$ and  $Y_1, Y_2, ..., Y_{n_y}$ is a sample of size $n_y$ (large) from a bernoulli distribution having an unknown parameter $P_y$.\n",
        "\n",
        "$X_1, X_2, ..., X_{n_x} \\sim Ber(P_x)$\n",
        "\n",
        "$Y_1, Y_2, ..., Y_{n_y} \\sim Ber(P_y)$\n",
        "\n",
        "$H_0 : P_x = P_y \\quad or \\quad P_x \\geq P_y$\n",
        "\n",
        "$H_1 : P_x < P_y$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Testing statistics:\n",
        "\n",
        "$ Z_0 \\equiv \\frac{\\overline{X}\\ -\\ \\overline{Y}\\ -\\ 0}{\\sqrt{\\frac{\\widehat{p}(1- \\widehat{p})}{n_x} + \\frac{\\widehat{p}(1-\\widehat{p})}{n_y}}}$\n",
        "\n",
        "$\\widehat{p} = \\frac{n_x \\overline{X}\\ +\\ n_y \\overline{Y}}{n_x\\ +\\ n_y}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "We accept $H_0$ if:\n",
        "\n",
        "1. $-\\ Z_{\\alpha}\\ <\\ Z_0 = \\frac{\\overline{X}\\ -\\ \\overline{Y}\\ -\\ 0}{\\sqrt{\\frac{\\widehat{p}(1- \\widehat{p})}{n_x} + \\frac{\\widehat{p}(1-\\widehat{p})}{n_y}}}\\ < \\infty$\n",
        "\n",
        "2. $-\\ Z_{\\alpha} \\sqrt{\\frac{\\widehat{p}(1- \\widehat{p})}{n_x} + \\frac{\\widehat{p}(1-\\widehat{p})}{n_y}} <\\ \\overline{X}\\ -\\ \\overline{Y}  < \\infty$\n",
        "\n",
        "3. P_value $ = P(Z \\leq Z_0) > \\alpha$"
      ],
      "metadata": {
        "id": "KICWomUlal8o"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "class p_bernoulli_two_populations:\n",
        "  \"\"\"\n",
        "  Parameters\n",
        "  ----------\n",
        "  null_mean : u0\n",
        "  population_sd : known standrad deviation of the population\n",
        "  n1 : optional, number of data1 members\n",
        "  n2 : optional, number of data2 members\n",
        "  alpha : significance level\n",
        "  type_t : 'two_tailed', 'right_tailed', 'left_tailed'\n",
        "  Sample_mean : mean of the sample\n",
        "  data : optional, if you do not know the sample mean, just pass the data\n",
        "  \"\"\"\n",
        "  def __init__(self, null_p, alpha, type_t, n1 = 0., n2 = 0., Sample_mean1 = 0., Sample_mean2 = 0., data1=None, data2=None):\n",
        "    self.Sample_mean1 = Sample_mean1\n",
        "    self.Sample_mean2 = Sample_mean2\n",
        "    self.null_p = null_p\n",
        "    self.type_t = type_t\n",
        "    self.n1 = n1\n",
        "    self.n2 = n2\n",
        "    self.alpha = alpha\n",
        "    self.data1 = data1\n",
        "    self.data2 = data2\n",
        "    if data1 is not None:\n",
        "      self.Sample_mean1 = np.mean(list(data1))\n",
        "      self.n1 = len(list(data1))\n",
        "    if data2 is not None:\n",
        "      self.Sample_mean2 = np.mean(list(data2))\n",
        "      self.n2 = len(list(data2))\n",
        "\n",
        "    p_bernoulli_two_populations.__test(self)\n",
        "  \n",
        "  def __test(self):\n",
        "    p_hat = ((self.n1 * self.Sample_mean1) + (self.n2 * self.Sample_mean2)) / (self.n1 + self.n2)\n",
        "    test_statistic = (self.Sample_mean1 - self.Sample_mean2) / np.sqrt((p_hat*(1-p_hat)/self.n1) + (p_hat*(1-p_hat)/self.n2))\n",
        "    print('test_statistic:', test_statistic, '\\n')\n",
        "\n",
        "    if self.type_t == 'two_tailed':\n",
        "      p_value = 2*(1-norm.cdf(abs(test_statistic)))\n",
        "      print('P_value:', p_value, '\\n')\n",
        "    elif self.type_t == 'left_tailed':\n",
        "      p_value = (norm.cdf(test_statistic))\n",
        "      print('P_value:', p_value, '\\n')\n",
        "    else:\n",
        "       p_value = (1-norm.cdf(test_statistic))\n",
        "       print('P_value:', p_value, '\\n')\n",
        "\n",
        "    if p_value < self.alpha:\n",
        "      print(f'Since p_value < {self.alpha}, reject null hypothesis.')\n",
        "    else:\n",
        "      print(f'Since p_value > {self.alpha}, the null hypothesis cannot be rejected.')"
      ],
      "metadata": {
        "id": "Q4xWEZaMa_vJ"
      },
      "execution_count": 29,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "p_bernoulli_two_populations(null_p = 0, n1 = 200, n2 = 200, alpha = 0.05, Sample_mean1 = 150/200,\n",
        "                            Sample_mean2 = 170/200, type_t = 'two_tailed');"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "wI-ZscfYcrAo",
        "outputId": "9e8308ec-6ab8-4252-f1cc-721e095547e3"
      },
      "execution_count": 30,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "test_statistic: -2.4999999999999996 \n",
            "\n",
            "P_value: 0.012419330651552318 \n",
            "\n",
            "Since p_value < 0.05, reject null hypothesis.\n"
          ]
        }
      ]
    }
  ]
}