Chapter 3 Confidence Intervals.ipynb

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  "cells": [
    {
      "cell_type": "markdown",
      "source": [
        "# **`Chapter 3: Confidence Intervals`**"
      ],
      "metadata": {
        "id": "dBRZ1hY24Rtd"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**Table of Content:**\n",
        "\n",
        "- [Import Libraries](#Import_Libraries)\n",
        "- [3.1. Confidence Interval for the Mean of a Normal Population](#Confidence_Interval_for_the_Mean_of_a_Normal_Population)\n",
        "  - [3.1.1. Known Standard Deviation](#Known_Standard_Deviation)\n",
        "  - [3.1.2. Unknown Standard Deviation](#Unknown_Standard_Deviation)\n",
        "\n",
        "- [3.2. Confidence Interval for the Variance of a Normal Population](#Confidence_Interval_for_the_Variance_of_a_Normal_Population)\n",
        "  - [3.2.1. Unknown Mean of the Population](#Unknown_Mean_of_the_Population)\n",
        "  - [3.2.2. Known Mean of the Population](#Known_Mean_of_the_Population)\n",
        "\n",
        "- [3.3. Confidence Interval for the Difference in Means of Two Normal Population](#Confidence_Interval_for_the_Difference_in_Means_of_Two_Normal_Populations)\n",
        "  - [3.3.1. Known Variances](#Known_Variances)\n",
        "  - [3.3.2. Unknown but Equal Variances](#Unknown_but_Equal_Variances)\n",
        "\n",
        "- [3.4. Confidence Interval for the Ratio of Variances of Two Normal Populations](#Confidence_Interval_for_the_Ratio_of_Variances_of_Two_Normal_Populations)\n",
        "- [3.5. Confidence Interval for the Mean of a Bernoulli Random Variable](#Confidence_Interval_for_the_Mean_of_a_Bernoulli_Random_Variable)\n"
      ],
      "metadata": {
        "id": "hCvB3MUGy4as"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Import_Libraries'></a>\n",
        "\n",
        "## **Import Libraries**"
      ],
      "metadata": {
        "id": "WJhmgDxsVHEO"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "!pip install --upgrade scipy"
      ],
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          "base_uri": "https://localhost:8080/"
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        "id": "vXStgb2JU6c0",
        "outputId": "5bc2ff25-f5a5-4054-de13-75841a76c9ff"
      },
      "execution_count": null,
      "outputs": [
        {
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          "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)"
      ],
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          "base_uri": "https://localhost:8080/"
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          "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='Confidence_Interval_for_the_Mean_of_a_Normal_Population'></a>\n",
        "\n",
        "## **3.1. Confidence Interval for the Mean of a Normal Population:**"
      ],
      "metadata": {
        "id": "rzplSNRK6pOR"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Known_Standard_Deviation'></a>\n",
        "\n",
        "### **3.1.1. Known Standard Deviation:**"
      ],
      "metadata": {
        "id": "JYXjgns760tV"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**A. Two-sided Confidence Interval:**\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",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "$P(-\\ Z_{\\frac{\\alpha}{2}}\\ \\leq\\ \\frac{\\overline{X}-\\mu}{\\frac{\\sigma}{\\sqrt{n}}}\\ \\leq\\ Z_{\\frac{\\alpha}{2}}) = 1-\\alpha$\n",
        "\n",
        "$P(\\overline{X}\\ -\\ Z_{\\frac{\\alpha}{2}} \\frac{\\sigma}{\\sqrt{n}}\\ \\leq\\ \\mu\\ \\leq\\ \\overline{X}\\ +\\ Z_{\\frac{\\alpha}{2}} \\frac{\\sigma}{\\sqrt{n}}) = 1-\\alpha$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Therefore, the $1-\\alpha$ confidence interval for the mean of a normal population is:\n",
        "\n",
        "$[\\overline{X}\\ -\\ Z_{\\frac{\\alpha}{2}} \\frac{\\sigma}{\\sqrt{n}},\\ \\overline{X}\\ +\\ Z_{\\frac{\\alpha}{2}} \\frac{\\sigma}{\\sqrt{n}}]$"
      ],
      "metadata": {
        "id": "10TKKoby6-D_"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**B. One-sided Lower Confidence Interval:**\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",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "$P(-\\infty\\ \\leq\\ \\frac{\\overline{X}-\\mu}{\\frac{\\sigma}{\\sqrt{n}}}\\ \\leq\\ Z_{\\alpha}) = 1-\\alpha$\n",
        "\n",
        "$P(-\\infty\\ \\leq\\ \\mu\\ \\leq\\ \\overline{X}\\ +\\ Z_{\\alpha} \\frac{\\sigma}{\\sqrt{n}}) = 1-\\alpha$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Therefore, the $1-\\alpha$ confidence interval for the mean of a normal population is:\n",
        "\n",
        "$[-\\infty,\\ \\overline{X}\\ +\\ Z_{\\alpha} \\frac{\\sigma}{\\sqrt{n}}]$"
      ],
      "metadata": {
        "id": "fGHYWk5y-JS2"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**C. One-sided Upper Confidence Interval:**\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",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "$P(-\\ Z_{\\alpha}\\ \\leq\\ \\frac{\\overline{X}-\\mu}{\\frac{\\sigma}{\\sqrt{n}}}\\ \\leq\\ \\infty) = 1-\\alpha$\n",
        "\n",
        "$P(\\overline{X}\\ -\\ Z_{\\alpha} \\frac{\\sigma}{\\sqrt{n}}\\ \\leq\\ \\mu\\ \\leq\\   \\infty) = 1-\\alpha$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Therefore, the $1-\\alpha$ confidence interval for the mean of a normal population is:\n",
        "\n",
        "$[\\overline{X}\\ -\\ Z_{\\alpha} \\frac{\\sigma}{\\sqrt{n}},\\ \\infty]$"
      ],
      "metadata": {
        "id": "yYGKzkCl_ws7"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "class confidence_interval_for_mean_with_known_variance:\n",
        "  \"\"\"\n",
        "  Parameters\n",
        "  ----------\n",
        "  population_sd : known standrad deviation of the population\n",
        "  n : optional, number of sample members\n",
        "  c_level : % confidence level\n",
        "  type_t : 'two_sided_confidence', 'lower_confidence', 'upper_confidence'\n",
        "  Sample_mean : mean of the sample\n",
        "  data : optional, if you do not know the Sample_mean and n, just pass the data\n",
        "  \"\"\"\n",
        "  def __init__(self, population_sd, c_level, type_c, Sample_mean = 0., n = 0., data=None):\n",
        "    self.Sample_mean = Sample_mean\n",
        "    self.population_sd = population_sd\n",
        "    self.type_c = type_c\n",
        "    self.n = n\n",
        "    self.c_level = c_level\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",
        "    confidence_interval_for_mean_with_known_variance.__test(self)\n",
        "  \n",
        "  def __test(self):\n",
        "    if self.type_c == 'two_sided_confidence':\n",
        "      c_u = self.Sample_mean + (-norm.ppf((1-self.c_level)/2)) * (self.population_sd/np.sqrt(self.n))\n",
        "      c_l = self.Sample_mean - (-norm.ppf((1-self.c_level)/2)) * (self.population_sd/np.sqrt(self.n))\n",
        "      display(Latex(f'${c_l} \\leq \\mu \\leq {c_u}$'))\n",
        "    elif self.type_c == 'lower_confidence':\n",
        "      c_u = self.Sample_mean + (-norm.ppf(1-self.c_level)) * (self.population_sd/np.sqrt(self.n))\n",
        "      display(Latex(f'$\\mu \\leq {c_u}$'))\n",
        "    elif self.type_c == 'upper_confidence':\n",
        "      c_l = self.Sample_mean - (-norm.ppf(1-self.c_level)) * (self.population_sd/np.sqrt(self.n))\n",
        "      display(Latex(f'${c_l} \\leq \\mu$'))"
      ],
      "metadata": {
        "id": "eai7LbK24ZjV"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "np.random.seed(1)\n",
        "data = np.random.normal(loc = 2, scale = 3, size = 20)\n",
        "confidence_interval_for_mean_with_known_variance(population_sd = 3, c_level = 0.95, type_c = 'two_sided_confidence', data=data);"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 39
        },
        "id": "9comBWfSWbuY",
        "outputId": "11da123a-7ee5-422f-f45e-2cf805331141"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<IPython.core.display.Latex object>"
            ],
            "text/latex": "$0.2851222797529398 \\leq \\mu \\leq 2.9146899014826846$"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "data = [5, 8.5, 12, 15, 7, 9, 7.5, 6.5, 10.5]\n",
        "confidence_interval_for_mean_with_known_variance(population_sd = 2, c_level = 0.95, type_c = 'lower_confidence', data=data);"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 39
        },
        "id": "rikL21RHdlwn",
        "outputId": "8d7db470-bec4-40f8-e699-33ec34abe3fc"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<IPython.core.display.Latex object>"
            ],
            "text/latex": "$\\mu \\leq 10.096569084634314$"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Unknown_Standard_Deviation'></a>\n",
        "\n",
        "### **3.1.2. Unknown Standard Deviation:**"
      ],
      "metadata": {
        "id": "j-dAKBdRgjMk"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**A. Two-sided Confidence Interval:**\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",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "$P(-\\ t_{\\frac{\\alpha}{2},n-1}\\ <\\ \\frac{\\overline{X}-\\mu}{\\frac{S}{\\sqrt{n}}} <\\ t_{\\frac{\\alpha}{2},n-1}) = 1-\\alpha$\n",
        "\n",
        "$P(\\overline{X}\\ -\\ t_{\\frac{\\alpha}{2},n-1} \\frac{S}{\\sqrt{n}}\\ <\\ \\mu\\ <\\ \\overline{X}\\ +\\ t_{\\frac{\\alpha}{2},n-1} \\frac{S}{\\sqrt{n}}) = 1- \\alpha$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Therefore, the $1-\\alpha$ confidence interval for the mean of a normal population is:\n",
        "\n",
        "$[\\overline{X}\\ -\\ t_{\\frac{\\alpha}{2},n-1} \\frac{S}{\\sqrt{n}},\\ \\overline{X}\\ +\\ t_{\\frac{\\alpha}{2},n-1} \\frac{S}{\\sqrt{n}}]$"
      ],
      "metadata": {
        "id": "vtKhaPgmij5l"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**B. One-sided Lower Confidence Interval:**\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",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "$P(-\\infty\\ \\leq\\ \\frac{\\overline{X}-\\mu}{\\frac{S}{\\sqrt{n}}}\\ \\leq\\ t_{\\alpha,n-1}) = 1-\\alpha$\n",
        "\n",
        "$P(-\\infty\\ \\leq\\ \\mu\\ \\leq\\ \\overline{X}\\ +\\ t_{\\alpha,n-1} \\frac{S}{\\sqrt{n}}) = 1-\\alpha$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Therefore, the $1-\\alpha$ confidence interval for the mean of a normal population is:\n",
        "\n",
        "$[-\\infty,\\ \\overline{X}\\ +\\ t_{\\alpha,n-1} \\frac{S}{\\sqrt{n}}]$"
      ],
      "metadata": {
        "id": "izR0qZWSjUv-"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**B. One-sided upper Confidence Interval:**\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",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "$P(-t_{\\alpha,n-1} \\leq\\ \\frac{\\overline{X}-\\mu}{\\frac{S}{\\sqrt{n}}}\\ \\leq\\ \\infty) = 1-\\alpha$\n",
        "\n",
        "$P(\\overline{X}\\ -\\ t_{\\alpha,n-1} \\frac{S}{\\sqrt{n}} \\leq\\ \\mu\\ \\leq\\ \\infty) = 1-\\alpha$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Therefore, the $1-\\alpha$ confidence interval for the mean of a normal population is:\n",
        "\n",
        "$[\\overline{X}\\ -\\ t_{\\alpha,n-1} \\frac{S}{\\sqrt{n}},\\ \\infty]$"
      ],
      "metadata": {
        "id": "OVn_SYxdkZDR"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "class confidence_interval_for_mean_with_unknown_variance:\n",
        "  \"\"\"\n",
        "  Parameters\n",
        "  ----------\n",
        "  n : optional, number of sample members\n",
        "  c_level : % confidence level\n",
        "  type_t : 'two_sided_confidence', 'lower_confidence', 'upper_confidence'\n",
        "  Sample_std : optional, std of the sample\n",
        "  Sample_mean : optional, mean of the sample\n",
        "  data : optional, if you do not know the Sample_mean and n, just pass the data\n",
        "  \"\"\"\n",
        "  def __init__(self, c_level, type_c, Sample_std = 0., Sample_mean = 0., n = 0.,  data=None):\n",
        "    self.Sample_mean = Sample_mean\n",
        "    self.Sample_std = Sample_std\n",
        "    self.type_c = type_c\n",
        "    self.n = n\n",
        "    self.c_level = c_level\n",
        "    self.data = data\n",
        "    if data is not None:\n",
        "      self.Sample_mean = np.mean(list(data))\n",
        "      self.Sample_std = np.std(list(data), ddof=1)\n",
        "      self.n = len(list(data))\n",
        "\n",
        "    confidence_interval_for_mean_with_unknown_variance.__test(self)\n",
        "  \n",
        "  def __test(self):\n",
        "    if self.type_c == 'two_sided_confidence':\n",
        "      c_u = self.Sample_mean + (t.isf((1-self.c_level)/2, self.n-1)) * (self.Sample_std/np.sqrt(self.n))\n",
        "      c_l = self.Sample_mean - (t.isf((1-self.c_level)/2, self.n-1)) * (self.Sample_std/np.sqrt(self.n))\n",
        "      display(Latex(f'${c_l} \\leq \\mu \\leq {c_u}$'))\n",
        "    elif self.type_c == 'lower_confidence':\n",
        "      c_u = self.Sample_mean + (t.isf(1-self.c_level, self.n-1)) * (self.Sample_std/np.sqrt(self.n))\n",
        "      display(Latex(f'$\\mu \\leq {c_u}$'))\n",
        "    elif self.type_c == 'upper_confidence':\n",
        "      c_l = self.Sample_mean - (t.isf(1-self.c_level, self.n-1)) * (self.Sample_std/np.sqrt(self.n))\n",
        "      display(Latex(f'${c_l} \\leq \\mu$'))"
      ],
      "metadata": {
        "id": "__wyr1psgzld"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "data = [5, 8.5, 12, 15, 7, 9, 7.5, 6.5, 10.5]\n",
        "confidence_interval_for_mean_with_unknown_variance(c_level = 0.95, type_c = 'two_sided_confidence', data=data);"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "BzJ6wQBHmvwI",
        "outputId": "01b45d36-41dc-4346-b701-5587ab239217"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<IPython.core.display.Latex object>"
            ],
            "text/latex": "$6.630805969849321 \\leq \\mu \\leq 11.369194030150679$"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Confidence_Interval_for_the_Variance_of_a_Normal_Population'></a>\n",
        "\n",
        "## **3.2. Confidence Interval for the Variance of a Normal Population:**"
      ],
      "metadata": {
        "id": "XqnRkA8t18kd"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Unknown_Mean_of_the_Population'></a>\n",
        "\n",
        "### **3.2.1. Unknown Mean of the Population:**"
      ],
      "metadata": {
        "id": "t8ukxT389zQi"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**A. Two-sided Confidence Interval:**\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",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "$\\ \\chi^2_{1-\\frac{\\alpha}{2}, n-1}\\ \\leq\\ \\frac{(n-1)\\ S^2}{\\sigma^2} \\ \\leq \\chi^2_{\\frac{\\alpha}{2}, n-1} $\n",
        "\n",
        "$\\ \\frac{(n-1)\\ S^2}{\\chi^2_{\\frac{\\alpha}{2}, n-1}} \\leq\\ \\sigma^2 \\leq\\ \\frac{(n-1)\\ S^2}{\\chi^2_{1-\\frac{\\alpha}{2}, n-1}}$\n",
        "\n",
        "$\\ \\sqrt{\\frac{(n-1)\\ S^2}{\\chi^2_{\\frac{\\alpha}{2}, n-1}}} \\leq\\ \\sigma \\leq\\ \\sqrt{\\frac{(n-1)\\ S^2}{\\chi^2_{1-\\frac{\\alpha}{2}, n-1}}}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Therefore, the $1-\\alpha$ confidence interval for the variance of a normal population is:\n",
        "\n",
        "$[\\frac{(n-1)\\ S^2}{\\chi^2_{\\frac{\\alpha}{2}, n-1}},\\ \\frac{(n-1)\\ S^2}{\\chi^2_{1-\\frac{\\alpha}{2}, n-1}}]$\n",
        "\n",
        "and the $1-\\alpha$ confidence interval for the standard deviation of a normal population is:\n",
        "\n",
        "$[\\sqrt{\\frac{(n-1)\\ S^2}{\\chi^2_{\\frac{\\alpha}{2}, n-1}}},\\ \\sqrt{\\frac{(n-1)\\ S^2}{\\chi^2_{1-\\frac{\\alpha}{2}, n-1}}}]$"
      ],
      "metadata": {
        "id": "fwZHtEgR2Jd6"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**B. One-sided Lower Confidence Interval:**\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",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "$\\ 0 \\leq\\ \\sigma^2 \\leq\\ \\frac{(n-1)\\ S^2}{\\chi^2_{1-\\alpha, n-1}}$\n",
        "\n",
        "$\\ 0 \\leq\\ \\sigma \\leq\\ \\sqrt{\\frac{(n-1)\\ S^2}{\\chi^2_{1-\\alpha, n-1}}}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Therefore, the $1-\\alpha$ confidence interval for the variance of a normal population is:\n",
        "\n",
        "$[0,\\ \\frac{(n-1)\\ S^2}{\\chi^2_{1-\\frac{\\alpha}{2}, n-1}}]$\n",
        "\n",
        "and the $1-\\alpha$ confidence interval for the standard deviation of a normal population is:\n",
        "\n",
        "$[0,\\ \\sqrt{\\frac{(n-1)\\ S^2}{\\chi^2_{1-\\alpha, n-1}}}]$"
      ],
      "metadata": {
        "id": "NdiBcaI24-PO"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**C. One-sided Upper Confidence Interval:**\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",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "$\\ \\frac{(n-1)\\ S^2}{\\chi^2_{\\alpha, n-1}} \\leq\\ \\sigma^2 \\leq\\ \\infty$\n",
        "\n",
        "$\\ \\sqrt{\\frac{(n-1)\\ S^2}{\\chi^2_{\\alpha, n-1}}} \\leq\\ \\sigma \\leq\\ \\infty$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Therefore, the $1-\\alpha$ confidence interval for the variance of a normal population is:\n",
        "\n",
        "$[\\frac{(n-1)\\ S^2}{\\chi^2_{\\alpha, n-1}},\\ \\infty]$\n",
        "\n",
        "and the $1-\\alpha$ confidence interval for the standard deviation of a normal population is:\n",
        "\n",
        "$[\\sqrt{\\frac{(n-1)\\ S^2}{\\chi^2_{\\alpha, n-1}}},\\ \\infty]$"
      ],
      "metadata": {
        "id": "jeU7NJdw6mlj"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "class confidence_interval_for_var_with_unknown_mean:\n",
        "  \"\"\"\n",
        "  Parameters\n",
        "  ----------\n",
        "  population_sd : known standrad deviation of the population\n",
        "  Sample_var : optional, variance of the sample\n",
        "  n : optional, number of sample members\n",
        "  c_level : % confidence level\n",
        "  type_t : 'two_sided_confidence', 'lower_confidence', 'upper_confidence'\n",
        "  data : optional, if you do not know the Sample_mean and n, just pass the data\n",
        "  \"\"\"\n",
        "  def __init__(self, c_level, type_c, Sample_var = 0., n = 0., data=None):\n",
        "    self.type_c = type_c\n",
        "    self.n = n\n",
        "    self.Sample_var = Sample_var\n",
        "    self.c_level = c_level\n",
        "    self.data = data\n",
        "    if data is not None:\n",
        "      self.n = len(list(data))\n",
        "      self.Sample_var = np.std(list(data), ddof=1)**2\n",
        "\n",
        "    confidence_interval_for_var_with_unknown_mean.__test(self)\n",
        "  \n",
        "  def __test(self):\n",
        "    if self.type_c == 'two_sided_confidence':\n",
        "      alpha = 1 - self.c_level\n",
        "      c_u = ((self.n-1) * self.Sample_var) / chi2.isf(1-(alpha/2), self.n-1)\n",
        "      c_l = ((self.n-1) * self.Sample_var) / chi2.isf(alpha/2, self.n-1)\n",
        "      c_u_r = np.sqrt(c_u)\n",
        "      c_l_r = np.sqrt(c_l)\n",
        "      display(Latex(f'${c_l} \\leq \\sigma^2 \\leq {c_u}$'))\n",
        "      display(Latex(f'${c_l_r} \\leq \\sigma \\leq {c_u_r}$'))\n",
        "    elif self.type_c == 'lower_confidence':\n",
        "      alpha = 1 - self.c_level\n",
        "      c_u = ((self.n-1) * self.Sample_var) / chi2.isf(1-alpha, self.n-1)\n",
        "      c_u_r = np.sqrt(c_u)\n",
        "      display(Latex(f'$0 \\leq \\sigma^2 \\leq {c_u}$'))\n",
        "      display(Latex(f'$0 \\leq \\sigma \\leq {c_u_r}$'))\n",
        "    elif self.type_c == 'upper_confidence':\n",
        "      alpha = 1 - self.c_level\n",
        "      c_l = ((self.n-1) * self.Sample_var) / chi2.isf((alpha), self.n-1)\n",
        "      c_l_r = np.sqrt(c_l)\n",
        "      display(Latex(f'${c_l} \\leq \\sigma^2$'))\n",
        "      display(Latex(f'${c_l_r} \\leq \\sigma$'))"
      ],
      "metadata": {
        "id": "5laNCDiU2FwW"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "data = [.123, .133, .124, .125, .126, .128, .120, .124, .130, .126]\n",
        "confidence_interval_for_var_with_unknown_mean(c_level = 0.9, type_c = 'two_sided_confidence', data=data);"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "R9guzCQHBGdH",
        "outputId": "7d590ce4-0021-41b5-b0fd-253838c365d5"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<IPython.core.display.Latex object>"
            ],
            "text/latex": "$7.2640323116473104e-06 \\leq \\sigma^2 \\leq 3.6961151636179396e-05$"
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<IPython.core.display.Latex object>"
            ],
            "text/latex": "$0.0026951868787984464 \\leq \\sigma \\leq 0.006079568375812496$"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Known_Mean_of_the_Population'></a>\n",
        "\n",
        "### **3.2.2. Known Mean of the Population:**"
      ],
      "metadata": {
        "id": "f_p42Mn-MC3E"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**A. Two-sided Confidence Interval:**\n",
        "\n",
        "Suppose that $X_1, X_2, ..., X_n$ is a sample of size $n$ from a normal distribution having an known mean $\\mu$ and a unknown variance $\\sigma^2$.\n",
        "\n",
        "$X_1, X_2, ..., X_n \\sim N( \\mu, \\sigma^2)$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "$S' = \\sqrt{\\frac{\\sum_{i=1}^n\\ (x_i\\ -\\ \\overline{x})^2}{n}}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "$\\ \\chi^2_{1-\\frac{\\alpha}{2}, n}\\ \\leq\\ \\frac{(n)\\ S'^2}{\\sigma^2} \\ \\leq \\chi^2_{\\frac{\\alpha}{2}, n} $\n",
        "\n",
        "$\\ \\frac{(n)\\ S'^2}{\\chi^2_{\\frac{\\alpha}{2}, n}} \\leq\\ \\sigma^2 \\leq\\ \\frac{(n)\\ S'^2}{\\chi^2_{1-\\frac{\\alpha}{2}, n}}$\n",
        "\n",
        "$\\ \\sqrt{\\frac{(n)\\ S'^2}{\\chi^2_{\\frac{\\alpha}{2}, n}}} \\leq\\ \\sigma \\leq\\ \\sqrt{\\frac{(n)\\ S'^2}{\\chi^2_{1-\\frac{\\alpha}{2}, n}}}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Therefore, the $1-\\alpha$ confidence interval for the variance of a normal population is:\n",
        "\n",
        "$[\\frac{(n)\\ S'^2}{\\chi^2_{\\frac{\\alpha}{2}, n}},\\ \\frac{(n)\\ S'^2}{\\chi^2_{1-\\frac{\\alpha}{2}, n}}]$\n",
        "\n",
        "and the $1-\\alpha$ confidence interval for the standard deviation of a normal population is:\n",
        "\n",
        "$[\\sqrt{\\frac{(n)\\ S'^2}{\\chi^2_{\\frac{\\alpha}{2}, n}}},\\ \\sqrt{\\frac{(n)\\ S'^2}{\\chi^2_{1-\\frac{\\alpha}{2}, n}}}]$"
      ],
      "metadata": {
        "id": "qDBOjmIFMQnw"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**B. One-sided Lower Confidence Interval:**\n",
        "\n",
        "Suppose that $X_1, X_2, ..., X_n$ is a sample of size $n$ from a normal distribution having an known mean $\\mu$ and a unknown variance $\\sigma^2$.\n",
        "\n",
        "$X_1, X_2, ..., X_n \\sim N( \\mu, \\sigma^2)$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "$\\ 0\\ \\leq\\ \\frac{(n)\\ S'^2}{\\sigma^2} \\ \\leq \\chi^2_{1-\\alpha, n} $\n",
        "\n",
        "$\\ 0 \\leq\\ \\sigma^2 \\leq\\ \\frac{(n)\\ S'^2}{\\chi^2_{1-\\alpha, n}}$\n",
        "\n",
        "$\\ 0 \\leq\\ \\sigma \\leq\\ \\sqrt{\\frac{(n)\\ S'^2}{\\chi^2_{1-\\alpha, n}}}$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Therefore, the $1-\\alpha$ confidence interval for the variance of a normal population is:\n",
        "\n",
        "$[0,\\ \\frac{(n)\\ S'^2}{\\chi^2_{1-\\frac{\\alpha}{2}, n}}]$\n",
        "\n",
        "and the $1-\\alpha$ confidence interval for the standard deviation of a normal population is:\n",
        "\n",
        "$[0,\\ \\sqrt{\\frac{(n)\\ S'^2}{\\chi^2_{1-\\alpha, n}}}]$"
      ],
      "metadata": {
        "id": "RxHYrQXhMQnx"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**C. One-sided Upper Confidence Interval:**\n",
        "\n",
        "Suppose that $X_1, X_2, ..., X_n$ is a sample of size $n$ from a normal distribution having an known mean $\\mu$ and a unknown variance $\\sigma^2$.\n",
        "\n",
        "$X_1, X_2, ..., X_n \\sim N( \\mu, \\sigma^2)$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "$\\ \\chi^2_{1-\\alpha, n}\\ \\leq\\ \\frac{(n)\\ S'^2}{\\sigma^2} \\ \\leq \\infty $\n",
        "\n",
        "$\\ \\frac{(n)\\ S'^2}{\\chi^2_{\\alpha, n}} \\leq\\ \\sigma^2 \\leq\\ \\infty$\n",
        "\n",
        "$\\ \\sqrt{\\frac{(n)\\ S'^2}{\\chi^2_{\\alpha, n}}} \\leq\\ \\sigma \\leq\\ \\infty$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Therefore, the $1-\\alpha$ confidence interval for the variance of a normal population is:\n",
        "\n",
        "$[\\frac{(n)\\ S'^2}{\\chi^2_{\\alpha, n}},\\ \\infty]$\n",
        "\n",
        "and the $1-\\alpha$ confidence interval for the standard deviation of a normal population is:\n",
        "\n",
        "$[\\sqrt{\\frac{(n)\\ S'^2}{\\chi^2_{\\alpha, n}}},\\ \\infty]$"
      ],
      "metadata": {
        "id": "pdrZXyj-MQny"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "class confidence_interval_for_var_with_known_mean:\n",
        "  \"\"\"\n",
        "  Parameters\n",
        "  ----------\n",
        "  population_sd : known standrad deviation of the population\n",
        "  Sample_var : optional, variance of the sample\n",
        "  n : optional, number of sample members\n",
        "  c_level : % confidence level\n",
        "  type_t : 'two_sided_confidence', 'lower_confidence', 'upper_confidence'\n",
        "  data : optional, if you do not know the Sample_mean and n, just pass the data\n",
        "  \"\"\"\n",
        "  def __init__(self, c_level, type_c, Sample_var = 0., n = 0., data=None):\n",
        "    self.type_c = type_c\n",
        "    self.n = n\n",
        "    self.Sample_var = Sample_var\n",
        "    self.c_level = c_level\n",
        "    self.data = data\n",
        "    if data is not None:\n",
        "      self.n = len(list(data))\n",
        "      self.Sample_var = np.std(list(data))**2\n",
        "\n",
        "    confidence_interval_for_var_with_known_mean.__test(self)\n",
        "  \n",
        "  def __test(self):\n",
        "    if self.type_c == 'two_sided_confidence':\n",
        "      alpha = 1 - self.c_level\n",
        "      c_u = ((self.n) * self.Sample_var) / chi2.isf(1-(alpha/2), self.n)\n",
        "      c_l = ((self.n) * self.Sample_var) / chi2.isf(alpha/2, self.n)\n",
        "      c_u_r = np.sqrt(c_u)\n",
        "      c_l_r = np.sqrt(c_l)\n",
        "      display(Latex(f'${c_l} \\leq \\sigma^2 \\leq {c_u}$'))\n",
        "      display(Latex(f'${c_l_r} \\leq \\sigma \\leq {c_u_r}$'))\n",
        "    elif self.type_c == 'lower_confidence':\n",
        "      alpha = 1 - self.c_level\n",
        "      c_u = ((self.n) * self.Sample_var) / chi2.isf(1-alpha, self.n)\n",
        "      c_u_r = np.sqrt(c_u)\n",
        "      display(Latex(f'$0 \\leq \\sigma^2 \\leq {c_u}$'))\n",
        "      display(Latex(f'$0 \\leq \\sigma \\leq {c_u_r}$'))\n",
        "    elif self.type_c == 'upper_confidence':\n",
        "      alpha = 1 - self.c_level\n",
        "      c_l = ((self.n) * self.Sample_var) / chi2.isf((alpha), self.n)\n",
        "      c_l_r = np.sqrt(c_l)\n",
        "      display(Latex(f'${c_l} \\leq \\sigma^2$'))\n",
        "      display(Latex(f'${c_l_r} \\leq \\sigma$'))"
      ],
      "metadata": {
        "id": "_53bdypaM_zk"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "data = [.123, .133, .124, .125, .126, .128, .120, .124, .130, .126]\n",
        "confidence_interval_for_var_with_known_mean(c_level = 0.9, type_c = 'two_sided_confidence', data=data);"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "HsAuQHyiNa7q",
        "outputId": "5cb6a1ae-1c2d-410d-c645-f345cc99cbc8"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<IPython.core.display.Latex object>"
            ],
            "text/latex": "$6.713265119259053e-06 \\leq \\sigma^2 \\leq 3.119052532672652e-05$"
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<IPython.core.display.Latex object>"
            ],
            "text/latex": "$0.002590996935401324 \\leq \\sigma \\leq 0.0055848478338023245$"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Confidence_Interval_for_the_Difference_in_Means_of_Two_Normal_Populations'></a>\n",
        "\n",
        "## **3.3. Confidence Interval for the Difference in Means of Two Normal Populations:**"
      ],
      "metadata": {
        "id": "LGrRkUrrO3qG"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Known_Variances'></a>\n",
        "\n",
        "### **3.3.1. Known Variances:**"
      ],
      "metadata": {
        "id": "fyBlrUYCPUzl"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**A. Two-sided Confidence Interval:**\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",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "$P(-\\ Z_{\\frac{\\alpha}{2}}\\ \\leq\\ \\frac{\\overline{X}-\\overline{Y} -\\ (\\mu_x - \\mu_y)}{\\sqrt{(\\frac{\\sigma^2_x}{n_x}) + (\\frac{\\sigma^2_y}{n_y})}}\\ \\leq\\ Z_{\\frac{\\alpha}{2}}) = 1- \\alpha$\n",
        "\n",
        "$P(\\overline{X} - \\overline{Y}-\\ Z_{\\frac{\\alpha}{2}} {\\sqrt{(\\frac{\\sigma^2_x}{n_x}) + (\\frac{\\sigma^2_y}{n_y})}}\\ \\leq\\ \\mu_x\\ - \\mu_y\\ \\leq\\ \\overline{X} - \\overline{Y}+\\ Z_{\\frac{\\alpha}{2}} {\\sqrt{(\\frac{\\sigma^2_x}{n_x}) + (\\frac{\\sigma^2_y}{n_y})}}) = 1- \\alpha$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Therefore, the $1-\\alpha$ confidence interval for the difference in Means of two normal populatios is:\n",
        "\n",
        "$[\\overline{X} - \\overline{Y}-\\ Z_{\\frac{\\alpha}{2}} {\\sqrt{(\\frac{\\sigma^2_x}{n_x}) + (\\frac{\\sigma^2_y}{n_y})}},\\ \\overline{X} - \\overline{Y}+\\ Z_{\\frac{\\alpha}{2}} {\\sqrt{(\\frac{\\sigma^2_x}{n_x}) + (\\frac{\\sigma^2_y}{n_y})}}]$"
      ],
      "metadata": {
        "id": "KwH7pIJtmLfS"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**B. One-sided Lower Confidence Interval:**\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",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "$P(-\\ \\infty\\ \\leq\\ \\frac{\\overline{X}-\\overline{Y} -\\ (\\mu_x - \\mu_y)}{\\sqrt{(\\frac{\\sigma^2_x}{n_x}) + (\\frac{\\sigma^2_y}{n_y})}}\\ \\leq\\ Z_{\\alpha}) = 1- \\alpha$\n",
        "\n",
        "$P(-\\ \\infty\\ \\leq\\ \\mu_x\\ - \\mu_y\\ \\leq\\ \\overline{X} - \\overline{Y}+\\ Z_{\\alpha} {\\sqrt{(\\frac{\\sigma^2_x}{n_x}) + (\\frac{\\sigma^2_y}{n_y})}}\\ ) = 1- \\alpha$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Therefore, the $1-\\alpha$ confidence interval for the difference in Means of two normal populatios is:\n",
        "\n",
        "$[-\\infty,\\ \\overline{X} - \\overline{Y}+\\ Z_{\\alpha} {\\sqrt{(\\frac{\\sigma^2_x}{n_x}) + (\\frac{\\sigma^2_y}{n_y})}}]$"
      ],
      "metadata": {
        "id": "QpffIAp_oC9K"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**C. One-sided Upper Confidence Interval:**\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",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "$P(-\\ Z_{\\alpha}\\ \\leq\\ \\frac{\\overline{X}-\\overline{Y} -\\ (\\mu_x - \\mu_y)}{\\sqrt{(\\frac{\\sigma^2_x}{n_x}) + (\\frac{\\sigma^2_y}{n_y})}}\\ \\leq\\ \\infty) = 1- \\alpha$\n",
        "\n",
        "$P(\\overline{X} - \\overline{Y}-\\ Z_{\\alpha} {\\sqrt{(\\frac{\\sigma^2_x}{n_x}) + (\\frac{\\sigma^2_y}{n_y})}}\\ \\leq\\ \\mu_x\\ - \\mu_y\\ \\leq\\ \\infty) = 1- \\alpha$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Therefore, the $1-\\alpha$ confidence interval for the difference in Means of two normal populatios is:\n",
        "\n",
        "$[\\overline{X} - \\overline{Y}-\\ Z_{\\alpha} {\\sqrt{(\\frac{\\sigma^2_x}{n_x}) + (\\frac{\\sigma^2_y}{n_y})}},\\ \\infty]$"
      ],
      "metadata": {
        "id": "xJ2xK9a6pNlQ"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "class confidence_interval_for_two_mean_with_known_variances:\n",
        "  \"\"\"\n",
        "  Parameters\n",
        "  ----------\n",
        "  population_sd1 : known standrad deviation of the population1\n",
        "  population_sd2 : known standrad deviation of the population2\n",
        "  n1 : optional, number of sample1 members\n",
        "  n2 : optional, number of sample2 members\n",
        "  c_level : % confidence level\n",
        "  type_t : 'two_sided_confidence', 'lower_confidence', 'upper_confidence'\n",
        "  Sample_mean1 : optional, mean of the sample1\n",
        "  Sample_mean2 : optional, mean of the sample2\n",
        "  data : optional, if you do not know the Sample_mean and n, just pass the data\n",
        "  \"\"\"\n",
        "  def __init__(self, population_sd1, population_sd2, c_level, type_c, Sample_mean1 = 0., Sample_mean2 = 0., n1 = 0., n2 = 0., data1=None, data2=None):\n",
        "    self.Sample_mean1 = Sample_mean1\n",
        "    self.Sample_mean2 = Sample_mean2\n",
        "    self.population_sd1 = population_sd1\n",
        "    self.population_sd2 = population_sd2\n",
        "    self.type_c = type_c\n",
        "    self.n1 = n1\n",
        "    self.n2 = n2\n",
        "    self.c_level = c_level\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",
        "    confidence_interval_for_two_mean_with_known_variances.__test(self)\n",
        "  \n",
        "  def __test(self):\n",
        "    if self.type_c == 'two_sided_confidence':\n",
        "      c_u = self.Sample_mean1 - self.Sample_mean2 + (-norm.ppf((1-self.c_level)/2)) * np.sqrt(self.population_sd1**2/self.n1 + self.population_sd2**2/self.n2)\n",
        "      c_l = self.Sample_mean1 - self.Sample_mean2 - (-norm.ppf((1-self.c_level)/2)) * np.sqrt(self.population_sd1**2/self.n1 + self.population_sd2**2/self.n2)\n",
        "      display(Latex(f'${c_l} \\leq \\mu_x - \\mu_y \\leq {c_u}$'))\n",
        "    elif self.type_c == 'lower_confidence':\n",
        "      c_u = self.Sample_mean1 - self.Sample_mean2 + (-norm.ppf(1-self.c_level)) * np.sqrt(self.population_sd1**2/self.n1 + self.population_sd2**2/self.n2)\n",
        "      display(Latex(f'$\\mu_x - \\mu_y \\leq {c_u}$'))\n",
        "    elif self.type_c == 'upper_confidence':\n",
        "      c_l = self.Sample_mean1 - self.Sample_mean2 - (-norm.ppf(1-self.c_level)) * np.sqrt(self.population_sd1**2/self.n1 + self.population_sd2**2/self.n2)\n",
        "      display(Latex(f'${c_l} \\leq \\mu_x - \\mu_y$'))"
      ],
      "metadata": {
        "id": "KccWG8REO5AV"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "data1 = [36,44,41,53,38,36,34,54,52,37,51,44,35,44]\n",
        "data2 = [52,64,38,68,66,52,60,44,48,46,70,62]\n",
        "confidence_interval_for_two_mean_with_known_variances(population_sd1 = np.sqrt(40), population_sd2 = 10, c_level = 0.95, \n",
        "                                                      type_c = 'two_sided_confidence', data1=data1, data2=data2);"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "3k4wDPQksf4c",
        "outputId": "ee22dfdb-a906-44e4-e43d-296ef38fad63"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<IPython.core.display.Latex object>"
            ],
            "text/latex": "$-19.604123716241833 \\leq \\mu_x - \\mu_y \\leq -6.491114378996268$"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Unknown_but_Equal_Variances'></a>\n",
        "\n",
        "### **3.3.2. Unknown but Equal Variances:**"
      ],
      "metadata": {
        "id": "KiM27pBRqGIU"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**A. Two-sided Confidence Interval:**\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",
        "$\\\\ $\n",
        "\n",
        "$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",
        "$P(-\\ t_{\\frac{\\alpha}{2}, n_x-n_y-2}\\ \\leq\\ \\frac{\\overline{X}-\\overline{Y} -\\ (\\mu_x - \\mu_y)}{{{S_p} \\sqrt{\\frac{1}{n_x} + \\frac{1}{n_y}}}}\\ \\leq\\ t_{\\frac{\\alpha}{2}, n_x-n_y-2}) = 1- \\alpha$\n",
        "\n",
        "$P(\\overline{X} - \\overline{Y}-\\ t_{\\frac{\\alpha}{2}, n_x-n_y-2}\\ {S_p} \\sqrt{\\frac{1}{n_x} + \\frac{1}{n_y}}\\ \\leq\\ \\mu_x\\ - \\mu_y\\ \\leq\\ \\overline{X} - \\overline{Y}+\\ t_{\\frac{\\alpha}{2}, n_x-n_y-2}\\ {S_p} \\sqrt{\\frac{1}{n_x} + \\frac{1}{n_y}}) = 1- \\alpha$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Therefore, the $1-\\alpha$ confidence interval for the difference in Means of two normal populatios is:\n",
        "\n",
        "$[\\overline{X} - \\overline{Y}-\\ t_{\\frac{\\alpha}{2}, n_x-n_y-2}\\ {S_p} \\sqrt{\\frac{1}{n_x} + \\frac{1}{n_y}},\\ \\overline{X} - \\overline{Y}+\\ t_{\\frac{\\alpha}{2}, n_x-n_y-2}\\ {S_p} \\sqrt{\\frac{1}{n_x} + \\frac{1}{n_y}}]$"
      ],
      "metadata": {
        "id": "mRzN17zcuBzG"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**B. One-sided Lower Confidence Interval:**\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",
        "$\\\\ $\n",
        "\n",
        "$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",
        "$P(-\\ \\infty\\ \\leq\\ \\frac{\\overline{X}-\\overline{Y} -\\ (\\mu_x - \\mu_y)}{{{S_p} \\sqrt{\\frac{1}{n_x} + \\frac{1}{n_y}}}}\\ \\leq\\ t_{\\alpha, n_x-n_y-2}) = 1- \\alpha$\n",
        "\n",
        "$P(-\\ \\infty\\ \\leq\\ \\mu_x\\ - \\mu_y\\ \\leq\\ \\overline{X} - \\overline{Y}+\\ t_{\\alpha, n_x-n_y-2}\\ {S_p} \\sqrt{\\frac{1}{n_x} + \\frac{1}{n_y}}) = 1- \\alpha$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Therefore, the $1-\\alpha$ confidence interval for the difference in Means of two normal populatios is:\n",
        "\n",
        "$[-\\infty,\\ \\overline{X} - \\overline{Y}+\\ t_{\\alpha, n_x-n_y-2}\\ {S_p} \\sqrt{\\frac{1}{n_x} + \\frac{1}{n_y}}]$"
      ],
      "metadata": {
        "id": "kbFCWqUE-qFS"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**C. One-sided Upper Confidence Interval:**\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",
        "$\\\\ $\n",
        "\n",
        "$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",
        "$P(-\\ t_{\\alpha, n_x-n_y-2}\\ \\leq\\ \\frac{\\overline{X}-\\overline{Y} -\\ (\\mu_x - \\mu_y)}{{{S_p} \\sqrt{\\frac{1}{n_x} + \\frac{1}{n_y}}}}\\ \\leq\\ \\infty) = 1- \\alpha$\n",
        "\n",
        "$P(\\overline{X} - \\overline{Y}-\\ t_{\\alpha, n_x-n_y-2}\\ {S_p} \\sqrt{\\frac{1}{n_x} + \\frac{1}{n_y}}\\ \\leq\\ \\mu_x\\ - \\mu_y\\ \\leq\\ \\infty) = 1- \\alpha$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Therefore, the $1-\\alpha$ confidence interval for the difference in Means of two normal populatios is:\n",
        "\n",
        "$[\\overline{X} - \\overline{Y}-\\ t_{\\alpha, n_x-n_y-2}\\ {S_p} \\sqrt{\\frac{1}{n_x} + \\frac{1}{n_y}}\\ ,\\ \\infty]$"
      ],
      "metadata": {
        "id": "155C-e99_8qt"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "class confidence_interval_for_two_mean_with_unknown_variances:\n",
        "  \"\"\"\n",
        "  Parameters\n",
        "  ----------\n",
        "  n1 : optional, number of sample1 members\n",
        "  n2 : optional, number of sample2 members\n",
        "  c_level : % confidence level\n",
        "  type_t : 'two_sided_confidence', 'lower_confidence', 'upper_confidence'\n",
        "  Sample_mean1 : optional, mean of the sample1\n",
        "  Sample_mean2 : optional, mean of the sample2\n",
        "  S1 : optional, std of the sample1\n",
        "  S2 : optional, std of the sample2\n",
        "  data : optional, if you do not know the Sample_mean and n, just pass the data\n",
        "  \"\"\"\n",
        "  def __init__(self, c_level, type_c, Sample_mean1 = 0., S1 = 0., S2 = 0., Sample_mean2 = 0., n1 = 0., n2 = 0., data1=None, data2=None):\n",
        "    self.Sample_mean1 = Sample_mean1\n",
        "    self.Sample_mean2 = Sample_mean2\n",
        "    self.S1 = S1\n",
        "    self.S2 = S2\n",
        "    self.type_c = type_c\n",
        "    self.n1 = n1\n",
        "    self.n2 = n2\n",
        "    self.c_level = c_level\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",
        "      self.S1 = np.std(list(data1), ddof = 1)\n",
        "    if data2 is not None:\n",
        "      self.Sample_mean2 = np.mean(list(data2))\n",
        "      self.n2 = len(list(data2))  \n",
        "      self.S2 = np.std(list(data2), ddof = 1)\n",
        "    \n",
        "    self.SP2 = ((self.n1-1)*(self.S1**2) + (self.n2-1)*(self.S2**2)) / (self.n1+self.n2-2)\n",
        "\n",
        "    confidence_interval_for_two_mean_with_unknown_variances.__test(self)\n",
        "  \n",
        "  def __test(self):\n",
        "    if self.type_c == 'two_sided_confidence':\n",
        "      alpha = 1-self.c_level\n",
        "      c_u = self.Sample_mean1 - self.Sample_mean2 + (t.isf(alpha/2, df = self.n1+self.n2-2)) * (np.sqrt(self.SP2)*np.sqrt(1/self.n1+1/self.n2))\n",
        "      c_l = self.Sample_mean1 - self.Sample_mean2 - (t.isf(alpha/2, df = self.n1+self.n2-2)) * (np.sqrt(self.SP2)*np.sqrt(1/self.n1+1/self.n2))\n",
        "      display(Latex(f'${c_l} \\leq \\mu_x - \\mu_y \\leq {c_u}$'))\n",
        "    elif self.type_c == 'lower_confidence':\n",
        "      alpha = 1-self.c_level\n",
        "      c_u = self.Sample_mean1 - self.Sample_mean2 + (t.isf(alpha, df = self.n1+self.n2-2)) * (np.sqrt(self.SP2)*np.sqrt(1/self.n1+1/self.n2))\n",
        "      display(Latex(f'$\\mu_x - \\mu_y \\leq {c_u}$'))\n",
        "    elif self.type_c == 'upper_confidence':\n",
        "      alpha = 1-self.c_level\n",
        "      c_l = self.Sample_mean1 - self.Sample_mean2 - (t.isf(alpha, df = self.n1+self.n2-2)) * (np.sqrt(self.SP2)*np.sqrt(1/self.n1+1/self.n2))\n",
        "      display(Latex(f'${c_l} \\leq \\mu_x - \\mu_y$'))"
      ],
      "metadata": {
        "id": "YsSOe8TEqRfy"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "data1 = [140,136,138,150,152,144,132,142,150,154,136,142]\n",
        "data2 = [144,132,136,140,128,150,130,134,130,146,128,131,137,135]\n",
        "confidence_interval_for_two_mean_with_unknown_variances(c_level = 0.9, type_c = 'two_sided_confidence', data1=data1, data2=data2);"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "_j0HozbIDoxI",
        "outputId": "78ae25b1-5bc7-400e-bf3f-d158715d4c8e"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<IPython.core.display.Latex object>"
            ],
            "text/latex": "$2.4981644975614827 \\leq \\mu_x - \\mu_y \\leq 11.930406931009962$"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Confidence_Interval_for_the_Ratio_of_Variances_of_Two_Normal_Populations'></a>\n",
        "## **3.4. Confidence Interval for the Ratio of Variances of Two Normal Populations:**"
      ],
      "metadata": {
        "id": "Bt-qEVhPYD5x"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**A. Two-sided Confidence Interval:**\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",
        "$X_1, X_2, ..., X_n \\sim N( \\mu_x, \\sigma^2_x)$\n",
        "\n",
        "$Y_1, Y_2, ..., Y_n \\sim N( \\mu_y, \\sigma^2_y)$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "$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",
        "$P(F_{{1-\\frac{\\alpha}{2}}, n_x-1, n_y-1}\\ \\leq\\  \\frac{S^2_x}{S^2_y}/\\frac{\\sigma^2_x}{\\sigma^2_y} \\ \\leq F_{{\\frac{\\alpha}{2}}, n_x-1, n_y-1}) = 1-\\alpha$\n",
        "\n",
        "$P(\\frac{S^2_x}{S^2_y}\\ \\times \\frac{1}{F_{\\frac{\\alpha}{2}, n_x-1, n_y-1}} \\leq\\ \\frac{\\sigma^2_x}{\\sigma^2_y} \\leq\\ \\frac{S^2_x}{S^2_y}\\ \\times \\frac{1}{F_{1-\\frac{\\alpha}{2}, n_x-1, n_y-1}}) = 1-\\alpha$\n",
        "\n",
        "$P(\\frac{S_x}{S_y}\\ \\times \\sqrt{\\frac{1}{F_{\\frac{\\alpha}{2}, n_x-1, n_y-1}}} \\leq\\ \\frac{\\sigma_x}{\\sigma_y} \\leq\\ \\frac{S_x}{S_y}\\ \\times \\sqrt{\\frac{1}{F_{1-\\frac{\\alpha}{2}, n_x-1, n_y-1}}}) = 1-\\alpha$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Therefore, the $1-\\alpha$ confidence interval for ratio of variances of two normal populatios is:\n",
        "\n",
        "$[\\frac{S^2_x}{S^2_y}\\ \\times \\frac{1}{F_{\\frac{\\alpha}{2}, n_x-1, n_y-1}}, \\frac{S^2_x}{S^2_y}\\ \\times \\frac{1}{F_{1-\\frac{\\alpha}{2}, n_x-1, n_y-1}}]$\n",
        "\n",
        "Therefore, the $1-\\alpha$ confidence interval for ratio of standard deviations of two normal populatios is:\n",
        "\n",
        "$[\\frac{S_x}{S_y}\\ \\times \\sqrt{\\frac{1}{F_{\\frac{\\alpha}{2}, n_x-1, n_y-1}}}\\ ,\\ \\frac{S_x}{S_y}\\ \\times \\sqrt{\\frac{1}{F_{1-\\frac{\\alpha}{2}, n_x-1, n_y-1}}}]$"
      ],
      "metadata": {
        "id": "yjRHUnfRaKnh"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**B. One-sided Lower Confidence Interval:**\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",
        "$X_1, X_2, ..., X_n \\sim N( \\mu_x, \\sigma^2_x)$\n",
        "\n",
        "$Y_1, Y_2, ..., Y_n \\sim N( \\mu_y, \\sigma^2_y)$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "$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",
        "$P(0 \\leq\\ \\frac{\\sigma^2_x}{\\sigma^2_y} \\leq\\ \\frac{S^2_x}{S^2_y}\\ \\times \\frac{1}{F_{1-\\alpha, n_x-1, n_y-1}}) = 1-\\alpha$\n",
        "\n",
        "$P(0 \\leq\\ \\frac{\\sigma_x}{\\sigma_y} \\leq\\ \\frac{S_x}{S_y}\\ \\times \\sqrt{\\frac{1}{F_{1-\\alpha, n_x-1, n_y-1}}}) = 1-\\alpha$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Therefore, the $1-\\alpha$ confidence interval for ratio of variances of two normal populatios is:\n",
        "\n",
        "$[0\\ ,\\ \\frac{S^2_x}{S^2_y}\\ \\times \\frac{1}{F_{1-\\alpha, n_x-1, n_y-1}}]$\n",
        "\n",
        "Therefore, the $1-\\alpha$ confidence interval for ratio of standard deviations of two normal populatios is:\n",
        "\n",
        "$[0\\ ,\\ \\frac{S_x}{S_y}\\ \\times \\sqrt{\\frac{1}{F_{1-\\alpha, n_x-1, n_y-1}}}]$"
      ],
      "metadata": {
        "id": "qZnUvRNddfbx"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**C. One-sided Upper Confidence Interval:**\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",
        "$X_1, X_2, ..., X_n \\sim N( \\mu_x, \\sigma^2_x)$\n",
        "\n",
        "$Y_1, Y_2, ..., Y_n \\sim N( \\mu_y, \\sigma^2_y)$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "$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",
        "$P(\\frac{S^2_x}{S^2_y}\\ \\times \\frac{1}{F_{\\alpha, n_x-1, n_y-1}} \\leq\\ \\frac{\\sigma^2_x}{\\sigma^2_y} \\leq\\ \\infty) = 1-\\alpha$\n",
        "\n",
        "$P(\\frac{S_x}{S_y}\\ \\times \\sqrt{\\frac{1}{F_{\\alpha, n_x-1, n_y-1}}} \\leq\\ \\frac{\\sigma_x}{\\sigma_y} \\leq\\ \\infty) = 1-\\alpha$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Therefore, the $1-\\alpha$ confidence interval for ratio of variances of two normal populatios is:\n",
        "\n",
        "$[\\frac{S^2_x}{S^2_y}\\ \\times \\frac{1}{F_{\\alpha, n_x-1, n_y-1}}, \\infty]$\n",
        "\n",
        "Therefore, the $1-\\alpha$ confidence interval for ratio of standard deviations of two normal populatios is:\n",
        "\n",
        "$[\\frac{S_x}{S_y}\\ \\times \\sqrt{\\frac{1}{F_{\\alpha, n_x-1, n_y-1}}}\\ ,\\ \\infty]$"
      ],
      "metadata": {
        "id": "l0gydmtze0L6"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "class confidence_interval_for_ratio_variances:\n",
        "  \"\"\"\n",
        "  Parameters\n",
        "  ----------\n",
        "  n1 : optional, number of sample1 members\n",
        "  n2 : optional, number of sample2 members\n",
        "  c_level : % confidence level\n",
        "  type_t : 'two_sided_confidence', 'lower_confidence', 'upper_confidence'\n",
        "  Sample_mean1 : optional, mean of the sample1\n",
        "  Sample_mean2 : optional, mean of the sample2\n",
        "  S1 : optional, std of the sample1\n",
        "  S2 : optional, std of the sample2\n",
        "  data : optional, if you do not know the Sample_mean and n, just pass the data\n",
        "  \"\"\"\n",
        "  def __init__(self, c_level, type_c, S1 = 0., S2 = 0., n1 = 0., n2 = 0., data1=None, data2=None):\n",
        "    self.S1 = S1\n",
        "    self.S2 = S2\n",
        "    self.type_c = type_c\n",
        "    self.n1 = n1\n",
        "    self.n2 = n2\n",
        "    self.c_level = c_level\n",
        "    self.data1 = data1\n",
        "    self.data2 = data2\n",
        "    if data1 is not None:\n",
        "      self.n1 = len(list(data1))\n",
        "      self.S1 = np.std(list(data1), ddof = 1)\n",
        "    if data2 is not None:\n",
        "      self.n2 = len(list(data2))  \n",
        "      self.S2 = np.std(list(data2), ddof = 1)\n",
        "    \n",
        "    confidence_interval_for_ratio_variances.__test(self)\n",
        "  \n",
        "  def __test(self):\n",
        "    if self.type_c == 'two_sided_confidence':\n",
        "      alpha = 1-self.c_level\n",
        "      c_u = ((self.S1**2)/(self.S2**2)) * (1/f.isf(1-alpha/2, self.n1-1, self.n1-1))\n",
        "      c_l = ((self.S1**2)/(self.S2**2)) * (1/f.isf(alpha/2, self.n1-1, self.n1-1))\n",
        "      c_u_r = np.sqrt(c_u)\n",
        "      c_l_r = np.sqrt(c_l)\n",
        "      display(Latex(f'${c_l} \\leq \\sigma^2_x/ \\sigma^2_y \\leq {c_u}$'))\n",
        "      display(Latex(f'${c_l_r} \\leq \\sigma_x/ \\sigma_y \\leq {c_u_r}$'))\n",
        "    elif self.type_c == 'lower_confidence':\n",
        "      alpha = 1-self.c_level\n",
        "      c_u = ((self.S1**2)/(self.S2**2)) * (1/f.isf(1-alpha, self.n1-1, self.n1-1))\n",
        "      c_u_r = np.sqrt(c_u)\n",
        "      display(Latex(f'$\\sigma^2_x/ \\sigma^2_y \\leq {c_u}$'))\n",
        "      display(Latex(f'$\\sigma_x/ \\sigma_y \\leq {c_u_r}$'))\n",
        "    elif self.type_c == 'upper_confidence':\n",
        "      alpha = 1-self.c_level\n",
        "      c_l = ((self.S1**2)/(self.S2**2)) * (1/f.isf(alpha, self.n1-1, self.n1-1))\n",
        "      c_l_r = np.sqrt(c_l)\n",
        "      display(Latex(f'${c_l} \\leq \\sigma^2_x/ \\sigma^2_y$'))\n",
        "      display(Latex(f'${c_l} \\leq \\sigma_x/ \\sigma_y$'))"
      ],
      "metadata": {
        "id": "tFM9PjrWYGr3"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "confidence_interval_for_ratio_variances(c_level = 0.95, type_c = 'two_sided_confidence', S1 = 2.51, S2 = 1.9, n1 = 10, n2 = 10);"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "HgzCfL8AjHoy",
        "outputId": "c1e36e7d-fa65-49e2-956f-ff3302e37557"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<IPython.core.display.Latex object>"
            ],
            "text/latex": "$0.4334780396566579 \\leq \\sigma^2_x/ \\sigma^2_y \\leq 7.026084708199053$"
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<IPython.core.display.Latex object>"
            ],
            "text/latex": "$0.6583904917726697 \\leq \\sigma_x/ \\sigma_y \\leq 2.6506762737458254$"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Confidence_Interval_for_the_Mean_of_a_Bernoulli_Random_Variable'></a>\n",
        "\n",
        "## **3.5. Confidence Interval for the Mean of a Bernoulli Random Variable:**"
      ],
      "metadata": {
        "id": "Djupsq6Kklgq"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**A. Two-sided Confidence Interval:**\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",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "We accept $H_0$ if:\n",
        "\n",
        "$P(-\\ Z_{\\frac{\\alpha}{2}}\\ \\leq\\ \\frac{\\overline{X}\\ -\\ P}{\\sqrt{\\frac{P(1-P)}{n}}}\\ \\leq Z_{\\frac{\\alpha}{2}}) = 1-\\alpha$\n",
        "\n",
        "$P(\\overline{X}\\ -\\ Z_{\\frac{\\alpha}{2}} \\sqrt{\\frac{P(1-P)}{n}} \\ \\leq\\ P \\ \\leq \\overline{X}\\ +\\ Z_{\\frac{\\alpha}{2}} \\sqrt{\\frac{P(1-P)}{n}}) = 1-\\alpha$\n",
        "\n",
        "Approximtely:\n",
        "\n",
        "$P(\\overline{X}\\ -\\ Z_{\\frac{\\alpha}{2}} \\sqrt{\\frac{\\overline{X}(1-\\overline{X})}{n}} \\ \\leq\\ P \\ \\leq \\overline{X}\\ +\\ Z_{\\frac{\\alpha}{2}} \\sqrt{\\frac{\\overline{X}(1-\\overline{X})}{n}}) = 1-\\alpha$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Therefore, the $1-\\alpha$ confidence interval for ratio of variances of two normal populatios is:\n",
        "\n",
        "$[\\overline{X}\\ -\\ Z_{\\frac{\\alpha}{2}} \\sqrt{\\frac{P\\overline{X}(1-\\overline{X})}{n}}\\ ,\\  \\overline{X}\\ +\\ Z_{\\frac{\\alpha}{2}} \\sqrt{\\frac{\\overline{X}(1-\\overline{X})}{n}}]$"
      ],
      "metadata": {
        "id": "QLb52JRhlaKx"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**B. One-sided Lower Confidence Interval:**\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",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "We accept $H_0$ if:\n",
        "\n",
        "$P(-\\infty\\ \\leq\\ \\frac{\\overline{X}\\ -\\ P}{\\sqrt{\\frac{P(1-P)}{n}}}\\ \\leq Z_{\\alpha}) = 1-\\alpha$\n",
        "\n",
        "$P(-\\ \\infty \\ \\leq\\ P \\ \\leq \\overline{X}\\ +\\ Z_{\\alpha} \\sqrt{\\frac{P(1-P)}{n}}) = 1-\\alpha$\n",
        "\n",
        "Approxiamtely:\n",
        "\n",
        "$P(-\\ \\infty \\ \\leq\\ P \\ \\leq \\overline{X}\\ +\\ Z_{\\alpha} \\sqrt{\\frac{\\overline{X}(1-\\overline{X})}{n}}) = 1-\\alpha$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Therefore, the $1-\\alpha$ confidence interval for ratio of variances of two normal populatios is:\n",
        "\n",
        "$[-\\infty\\ ,\\ \\overline{X}\\ +\\ Z_{\\alpha} \\sqrt{\\frac{\\overline{X}(1-\\overline{X})}{n}}]$"
      ],
      "metadata": {
        "id": "OeVw7qYlnL5Z"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**C. One-sided Upper Confidence Interval:**\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",
        "$\\\\ $\n",
        "\n",
        "Significance level = $\\alpha$\n",
        "\n",
        "We accept $H_0$ if:\n",
        "\n",
        "$P(-\\ Z_{\\alpha}\\ \\leq\\ \\frac{\\overline{X}\\ -\\ P}{\\sqrt{\\frac{P(1-P)}{n}}}\\ \\leq \\infty) = 1-\\alpha$\n",
        "\n",
        "$P(\\overline{X}\\ -\\ Z_{\\alpha} \\sqrt{\\frac{P(1-P)}{n}} \\ \\leq\\ P \\ \\leq \\infty) = 1-\\alpha$\n",
        "\n",
        "Approxiamtely:\n",
        "\n",
        "$P(\\overline{X}\\ -\\ Z_{\\alpha} \\sqrt{\\frac{\\overline{X}(1-\\overline{X})}{n}} \\ \\leq\\ P \\ \\leq \\infty) = 1-\\alpha$\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "Therefore, the $1-\\alpha$ confidence interval for ratio of variances of two normal populatios is:\n",
        "\n",
        "$[\\overline{X}\\ -\\ Z_{\\alpha} \\sqrt{\\frac{\\overline{X}(1-\\overline{X})}{n}}\\ ,\\ \\infty]$"
      ],
      "metadata": {
        "id": "wSlhJ0A6nwyW"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "class confidence_interval_for_p_bernoulli:\n",
        "  \"\"\"\n",
        "  Parameters\n",
        "  ----------\n",
        "  n : optional, number of sample members\n",
        "  c_level : % confidence level\n",
        "  type_t : 'two_sided_confidence', 'lower_confidence', 'upper_confidence'\n",
        "  Sample_mean : mean of the sample\n",
        "  data : optional, if you do not know the Sample_mean and n, just pass the data\n",
        "  \"\"\"\n",
        "  def __init__(self, c_level, type_c, Sample_mean = 0., n = 0., data=None):\n",
        "    self.Sample_mean = Sample_mean\n",
        "    self.type_c = type_c\n",
        "    self.n = n\n",
        "    self.c_level = c_level\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",
        "    confidence_interval_for_p_bernoulli.__test(self)\n",
        "  \n",
        "  def __test(self):\n",
        "    if self.type_c == 'two_sided_confidence':\n",
        "      c_u = self.Sample_mean + (-norm.ppf((1-self.c_level)/2)) * (np.sqrt(self.Sample_mean*(1-self.Sample_mean)/self.n))\n",
        "      c_l = self.Sample_mean - (-norm.ppf((1-self.c_level)/2)) * (np.sqrt(self.Sample_mean*(1-self.Sample_mean)/self.n))\n",
        "      display(Latex(f'${c_l} \\leq P \\leq {c_u}$'))\n",
        "    elif self.type_c == 'lower_confidence':\n",
        "      c_u = self.Sample_mean + (-norm.ppf(1-self.c_level)) * (np.sqrt(self.Sample_mean*(1-self.Sample_mean)/self.n))\n",
        "      display(Latex(f'$P \\leq {c_u}$'))\n",
        "    elif self.type_c == 'upper_confidence':\n",
        "      c_l = self.Sample_mean - (-norm.ppf(1-self.c_level)) * (np.sqrt(self.Sample_mean*(1-self.Sample_mean)/self.n))\n",
        "      display(Latex(f'${c_l} \\leq P$'))"
      ],
      "metadata": {
        "id": "jUyHpFqalR6l"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "confidence_interval_for_p_bernoulli(c_level = 0.95, type_c = 'two_sided_confidence', Sample_mean = 0.2, n = 100);"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "9mxXutjYqTUi",
        "outputId": "c9ed9318-7efd-4468-8d1a-8410d648f081"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<IPython.core.display.Latex object>"
            ],
            "text/latex": "$0.12160144061839785 \\leq P \\leq 0.2783985593816022$"
          },
          "metadata": {}
        }
      ]
    }
  ]
}