{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "from ipywidgets import *\n",
    "import matplotlib.pyplot as plt\n",
    "from IPython.display import set_matplotlib_formats\n",
    "set_matplotlib_formats('svg')\n",
    "import numpy as np\n",
    "import scipy.stats as stats\n",
    "import matplotlib.patches as mpatches"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def fill(lower, upper, mu, sigma, col='#add8e6', ax=None):    \n",
    "    x = np.linspace(lower, upper, 100)\n",
    "    y = stats.norm.pdf(x,mu,sigma)\n",
    "    if ax:\n",
    "        ax.fill_between(x, y, color=col, alpha=0.5)\n",
    "    else:\n",
    "        plt.fill_between(x, y, color=col, alpha=0.5)\n",
    "    \n",
    "def norm_filled(mu, sigma, filled=False, ax=None):\n",
    "    x_min = mu-3.5*sigma\n",
    "    x_max = mu+3.5*sigma\n",
    "    x = np.linspace(x_min, x_max, 100)\n",
    "    if not ax:\n",
    "        plt.plot(x, stats.norm.pdf(x, mu, sigma))\n",
    "    else:\n",
    "        ax.plot(x, stats.norm.pdf(x, mu, sigma))\n",
    "    if filled:\n",
    "        fill(mu-3.5*sigma, mu+3.5*sigma, mu, sigma, ax=ax)\n",
    "    return x_min, x_max\n",
    "    \n",
    "def iq_sample_mean_dist(x_bar=101, normalized=False, c_crit=False, n=30):\n",
    "    mu = 0 if normalized else 100\n",
    "    sigma = 1 if normalized else 15/np.sqrt(n)\n",
    "    norm_filled(mu, sigma, True)\n",
    "    plt.xlabel(r\"$\\bar{X}_{IQ}$\")\n",
    "    plt.xlim(mu-3.25*sigma,mu+3.25*sigma)\n",
    "    if normalized:\n",
    "        x_bar = (x_bar-100)/(15/np.sqrt(n))\n",
    "    plt.vlines(x=x_bar, ymin=0, ymax=stats.norm.pdf(x_bar, mu, sigma))\n",
    "    plt.gca().set_ylim(bottom=0)\n",
    "    title = r\"$z=$\" if normalized else r\"$\\bar{x}=$\"\n",
    "    title += str(round(x_bar,2))\n",
    "    if c_crit:\n",
    "        z_crit = 1.96\n",
    "        if not normalized:\n",
    "            z_crit = (z_crit*sigma+mu)\n",
    "        fill(z_crit, mu+3.5*sigma, mu, sigma, \"#ff4c4c\")\n",
    "    plt.title(title)\n",
    "\n",
    "def critical_region(test_type=\"right\"):\n",
    "    mu = 0\n",
    "    sigma = 1\n",
    "    x_min, x_max = norm_filled(mu, sigma)\n",
    "    alpha = 0.05\n",
    "    if test_type == \"left\":\n",
    "        z_crit = stats.norm.ppf(alpha, mu, sigma)\n",
    "        fill(x_min, z_crit, mu, sigma, \"#ff4c4c\")\n",
    "        fill(z_crit, x_max, mu, sigma)\n",
    "        plt.title(r\"Test lewostronny: $C_{kr} = (-\\infty, -z_{kr})$\")\n",
    "        plt.text(-2.25, 0.02, r\"$\\alpha\\%$\", fontsize=12)\n",
    "    elif test_type == \"right\":\n",
    "        z_crit = stats.norm.ppf(1-alpha, mu, sigma)\n",
    "        fill(x_min, z_crit, mu, sigma)\n",
    "        fill(z_crit, x_max, mu, sigma, \"#ff4c4c\")\n",
    "        plt.title(r\"Test prawostronny: $C_{kr} = (z_{kr},\\infty)$\")\n",
    "        plt.text(1.75, 0.02, r\"$\\alpha\\%$\", fontsize=12)\n",
    "    else:\n",
    "        z_crit = stats.norm.ppf(alpha/2, mu, sigma)\n",
    "        fill(z_crit, -1*z_crit, mu, sigma)\n",
    "        fill(x_min, z_crit, mu, sigma, \"#ff4c4c\")\n",
    "        fill(-1*z_crit, x_max, mu, sigma, \"#ff4c4c\")\n",
    "        plt.title(r\"Test: dwustronny: $C_{kr} = (-\\infty, -z_{kr}) \\cup  (z_{kr},\\infty)$\")\n",
    "        plt.text(2, 0.005, r\"$\\alpha/2\\%$\", fontsize=10)\n",
    "        plt.text(-2.5, 0.005, r\"$\\alpha/2\\%$\", fontsize=10)\n",
    "\n",
    "def p_value(x_bar=106):\n",
    "    fig, axes = plt.subplots(nrows=1, ncols=2, figsize=(12,5))\n",
    "    mu = 0\n",
    "    sigma = 1\n",
    "    alpha = 0.05\n",
    "    z_crit = stats.norm.ppf(1-alpha, mu, sigma)\n",
    "    for ax in axes:\n",
    "        norm_filled(mu, sigma, True, ax)\n",
    "    fill(z_crit, 3.5, mu, sigma, \"#ff4c4c\", axes[0])\n",
    "    x_bar = (x_bar-100)/(15/np.sqrt(30))\n",
    "    axes[0].vlines(x=x_bar, ymin=0, ymax=stats.norm.pdf(x_bar, mu, sigma))\n",
    "    axes[0].set_title(r\"z=\"+str(round(x_bar,2)))\n",
    "    axes[1].vlines(x=x_bar, ymin=0, ymax=stats.norm.pdf(x_bar, mu, sigma))\n",
    "    fill(x_bar, 3.5, mu, sigma, \"#ffa500\", axes[1])\n",
    "    p_val = 1-stats.norm.cdf(x_bar, mu, sigma)\n",
    "    axes[1].set_title(r\"p-value=\"+str(round(p_val,4)))\n",
    "\n",
    "def alpha_beta(alpha=0.05, mu2=107, n=30):\n",
    "    mu1 = 100\n",
    "    sigma = 15/np.sqrt(n)\n",
    "    xs = [np.linspace(m-3.5*sigma, m+3.5*sigma, 100) for m in [mu1, mu2]]\n",
    "    for x, m in zip(xs,[mu1, mu2]):\n",
    "        plt.plot(x, stats.norm.pdf(x, m, sigma))\n",
    "    z_crit = stats.norm.ppf(1-alpha, mu1, sigma)\n",
    "    fill(z_crit, mu1+3.5*sigma, mu1, sigma, \"#ff4c4c\")\n",
    "    fill(mu2-3.5*sigma, z_crit, mu2, sigma, \"#98fb98\")\n",
    "    plt.gca().set_ylim(bottom=0)\n",
    "    beta = stats.norm.cdf(z_crit, mu2, sigma)\n",
    "    plt.title(r\"$\\beta=$\"+str(round(beta,2)))\n",
    "    a = mpatches.Patch(color='#ff4c4c', label=r'$\\alpha$', alpha=0.5)\n",
    "    b = mpatches.Patch(color='#98fb98', label=r'$\\beta$', alpha=0.5)\n",
    "    plt.legend(handles=[a, b])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Przykład: IQ"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![iq](https://upload.wikimedia.org/wikipedia/commons/thumb/3/39/IQ_distribution.svg/800px-IQ_distribution.svg.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[Źródło](https://upload.wikimedia.org/wikipedia/commons/thumb/3/39/IQ_distribution.svg/800px-IQ_distribution.svg.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$IQ \\sim N(100, 15)$\n",
    "\n",
    "$\\bar{IQ}_{30} \\sim N(100, \\frac{15}{\\sqrt{30}})$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "02b4d5bab7cd49618259cb72bda6d350",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(IntSlider(value=101, description='x_bar', max=110, min=100), Checkbox(value=False, descr…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<function __main__.iq_sample_mean_dist(x_bar=101, normalized=False, c_crit=False, n=30)>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "interact(iq_sample_mean_dist, x_bar=(100, 110, 1), normalized=False, c_crit=False, n=fixed(30))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Schemat postępowania"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "-  Określenie badanej cechy, jej rozkładu i badanego parametru jej rozkładu\n",
    "-  Określenie hipotezy zerowej $H_0$ i hipotezy alternatywnej $H_1$\n",
    "-  Identyfikacja testu statystycznego i statystyki testowej\n",
    "-  Wybór poziomu istotności $\\alpha$\n",
    "-  Sformułowanie reguły decyzyjnej: określenie obszarów krytycznych i zasad odrzucenia hipotezy $H_0$\n",
    "-  Pobranie próby losowej prostej. Obliczenie na podstawie próbki wartości statystyki testowej\n",
    "-  Podjęcie decyzji"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Określenie badanej cechy, jej rozkładu i badanego parametru jej rozkładu"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Przykład: $X \\sim N(\\mu, 15)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Określenie hipotezy zerowej i alternatywnej"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$H_0: \\theta = \\theta_0$\n",
    "<br>$H_1: \\theta > \\theta_0$\n",
    "\n",
    "---\n",
    "\n",
    "$H_0: \\theta = \\theta_0$\n",
    "<br>$H_1: \\theta < \\theta_0$\n",
    "\t\n",
    "---    \n",
    "    \n",
    "$H_0: \\theta = \\theta_0$\n",
    "<br>$H_1: \\theta \\neq \\theta_0$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Przykład:\n",
    "<br>$\\mu_0 = 100$\n",
    "\t\t\n",
    "Układ hipotez:\n",
    "<br>$H_0: \\mu = 100$\n",
    "<br>$H_1: \\mu > 100$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Identyfikacja testu statystycznego i statystyki testowej"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Przykład: zmienna o rozkładzie normalnym ze znaną wariancją:\n",
    "\n",
    "$Z=\\frac{\\bar{X}-\\mu_0}{\\frac{\\sigma}{\\sqrt{n}}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Sformułowanie reguły decyzyjnej: określenie obszarów krytycznych i zasad odrzucenia hipotezy $H_0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "39d27f6851f24f969f8a5351362f34f5",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(Dropdown(description='test_type', index=1, options=('left', 'right', 'two-sided'), value…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<function __main__.critical_region(test_type='right')>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "interact(critical_region, test_type=[\"left\", \"right\", \"two-sided\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$Z \\in C_{kr}: \\textrm{ Odrzucamy } H_0 \\textrm{ na rzecz } H_1$\n",
    "<br>$Z \\notin C_{kr}: \\textrm{ Brak podstaw do odrzucenia } H_0$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Pobranie próby losowej prostej, obliczenie na podstawie próbki wartości statystyki testowej, podjęcie decyzji"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Przykład: test prawostronny, $\\alpha=0.05$\n",
    "\n",
    "$C_{kr} = (1.644854, \\infty)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dla $\\bar{x} = 101$:\n",
    "\n",
    "$Z = \\frac{101-100}{\\frac{15}{\\sqrt{30}}} = 0.3651484$\n",
    "\n",
    "$Z \\notin C_{kr}$\n",
    "\t\n",
    "$ \\textrm{ Brak podstaw do odrzucenia } H_0$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dla $\\bar{x} = 106$:\n",
    "\n",
    "$Z = \\frac{106-100}{\\frac{15}{\\sqrt{30}}} = 2.19089$\n",
    "\n",
    "$Z \\in C_{kr}$\n",
    "\n",
    "$ \\textrm{ Odrzucamy } H_0 \\textrm{ na rzecz } H_1$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### p-wartość"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "0cbbd7ac39e64fbea812ca2857f5a593",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(IntSlider(value=106, description='x_bar', max=110, min=100), Output()), _dom_classes=('w…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<function __main__.p_value(x_bar=106)>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "interact(p_value, x_bar=(100, 110, 1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Przykład:\n",
    "\n",
    "$\\alpha = 0.05$\n",
    "\n",
    "$P(\\bar{X}>106|\\mu=100)$\n",
    "\n",
    "$=P(Z>\\frac{106-100}{{\\frac{15}{\\sqrt{30}}}})$\n",
    "\n",
    "$=P(Z>2.19089) \\approx 0.014$\n",
    "\n",
    "$p < \\alpha$\n",
    "\n",
    "$ \\textrm{ Odrzucamy } H_0 \\textrm{ na rzecz } H_1$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Testowanie hipotez dla jednej zbiorowości (rozkład normalny ze znaną wariancją): podsumowanie"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$H_0: \\mu = \\mu_0$\n",
    "\n",
    "$H_1: \\mu > / \\neq / < \\mu_0$\n",
    "\n",
    "- założenia: rozkład normalny, odchylenie standardowe jest znane \n",
    "- statystyka:\n",
    "$$Z=\\frac{\\bar{X}-\\mu_0}{\\frac{\\sigma}{\\sqrt{n}}}$$\n",
    "- zbiór krytyczny, wartość krytyczna, zbiór przyjęć\n",
    "- odrzucenie $H_0$:\n",
    "     - test prawostronny: $Z \\in (z_{kr},\\infty)$; $p \\leq \\alpha$\n",
    "     - test lewostronny: $Z \\in (\\infty, -z_{kr})$; $p \\leq \\alpha$\n",
    "     - test dwustronny: $Z \\in (-\\infty, -z_{kr}) \\cup  (z_{kr},\\infty)$; $2p \\leq \\alpha$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Decyzje względem $H_0$ i rodzaje błędów"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "| | Odrzucamy $H_0$ | Nie odrzucamy $H_0$ |\n",
    "| --- | --- | --- |\n",
    "| $H_0$ prawdziwa | Błąd I rodzaju $\\alpha$ | Decyzja właściwa |\n",
    "| $H_0$ fałszywa | Decyzja właściwa ($1-\\beta$) | Błąd II rodzaju ($\\beta$) |"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- $\\alpha$ poziom istotności testu\n",
    "- $1-\\beta$ moc testu \n",
    "- $\\alpha \\downarrow \\rightarrow \\beta \\uparrow$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![błędy](https://effectsizefaq.files.wordpress.com/2010/05/type-i-and-type-ii-errors.jpg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[Źródło](https://effectsizefaq.files.wordpress.com/2010/05/type-i-and-type-ii-errors.jpg)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "c2e139ee39bd4326936f7a734ae0c240",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(FloatSlider(value=0.05, description='alpha', max=0.25, min=0.01, step=0.01), IntSlider(v…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<function __main__.alpha_beta(alpha=0.05, mu2=107, n=30)>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "interact(alpha_beta, alpha=(0.01, 0.25, 0.01), mu2=(100, 110, 1), n=(30, 60, 5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Test w rozkładzie dwupunktowym"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Hipoteza zerowa:\n",
    "\n",
    "$H_0: p = p_0$\n",
    "\n",
    "- Hipoteza alternatywna, np.:\n",
    "\n",
    "$H_1: p < p_0 $\n",
    "\n",
    "- Statystyka\n",
    "\n",
    "$ S_n = \\sum_{i=1}^{n}X_i \\sim B_n(p) $"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Statystyka testowa – standaryzacja"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$ S_n = \\sum_{i=1}^{n}X_i \\sim B_n(p) $\n",
    "\n",
    "$E[S_n] = n \\cdot p_0$\n",
    "\n",
    "$D^2[S_n] = n \\cdot p_0 \\cdot(1-p_0)$\n",
    "\n",
    "$D[S_n] = \\sqrt{n \\cdot p_0 \\cdot(1-p_0)}$\n",
    "\n",
    "$Z = \\frac{S_n - E[S_n]}{D[S_n]} = \\frac{S_n - n \\cdot p_0}{\\sqrt{n \\cdot p_0 \\cdot (1 - p_0)}}$\n",
    "\n",
    "- dla $n \\cdot p \\geq 5$ i $n \\cdot (1 - p) \\geq 5$:\n",
    "\n",
    "$ Z \\sim N(0,1) $"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### lub"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Statystyka\n",
    "\n",
    "$ \\bar{X} = \\frac{\\textrm{liczba sukcesów}}{n} $\n",
    "\n",
    "- Standaryzacja\n",
    "\n",
    "$E[\\bar{X}] = p_0$\n",
    "\n",
    "$D[\\bar{X}] = \\sqrt{\\frac{{p_0(1-p_0)}}{n}}$\n",
    "\n",
    "$Z = \\frac{\\bar{X}-E[\\bar{X}]}{D[\\bar{X}]}=\\frac{\\bar{X}-p_0}{\\sqrt{\\frac{{p_0(1-p_0)}}{n}}}$"
   ]
  }
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   "labels_anchors": false,
   "latex_user_defs": false,
   "report_style_numbering": false,
   "user_envs_cfg": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}