Statistics_Lab2_(Probability Distributions & CIs)_18_Feb_2022.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "from scipy import stats\n",
    "import seaborn as sns\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Binomial Distribution\n",
    "*Source: Wackerly, D., Mendenhall, W., & Scheaffer, R. L. (2014). Mathematical statistics with applications. Cengage Learning.*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 3.40**\n",
    "\n",
    "The probability that a patient recovers from a stomach disease is 0.8. \n",
    "\n",
    "Suppose 20 people are known to have contracted this disease. \n",
    "\n",
    "What is the probability that"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "a. exactly 14 recover?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "this is P(X=14)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 20  #the number of trials\n",
    "p = 0.8 #the probability of success\n",
    "x = 14  #R.V."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.10909970097304983"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.binom.pmf(x, n, p)  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "16.0"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.binom.mean(n, p)  #the expected value"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b. at least 10 recover?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "this is P(X≥10)=P(10)+P(11)+P(12)+...+P(20)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = 9"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9994365863023398"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1 - stats.binom.cdf(k=9, n=20, p=0.8)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c. at least 14 but not more than 18 recover?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "this is P(14≤X≤18)=P(14)+P(15)+P(16)+P(17)+P(18)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "x1 = 14\n",
    "x2 = 18"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.844132196156189"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.binom.cdf(18, 20, 0.8) - stats.binom.cdf(13, 20, 0.8)   #cdf method"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.8441321961561871"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#pmf method\n",
    "stats.binom.pmf(14,20,0.8)+stats.binom.pmf(15,20,0.8)+stats.binom.pmf(16,20,0.8)+stats.binom.pmf(17,20,0.8)+stats.binom.pmf(18,20,0.8)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "d. at most 16 recover?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "this is P(X≤16)=1−P(X>16)=P(17)+P(18)+P(19)+P(20)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = 16"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5885511380434305"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.binom.cdf(x,n,p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5885511380434305"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.binom.cdf(16,20,0.8)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "e. Construct discrete probability distribution table and draw a graph"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0\t1.048576000000001e-14\n",
      "1\t8.388607999999975e-13\n",
      "2\t3.187671040000001e-11\n",
      "3\t7.650410495999974e-10\n",
      "4\t1.3005697843199945e-08\n",
      "5\t1.6647293239295892e-07\n",
      "6\t1.664729323929588e-06\n",
      "7\t1.3317834591436803e-05\n",
      "8\t8.656592484433902e-05\n",
      "9\t0.000461684932503141\n",
      "10\t0.002031413703013826\n",
      "11\t0.007386958920050259\n",
      "12\t0.022160876760150803\n",
      "13\t0.0545498504865252\n",
      "14\t0.10909970097304983\n",
      "15\t0.17455952155687976\n",
      "16\t0.21819940194610007\n",
      "17\t0.2053641430080944\n",
      "18\t0.13690942867206304\n",
      "19\t0.05764607523034236\n",
      "20\t0.011529215046068481\n"
     ]
    }
   ],
   "source": [
    "dist = []\n",
    "\n",
    "for i in range(n+1):\n",
    "    dist.append(stats.binom.pmf(i,n,p))\n",
    "    print(str(i) + \"\\t\" + str(stats.binom.pmf(i,n,p)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "list(range(n+1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "plt.bar(list(range(n+1)), dist)\n",
    "plt.show;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Poisson Distribution\n",
    "*Source: Wackerly, D., Mendenhall, W., & Scheaffer, R. L. (2014). Mathematical statistics with applications. Cengage Learning.*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Suppose that a random system of police patrol is devised so that a patrol officer may visit a given beat location Y = 0, 1, 2, 3, . . . times per half-hour period, with each location being visited an average of once per time period. \n",
    "\n",
    "Assume that Y possesses, approximately, a Poisson probability distribution. \n",
    "\n",
    "Calculate the probability that the patrol officer will miss a given location during a half-hour period. \n",
    "\n",
    "What is the probability that it will be visited once? \n",
    "\n",
    "Twice? \n",
    "\n",
    "At least once?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "#For this example the time period is a half-hour, and the mean number of visits per half-hour interval is λ = 1. Then\n",
    "\n",
    "mu = 1  #rate"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "P(Y = 0) = ?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.36787944117144233"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.poisson.pmf(k=0, mu=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "P(Y = 1) = ?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.36787944117144233"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.poisson.pmf(k=1, mu=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "P(Y = 2) = ?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.18393972058572114"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.poisson.pmf(k=2, mu=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "P(Y ≥ 1) = ?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.6321205588285576"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1 - stats.poisson.cdf(k=0, mu=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.6321205588285577"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1 - stats.poisson.pmf(k=0, mu=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "P(Y ≥ 3) = ?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.08030139707139416"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1 - stats.poisson.cdf(k=2, mu=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Normal Distribution\n",
    "*Source: Wackerly, D., Mendenhall, W., & Scheaffer, R. L. (2014). Mathematical statistics with applications. Cengage Learning.*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 4.74**\n",
    "\n",
    "Scores on an examination are assumed to be normally distributed with mean 78 and variance 36."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "a. What is the probability that a person taking the examination scores higher than 72?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "this is P(X>72)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "mu = 78\n",
    "sigma = np.sqrt(36)   #std is 6\n",
    "x = 72"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.8413447460685429"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1 - stats.norm.cdf(x=72, loc=78, scale=6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b. Suppose that students scoring in the top 10% of this distribution are to receive an A grade.\n",
    "What is the minimum score a student must achieve to earn an A grade?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "q = 1 - 0.10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "85.6893093932676"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.norm.ppf(q=0.9, loc=78, scale=6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c. What must be the cutoff point for passing the examination if the examiner wants only the\n",
    "top 28.1% of all scores to be passing?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "q = 1 - 0.281"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "81.47924035456623"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.norm.ppf(q=1 - 0.281, loc=78, scale=6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "d. Approximately what proportion of students have scores 5 or more points above the score\n",
    "that cuts off the lowest 25%?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "73.95306149882352"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = stats.norm.ppf(0.25, loc=78, scale=6)\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.43689605516226615"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1 - stats.norm.cdf(x+5, loc=78, scale=6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*OPTIONAL*\n",
    "\n",
    "f. If it is known that a student’s score exceeds 72, what is the probability that his or her score\n",
    "exceeds 84? "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.18857341734506025"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#P(X=84 | X=72) = P(X=84) / P(X=72)\n",
    "\n",
    "(1 - stats.norm.cdf(x=84, loc=78, scale=6)) / (1 - stats.norm.cdf(x=72, loc=78, scale=6))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## t Distribution\n",
    "*Lane, D. (2006). Online statistics education: a multimedia course of study (http://onlinestatbook.com/). Rice University.*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For a t distribution with 15 degrees of freedom, 90% of the distribution is within how many standard deviations of the mean?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "dof = 15\n",
    "q = 0.95"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.7530503556925547"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.t.ppf(q=0.95, df=15)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Confidence Intervals Using the Normal Distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we’re working with larger samples (n≥30), we can assume that the sampling distribution of the sample mean is normally distributed (thanks to the Central Limit Theorem) and can instead use the norm.interval() function from the scipy.stats library."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['anagrams',\n",
       " 'anscombe',\n",
       " 'attention',\n",
       " 'brain_networks',\n",
       " 'car_crashes',\n",
       " 'diamonds',\n",
       " 'dots',\n",
       " 'exercise',\n",
       " 'flights',\n",
       " 'fmri',\n",
       " 'gammas',\n",
       " 'geyser',\n",
       " 'iris',\n",
       " 'mpg',\n",
       " 'penguins',\n",
       " 'planets',\n",
       " 'taxis',\n",
       " 'tips',\n",
       " 'titanic']"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sns.get_dataset_names()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. Load \"tips\" dataset from Seaborn library"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>total_bill</th>\n",
       "      <th>tip</th>\n",
       "      <th>sex</th>\n",
       "      <th>smoker</th>\n",
       "      <th>day</th>\n",
       "      <th>time</th>\n",
       "      <th>size</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>16.99</td>\n",
       "      <td>1.01</td>\n",
       "      <td>Female</td>\n",
       "      <td>No</td>\n",
       "      <td>Sun</td>\n",
       "      <td>Dinner</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>10.34</td>\n",
       "      <td>1.66</td>\n",
       "      <td>Male</td>\n",
       "      <td>No</td>\n",
       "      <td>Sun</td>\n",
       "      <td>Dinner</td>\n",
       "      <td>3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>21.01</td>\n",
       "      <td>3.50</td>\n",
       "      <td>Male</td>\n",
       "      <td>No</td>\n",
       "      <td>Sun</td>\n",
       "      <td>Dinner</td>\n",
       "      <td>3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>23.68</td>\n",
       "      <td>3.31</td>\n",
       "      <td>Male</td>\n",
       "      <td>No</td>\n",
       "      <td>Sun</td>\n",
       "      <td>Dinner</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>24.59</td>\n",
       "      <td>3.61</td>\n",
       "      <td>Female</td>\n",
       "      <td>No</td>\n",
       "      <td>Sun</td>\n",
       "      <td>Dinner</td>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   total_bill   tip     sex smoker  day    time  size\n",
       "0       16.99  1.01  Female     No  Sun  Dinner     2\n",
       "1       10.34  1.66    Male     No  Sun  Dinner     3\n",
       "2       21.01  3.50    Male     No  Sun  Dinner     3\n",
       "3       23.68  3.31    Male     No  Sun  Dinner     2\n",
       "4       24.59  3.61  Female     No  Sun  Dinner     4"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tips = sns.load_dataset(\"tips\")\n",
    "tips.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. Draw a set of vertical bar plots (based on \"total_bill\") grouped by \"day\" variable:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "sns.barplot(x=\"day\", y=\"total_bill\", data=tips, ci=95);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3. Subset a dataframe just including Sunday tips"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "tipsSun = tips[tips['day'] == \"Sun\"]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "4. Calculate mean and std"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "21.410000000000004"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tipsSun[\"total_bill\"].mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8.832121828869889"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tipsSun[\"total_bill\"].std()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "5. Calculate standart error of the mean"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0131138555021968"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sem = tipsSun[\"total_bill\"].std() / np.sqrt(tipsSun.shape[0])\n",
    "sem"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0131138555021968"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tipsSun[\"total_bill\"].sem()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "6. Calculate the margin of error (95%)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.9857031567843058"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "moe = 1.96 * sem\n",
    "moe"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.9856666690228222"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#more sensitive calculation\n",
    "moe = stats.norm.ppf(0.975) * sem\n",
    "moe"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "7. Calculate the upper confidence limit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "23.395666669022827"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tipsSun[\"total_bill\"].mean() + moe"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "8. Calculate the lower confidence limit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "19.42433333097718"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tipsSun[\"total_bill\"].mean() - moe"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "9. Calculate 95% Confidence Interval around the mean by using scipy.stats"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(19.42433333097718, 23.395666669022827)"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.norm.interval(0.95, loc = tipsSun[\"total_bill\"].mean(), scale = tipsSun[\"total_bill\"].sem())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 8.7 (Large Samples)\n",
    "*Source: Wackerly, D., Mendenhall, W., & Scheaffer, R. L. (2014). Mathematical statistics with applications. Cengage Learning.*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- The shopping times of n = 64 randomly selected customers at a local supermarket were recorded. \n",
    "\n",
    "- The average and variance of the 64 shopping times were 33 $minutes$ and 256 $minutes^2$, respectively. \n",
    "\n",
    "- Estimate μ, the true average shopping time per customer, with a confidence coefficient of 1 − α = .90."
   ]
  },
  {
   "attachments": {
    "image.png": {
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    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![image.png](attachment:image.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 64\n",
    "xbar = 33\n",
    "var = 256\n",
    "ci = 0.90\n",
    "s = np.sqrt(var)\n",
    "\n",
    "sem = s/np.sqrt(64)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(29.710292746097053, 36.28970725390295)"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Using Normal Distribution\n",
    "stats.norm.interval(0.90, loc = xbar, scale = sem)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(29.661195556984076, 36.338804443015924)"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Using t Distribution\n",
    "stats.t.interval(0.90, df=n-1, loc = xbar, scale = sem)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 8.11 (Small Samples)\n",
    "*Source: Wackerly, D., Mendenhall, W., & Scheaffer, R. L. (2014). Mathematical statistics with applications. Cengage Learning.*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A manufacturer of gunpowder has developed a new powder, which was tested in eight shells. The resulting muzzle velocities, in feet per second, were as follows:\n",
    "\n",
    "3005 2925 2935 2965\n",
    "\n",
    "2995 3005 2937 2905\n",
    "\n",
    "Find a 95% confidence interval for the true average velocity μ for shells of this type. Assume that muzzle velocities are approximately normally distributed."
   ]
  },
  {
   "attachments": {
    "image.png": {
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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![image.png](attachment:image.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = [3005, 2925, 2935, 2965, 2995, 3005, 2937, 2905]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Sample size\n",
    "n = len(x)\n",
    "\n",
    "# Sample mean\n",
    "xbar = np.mean(x)\n",
    "\n",
    "# Sample standard deviation\n",
    "s = np.std(x, ddof=1)\n",
    "\n",
    "# Confidence level\n",
    "cl = 0.95"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2926.3202426713874, 2991.6797573286126)"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.t.interval(cl, df=n-1, loc=xbar, scale=s/np.sqrt(n))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.3646242510102993"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#confidence multiplier\n",
    "t = stats.t.ppf(0.975, df=n-1)\n",
    "t"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13.820274961085254"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#standard error of the mean\n",
    "sem = s/np.sqrt(n)\n",
    "sem"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "32.67975732861261"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#margin of error\n",
    "moe = t*sem\n",
    "moe"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2926.3202426713874"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#lower confidence limit\n",
    "low_limit = xbar - moe\n",
    "low_limit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2991.6797573286126"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#higher confidence limit\n",
    "high_limit = xbar + moe\n",
    "high_limit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.11"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}