Salinee_Kingbaisomboon-L04-AutoHypothesis.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lesson 4 Assignment - Automobile Price Hypothesis Testing\n",
    "## Author - Salinee Kingbaisomboon\n",
    "### UW NetID: 1950831"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Instructions\n",
    "Test hypotheses for the price of automobiles:\n",
    "\n",
    "1. Compare and test the normality of the distributions of **price** and **log price**. <br/>\n",
    "Use both a graphical method and a formal test. Hint: remember these relationships will always be approximate.\n",
    "\n",
    "2. Test significance of **price (or log price)** stratified by \n",
    "a) **fuel type**, b) **aspiration**, and c) **rear vs. front wheel drive**. <br />Use both graphical methods and the formal test.\n",
    "\n",
    "3. Apply ANOVA and Tukey's HSD test to the auto price data to compare the **price (or log price** if closer to a normal distribution) of autos stratified by **body style**.\n",
    "\n",
    "4. Graphically explore the differences between the price conditioned by the categories of each variable. <br/>\n",
    "Hint: Make sure you have enough data for each category and drop categories with insufficient data.\n",
    "\n",
    "5. Use standard ANOVA and Tukey HSD to test the differences of these groups.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Note: \n",
    "Please clearly outline the results of these tests with markdown text. Be sure your discussion includes narrative for your figures and tables so it is clear to the reader the importance of each bit of evidence."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Import libraries\n",
    "import pandas as pd\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy import stats\n",
    "from scipy.stats import kstest\n",
    "from sklearn.preprocessing import scale\n",
    "import statsmodels.stats.weightstats as ws\n",
    "from statsmodels.stats.multicomp import pairwise_tukeyhsd\n",
    "import statsmodels.stats.power as smsp\n",
    "from statistics import mean\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Cleaned up Dataset location\n",
    "fileName = \"https://library.startlearninglabs.uw.edu/DATASCI410/Datasets/Automobile%20price%20data%20_Raw_.csv\"\n",
    "\n",
    "# Read the .csv file with the pandas read_csv method\n",
    "auto_price = pd.read_csv(fileName)\n",
    "\n",
    "# Remove rows with missing values for price, accounting for mising values coded as '?'\n",
    "auto_price.loc[auto_price['price'] == '?', 'price'] = np.nan\n",
    "auto_price.dropna(axis = 0, inplace = True)\n",
    "\n",
    "# Convert price's column to numeric values\n",
    "auto_price['price'] = pd.to_numeric(auto_price['price'])\n",
    "\n",
    "# Create a new column for log price\n",
    "auto_price['log_price'] = auto_price.loc[:, 'price'].apply(np.log)\n",
    "\n",
    "# Create a new column used for counting the occurance\n",
    "auto_price['counts'] = 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
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       "\n",
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       "\n",
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       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>symboling</th>\n",
       "      <th>normalized-losses</th>\n",
       "      <th>make</th>\n",
       "      <th>fuel-type</th>\n",
       "      <th>aspiration</th>\n",
       "      <th>num-of-doors</th>\n",
       "      <th>body-style</th>\n",
       "      <th>drive-wheels</th>\n",
       "      <th>engine-location</th>\n",
       "      <th>wheel-base</th>\n",
       "      <th>...</th>\n",
       "      <th>bore</th>\n",
       "      <th>stroke</th>\n",
       "      <th>compression-ratio</th>\n",
       "      <th>horsepower</th>\n",
       "      <th>peak-rpm</th>\n",
       "      <th>city-mpg</th>\n",
       "      <th>highway-mpg</th>\n",
       "      <th>price</th>\n",
       "      <th>log_price</th>\n",
       "      <th>counts</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>3</td>\n",
       "      <td>?</td>\n",
       "      <td>alfa-romero</td>\n",
       "      <td>gas</td>\n",
       "      <td>std</td>\n",
       "      <td>two</td>\n",
       "      <td>convertible</td>\n",
       "      <td>rwd</td>\n",
       "      <td>front</td>\n",
       "      <td>88.6</td>\n",
       "      <td>...</td>\n",
       "      <td>3.47</td>\n",
       "      <td>2.68</td>\n",
       "      <td>9.0</td>\n",
       "      <td>111</td>\n",
       "      <td>5000</td>\n",
       "      <td>21</td>\n",
       "      <td>27</td>\n",
       "      <td>13495</td>\n",
       "      <td>9.510075</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>3</td>\n",
       "      <td>?</td>\n",
       "      <td>alfa-romero</td>\n",
       "      <td>gas</td>\n",
       "      <td>std</td>\n",
       "      <td>two</td>\n",
       "      <td>convertible</td>\n",
       "      <td>rwd</td>\n",
       "      <td>front</td>\n",
       "      <td>88.6</td>\n",
       "      <td>...</td>\n",
       "      <td>3.47</td>\n",
       "      <td>2.68</td>\n",
       "      <td>9.0</td>\n",
       "      <td>111</td>\n",
       "      <td>5000</td>\n",
       "      <td>21</td>\n",
       "      <td>27</td>\n",
       "      <td>16500</td>\n",
       "      <td>9.711116</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1</td>\n",
       "      <td>?</td>\n",
       "      <td>alfa-romero</td>\n",
       "      <td>gas</td>\n",
       "      <td>std</td>\n",
       "      <td>two</td>\n",
       "      <td>hatchback</td>\n",
       "      <td>rwd</td>\n",
       "      <td>front</td>\n",
       "      <td>94.5</td>\n",
       "      <td>...</td>\n",
       "      <td>2.68</td>\n",
       "      <td>3.47</td>\n",
       "      <td>9.0</td>\n",
       "      <td>154</td>\n",
       "      <td>5000</td>\n",
       "      <td>19</td>\n",
       "      <td>26</td>\n",
       "      <td>16500</td>\n",
       "      <td>9.711116</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>2</td>\n",
       "      <td>164</td>\n",
       "      <td>audi</td>\n",
       "      <td>gas</td>\n",
       "      <td>std</td>\n",
       "      <td>four</td>\n",
       "      <td>sedan</td>\n",
       "      <td>fwd</td>\n",
       "      <td>front</td>\n",
       "      <td>99.8</td>\n",
       "      <td>...</td>\n",
       "      <td>3.19</td>\n",
       "      <td>3.40</td>\n",
       "      <td>10.0</td>\n",
       "      <td>102</td>\n",
       "      <td>5500</td>\n",
       "      <td>24</td>\n",
       "      <td>30</td>\n",
       "      <td>13950</td>\n",
       "      <td>9.543235</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>2</td>\n",
       "      <td>164</td>\n",
       "      <td>audi</td>\n",
       "      <td>gas</td>\n",
       "      <td>std</td>\n",
       "      <td>four</td>\n",
       "      <td>sedan</td>\n",
       "      <td>4wd</td>\n",
       "      <td>front</td>\n",
       "      <td>99.4</td>\n",
       "      <td>...</td>\n",
       "      <td>3.19</td>\n",
       "      <td>3.40</td>\n",
       "      <td>8.0</td>\n",
       "      <td>115</td>\n",
       "      <td>5500</td>\n",
       "      <td>18</td>\n",
       "      <td>22</td>\n",
       "      <td>17450</td>\n",
       "      <td>9.767095</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>5 rows × 28 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "   symboling normalized-losses         make fuel-type aspiration num-of-doors  \\\n",
       "0          3                 ?  alfa-romero       gas        std          two   \n",
       "1          3                 ?  alfa-romero       gas        std          two   \n",
       "2          1                 ?  alfa-romero       gas        std          two   \n",
       "3          2               164         audi       gas        std         four   \n",
       "4          2               164         audi       gas        std         four   \n",
       "\n",
       "    body-style drive-wheels engine-location  wheel-base  ...  bore  stroke  \\\n",
       "0  convertible          rwd           front        88.6  ...  3.47    2.68   \n",
       "1  convertible          rwd           front        88.6  ...  3.47    2.68   \n",
       "2    hatchback          rwd           front        94.5  ...  2.68    3.47   \n",
       "3        sedan          fwd           front        99.8  ...  3.19    3.40   \n",
       "4        sedan          4wd           front        99.4  ...  3.19    3.40   \n",
       "\n",
       "   compression-ratio  horsepower peak-rpm city-mpg  highway-mpg  price  \\\n",
       "0                9.0         111     5000       21           27  13495   \n",
       "1                9.0         111     5000       21           27  16500   \n",
       "2                9.0         154     5000       19           26  16500   \n",
       "3               10.0         102     5500       24           30  13950   \n",
       "4                8.0         115     5500       18           22  17450   \n",
       "\n",
       "  log_price counts  \n",
       "0  9.510075      1  \n",
       "1  9.711116      1  \n",
       "2  9.711116      1  \n",
       "3  9.543235      1  \n",
       "4  9.767095      1  \n",
       "\n",
       "[5 rows x 28 columns]"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# View the first 5 rows of this dataframe\n",
    "auto_price.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>symboling</th>\n",
       "      <th>wheel-base</th>\n",
       "      <th>length</th>\n",
       "      <th>width</th>\n",
       "      <th>height</th>\n",
       "      <th>curb-weight</th>\n",
       "      <th>engine-size</th>\n",
       "      <th>compression-ratio</th>\n",
       "      <th>city-mpg</th>\n",
       "      <th>highway-mpg</th>\n",
       "      <th>price</th>\n",
       "      <th>log_price</th>\n",
       "      <th>counts</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>count</th>\n",
       "      <td>201.000000</td>\n",
       "      <td>201.000000</td>\n",
       "      <td>201.000000</td>\n",
       "      <td>201.000000</td>\n",
       "      <td>201.000000</td>\n",
       "      <td>201.000000</td>\n",
       "      <td>201.000000</td>\n",
       "      <td>201.000000</td>\n",
       "      <td>201.000000</td>\n",
       "      <td>201.000000</td>\n",
       "      <td>201.000000</td>\n",
       "      <td>201.000000</td>\n",
       "      <td>201.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean</th>\n",
       "      <td>0.840796</td>\n",
       "      <td>98.797015</td>\n",
       "      <td>174.200995</td>\n",
       "      <td>65.889055</td>\n",
       "      <td>53.766667</td>\n",
       "      <td>2555.666667</td>\n",
       "      <td>126.875622</td>\n",
       "      <td>10.164279</td>\n",
       "      <td>25.179104</td>\n",
       "      <td>30.686567</td>\n",
       "      <td>13207.129353</td>\n",
       "      <td>9.350115</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>std</th>\n",
       "      <td>1.254802</td>\n",
       "      <td>6.066366</td>\n",
       "      <td>12.322175</td>\n",
       "      <td>2.101471</td>\n",
       "      <td>2.447822</td>\n",
       "      <td>517.296727</td>\n",
       "      <td>41.546834</td>\n",
       "      <td>4.004965</td>\n",
       "      <td>6.423220</td>\n",
       "      <td>6.815150</td>\n",
       "      <td>7947.066342</td>\n",
       "      <td>0.502251</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>min</th>\n",
       "      <td>-2.000000</td>\n",
       "      <td>86.600000</td>\n",
       "      <td>141.100000</td>\n",
       "      <td>60.300000</td>\n",
       "      <td>47.800000</td>\n",
       "      <td>1488.000000</td>\n",
       "      <td>61.000000</td>\n",
       "      <td>7.000000</td>\n",
       "      <td>13.000000</td>\n",
       "      <td>16.000000</td>\n",
       "      <td>5118.000000</td>\n",
       "      <td>8.540519</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25%</th>\n",
       "      <td>0.000000</td>\n",
       "      <td>94.500000</td>\n",
       "      <td>166.800000</td>\n",
       "      <td>64.100000</td>\n",
       "      <td>52.000000</td>\n",
       "      <td>2169.000000</td>\n",
       "      <td>98.000000</td>\n",
       "      <td>8.600000</td>\n",
       "      <td>19.000000</td>\n",
       "      <td>25.000000</td>\n",
       "      <td>7775.000000</td>\n",
       "      <td>8.958669</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50%</th>\n",
       "      <td>1.000000</td>\n",
       "      <td>97.000000</td>\n",
       "      <td>173.200000</td>\n",
       "      <td>65.500000</td>\n",
       "      <td>54.100000</td>\n",
       "      <td>2414.000000</td>\n",
       "      <td>120.000000</td>\n",
       "      <td>9.000000</td>\n",
       "      <td>24.000000</td>\n",
       "      <td>30.000000</td>\n",
       "      <td>10295.000000</td>\n",
       "      <td>9.239414</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>75%</th>\n",
       "      <td>2.000000</td>\n",
       "      <td>102.400000</td>\n",
       "      <td>183.500000</td>\n",
       "      <td>66.600000</td>\n",
       "      <td>55.500000</td>\n",
       "      <td>2926.000000</td>\n",
       "      <td>141.000000</td>\n",
       "      <td>9.400000</td>\n",
       "      <td>30.000000</td>\n",
       "      <td>34.000000</td>\n",
       "      <td>16500.000000</td>\n",
       "      <td>9.711116</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>max</th>\n",
       "      <td>3.000000</td>\n",
       "      <td>120.900000</td>\n",
       "      <td>208.100000</td>\n",
       "      <td>72.000000</td>\n",
       "      <td>59.800000</td>\n",
       "      <td>4066.000000</td>\n",
       "      <td>326.000000</td>\n",
       "      <td>23.000000</td>\n",
       "      <td>49.000000</td>\n",
       "      <td>54.000000</td>\n",
       "      <td>45400.000000</td>\n",
       "      <td>10.723267</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "        symboling  wheel-base      length       width      height  \\\n",
       "count  201.000000  201.000000  201.000000  201.000000  201.000000   \n",
       "mean     0.840796   98.797015  174.200995   65.889055   53.766667   \n",
       "std      1.254802    6.066366   12.322175    2.101471    2.447822   \n",
       "min     -2.000000   86.600000  141.100000   60.300000   47.800000   \n",
       "25%      0.000000   94.500000  166.800000   64.100000   52.000000   \n",
       "50%      1.000000   97.000000  173.200000   65.500000   54.100000   \n",
       "75%      2.000000  102.400000  183.500000   66.600000   55.500000   \n",
       "max      3.000000  120.900000  208.100000   72.000000   59.800000   \n",
       "\n",
       "       curb-weight  engine-size  compression-ratio    city-mpg  highway-mpg  \\\n",
       "count   201.000000   201.000000         201.000000  201.000000   201.000000   \n",
       "mean   2555.666667   126.875622          10.164279   25.179104    30.686567   \n",
       "std     517.296727    41.546834           4.004965    6.423220     6.815150   \n",
       "min    1488.000000    61.000000           7.000000   13.000000    16.000000   \n",
       "25%    2169.000000    98.000000           8.600000   19.000000    25.000000   \n",
       "50%    2414.000000   120.000000           9.000000   24.000000    30.000000   \n",
       "75%    2926.000000   141.000000           9.400000   30.000000    34.000000   \n",
       "max    4066.000000   326.000000          23.000000   49.000000    54.000000   \n",
       "\n",
       "              price   log_price  counts  \n",
       "count    201.000000  201.000000   201.0  \n",
       "mean   13207.129353    9.350115     1.0  \n",
       "std     7947.066342    0.502251     0.0  \n",
       "min     5118.000000    8.540519     1.0  \n",
       "25%     7775.000000    8.958669     1.0  \n",
       "50%    10295.000000    9.239414     1.0  \n",
       "75%    16500.000000    9.711116     1.0  \n",
       "max    45400.000000   10.723267     1.0  "
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# View the summary statistics for this dataframe\n",
    "auto_price.describe()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Declare Functions used in this assignment"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Function to do Stratified Sampling\n",
    "def stratify(dat, p):\n",
    "    groups = dat.groupby('group') # Create the groups\n",
    "    nums = min(groups.size()) # Find the size of the smallest group\n",
    "    num = int(p * dat.shape[0]) # Compute the desired number of samples per group\n",
    "    if num <= nums: # If sufficient group size, sample each group and return. \n",
    "        return groups.apply(lambda x: x.sample(n=num)) # used to get a random sample of items from an axis of object\n",
    "    else: # Oops. p is to large\n",
    "        pmax = nums / dat.shape[0]\n",
    "        print('The maximum value of p = ' + str(pmax))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Function to get some statistics (counts and mean) of the input data\n",
    "def count_mean(dat):\n",
    "    groups = dat.groupby('group') # Create the groups\n",
    "    ## Create a data frame with the counts and the means of the groups\n",
    "    return pd.DataFrame({'count': groups.size(), # size of each group\n",
    "                        'mean': groups.aggregate(np.mean).loc[:, 'var']}) #mean on var column"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Function to do the Kolmogorov-Smirnov Test for Distributions\n",
    "def ks_test(dat, dist = 'norm'):\n",
    "    from scipy.stats import kstest \n",
    "    ks_statisic, pvalue = kstest(dat, dist)\n",
    "    print('KS-statistic = ' + str(ks_statisic))\n",
    "    print('P-value = ' + str(pvalue))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Function to plot the histogram with mean value (as a red vertcal line)\n",
    "def hist_plot(vec, bins):\n",
    "    plt.hist(vec, bins = bins)\n",
    "    plt.axvline(mean(vec), color = 'red') # Add a vertical line across the axes\n",
    "    plt.ylabel('Frequency')\n",
    "    plt.xlabel('Value')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Function to plot the histogram with confident inverval as the red vertical lines\n",
    "def hist_mean_ci(vec, t_test, bins):\n",
    "    upper = mean(vec) + t_test[5] - t_test[1]\n",
    "    lower = mean(vec) + t_test[4] - t_test[1]\n",
    "    plt.hist(vec, bins = bins)\n",
    "    plt.axvline(mean(vec), color = 'red')\n",
    "    plt.axvline(upper, color = 'red', linestyle='--')\n",
    "    plt.axvline(lower, color = 'red', linestyle='--')\n",
    "    plt.ylabel('Frequency')\n",
    "    plt.xlabel('Value')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Function use to calculate the Kolmogorov-Smirnov Statistic\n",
    "# KS Statistic = Max(CDF(Test)-CDF(Sample))\n",
    "def plot_cums(dist1, dist2):\n",
    "    ## sort the first data sample:\n",
    "    data_sorted = np.sort(dist1)\n",
    "    # calculate the proportional values of samples\n",
    "    p = 1. * np.arange(len(dist1)) / (len(dist1) - 1) # similar to quantile but it's just a proportion\n",
    "\n",
    "    # Now plot as line and scatter plot. \n",
    "    ax = plt.figure(figsize=(6, 6)).gca() # define axis\n",
    "    ax.set_ylabel('Standard deviation estimate')\n",
    "    ax.set_xlabel('Number of samples')\n",
    "    ax.set_title('Standard deviation estimate vs. number of samples')\n",
    "    \n",
    "    plt.plot(data_sorted, p)\n",
    "    plt.scatter(data_sorted, p, color = 'red')\n",
    "    \n",
    "    # sort the second data sample:\n",
    "    data_sorted = np.sort(dist2)\n",
    "    \n",
    "    # And the second plot\n",
    "    plt.plot(data_sorted, p, color = 'yellow')\n",
    "    plt.scatter(data_sorted, p, color = 'green')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Function to plot the histogram of the t-test\n",
    "def plot_t(a, b, t_test, cols, nbins = 20):\n",
    "    minx = min([min(a), min(b)])\n",
    "    maxx = max([max(a), max(b)])\n",
    "    stepx = (maxx - minx)/(nbins + 1)\n",
    "    bins = [minx + i * stepx for i in range(nbins + 1)]\n",
    "    plt.subplot(2, 1, 1) # The first plot\n",
    "    hist_mean_ci(a, t_test, bins)\n",
    "    plt.title('Histograme of ' + cols[0] + ' and ' + cols[1])\n",
    "    plt.subplot(2, 1, 2) # The next plot\n",
    "    hist_plot(b, bins)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Function to calculate the t-test statistics\n",
    "def t_test(a, b, alpha, alternative='two-sided'):\n",
    "    diff = a.mean() - b.mean() # the mean different of two samples\n",
    "\n",
    "    res = stats.ttest_ind(a, b, equal_var=False) # return The calculated t-statistic and the p-value\n",
    "      \n",
    "    means = ws.CompareMeans(ws.DescrStatsW(a), ws.DescrStatsW(b))\n",
    "    confint = means.tconfint_diff(alpha=alpha, alternative=alternative, usevar='unequal') # computed the confident interval\n",
    "    degfree = means.dof_satt() # use Welch’s T-test to get the degree of freedom\n",
    "\n",
    "    index = ['DegFreedom', 'Difference', 'Statistic', 'PValue', 'Low95CI', 'High95CI'] # upper & local confident interval\n",
    "    return pd.Series([degfree, diff, res[0], res[1], confint[0], confint[1]], index = index) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 1. Compare and test Normality of the distribution of price and log price"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "price_scale = scale(auto_price['price'].tolist()) # scale the price\n",
    "log_price_scale = scale(auto_price['log_price'].tolist()) # scale the log price\n",
    "\n",
    "plot_cums(price_scale, log_price_scale)  # plot the cumulative distribution function between those two lists"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[1m\u001b[4mKS Test of price\u001b[0m\n",
      "KS-statistic = 0.1542761480184771\n",
      "P-value = 0.00012108611006800808\n",
      "----------------------------------\n",
      "\u001b[1m\u001b[4mKS Test of log price\u001b[0m\n",
      "KS-statistic = 0.09781904713791056\n",
      "P-value = 0.03984598613097214\n"
     ]
    }
   ],
   "source": [
    "# Compute the K-S statistic and p-value for both price and log price\n",
    "print('\\033[1m\\033[4mKS Test of price\\033[0m')\n",
    "ks_test(price_scale)\n",
    "print('----------------------------------')\n",
    "print('\\033[1m\\033[4mKS Test of log price\\033[0m')\n",
    "ks_test(log_price_scale)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***\n",
    "### Summary\n",
    "\n",
    "- **From K-S statistics:** We found that **p-values are small for both price and log price.** We **can't reject the null hypothesis** since the chance is low for the difference to arise from random variation.\n",
    "- **From the plot:** The two cumulative density function are quite similar. This indicates the two samples are drawn from the same distribution.\n",
    "- These two samples have the same distribution\n",
    "***"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2. Test significance of price stratified by:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## a) Fuel Type"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
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       "    }\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></th>\n",
       "      <th>var</th>\n",
       "      <th>group</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>group</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th rowspan=\"10\" valign=\"top\">diesel</th>\n",
       "      <th>159</th>\n",
       "      <td>7788</td>\n",
       "      <td>diesel</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>192</th>\n",
       "      <td>13845</td>\n",
       "      <td>diesel</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>112</th>\n",
       "      <td>16900</td>\n",
       "      <td>diesel</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>116</th>\n",
       "      <td>17950</td>\n",
       "      <td>diesel</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>187</th>\n",
       "      <td>9495</td>\n",
       "      <td>diesel</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>69</th>\n",
       "      <td>28176</td>\n",
       "      <td>diesel</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>158</th>\n",
       "      <td>7898</td>\n",
       "      <td>diesel</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>66</th>\n",
       "      <td>18344</td>\n",
       "      <td>diesel</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>182</th>\n",
       "      <td>7775</td>\n",
       "      <td>diesel</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>114</th>\n",
       "      <td>17075</td>\n",
       "      <td>diesel</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th rowspan=\"10\" valign=\"top\">gas</th>\n",
       "      <th>15</th>\n",
       "      <td>30760</td>\n",
       "      <td>gas</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>161</th>\n",
       "      <td>8358</td>\n",
       "      <td>gas</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>133</th>\n",
       "      <td>12170</td>\n",
       "      <td>gas</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>145</th>\n",
       "      <td>11259</td>\n",
       "      <td>gas</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>190</th>\n",
       "      <td>9980</td>\n",
       "      <td>gas</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>77</th>\n",
       "      <td>6189</td>\n",
       "      <td>gas</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>151</th>\n",
       "      <td>6338</td>\n",
       "      <td>gas</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>39</th>\n",
       "      <td>8845</td>\n",
       "      <td>gas</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>47</th>\n",
       "      <td>32250</td>\n",
       "      <td>gas</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>125</th>\n",
       "      <td>22018</td>\n",
       "      <td>gas</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "              var   group\n",
       "group                    \n",
       "diesel 159   7788  diesel\n",
       "       192  13845  diesel\n",
       "       112  16900  diesel\n",
       "       116  17950  diesel\n",
       "       187   9495  diesel\n",
       "       69   28176  diesel\n",
       "       158   7898  diesel\n",
       "       66   18344  diesel\n",
       "       182   7775  diesel\n",
       "       114  17075  diesel\n",
       "gas    15   30760     gas\n",
       "       161   8358     gas\n",
       "       133  12170     gas\n",
       "       145  11259     gas\n",
       "       190   9980     gas\n",
       "       77    6189     gas\n",
       "       151   6338     gas\n",
       "       39    8845     gas\n",
       "       47   32250     gas\n",
       "       125  22018     gas"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Startify price by fuel type\n",
    "data_by_fuel_type = pd.DataFrame({\"var\": auto_price['price'],\n",
    "                     \"group\":auto_price['fuel-type']}) \n",
    "\n",
    "stratified_by_fuel_type = stratify(data_by_fuel_type, 0.05)\n",
    "stratified_by_fuel_type"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "\n",
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       "    }\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>count</th>\n",
       "      <th>mean</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>group</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>diesel</th>\n",
       "      <td>10</td>\n",
       "      <td>14524.6</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>gas</th>\n",
       "      <td>10</td>\n",
       "      <td>14816.7</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "        count     mean\n",
       "group                 \n",
       "diesel     10  14524.6\n",
       "gas        10  14816.7"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Count the fuel type and mean for each strata\n",
    "stratified_by_fuel_type.reset_index(inplace=True, drop=True)\n",
    "count_mean(stratified_by_fuel_type)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [],
   "source": [
    "diesel = stratified_by_fuel_type.loc[stratified_by_fuel_type['group'] == 'diesel'] # get the group 1 (diesel)\n",
    "gas = stratified_by_fuel_type.loc[stratified_by_fuel_type['group'] == 'gas'] # get the group 2 (gas)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "DegFreedom      15.631662\n",
       "Difference    -292.100000\n",
       "Statistic       -0.078000\n",
       "PValue           0.938818\n",
       "Low95CI      -8246.096389\n",
       "High95CI      7661.896389\n",
       "dtype: float64"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Compute the t-statistics for these two groups\n",
    "test_fuel = t_test(diesel.loc[:, 'var'], gas.loc[:, 'var'], 0.05)\n",
    "test_fuel"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_t(diesel.loc[:, 'var'], gas.loc[:, 'var'], test_fuel, ['gas', 'diesel']) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***\n",
    "### Examine these results:\n",
    "1. The t-statistic is small, the **p-value is large**, and the **confidence interval of the difference of the means overlap zero**. We have to **accept the null hypothesis** that the means are the same between gas and diesel. There is a high chance that the difference in means is from random variation alone.\n",
    "2. The **mean of the second group (gas) falls inside the confidence interval**, confirming that **there is a high chance that the difference in means is from random variation alone**.\n",
    "3. We can see from both histograms that the **distribution of fuel type are skewed to the left (asymmetric)**.\n",
    "***"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## b) Aspiration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
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       "        vertical-align: middle;\n",
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       "\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></th>\n",
       "      <th>var</th>\n",
       "      <th>group</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>group</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th rowspan=\"10\" valign=\"top\">std</th>\n",
       "      <th>66</th>\n",
       "      <td>18344</td>\n",
       "      <td>std</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>81</th>\n",
       "      <td>8499</td>\n",
       "      <td>std</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>53</th>\n",
       "      <td>6695</td>\n",
       "      <td>std</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>92</th>\n",
       "      <td>6849</td>\n",
       "      <td>std</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>144</th>\n",
       "      <td>9233</td>\n",
       "      <td>std</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>28</th>\n",
       "      <td>8921</td>\n",
       "      <td>std</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>139</th>\n",
       "      <td>7053</td>\n",
       "      <td>std</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>6377</td>\n",
       "      <td>std</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>33</th>\n",
       "      <td>6529</td>\n",
       "      <td>std</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>143</th>\n",
       "      <td>9960</td>\n",
       "      <td>std</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th rowspan=\"10\" valign=\"top\">turbo</th>\n",
       "      <th>149</th>\n",
       "      <td>11694</td>\n",
       "      <td>turbo</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>114</th>\n",
       "      <td>17075</td>\n",
       "      <td>turbo</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>23875</td>\n",
       "      <td>turbo</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>116</th>\n",
       "      <td>17950</td>\n",
       "      <td>turbo</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>67</th>\n",
       "      <td>25552</td>\n",
       "      <td>turbo</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>82</th>\n",
       "      <td>12629</td>\n",
       "      <td>turbo</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>136</th>\n",
       "      <td>18150</td>\n",
       "      <td>turbo</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>119</th>\n",
       "      <td>7957</td>\n",
       "      <td>turbo</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>112</th>\n",
       "      <td>16900</td>\n",
       "      <td>turbo</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>203</th>\n",
       "      <td>22470</td>\n",
       "      <td>turbo</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "             var  group\n",
       "group                  \n",
       "std   66   18344    std\n",
       "      81    8499    std\n",
       "      53    6695    std\n",
       "      92    6849    std\n",
       "      144   9233    std\n",
       "      28    8921    std\n",
       "      139   7053    std\n",
       "      22    6377    std\n",
       "      33    6529    std\n",
       "      143   9960    std\n",
       "turbo 149  11694  turbo\n",
       "      114  17075  turbo\n",
       "      8    23875  turbo\n",
       "      116  17950  turbo\n",
       "      67   25552  turbo\n",
       "      82   12629  turbo\n",
       "      136  18150  turbo\n",
       "      119   7957  turbo\n",
       "      112  16900  turbo\n",
       "      203  22470  turbo"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Startify price by aspiration\n",
    "data_by_aspiration = pd.DataFrame({\"var\": auto_price['price'],\n",
    "                     \"group\":auto_price['aspiration']}) \n",
    "\n",
    "stratified_by_aspiration = stratify(data_by_aspiration, 0.05)\n",
    "stratified_by_aspiration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>count</th>\n",
       "      <th>mean</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>group</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>std</th>\n",
       "      <td>10</td>\n",
       "      <td>8846.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>turbo</th>\n",
       "      <td>10</td>\n",
       "      <td>17425.2</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "       count     mean\n",
       "group                \n",
       "std       10   8846.0\n",
       "turbo     10  17425.2"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Count the aspiration and mean for each strata\n",
    "stratified_by_aspiration.reset_index(inplace=True, drop=True)\n",
    "count_mean(stratified_by_aspiration)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [],
   "source": [
    "std = stratified_by_aspiration.loc[stratified_by_aspiration['group'] == 'std'] # get the group 1 (standard)\n",
    "turbo = stratified_by_aspiration.loc[stratified_by_aspiration['group'] == 'turbo'] # get the group 2 (turbo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "DegFreedom       15.330040\n",
       "Difference    -8579.200000\n",
       "Statistic        -4.096112\n",
       "PValue            0.000915\n",
       "Low95CI      -13035.109339\n",
       "High95CI      -4123.290661\n",
       "dtype: float64"
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Compute the t-statistics for these two groups\n",
    "test_aspiration = t_test(std.loc[:, 'var'], turbo.loc[:, 'var'], 0.05)\n",
    "test_aspiration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_t(std.loc[:, 'var'], turbo.loc[:, 'var'], test_aspiration, ['standard', 'turbo']) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***\n",
    "### Examine these results:\n",
    "1. The t-statistic is small, the **p-value is large**, and the **confidence interval of the difference of the means overlap zero**. We have to **accept the null hypothesis** that the means are the same between standard and turbo. There is a high chance that the difference in means is from random variation alone.\n",
    "2. The **mean of the second group (gas) falls inside the confidence interval**, confirming that **there is a high chance that the difference in means is from random variation alone**.\n",
    "3. We can see from the histograms that the **distribution of the aspiration are sysmetric**.\n",
    "4. **Note!** There is an outlier in the **standard** aspiration (from Lesson 1), so we might have a high chance that sometime we stratify those outliers into the sample which make the p-value confident interval redults different from 1 & 2 and we have to reject null hypothesis. We need to stratify many times to see these effects.\n",
    "***"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## c) Rear vs. Front Wheel Drive"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "outputs": [],
   "source": [
    "auto_price = auto_price[auto_price['drive-wheels'] != '4wd'] # drop the '4wd' drive wheels's rows from the data frame"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array(['rwd', 'fwd'], dtype=object)"
      ]
     },
     "execution_count": 96,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "auto_price['drive-wheels'].unique() # check to make sure there are two type of drive wheels left in the data frame"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "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></th>\n",
       "      <th>var</th>\n",
       "      <th>group</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>group</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th rowspan=\"9\" valign=\"top\">fwd</th>\n",
       "      <th>156</th>\n",
       "      <td>6938</td>\n",
       "      <td>fwd</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>34</th>\n",
       "      <td>7129</td>\n",
       "      <td>fwd</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>193</th>\n",
       "      <td>12290</td>\n",
       "      <td>fwd</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>161</th>\n",
       "      <td>8358</td>\n",
       "      <td>fwd</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>79</th>\n",
       "      <td>7689</td>\n",
       "      <td>fwd</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>187</th>\n",
       "      <td>9495</td>\n",
       "      <td>fwd</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>94</th>\n",
       "      <td>7299</td>\n",
       "      <td>fwd</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>85</th>\n",
       "      <td>6989</td>\n",
       "      <td>fwd</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>97</th>\n",
       "      <td>7999</td>\n",
       "      <td>fwd</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th rowspan=\"9\" valign=\"top\">rwd</th>\n",
       "      <th>115</th>\n",
       "      <td>16630</td>\n",
       "      <td>rwd</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>202</th>\n",
       "      <td>21485</td>\n",
       "      <td>rwd</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>171</th>\n",
       "      <td>11549</td>\n",
       "      <td>rwd</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>16430</td>\n",
       "      <td>rwd</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>201</th>\n",
       "      <td>19045</td>\n",
       "      <td>rwd</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>66</th>\n",
       "      <td>18344</td>\n",
       "      <td>rwd</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>196</th>\n",
       "      <td>15985</td>\n",
       "      <td>rwd</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>20970</td>\n",
       "      <td>rwd</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>70</th>\n",
       "      <td>31600</td>\n",
       "      <td>rwd</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "             var group\n",
       "group                 \n",
       "fwd   156   6938   fwd\n",
       "      34    7129   fwd\n",
       "      193  12290   fwd\n",
       "      161   8358   fwd\n",
       "      79    7689   fwd\n",
       "      187   9495   fwd\n",
       "      94    7299   fwd\n",
       "      85    6989   fwd\n",
       "      97    7999   fwd\n",
       "rwd   115  16630   rwd\n",
       "      202  21485   rwd\n",
       "      171  11549   rwd\n",
       "      10   16430   rwd\n",
       "      201  19045   rwd\n",
       "      66   18344   rwd\n",
       "      196  15985   rwd\n",
       "      12   20970   rwd\n",
       "      70   31600   rwd"
      ]
     },
     "execution_count": 97,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Startify price by fuel type\n",
    "data_by_drive_wheels = pd.DataFrame({\"var\": auto_price['price'],\n",
    "                     \"group\":auto_price['drive-wheels']}) \n",
    "\n",
    "stratified_by_drive_wheels = stratify(data_by_drive_wheels, 0.05)\n",
    "stratified_by_drive_wheels"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>count</th>\n",
       "      <th>mean</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>group</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>fwd</th>\n",
       "      <td>9</td>\n",
       "      <td>8242.888889</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>rwd</th>\n",
       "      <td>9</td>\n",
       "      <td>19115.333333</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "       count          mean\n",
       "group                     \n",
       "fwd        9   8242.888889\n",
       "rwd        9  19115.333333"
      ]
     },
     "execution_count": 98,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Count the drive wheels and mean for each strata\n",
    "stratified_by_drive_wheels.reset_index(inplace=True, drop=True)\n",
    "count_mean(stratified_by_drive_wheels)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "outputs": [],
   "source": [
    "rwd = stratified_by_drive_wheels.loc[stratified_by_drive_wheels['group'] == 'rwd'] # get the group 1 (rear wheels)\n",
    "fwd = stratified_by_drive_wheels.loc[stratified_by_drive_wheels['group'] == 'fwd'] # get the group 2 (front wheels)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "DegFreedom        9.529634\n",
       "Difference    10872.444444\n",
       "Statistic         5.617719\n",
       "PValue            0.000265\n",
       "Low95CI        6531.113685\n",
       "High95CI      15213.775204\n",
       "dtype: float64"
      ]
     },
     "execution_count": 100,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Compute the t-statistics for these two groups\n",
    "test_drive_wheels = t_test(rwd.loc[:, 'var'], fwd.loc[:, 'var'], 0.05)\n",
    "test_drive_wheels"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_t(rwd.loc[:, 'var'], fwd.loc[:, 'var'], test_drive_wheels, ['rear wheels', 'front wheels']) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Examine these results:\n",
    "1. The t-statistic is large, the **p-value is small, and the confidence interval of the difference of the means does not overlap zero**. We can **reject the null hypothesis that the means are the same between rear and front wheels**. There is a low chance that the difference in means is from random variation alone.\n",
    "2. The **mean of the second sample falls outside the confidence interval**, confirming that there is a low chance that the difference in means is from random variation alone.\n",
    "3. We can see from the histogram that rear wheels has symmetric dustribution while front wheels has asymmetric distribution (skewed to the left)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3. Use the Kolmogorov-Smirnov Test for Distributions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "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": [
    "hist_plot(auto_price['price'], 20)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "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": [
    "hist_plot(auto_price['log_price'], 20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***\n",
    "### Based on both histogram, I decide to test the auto price data compare to the log price since it's closer to a Normal Distribution.\n",
    "***"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "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>counts</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>body-style</th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>convertible</th>\n",
       "      <td>6</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>hardtop</th>\n",
       "      <td>8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>hatchback</th>\n",
       "      <td>67</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>sedan</th>\n",
       "      <td>91</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>wagon</th>\n",
       "      <td>21</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "             counts\n",
       "body-style         \n",
       "convertible       6\n",
       "hardtop           8\n",
       "hatchback        67\n",
       "sedan            91\n",
       "wagon            21"
      ]
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Check the counts for each body types, we will drop categories with insufficient data\n",
    "auto_price[['counts', 'body-style']].groupby(['body-style']).agg('count')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Bsed on the above counts, I decide to drop convertible and hardtop since both have insufficient data\n",
    "auto_price = auto_price[auto_price['body-style'] != 'convertible']\n",
    "auto_price = auto_price[auto_price['body-style'] != 'hardtop']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Create 3 groups of body style\n",
    "df1 = auto_price[auto_price['body-style'] == 'hatchback']\n",
    "df2 = auto_price[auto_price['body-style'] == 'sedan']\n",
    "df3 = auto_price[auto_price['body-style'] == 'wagon']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "F statistic = 8.976626457244592\n",
      "P-value = 0.00019398707681679667\n"
     ]
    }
   ],
   "source": [
    "# Compute the statistics for  these three groups\n",
    "f_statistic, p_value = stats.f_oneway(df1['log_price'], df2['log_price'], df3['log_price']) \n",
    "print('F statistic = ' + str(f_statistic))\n",
    "print('P-value = ' + str(p_value))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The **F statistic is fairly large and the p-value is small**. We can **reject the null hypothesis** that the 3 variables have the same mean, as the probability of the differences arising from random chance is quite low."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
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       "\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>vals</th>\n",
       "      <th>group_num</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>9.711116</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>8.546946</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>8.747511</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>8.625509</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>8.760453</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>174</th>\n",
       "      <td>9.664596</td>\n",
       "      <td>3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>175</th>\n",
       "      <td>9.416541</td>\n",
       "      <td>3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>176</th>\n",
       "      <td>9.504129</td>\n",
       "      <td>3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>177</th>\n",
       "      <td>9.712024</td>\n",
       "      <td>3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>178</th>\n",
       "      <td>9.849559</td>\n",
       "      <td>3</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>179 rows × 2 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "         vals group_num\n",
       "0    9.711116         1\n",
       "1    8.546946         1\n",
       "2    8.747511         1\n",
       "3    8.625509         1\n",
       "4    8.760453         1\n",
       "..        ...       ...\n",
       "174  9.664596         3\n",
       "175  9.416541         3\n",
       "176  9.504129         3\n",
       "177  9.712024         3\n",
       "178  9.849559         3\n",
       "\n",
       "[179 rows x 2 columns]"
      ]
     },
     "execution_count": 108,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Concatenate those three body styles with the group number into a data frame\n",
    "df = pd.DataFrame({'vals': np.concatenate(( df1['log_price'],df2['log_price'], df3['log_price'])).flatten(),\n",
    "                   'group_num': ['1'] * 67 + ['2'] * 91 + ['3'] * 21})\n",
    "df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Multiple Comparison of Means - Tukey HSD, FWER=0.05\n",
      "===================================================\n",
      "group1 group2 meandiff p-adj   lower  upper  reject\n",
      "---------------------------------------------------\n",
      "     1      2    0.309  0.001  0.1332 0.4848   True\n",
      "     1      3   0.2673 0.0566 -0.0058 0.5405  False\n",
      "     2      3  -0.0417    0.9 -0.3061 0.2227  False\n",
      "---------------------------------------------------\n"
     ]
    }
   ],
   "source": [
    "# Compute Turkey's ANOVA\n",
    "Tukey_HSD = pairwise_tukeyhsd(df.vals, df.group_num)\n",
    "print(Tukey_HSD)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***\n",
    "### Summary\n",
    "\n",
    "- **From Tukey HSD statistics:** We found that **group 1 (hatchback) and group 2 (sedan)** have a significant difference at the 95% confidence level (the reject column on the right is True).\n",
    "- The difference in means between the **group 1 (hatchback) and group 2 (sedan)** is significant since the confidence interval (lower and upper) are not include zero.\n",
    "***"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 4. Graphically explore the differences between the price conditioned by the categories of body style"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Box plot of variables')"
      ]
     },
     "execution_count": 110,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "plt.boxplot([df1['log_price'], df2['log_price'], df3['log_price']])\n",
    "plt.ylabel('Value')\n",
    "plt.xlabel('Variable: 1 = hatchback, 2 = sedan, 3 = wagon')\n",
    "plt.title('Box plot of variables')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 5. Use standard ANOVA and Tukey HSD to test the differences of these groups"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x432 with 1 Axes>"
      ]
     },
     "execution_count": 112,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "Tukey_HSD.plot_simultaneous()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***\n",
    "### Summary\n",
    "\n",
    "- **From the test result plots:** We found that the different in mean between **group 1 (hatchback) and group 2 (sedan)** is significant at the confidence level since the confidence intervals didn't not overlap at 95% confident level. We can **reject the null hypothesis** for group 1 (hatchback) and group 2 (sedan) and said both groups are not the same.\n",
    "***"
   ]
  },
  {
   "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": "3.7.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}