Intro_Hypothesis_Testing.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Hypothesis Testing - simple examples\n",
    "### Dr. Tirthajyoti Sarkar, Fremont, 94536\n",
    "\n",
    "Hypothesis testing is a critical tool in inferential statistics, for determing what the value of a population parameter could be. We often draw this conclusion based on a sample data analysis.\n",
    "\n",
    "Check this document for a [quick and comprehensive link](https://github.com/tirthajyoti/Stats-Maths-with-Python/blob/master/Resources/Introduction%20to%20Hypothesis%20Testing.pdf).\n",
    "\n",
    "The basis of hypothesis testing has two attributes:\n",
    "\n",
    "**Null Hypothesis: $H_0$**\n",
    "<br>**Alternative Hypothesis: $H_a$**\n",
    "\n",
    "The tests we will discuss in this notebook are:\n",
    "\n",
    "- One Population Proportion\n",
    "- Difference in Population Proportions\n",
    "- One Population Mean\n",
    "- Difference in Population Means\n",
    "\n",
    "In this notebook, we will also introduce some functions (from the `statsmodels` Python package) that are extremely useful when calculating a t-statistic, or a z-statistic, and corresponding p-values for a hypothesis test.\n",
    "\n",
    "Let's quickly review the following ways to calculate a test statistic for the tests listed above.\n",
    "\n",
    "The equation is:\n",
    "\n",
    "$$\\frac{Best\\ Estimate - Hypothesized\\ Estimate}{Standard\\ Error\\ of\\ Estimate}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import statsmodels.api as sm\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import pandas as pd"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## One Population Proportion\n",
    "\n",
    "### Research Question \n",
    "\n",
    "In previous years, 52% of parents believed that electronics and social media was the cause of their teenager’s lack of sleep. Do more parents today believe that their teenager’s lack of sleep is caused due to electronics and social media? \n",
    "\n",
    "**Population**: Parents with a teenager (age 13-18)  \n",
    "**Parameter of Interest**: p  \n",
    "\n",
    "**Null Hypothesis:** p = 0.52  \n",
    "**Alternative Hypthosis:** p > 0.52 (note that this is a one-sided test)\n",
    "\n",
    "**Data**: 1018 people were surveyed. 56% of those who were surveyed believe that their teenager’s lack of sleep is caused due to electronics and social media."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Use of `proportions_ztest()` from `statsmodels`\n",
    "\n",
    "Note the argument `alternative=\"larger\"` indicating a one-sided test. The function returns two values - the z-statistic and the corresponding p-value."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 1018\n",
    "pnull = .52\n",
    "phat = .56"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2.571067795759113, 0.005069273865860533)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sm.stats.proportions_ztest(phat * n, n, pnull, alternative='larger')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Conclusion of the hypothesis test\n",
    "Since the calculated p-value of the z-test is pretty small, we can reject the Null hypothesis that the percentage of parents, who believe that their teenager’s lack of sleep is caused due to electronics and social media, is as same as previous years' estimate i.e. 52%.\n",
    "\n",
    "Although, we do not accept the alternate hypothesis, this informally means that there is a good chance of this proportion being more than 52%."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Difference in Population Proportions\n",
    "\n",
    "### Research Question\n",
    "\n",
    "Is there a significant difference between the population proportions of parents of black children and parents of Hispanic children who report that their child has had some swimming lessons?\n",
    "\n",
    "**Populations**: All parents of black children age 6-18 and all parents of Hispanic children age 6-18  \n",
    "**Parameter of Interest**: p1 - p2, where p1 = black and p2 = hispanic  \n",
    "\n",
    "**Null Hypothesis:** p1 - p2 = 0  \n",
    "**Alternative Hypthosis:** p1 - p2 $\\neq$ = 0  \n",
    "\n",
    "**Data**: 247 Parents of Black Children. 36.8% of parents report that their child has had some swimming lessons. \n",
    "<br>308 Parents of Hispanic Children. 38.9% of parents report that their child has had some swimming lessons."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Use of `ttest_ind()` from `statsmodels`\n",
    "Difference in population proportion needs t-test. Also, the population follow a binomial distribution here. We can just pass on the two population quantities with the appropriate binomial distribution parameters to the t-test function.\n",
    "\n",
    "The function returns three values: (a) test statisic, (b) p-value of the t-test, and (c) degrees of freedom used in the t-test."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "n1 = 247\n",
    "p1 = .37\n",
    "\n",
    "n2 = 308\n",
    "p2 = .39\n",
    "\n",
    "population1 = np.random.binomial(1, p1, n1)\n",
    "population2 = np.random.binomial(1, p2, n2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.9309838177540464, 0.3522681761633615, 553.0)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sm.stats.ttest_ind(population1, population2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Conclusion of the hypothesis test\n",
    "Since the p-value is quite high ~0.768, we cannot reject the Null hypothesis in this case i.e. the difference in the population proportions are not statistically significant.\n",
    "\n",
    "### But what happens if we could survey much higher number of people?\n",
    "We do not chnage the proportions, just the number of survey participants in the two population. The slight difference in the proportion could become statistically significant in this situation. There is no guarantee that when you run the code, you will get a p-value < 0.05 all the time as the samples are randomly generated each itme. But if you run it a few times, you will notice some p-values < 0.05 for sure."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "n1 = 5000\n",
    "p1 = .37\n",
    "\n",
    "n2 = 5000\n",
    "p2 = .39\n",
    "\n",
    "population1 = np.random.binomial(1, p1, n1)\n",
    "population2 = np.random.binomial(1, p2, n2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-3.2892837551514322, 0.0010079001031011582, 9998.0)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sm.stats.ttest_ind(population1, population2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## One Population Mean\n",
    "\n",
    "### Research Question \n",
    "\n",
    "Let's say a cartwheeling competition was organized for some adults. The data looks like following,\n",
    "\n",
    "(80.57, 98.96, 85.28, 83.83, 69.94, 89.59, 91.09, 66.25, 91.21, 82.7 , 73.54, 81.99, 54.01, 82.89, 75.88, 98.32, 107.2 , 85.53, 79.08, 84.3 , 89.32, 86.35, 78.98, 92.26, 87.01)\n",
    "\n",
    "Is distance Is the average cartwheel distance (in inches) for adults more than 80 inches?\n",
    "\n",
    "**Population**: All adults  \n",
    "**Parameter of Interest**: $\\mu$, population mean cartwheel distance.\n",
    "\n",
    "**Null Hypothesis:** $\\mu$ = 80 \n",
    "<br>**Alternative Hypthosis**: $\\mu$ > 80\n",
    "\n",
    "**Data**:\n",
    "<br>25 adult participants. \n",
    "<br>$\\mu = 83.84$\n",
    "<br>$\\sigma = 10.72$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "cwdata = np.array([80.57, 98.96, 85.28, 83.83, 69.94, 89.59, 91.09, 66.25, 91.21, 82.7 , 73.54, 81.99, 54.01, \n",
    "                 82.89, 75.88, 98.32, 107.2 , 85.53, 79.08, 84.3 , 89.32, 86.35, 78.98, 92.26, 87.01])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(25, 83.84320000000001, 10.716018932420752)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = len(cwdata)\n",
    "mean = cwdata.mean()\n",
    "sd = cwdata.std()\n",
    "(n, mean, sd)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1.756973189172546, 0.039461189601168366)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sm.stats.ztest(cwdata, value = 80, alternative = \"larger\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Conclusion of the hypothesis test\n",
    "Since the p-value  (0.0394) is lower than the standard confidence level 0.05, we can reject the Null hypothesis that the mean cartwheel distance for adults (a population quantity) is equal to 80 inches. There is strong evidence in support for the alternatine hypothesis that the mean cartwheel distance is, in fact, higher than 80 inches. Note, we used `alternative=\"larger\"` in the z-test.\n",
    "\n",
    "We can also plot the histogram of the data to check if it approximately follows a Normal distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(cwdata,bins=5,edgecolor='k')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Difference in Population Means\n",
    "\n",
    "### Research Question \n",
    "\n",
    "Considering adults in the [NHANES data](https://www.cdc.gov/nchs/nhanes/index.htm), do males have a significantly higher mean [Body Mass Index](https://www.cdc.gov/healthyweight/assessing/bmi/index.html) than females?\n",
    "\n",
    "**Population**: Adults in the NHANES data.  \n",
    "**Parameter of Interest**: $\\mu_1 - \\mu_2$, Body Mass Index.  \n",
    "\n",
    "**Null Hypothesis:** $\\mu_1 = \\mu_2$  \n",
    "**Alternative Hypthosis:** $\\mu_1 \\neq \\mu_2$\n",
    "\n",
    "**Data:**\n",
    "\n",
    "2976 Females \n",
    "$\\mu_1 = 29.94$  \n",
    "$\\sigma_1 = 7.75$  \n",
    "\n",
    "2759 Male Adults  \n",
    "$\\mu_2 = 28.78$  \n",
    "$\\sigma_2 = 6.25$  \n",
    "\n",
    "$\\mu_1 - \\mu_2 = 1.16$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
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       "      <th>BMXBMI</th>\n",
       "      <th>BMXLEG</th>\n",
       "      <th>BMXARML</th>\n",
       "      <th>BMXARMC</th>\n",
       "      <th>BMXWAIST</th>\n",
       "      <th>HIQ210</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>83732</td>\n",
       "      <td>1.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>62</td>\n",
       "      <td>3</td>\n",
       "      <td>1.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>...</td>\n",
       "      <td>124.0</td>\n",
       "      <td>64.0</td>\n",
       "      <td>94.8</td>\n",
       "      <td>184.5</td>\n",
       "      <td>27.8</td>\n",
       "      <td>43.3</td>\n",
       "      <td>43.6</td>\n",
       "      <td>35.9</td>\n",
       "      <td>101.1</td>\n",
       "      <td>2.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>83733</td>\n",
       "      <td>1.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>6.0</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>53</td>\n",
       "      <td>3</td>\n",
       "      <td>2.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>...</td>\n",
       "      <td>140.0</td>\n",
       "      <td>88.0</td>\n",
       "      <td>90.4</td>\n",
       "      <td>171.4</td>\n",
       "      <td>30.8</td>\n",
       "      <td>38.0</td>\n",
       "      <td>40.0</td>\n",
       "      <td>33.2</td>\n",
       "      <td>107.9</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>83734</td>\n",
       "      <td>1.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>78</td>\n",
       "      <td>3</td>\n",
       "      <td>1.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>...</td>\n",
       "      <td>132.0</td>\n",
       "      <td>44.0</td>\n",
       "      <td>83.4</td>\n",
       "      <td>170.1</td>\n",
       "      <td>28.8</td>\n",
       "      <td>35.6</td>\n",
       "      <td>37.0</td>\n",
       "      <td>31.0</td>\n",
       "      <td>116.5</td>\n",
       "      <td>2.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>83735</td>\n",
       "      <td>2.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "      <td>56</td>\n",
       "      <td>3</td>\n",
       "      <td>1.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>...</td>\n",
       "      <td>134.0</td>\n",
       "      <td>68.0</td>\n",
       "      <td>109.8</td>\n",
       "      <td>160.9</td>\n",
       "      <td>42.4</td>\n",
       "      <td>38.5</td>\n",
       "      <td>37.7</td>\n",
       "      <td>38.3</td>\n",
       "      <td>110.1</td>\n",
       "      <td>2.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>83736</td>\n",
       "      <td>2.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "      <td>42</td>\n",
       "      <td>4</td>\n",
       "      <td>1.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>...</td>\n",
       "      <td>114.0</td>\n",
       "      <td>54.0</td>\n",
       "      <td>55.2</td>\n",
       "      <td>164.9</td>\n",
       "      <td>20.3</td>\n",
       "      <td>37.4</td>\n",
       "      <td>36.0</td>\n",
       "      <td>27.2</td>\n",
       "      <td>80.4</td>\n",
       "      <td>2.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>5 rows × 28 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "    SEQN  ALQ101  ALQ110  ALQ130  SMQ020  RIAGENDR  RIDAGEYR  RIDRETH1  \\\n",
       "0  83732     1.0     NaN     1.0       1         1        62         3   \n",
       "1  83733     1.0     NaN     6.0       1         1        53         3   \n",
       "2  83734     1.0     NaN     NaN       1         1        78         3   \n",
       "3  83735     2.0     1.0     1.0       2         2        56         3   \n",
       "4  83736     2.0     1.0     1.0       2         2        42         4   \n",
       "\n",
       "   DMDCITZN  DMDEDUC2  ...  BPXSY2  BPXDI2  BMXWT  BMXHT  BMXBMI  BMXLEG  \\\n",
       "0       1.0       5.0  ...   124.0    64.0   94.8  184.5    27.8    43.3   \n",
       "1       2.0       3.0  ...   140.0    88.0   90.4  171.4    30.8    38.0   \n",
       "2       1.0       3.0  ...   132.0    44.0   83.4  170.1    28.8    35.6   \n",
       "3       1.0       5.0  ...   134.0    68.0  109.8  160.9    42.4    38.5   \n",
       "4       1.0       4.0  ...   114.0    54.0   55.2  164.9    20.3    37.4   \n",
       "\n",
       "   BMXARML  BMXARMC  BMXWAIST  HIQ210  \n",
       "0     43.6     35.9     101.1     2.0  \n",
       "1     40.0     33.2     107.9     NaN  \n",
       "2     37.0     31.0     116.5     2.0  \n",
       "3     37.7     38.3     110.1     2.0  \n",
       "4     36.0     27.2      80.4     2.0  \n",
       "\n",
       "[5 rows x 28 columns]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "url = \"https://raw.githubusercontent.com/kshedden/statswpy/master/NHANES/merged/nhanes_2015_2016.csv\"\n",
    "da = pd.read_csv(url)\n",
    "da.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "females = da[da[\"RIAGENDR\"] == 2]\n",
    "male = da[da[\"RIAGENDR\"] == 1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2976, 29.93994565217392, 7.753318809545674)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n1 = len(females)\n",
    "mu1 = females[\"BMXBMI\"].mean()\n",
    "sd1 = females[\"BMXBMI\"].std()\n",
    "\n",
    "(n1, mu1, sd1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2759, 28.778072111846942, 6.2525676168014614)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n2 = len(male)\n",
    "mu2 = male[\"BMXBMI\"].mean()\n",
    "sd2 = male[\"BMXBMI\"].std()\n",
    "\n",
    "(n2, mu2, sd2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(6.1755933531383205, 6.591544431126401e-10)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sm.stats.ztest(females[\"BMXBMI\"].dropna(), male[\"BMXBMI\"].dropna(),alternative='two-sided')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Conclusion of the hypothesis test\n",
    "Since the p-value  (6.59e-10) is extremely small, we can reject the Null hypothesis that the mean BMI of males is same as that of females. Note, we used `alternative=\"two-sided\"` in the z-test because here we are checking for inequality.\n",
    "\n",
    "We can also plot the histogram of the data to check if it approximately follows a Normal distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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n2iuA69rHa4GTkyxOcghwKPCN4TVZkjTq+plteDRwCvCdJOvbsncAr06ykmZI8GbgDQBVdX2SC4Hv0sxUPN2ZhpKkYZoxvKrqa0x9HOuSnWxzJnDmHNolSdK0PMOGJKlzDC9JUucYXpKkzjG8JEmdY3hJkjrH8JIkdY7hJUnqHMNLktQ5hpckqXMML0lS5xhekqTOMbwkSZ1jeEmSOsfwkiR1juElSeocw0t7jMWLHkuSWd/GVhy80E2XNEv9XElZ6oTtD9xPXXn1rLfLMUfMQ2skzSd7XpKkzjG8JEmdY3hJkjrH8JIkdc6M4ZVkeZIrkmxIcn2SN7Xl+ye5LMn32/v92vIk+WCSjUmuTfLc+X4TkqTR0k/P60HgrVV1GHAUcHqSZwJnAJdX1aHA5e0ywEuBQ9vbauCcobdakjTSZgyvqtpaVd9sH98DbAAOAk4A1rTV1gAnto9PAD5Wja8D+yZZOvSWS5JG1qyOeSUZAw4HrgKeUlVboQk44MC22kHA5p7NtrRlk59rdZJ1SdaNj4/PvuWSpJHVd3gleQLwWeDNVXX3zqpOUVaPKqg6t6pWVdWqJUuW9NsMSZL6C68ki2iC65NVdXFbfNvEcGB7v60t3wIs79l8GXDrcJorSVJ/sw0DnAdsqKr39axaC5zWPj4N+HxP+antrMOjgLsmhhclSRqGfs5teDRwCvCdJOvbsncA7wUuTPI6YBNwUrvuEuB4YCNwH/DaobZYkjTyZgyvqvoaUx/HAnjhFPULOH2O7ZIkaVqeYUOS1DmGlySpcwwvSVLnGF6SpM4xvCRJnWN4SZI6x/Cag7EVB5NkoJskaXD9fElZ07hl8ybqyqsH2jbHHDHk1kjS6LDnJUnqHMNLktQ5hpckqXMML0lS5xhekqTOMbwkSZ1jeEmSOsfwkiR1juElSeocw0uS1DmGlySpcwwvSVLnzBheST6aZFuS63rK3pXkR0nWt7fje9a9PcnGJDcmecl8NVySNLr66XmdDxw3RfnZVbWyvV0CkOSZwMnAs9pt/nuSvYbVWGk+LF702IEuazO24uCFbro0sma8JEpVfSXJWJ/PdwJwQVVtB25KshE4Evi/A7dQmmfbH7h/oEvbeFkbaeHM5ZjXG5Nc2w4r7teWHQRs7qmzpS2TJGloBg2vc4CnASuBrcBfteVTXSK4pnqCJKuTrEuybnx8fMBmSJJG0UDhVVW3VdVDVfUw8GGaoUFoelrLe6ouA26d5jnOrapVVbVqyZIlgzRDkjSiBgqvJEt7Fl8BTMxEXAucnGRxkkOAQ4FvzK2JkiQ90owTNpJ8GjgGOCDJFuDPgWOSrKQZErwZeANAVV2f5ELgu8CDwOlV9dD8NF2SNKr6mW346imKz9tJ/TOBM+fSKEmSdsYzbEiSOsfwkiR1juElSeocw0uS1DmGlzQgz4koLZwZZxtKmprnRJQWjj0vSVLnGF6SpM4xvCRJnWN4SZI6x/CSJHWO4SVJ6hzDS5LUOYaXJKlzDC9JUucYXpKkzjG8JEmdY3hJkjrH8JIkdY7hJUnqHMNLktQ5M4ZXko8m2Zbkup6y/ZNcluT77f1+bXmSfDDJxiTXJnnufDZekjSa+ul5nQ8cN6nsDODyqjoUuLxdBngpcGh7Ww2cM5xmSpK0w4zhVVVfAW6fVHwCsKZ9vAY4saf8Y9X4OrBvkqXDaqwkSTD4Ma+nVNVWgPb+wLb8IGBzT70tbdmjJFmdZF2SdePj4wM2Q5I0ioY9YSNTlNVUFavq3KpaVVWrlixZMuRmSJL2ZIOG120Tw4Ht/ba2fAuwvKfeMuDWwZsnSdKjDRpea4HT2senAZ/vKT+1nXV4FHDXxPCiJEnDsvdMFZJ8GjgGOCDJFuDPgfcCFyZ5HbAJOKmtfglwPLARuA947Ty0WZI04mYMr6p69TSrXjhF3QJOn2ujJEnaGc+wIUnqHMNLktQ5hpckqXMML0lS5xhekqTOMbwkSZ1jeEmSOsfwkiR1juEl7WKLFz2WJAPdxlYcvNDNl3YLM55hQ9JwbX/gfurKqwfaNsccMeTWSN1kz0uS1DmGlySpcwwvSVLnGF5Shww62cOJHtrTOGFD6pBBJ3s40UN7GntekqTOMbwkSZ1jeEmSOsfwkiR1juEFjK04eKAZXJKkheFsQ+CWzZucwSVJHTKn8EpyM3AP8BDwYFWtSrI/8HfAGHAz8O+r6o65NVOSpB2GMWz4gqpaWVWr2uUzgMur6lDg8nZZkqShmY9jXicAa9rHa4AT5+E1JEkjbK7hVcA/Jbkmyeq27ClVtRWgvT9wqg2TrE6yLsm68fHxOTZDkjRK5jph4+iqujXJgcBlSW7od8OqOhc4F2DVqlU1x3ZIkkbInHpeVXVre78N+BxwJHBbkqUA7f22uTZS0tx4Ql/taQbueSV5PPCYqrqnffxi4N3AWuA04L3t/eeH0VBJg/OEvtrTzGXY8CnA59ov6+4NfKqqLk1yNXBhktcBm4CT5t5MSZJ2GDi8quqHwHOmKP8J8MK5NEqSpJ3x9FCSpM4xvCRJnWN4SZI6x/CSJHWO4SVp6Aa9zJDfK1O/vCSKpKHzMkOab/a8JEmdY89L0rQmTisl7W4ML0nT8rRS2l05bChJ6hzDS5LUOYaXJKlzDC9JUucYXpKkzjG8JEmdY3hJkjrH8JIkdc4eFV6DngxU0ujyJMLdtEedYcOTgUqja2zFwdyyedNA2/q50T17VHhJGl3+8zpa9qhhQ0ndNnEi4C4M/w/aVocbh2Peel5JjgM+AOwFfKSq3jtfryVpzzDoiYBh1/egBm3rPr9+9EBhe/DyFdy86ZZZbweDD6nO5TXn27yEV5K9gA8Bvw5sAa5OsraqvjsfrydJXbGrQ2/CnjakOl89ryOBjVX1Q4AkFwAnAIaXJA1gIXqlg17PbVf02FJVw3/S5JXAcVX1+nb5FOCXq+qNPXVWA6vbxWcANw69IbuHA4AfL3QjdhPuix3cF4/k/tjBfbHDwVW1ZKoV89XzmiqqH5GSVXUucO48vf5uI8m6qlq10O3YHbgvdnBfPJL7Ywf3RX/ma7bhFmB5z/Iy4NZ5ei1J0oiZr/C6Gjg0ySFJHgucDKydp9eSJI2YeRk2rKoHk7wR+EeaqfIfrarr5+O1OmCPHxqdBffFDu6LR3J/7OC+6MO8TNiQJGk+eYYNSVLnGF6SpM4xvIYkyfIkVyTZkOT6JG9qy/dPclmS77f3+y10W3eFJPsk+UaSb7f74y/a8kOSXNXuj79rJ/SMhCR7JflWki+0yyO5L5LcnOQ7SdYnWdeWjerfyb5JLkpyQ/vZ8Sujui9my/AangeBt1bVYcBRwOlJngmcAVxeVYcCl7fLo2A7cGxVPQdYCRyX5CjgLODsdn/cAbxuAdu4q70J2NCzPMr74gVVtbLn+0yj+nfyAeDSqvoF4Dk0vx+jui9mxfAakqraWlXfbB/fQ/NLeBDNabHWtNXWACcuTAt3rWrc2y4uam8FHAtc1JaPzP5Isgz4DeAj7XIY0X0xjZH7O0nyJOB5wHkAVXV/Vd3JCO6LQRhe8yDJGHA4cBXwlKraCk3AAQcuXMt2rXaYbD2wDbgM+AFwZ1U92FbZQhPwo+D9wJ8CD7fLT2Z090UB/5TkmvY0cTCafydPBcaBv22Hkz+S5PGM5r6YNcNryJI8Afgs8Oaqunuh27OQquqhqlpJc4aVI4HDpqq2a1u16yV5GbCtqq7pLZ6i6h6/L1pHV9VzgZfSDK8/b6EbtED2Bp4LnFNVhwM/xSHCvhleQ5RkEU1wfbKqLm6Lb0uytF2/lKYXMlLaoZAraY4F7ptk4svxo3LasKOBlye5GbiAZrjw/YzmvqCqbm3vtwGfo/nHZhT/TrYAW6rqqnb5IpowG8V9MWuG15C0xzDOAzZU1ft6Vq0FTmsfnwZ8fle3bSEkWZJk3/bx44AX0RwHvAJ4ZVttJPZHVb29qpZV1RjNqdK+VFW/wwjuiySPT/LEicfAi4HrGMG/k6r6Z2Bzkme0RS+kuWzUyO2LQXiGjSFJ8mvAV4HvsOO4xjtojntdCKwANgEnVdXtC9LIXSjJL9IcbN6L5p+kC6vq3UmeStP72B/4FvC7VbV94Vq6ayU5BvjjqnrZKO6L9j1/rl3cG/hUVZ2Z5MmM5t/JSppJPI8Ffgi8lvbvhRHbF7NleEmSOsdhQ0lS5xhekqTOMbwkSZ1jeEmSOsfwkiR1juElSeocw0uS1Dn/D57Qn7WdL19XAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 504x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 504x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(7,4))\n",
    "plt.title(\"Female BMI histogram\",fontsize=16)\n",
    "plt.hist(females[\"BMXBMI\"].dropna(),edgecolor='k',color='pink',bins=25)\n",
    "plt.show()\n",
    "\n",
    "plt.figure(figsize=(7,4))\n",
    "plt.title(\"Male BMI histogram\",fontsize=16)\n",
    "plt.hist(male[\"BMXBMI\"].dropna(),edgecolor='k',color='blue',bins=25)\n",
    "plt.show()"
   ]
  }
 ],
 "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": "3.7.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}