Analyze_ab_test_results_notebook.ipynb

{
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  {
   "cell_type": "markdown",
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   "source": [
    "## Analyze A/B Test Results\n",
    "\n",
    "This project will assure you have mastered the subjects covered in the statistics lessons.  The hope is to have this project be as comprehensive of these topics as possible.  Good luck!\n",
    "\n",
    "## Table of Contents\n",
    "- [Introduction](#intro)\n",
    "- [Part I - Probability](#probability)\n",
    "- [Part II - A/B Test](#ab_test)\n",
    "- [Part III - Regression](#regression)\n",
    "\n",
    "\n",
    "<a id='intro'></a>\n",
    "### Introduction\n",
    "\n",
    "A/B tests are very commonly performed by data analysts and data scientists.  It is important that you get some practice working with the difficulties of these \n",
    "\n",
    "For this project, you will be working to understand the results of an A/B test run by an e-commerce website.  Your goal is to work through this notebook to help the company understand if they should implement the new page, keep the old page, or perhaps run the experiment longer to make their decision.\n",
    "\n",
    "**As you work through this notebook, follow along in the classroom and answer the corresponding quiz questions associated with each question.** The labels for each classroom concept are provided for each question.  This will assure you are on the right track as you work through the project, and you can feel more confident in your final submission meeting the criteria.  As a final check, assure you meet all the criteria on the [RUBRIC](https://review.udacity.com/#!/projects/37e27304-ad47-4eb0-a1ab-8c12f60e43d0/rubric).\n",
    "\n",
    "<a id='probability'></a>\n",
    "#### Part I - Probability\n",
    "\n",
    "To get started, let's import our libraries."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "import random\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "#We are setting the seed to assure you get the same answers on quizzes as we set up\n",
    "random.seed(42)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`1.` Now, read in the `ab_data.csv` data. Store it in `df`.  **Use your dataframe to answer the questions in Quiz 1 of the classroom.**\n",
    "\n",
    "a. Read in the dataset and take a look at the top few rows here:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>user_id</th>\n",
       "      <th>timestamp</th>\n",
       "      <th>group</th>\n",
       "      <th>landing_page</th>\n",
       "      <th>converted</th>\n",
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       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>851104</td>\n",
       "      <td>2017-01-21 22:11:48.556739</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
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       "      <th>1</th>\n",
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       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>661590</td>\n",
       "      <td>2017-01-11 16:55:06.154213</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
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       "    <tr>\n",
       "      <th>3</th>\n",
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       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
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       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>864975</td>\n",
       "      <td>2017-01-21 01:52:26.210827</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>1</td>\n",
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      "text/plain": [
       "   user_id                   timestamp      group landing_page  converted\n",
       "0   851104  2017-01-21 22:11:48.556739    control     old_page          0\n",
       "1   804228  2017-01-12 08:01:45.159739    control     old_page          0\n",
       "2   661590  2017-01-11 16:55:06.154213  treatment     new_page          0\n",
       "3   853541  2017-01-08 18:28:03.143765  treatment     new_page          0\n",
       "4   864975  2017-01-21 01:52:26.210827    control     old_page          1"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = pd.read_csv('ab_data.csv')\n",
    "df.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b. Use the below cell to find the number of rows in the dataset."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(294478, 5)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c. The number of unique users in the dataset."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "290584"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df['user_id'].nunique()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "d. The proportion of users converted."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.12104245244060237"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df[df['converted'] == 1]['user_id'].nunique() / df['user_id'].nunique()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "e. The number of times the `new_page` and `treatment` don't line up."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/amalalmutairi/opt/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:1: UserWarning: Boolean Series key will be reindexed to match DataFrame index.\n",
      "  \"\"\"Entry point for launching an IPython kernel.\n",
      "/Users/amalalmutairi/opt/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:2: UserWarning: Boolean Series key will be reindexed to match DataFrame index.\n",
      "  \n"
     ]
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       "3893"
      ]
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     "execution_count": 6,
     "metadata": {},
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   ],
   "source": [
    "len(df[df['group'] == 'treatment'][df['landing_page'] == 'old_page']) + \\\n",
    "    len(df[df['group'] == 'control'][df['landing_page'] == 'new_page'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "f. Do any of the rows have missing values?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "<class 'pandas.core.frame.DataFrame'>\n",
      "RangeIndex: 294478 entries, 0 to 294477\n",
      "Data columns (total 5 columns):\n",
      " #   Column        Non-Null Count   Dtype \n",
      "---  ------        --------------   ----- \n",
      " 0   user_id       294478 non-null  int64 \n",
      " 1   timestamp     294478 non-null  object\n",
      " 2   group         294478 non-null  object\n",
      " 3   landing_page  294478 non-null  object\n",
      " 4   converted     294478 non-null  int64 \n",
      "dtypes: int64(2), object(3)\n",
      "memory usage: 11.2+ MB\n"
     ]
    }
   ],
   "source": [
    "df.info()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### No missing Values"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`2.` For the rows where **treatment** is not aligned with **new_page** or **control** is not aligned with **old_page**, we cannot be sure if this row truly received the new or old page.  Use **Quiz 2** in the classroom to provide how we should handle these rows.  \n",
    "\n",
    "a. Now use the answer to the quiz to create a new dataset that meets the specifications from the quiz.  Store your new dataframe in **df2**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/amalalmutairi/opt/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:1: UserWarning: Boolean Series key will be reindexed to match DataFrame index.\n",
      "  \"\"\"Entry point for launching an IPython kernel.\n",
      "/Users/amalalmutairi/opt/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:2: UserWarning: Boolean Series key will be reindexed to match DataFrame index.\n",
      "  \n"
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       "      <td>0</td>\n",
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      "text/plain": [
       "        user_id                   timestamp    group landing_page  converted\n",
       "294471   718310  2017-01-21 22:44:20.378320  control     old_page          0\n",
       "294473   751197  2017-01-03 22:28:38.630509  control     old_page          0\n",
       "294474   945152  2017-01-12 00:51:57.078372  control     old_page          0\n",
       "294475   734608  2017-01-22 11:45:03.439544  control     old_page          0\n",
       "294476   697314  2017-01-15 01:20:28.957438  control     old_page          0"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_new = df[df['group'] == 'treatment'][df['landing_page'] == 'new_page']\n",
    "df_old = df[df['group'] == 'control'][df['landing_page'] == 'old_page']\n",
    "df2 = pd.concat([df_new, df_old])\n",
    "df2.tail()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`3.` Use **df2** and the cells below to answer questions for **Quiz3** in the classroom."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "a. How many unique **user_id**s are in **df2**?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "290584"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2['user_id'].nunique()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "290585"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2['user_id'].count()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "b. There is one **user_id** repeated in **df2**.  What is it?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>user_id</th>\n",
       "      <th>timestamp</th>\n",
       "      <th>group</th>\n",
       "      <th>landing_page</th>\n",
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       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
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      "text/plain": [
       "      user_id                   timestamp      group landing_page  converted\n",
       "2893   773192  2017-01-14 02:55:59.590927  treatment     new_page          0"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2[df2['user_id'].duplicated()]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c. What is the row information for the repeat **user_id**? "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
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       "      <td>treatment</td>\n",
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       "      <td>773192</td>\n",
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       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
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      "text/plain": [
       "      user_id                   timestamp      group landing_page  converted\n",
       "1899   773192  2017-01-09 05:37:58.781806  treatment     new_page          0\n",
       "2893   773192  2017-01-14 02:55:59.590927  treatment     new_page          0"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2.query('user_id == \"773192\"')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "d. Remove **one** of the rows with a duplicate **user_id**, but keep your dataframe as **df2**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "df2 = df2.drop([2893])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
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     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "df2['user_id'].count()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(df2.duplicated())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`4.` Use **df2** in the below cells to answer the quiz questions related to **Quiz 4** in the classroom.\n",
    "\n",
    "a. What is the probability of an individual converting regardless of the page they receive?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11959708724499628"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2['converted'].mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b. Given that an individual was in the `control` group, what is the probability they converted?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.1203863045004612"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2[df2['group'] == 'control']['converted'].mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c. Given that an individual was in the `treatment` group, what is the probability they converted?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11880806551510564"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2[df2['group'] == 'treatment']['converted'].mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "d. What is the probability that an individual received the new page?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5000619442226688"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(df2[df2['landing_page'] == 'new_page']) / len(df2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "e. Consider your results from a. through d. above, and explain below whether you think there is sufficient evidence to say that the new treatment page leads to more conversions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Probability of individual converting given individual is in control group is 0.1203863045004612. Probability of individual converting given individual is in treatment group is 0.11880724790277405. According to the analysis this is clear that there is no more conversion between new page and old page. As the converting rate is similar in both cases so it is important to consider other factors."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='ab_test'></a>\n",
    "### Part II - A/B Test\n",
    "\n",
    "Notice that because of the time stamp associated with each event, you could technically run a hypothesis test continuously as each observation was observed.  \n",
    "\n",
    "However, then the hard question is do you stop as soon as one page is considered significantly better than another or does it need to happen consistently for a certain amount of time?  How long do you run to render a decision that neither page is better than another?  \n",
    "\n",
    "These questions are the difficult parts associated with A/B tests in general.  \n",
    "\n",
    "\n",
    "`1.` For now, consider you need to make the decision just based on all the data provided.  If you want to assume that the old page is better unless the new page proves to be definitely better at a Type I error rate of 5%, what should your null and alternative hypotheses be?  You can state your hypothesis in terms of words or in terms of **$p_{old}$** and **$p_{new}$**, which are the converted rates for the old and new pages."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*$H_{Null}$ : $p_{old}$ >= $p_{new}$*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$H_{alternative}$ : $p_{old}$ < $p_{new}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`2.` Assume under the null hypothesis, $p_{new}$ and $p_{old}$ both have \"true\" success rates equal to the **converted** success rate regardless of page - that is $p_{new}$ and $p_{old}$ are equal. Furthermore, assume they are equal to the **converted** rate in **ab_data.csv** regardless of the page. <br><br>\n",
    "\n",
    "Use a sample size for each page equal to the ones in **ab_data.csv**.  <br><br>\n",
    "\n",
    "Perform the sampling distribution for the difference in **converted** between the two pages over 10,000 iterations of calculating an estimate from the null.  <br><br>\n",
    "\n",
    "Use the cells below to provide the necessary parts of this simulation.  If this doesn't make complete sense right now, don't worry - you are going to work through the problems below to complete this problem.  You can use **Quiz 5** in the classroom to make sure you are on the right track.<br><br>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "a. What is the **convert rate** for $p_{new}$ under the null? "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11959708724499628"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_new = df2['converted'].mean()\n",
    "p_new"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b. What is the **convert rate** for $p_{old}$ under the null? <br><br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11959708724499628"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_old = df2['converted'].mean()\n",
    "p_old"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c. What is $n_{new}$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "145310"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n_new = len(df2[df2['landing_page'] == 'new_page'])\n",
    "n_new"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "d. What is $n_{old}$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "145274"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n_old = len(df2[df2['landing_page'] == 'old_page'])\n",
    "n_old"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "e. Simulate $n_{new}$ transactions with a convert rate of $p_{new}$ under the null.  Store these $n_{new}$ 1's and 0's in **new_page_converted**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "new_page_converted = np.random.choice([0, 1], n_new, p = [p_new, 1-p_new])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "f. Simulate $n_{old}$ transactions with a convert rate of $p_{old}$ under the null.  Store these $n_{old}$ 1's and 0's in **old_page_converted**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "old_page_converted = np.random.choice([0, 1], n_old, p = [p_old, 1-p_old])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "g. Find $p_{new}$ - $p_{old}$ for your simulated values from part (e) and (f)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.0001836550056819597"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff = new_page_converted.mean() - old_page_converted.mean()# differences computed in from p_new and p_old\n",
    "diff"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "h. Simulate 10,000 $p_{new}$ - $p_{old}$ values using this same process similarly to the one you calculated in parts **a. through g.** above.  Store all 10,000 values in a numpy array called **p_diffs**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "p_diffs = []\n",
    "for i in range(10000):\n",
    "    \n",
    "    new_page_converted = np.random.choice([1, 0],n_new,replace = True,p = [p_new, 1-p_new])\n",
    "    old_page_converted = np.random.choice([1, 0],n_old,replace = True,p = [p_old, 1-p_old])\n",
    "    p_diff = new_page_converted.mean() - old_page_converted.mean()\n",
    "    p_diffs.append(p_diff)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "i. Plot a histogram of the **p_diffs**.  Does this plot look like what you expected?  Use the matching problem in the classroom to assure you fully understand what was computed here."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'Count')"
      ]
     },
     "execution_count": 28,
     "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": [
    "p_diffs=np.array(p_diffs)\n",
    "plt.hist(p_diffs)\n",
    "plt.title(\"Simulated Differences in Conversion Rates\", fontsize=14)\n",
    "plt.xlabel(\"Difference in Probability\", fontsize=12)\n",
    "plt.ylabel('Count', fontsize=12)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "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": [
    "p_diffs=np.array(p_diffs)\n",
    "\n",
    "plt.hist(p_diffs)\n",
    "plt.title(\"Simulated Differences in Conversion Rates\", fontsize=14)\n",
    "plt.xlabel(\"Difference in Probability\", fontsize=12)\n",
    "plt.ylabel('Count', fontsize=12)\n",
    "plt.axvline(diff, color='r');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "j. What proportion of the **p_diffs** are greater than the actual difference observed in **ab_data.csv**?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.9023\n"
     ]
    }
   ],
   "source": [
    "var1 = df2[df2['landing_page'] == 'new_page']\n",
    "var1=var1['converted'].mean()\n",
    "var2 = df2[df2['landing_page'] == 'old_page']\n",
    "var2 = var2['converted'].mean()\n",
    "actual_diff = var1-var2\n",
    "count = 0\n",
    "for i in p_diffs:\n",
    "    if i> actual_diff:\n",
    "        count = count+1\n",
    "        \n",
    "print (count/(len(p_diffs)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "k. In words, explain what you just computed in part **j.**  What is this value called in scientific studies?  What does this value mean in terms of whether or not there is a difference between the new and old pages?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I just computed the proportion of the p_diffs are the greated than the actual difference. In scientific studies, this value is called p-value. The result shows above is 0.9061, which is far greated than than 0.05. Hence, we can not reject the null hypothesis. We think there is no difference between the new and old pages.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "l. We could also use a built-in to achieve similar results.  Though using the built-in might be easier to code, the above portions are a walkthrough of the ideas that are critical to correctly thinking about statistical significance. Fill in the below to calculate the number of conversions for each page, as well as the number of individuals who received each page. Let `n_old` and `n_new` refer the the number of rows associated with the old page and new pages, respectively."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "145310"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import statsmodels.api as sm\n",
    "\n",
    "convert_old = len(df2.query('converted==1 and landing_page==\"old_page\"'))\n",
    "convert_new = len(df2.query('converted==1 and landing_page==\"new_page\"'))\n",
    "n_old = len(df2.query('landing_page== \"old_page\"'))\n",
    "n_new = len(df2.query('landing_page== \"new_page\"'))\n",
    "n_new"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "m. Now use `stats.proportions_ztest` to compute your test statistic and p-value.  [Here](http://knowledgetack.com/python/statsmodels/proportions_ztest/) is a helpful link on using the built in."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.3109241984234394 0.9050583127590245\n"
     ]
    }
   ],
   "source": [
    "z_score, p_value = sm.stats.proportions_ztest([convert_old,convert_new], [n_old, n_new],alternative='smaller') \n",
    "\n",
    "\n",
    "print(z_score,p_value)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9050583127590245"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy.stats import norm\n",
    "norm.cdf(z_score)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.959963984540054"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "norm.ppf(1-(0.05/2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "n. What do the z-score and p-value you computed in the previous question mean for the conversion rates of the old and new pages?  Do they agree with the findings in parts **j.** and **k.**?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since the z-score of 1.3109241984234394 does not exceed the critical value of 1.959963984540054, we fail to reject the null hypothesis that old page users has a better or equal converted rate than old page users. Therefore, the converted rate for new page and old page have no difference. The p_value does not match as parts j and k ,however, this conclusion is the same.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='regression'></a>\n",
    "### Part III - A regression approach\n",
    "\n",
    "`1.` In this final part, you will see that the result you acheived in the previous A/B test can also be acheived by performing regression.<br><br>\n",
    "\n",
    "a. Since each row is either a conversion or no conversion, what type of regression should you be performing in this case?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Logistic Regression"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b. The goal is to use **statsmodels** to fit the regression model you specified in part **a.** to see if there is a significant difference in conversion based on which page a customer receives.  However, you first need to create a column for the intercept, and create a dummy variable column for which page each user received.  Add an **intercept** column, as well as an **ab_page** column, which is 1 when an individual receives the **treatment** and 0 if **control**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>user_id</th>\n",
       "      <th>timestamp</th>\n",
       "      <th>group</th>\n",
       "      <th>landing_page</th>\n",
       "      <th>converted</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>661590</td>\n",
       "      <td>2017-01-11 16:55:06.154213</td>\n",
       "      <td>treatment</td>\n",
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       "      <th>3</th>\n",
       "      <td>853541</td>\n",
       "      <td>2017-01-08 18:28:03.143765</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
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       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>679687</td>\n",
       "      <td>2017-01-19 03:26:46.940749</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>1</td>\n",
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       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>817355</td>\n",
       "      <td>2017-01-04 17:58:08.979471</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>839785</td>\n",
       "      <td>2017-01-15 18:11:06.610965</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   user_id                   timestamp      group landing_page  converted\n",
       "2   661590  2017-01-11 16:55:06.154213  treatment     new_page          0\n",
       "3   853541  2017-01-08 18:28:03.143765  treatment     new_page          0\n",
       "6   679687  2017-01-19 03:26:46.940749  treatment     new_page          1\n",
       "8   817355  2017-01-04 17:58:08.979471  treatment     new_page          1\n",
       "9   839785  2017-01-15 18:11:06.610965  treatment     new_page          1"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "      <th></th>\n",
       "      <th>user_id</th>\n",
       "      <th>timestamp</th>\n",
       "      <th>group</th>\n",
       "      <th>landing_page</th>\n",
       "      <th>converted</th>\n",
       "      <th>intercept</th>\n",
       "      <th>ab_page</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>661590</td>\n",
       "      <td>2017-01-11 16:55:06.154213</td>\n",
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       "      <td>1</td>\n",
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       "      <th>3</th>\n",
       "      <td>853541</td>\n",
       "      <td>2017-01-08 18:28:03.143765</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
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       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>679687</td>\n",
       "      <td>2017-01-19 03:26:46.940749</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
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       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>817355</td>\n",
       "      <td>2017-01-04 17:58:08.979471</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
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       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>839785</td>\n",
       "      <td>2017-01-15 18:11:06.610965</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   user_id                   timestamp      group landing_page  converted  \\\n",
       "2   661590  2017-01-11 16:55:06.154213  treatment     new_page          0   \n",
       "3   853541  2017-01-08 18:28:03.143765  treatment     new_page          0   \n",
       "6   679687  2017-01-19 03:26:46.940749  treatment     new_page          1   \n",
       "8   817355  2017-01-04 17:58:08.979471  treatment     new_page          1   \n",
       "9   839785  2017-01-15 18:11:06.610965  treatment     new_page          1   \n",
       "\n",
       "   intercept  ab_page  \n",
       "2          1        1  \n",
       "3          1        1  \n",
       "6          1        1  \n",
       "8          1        1  \n",
       "9          1        1  "
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2['intercept'] = 1\n",
    "df2[['drop', 'ab_page']] = pd.get_dummies(df2['group'])\n",
    "df2.drop('drop', axis=1, inplace=True)\n",
    "df2.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c. Use **statsmodels** to import your regression model.  Instantiate the model, and fit the model using the two columns you created in part **b.** to predict whether or not an individual converts."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 0.366118\n",
      "         Iterations 6\n"
     ]
    }
   ],
   "source": [
    "from scipy import stats\n",
    "stats.chisqprob = lambda chisq, df:stats.chi2.sf(chisq, df)\n",
    "result = sm.Logit(df2['converted'], df2[['intercept', 'ab_page']]).fit()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "d. Provide the summary of your model below, and use it as necessary to answer the following questions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>Logit Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>       <td>converted</td>    <th>  No. Observations:  </th>   <td>290584</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                 <td>Logit</td>      <th>  Df Residuals:      </th>   <td>290582</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>                 <td>MLE</td>       <th>  Df Model:          </th>   <td>     1</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>            <td>Thu, 11 Jun 2020</td> <th>  Pseudo R-squ.:     </th>  <td>8.077e-06</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                <td>11:03:07</td>     <th>  Log-Likelihood:    </th> <td>-1.0639e+05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>converged:</th>             <td>True</td>       <th>  LL-Null:           </th> <td>-1.0639e+05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>     <td>nonrobust</td>    <th>  LLR p-value:       </th>   <td>0.1899</td>   \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "      <td></td>         <th>coef</th>     <th>std err</th>      <th>z</th>      <th>P>|z|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>intercept</th> <td>   -1.9888</td> <td>    0.008</td> <td> -246.669</td> <td> 0.000</td> <td>   -2.005</td> <td>   -1.973</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ab_page</th>   <td>   -0.0150</td> <td>    0.011</td> <td>   -1.311</td> <td> 0.190</td> <td>   -0.037</td> <td>    0.007</td>\n",
       "</tr>\n",
       "</table>"
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                           Logit Regression Results                           \n",
       "==============================================================================\n",
       "Dep. Variable:              converted   No. Observations:               290584\n",
       "Model:                          Logit   Df Residuals:                   290582\n",
       "Method:                           MLE   Df Model:                            1\n",
       "Date:                Thu, 11 Jun 2020   Pseudo R-squ.:               8.077e-06\n",
       "Time:                        11:03:07   Log-Likelihood:            -1.0639e+05\n",
       "converged:                       True   LL-Null:                   -1.0639e+05\n",
       "Covariance Type:            nonrobust   LLR p-value:                    0.1899\n",
       "==============================================================================\n",
       "                 coef    std err          z      P>|z|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "intercept     -1.9888      0.008   -246.669      0.000      -2.005      -1.973\n",
       "ab_page       -0.0150      0.011     -1.311      0.190      -0.037       0.007\n",
       "==============================================================================\n",
       "\"\"\""
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "result.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "e. What is the p-value associated with **ab_page**? Why does it differ from the value you found in **Part II**?<br><br>  **Hint**: What are the null and alternative hypotheses associated with your regression model, and how do they compare to the null and alternative hypotheses in the **Part II**?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The p-value associated with ab_page is 0.190. It differs from the value I found in the Part II for the reason that they maintain different hypothesises, which can be easily represented by symbols as follows:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Part II hypothesis:\n",
    "\n",
    "*$H_{Null}$ : $p_{old}$ >= $p_{new}$*\n",
    "\n",
    "$H_{alternative}$ : $p_{old}$ < $p_{new}$\n",
    "\n",
    "\n",
    "Part Ⅲ hypothesis:\n",
    "\n",
    "*$H_{Null}$ : $p_{old}$ = $p_{new}$*\n",
    "\n",
    "$H_{alternative}$ : $p_{old}$ =! $p_{new}$\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "f. Now, you are considering other things that might influence whether or not an individual converts. Discuss why it is a good idea to consider other factors to add into your regression model. Are there any disadvantages to adding additional terms into your regression model?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is a good idea to consider other factors to add into my regression model. Obvisously, there are many factors may effect result. If we do not consider them, our result will be not correct. Every coin has two sides. There are some disadvantages to adding additional terms into my regression model, one of them is multicollinearility, which means that one factor is related to another.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "g. Now along with testing if the conversion rate changes for different pages, also add an effect based on which country a user lives. You will need to read in the **countries.csv** dataset and merge together your datasets on the approporiate rows.  [Here](https://pandas.pydata.org/pandas-docs/stable/generated/pandas.DataFrame.join.html) are the docs for joining tables. \n",
    "\n",
    "Does it appear that country had an impact on conversion?  Don't forget to create dummy variables for these country columns - **Hint: You will need two columns for the three dummy variables.** Provide the statistical output as well as a written response to answer this question."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
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       "      <th>country</th>\n",
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       "      <th>2</th>\n",
       "      <td>822059</td>\n",
       "      <td>UK</td>\n",
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       "      <th>3</th>\n",
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       "      <td>UK</td>\n",
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       "      <td>UK</td>\n",
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      ],
      "text/plain": [
       "   user_id country\n",
       "0   834778      UK\n",
       "1   928468      US\n",
       "2   822059      UK\n",
       "3   711597      UK\n",
       "4   710616      UK"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "country_df = pd.read_csv('./countries.csv')\n",
    "country_df.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "      <td>1</td>\n",
       "      <td>1</td>\n",
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       "      <td>2017-01-15 18:11:06.610965</td>\n",
       "      <td>treatment</td>\n",
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       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>CA</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   user_id                   timestamp      group landing_page  converted  \\\n",
       "0   661590  2017-01-11 16:55:06.154213  treatment     new_page          0   \n",
       "1   853541  2017-01-08 18:28:03.143765  treatment     new_page          0   \n",
       "2   679687  2017-01-19 03:26:46.940749  treatment     new_page          1   \n",
       "3   817355  2017-01-04 17:58:08.979471  treatment     new_page          1   \n",
       "4   839785  2017-01-15 18:11:06.610965  treatment     new_page          1   \n",
       "\n",
       "   intercept  ab_page country  \n",
       "0          1        1      US  \n",
       "1          1        1      US  \n",
       "2          1        1      CA  \n",
       "3          1        1      UK  \n",
       "4          1        1      CA  "
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_new = df2.merge(country_df, on='user_id', how='inner')\n",
    "df_new.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
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       "      <th>user_id</th>\n",
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       "      <th>group</th>\n",
       "      <th>landing_page</th>\n",
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       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>CA</td>\n",
       "      <td>0</td>\n",
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       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>UK</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
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       "      <th>4</th>\n",
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       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>CA</td>\n",
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      ],
      "text/plain": [
       "   user_id                   timestamp      group landing_page  converted  \\\n",
       "0   661590  2017-01-11 16:55:06.154213  treatment     new_page          0   \n",
       "1   853541  2017-01-08 18:28:03.143765  treatment     new_page          0   \n",
       "2   679687  2017-01-19 03:26:46.940749  treatment     new_page          1   \n",
       "3   817355  2017-01-04 17:58:08.979471  treatment     new_page          1   \n",
       "4   839785  2017-01-15 18:11:06.610965  treatment     new_page          1   \n",
       "\n",
       "   intercept  ab_page country  US  UK  \n",
       "0          1        1      US   1   0  \n",
       "1          1        1      US   1   0  \n",
       "2          1        1      CA   0   0  \n",
       "3          1        1      UK   0   1  \n",
       "4          1        1      CA   0   0  "
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "### Create the necessary dummy variables\n",
    "df_new[['US', 'UK']] = pd.get_dummies(df_new['country'])[['US', \"UK\"]]\n",
    "df_new.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "h. Though you have now looked at the individual factors of country and page on conversion, we would now like to look at an interaction between page and country to see if there significant effects on conversion.  Create the necessary additional columns, and fit the new model.  \n",
    "\n",
    "Provide the summary results, and your conclusions based on the results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
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       "      <th></th>\n",
       "      <th>user_id</th>\n",
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       "      <td>0</td>\n",
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       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>CA</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
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       "      <th>3</th>\n",
       "      <td>817355</td>\n",
       "      <td>2017-01-04 17:58:08.979471</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>UK</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
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       "      <td>839785</td>\n",
       "      <td>2017-01-15 18:11:06.610965</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>CA</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
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      ],
      "text/plain": [
       "   user_id                   timestamp      group landing_page  converted  \\\n",
       "0   661590  2017-01-11 16:55:06.154213  treatment     new_page          0   \n",
       "1   853541  2017-01-08 18:28:03.143765  treatment     new_page          0   \n",
       "2   679687  2017-01-19 03:26:46.940749  treatment     new_page          1   \n",
       "3   817355  2017-01-04 17:58:08.979471  treatment     new_page          1   \n",
       "4   839785  2017-01-15 18:11:06.610965  treatment     new_page          1   \n",
       "\n",
       "   intercept  ab_page country  US  UK  US_ab_page  UK_ab_page  \n",
       "0          1        1      US   1   0           1           0  \n",
       "1          1        1      US   1   0           1           0  \n",
       "2          1        1      CA   0   0           0           0  \n",
       "3          1        1      UK   0   1           0           1  \n",
       "4          1        1      CA   0   0           0           0  "
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "### Fit Your Linear Model And Obtain the Results\n",
    "df_new['US_ab_page'] = df_new['US'] * df_new['ab_page']\n",
    "df_new['UK_ab_page'] = df_new['UK'] * df_new['ab_page']\n",
    "df_new.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 0.366109\n",
      "         Iterations 6\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>Logit Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>       <td>converted</td>    <th>  No. Observations:  </th>   <td>290584</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                 <td>Logit</td>      <th>  Df Residuals:      </th>   <td>290578</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>                 <td>MLE</td>       <th>  Df Model:          </th>   <td>     5</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>            <td>Thu, 11 Jun 2020</td> <th>  Pseudo R-squ.:     </th>  <td>3.482e-05</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                <td>11:03:08</td>     <th>  Log-Likelihood:    </th> <td>-1.0639e+05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>converged:</th>             <td>True</td>       <th>  LL-Null:           </th> <td>-1.0639e+05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>     <td>nonrobust</td>    <th>  LLR p-value:       </th>   <td>0.1920</td>   \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "       <td></td>         <th>coef</th>     <th>std err</th>      <th>z</th>      <th>P>|z|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>intercept</th>  <td>   -2.0040</td> <td>    0.036</td> <td>  -55.008</td> <td> 0.000</td> <td>   -2.075</td> <td>   -1.933</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ab_page</th>    <td>   -0.0674</td> <td>    0.052</td> <td>   -1.297</td> <td> 0.195</td> <td>   -0.169</td> <td>    0.034</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>US</th>         <td>    0.0175</td> <td>    0.038</td> <td>    0.465</td> <td> 0.642</td> <td>   -0.056</td> <td>    0.091</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>UK</th>         <td>    0.0118</td> <td>    0.040</td> <td>    0.296</td> <td> 0.767</td> <td>   -0.066</td> <td>    0.090</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>US_ab_page</th> <td>    0.0469</td> <td>    0.054</td> <td>    0.872</td> <td> 0.383</td> <td>   -0.059</td> <td>    0.152</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>UK_ab_page</th> <td>    0.0783</td> <td>    0.057</td> <td>    1.378</td> <td> 0.168</td> <td>   -0.033</td> <td>    0.190</td>\n",
       "</tr>\n",
       "</table>"
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                           Logit Regression Results                           \n",
       "==============================================================================\n",
       "Dep. Variable:              converted   No. Observations:               290584\n",
       "Model:                          Logit   Df Residuals:                   290578\n",
       "Method:                           MLE   Df Model:                            5\n",
       "Date:                Thu, 11 Jun 2020   Pseudo R-squ.:               3.482e-05\n",
       "Time:                        11:03:08   Log-Likelihood:            -1.0639e+05\n",
       "converged:                       True   LL-Null:                   -1.0639e+05\n",
       "Covariance Type:            nonrobust   LLR p-value:                    0.1920\n",
       "==============================================================================\n",
       "                 coef    std err          z      P>|z|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "intercept     -2.0040      0.036    -55.008      0.000      -2.075      -1.933\n",
       "ab_page       -0.0674      0.052     -1.297      0.195      -0.169       0.034\n",
       "US             0.0175      0.038      0.465      0.642      -0.056       0.091\n",
       "UK             0.0118      0.040      0.296      0.767      -0.066       0.090\n",
       "US_ab_page     0.0469      0.054      0.872      0.383      -0.059       0.152\n",
       "UK_ab_page     0.0783      0.057      1.378      0.168      -0.033       0.190\n",
       "==============================================================================\n",
       "\"\"\""
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "logit3 = sm.Logit(df_new['converted'], df_new[['intercept', 'ab_page', 'US', 'UK' , 'US_ab_page', 'UK_ab_page']])\n",
    "result3 = logit3.fit()\n",
    "result3.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Conclusions:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " None of the variables have significant p-values. Therefore, we will fail to reject the null and conclude that there is not sufficient evidence to suggest that there is an interaction between country and page received that will predict whether a user converts or not.\n",
    "In the larger picture, based on the available information, we do not have sufficient evidence to suggest that the new page results in more conversions than the old page."
   ]
  }
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