StroopEffect/Stroop_Effect_Perceptual_Phenomenon.ipynb

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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# The Stroop Effect\n",
    "\n",
    "In this notebook, we perform an statistical test on a [dataset](./data/stroopdata.csv) from Udacity, which contains data of participants involved in a perceptual test. The name of the test is Stroop test and it know percetual phenomenon in human nature called Stroop effect.\n",
    "\n",
    "\n",
    "## Content\n",
    "\n",
    "<ul>\n",
    "<li><a href=\"#background\">Background on Stroop Effect</a></li>\n",
    "<li><a href=\"#variables\">Variables Identification</a></li>\n",
    "<li><a href=\"#hypothesis\">Hypothesis</a></li>\n",
    "<li><a href=\"#analysis\">Data Analysis</a></li>\n",
    "    <ul>\n",
    "    <li><a href=\"#descriptive\">Descriptive Statistics</a></li>\n",
    "    <li><a href=\"#eda\">Exploratory Data Analysis (EDA)</a></li>\n",
    "    <li><a href=\"#stats_test\">Statistical Test</a></li>\n",
    "        <ul>\n",
    "           <li><a href='#bootstrapping'>Boostrapping</a></li>\n",
    "           <li><a href='#paired_t-test'>Two-sample Paired t-test</a></li>\n",
    "        </ul>\n",
    "    </ul>\n",
    "<li><a href=\"#conclusions\">Conclusions</a></li>\n",
    "<li><a href=\"#further\">Further Investigation</a></li>\n",
    "<li><a href=\"#reference\">References</a></li>\n",
    "<li><a href='#appendix'>Appendix</a></li>\n",
    "    <ul>\n",
    "        <li><a href='#one-sample_t-test'>One-sample t-test</a></li>\n",
    "        <li><a href='#t-test_equation'>t-test Statistcs Equation</a></li>\n",
    "    </ul>\n",
    "</ul>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='background'></a>\n",
    "## Background on Stroop Effect \n",
    "\n",
    "\n",
    "The Stroop effect[[1](https://docs.google.com/document/d/1-OkpZLjG_kX9J6LIQ5IltsqMzVWjh36QpnP2RYpVdPU/pub?embedded=True)] is a psycological phenomenon shown in human reaction to relate concepts (words) to vision (images, colors). In order to understand such relation of concepts and images, psycologists test a participant with two lists of words, which display words in different color of ink. The participant's task is to say the color of the ink of the word shown.   \n",
    "\n",
    "The task has two scenarios with equally-size lists of words: \n",
    "1. **Congruent** words:\n",
    "Shown words are colored matching the colors in which they are displayed. _Example: <font color=\"red\">RED</font>, <font color=\"blue\">BLUE</font>._\n",
    "<p></p>\n",
    "2. **Incongruent** words: \n",
    "Shown words are colored, but this time the color of words does not match the word. Thus, the name of incongruent. _Example: <font color=\"orange\">PURPLE</font>, <font color=\"purple\">ORANGE</font>._\n",
    "\n",
    "\n",
    "For each scenario, one measures the time it takes each participant to go through all words. \n",
    "\n",
    "It is important to notice that the colored words congruent or incongruent are stimuli, while the participant's response is a reaction.\n",
    "\n",
    ">**Note:** The dataset in this notebook contains the times of participants taking a Stroop test.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id=\"variables\"></a>\n",
    "## Variables Identification\n",
    "\n",
    "In order to start working on our statistical test and choose a hypothesis to test for, we need to define the variables we will investigate and wether if they are independent from or dependent to other factors."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus, we ask ourselves the questions: What is the independent variable? What is the dependent variable?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To identify our independent and dependent variables we must ask which facts influence the outcome. In this case we know the words and colors are stimuli and the particpant's answer is a reaction. Thus, we can say that the words and its colors influence the reaction of the participant measured in time units. As a result, we can classify our variables as follow:\n",
    "\n",
    "**Independent variables**\n",
    "1. Color of word\n",
    "2. Word\n",
    "3. Person (Participant)\n",
    "\n",
    "Basically, independent variables 1 and 2 refer to our **Congruent/Incongruent** words. \n",
    "\n",
    "**Dependent variables**\n",
    "1. Time, the _reaction time_ of a participant"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id=\"hypothesis\"></a>\n",
    "## Hypothesis\n",
    "\n",
    "Once we identified our independent and dependent variables, we want to make a hypothesis on the outcome of our dependent variables which is the result of our test. Thus, we can ask: What is an appropriate set of hypotheses for this task?\n",
    "\n",
    "For a Stroop test, the reaction time is our tests' result and dependent variable. In fact, that is what we are interested in comparing and the purpose of having 2 scenarios on the test. We want to look at the reaction time of people in both scenarios and make conclusions on participants' reactions. Therefore, we will compare the reaction time of **Congruent** and **Incongruent** scenarios. Since we have a small sample size for our hypothesis test, we will consider the average of our congruent's and incongruent's time to be our statistic for the test. The statistical test has to be a t-test since **sample size is smaller than 30**. Moreover, we know that the same participant did a congruent and incongruent test, thus these two times are related. A comparison between such times lead us to define we need to do a **paired t-test (two-sample paired t-test)** to compare the congruent mean and incongruent mean.[3]\n",
    "\n",
    ">**Note:** If the a two-sample paired t-test gives the same result of a one-sample t-test if the substract group A's mean - group B's mean and we assume their difference is center in 0. This is another approach to the hypothesis test, so it is important to keep in mind.\n",
    "\n",
    "For the hypothesis, the traditional thinking believes that the participants' reaction time is the same or maybe even slower when they have congruent words. However, we want to challenge that idea by saying that we believe the participants' have a faster reaction time when color and words match (congruent) than when color and words do not match (incongruent). Thus, we can stablish a hypothesis as follow:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$H_0: \\mu_0 \\geq \\mu_1$$\n",
    "\n",
    "$$H_1: \\mu_0 < \\mu_1$$\n",
    "\n",
    "where $\\mu_0$ is the mean of the time it took people on the sample to read **congruent** words (labels consistent with color)\n",
    "\n",
    "and $\\mu_1$ is the mean of the time it took people on the sample to read **incongruent** words (labels differ from color)\n",
    "\n",
    "In other words, our hypotheses are:\n",
    "* **Null hypothesis** ($H_0$). The reaction time of participants when reading congruent words is equal or greater than when they read incongruent words.\n",
    "* **Alternative hypothesis** ($H_1$). The reaction time of participants when reading congruent words is faster then when reading incongruent words.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='analysis'></a>\n",
    "## Data Analysis\n",
    "\n",
    "In this section, we will concentrate on obtaining some statistics of our data and doing some exploration of it. The goal will be to perform the necessary analyses and tests to determine if we reject or show that there is not enough information to reject the null hypothesis $H_0$.\n",
    "\n",
    "<a id='descriptive'></a>\n",
    "### Descriptive Statistics\n",
    "\n",
    "First, we need to understand our data and see what kind of data we have. It is also important to check if we need to clean our dataset or if we were lucky to have gotten one dataset that was preprocessed by someone else and is clean.\n",
    "Then, we want to verify if our data is close enough to be normally distributed using descriptive statistics such as two central tendency measures (median, mean) and checking for the variability of our data ( we will need to consider the size of the sample to judge this point)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Packages import\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "import scipy.stats as stats\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "<div>\n",
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       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Congruent</th>\n",
       "      <th>Incongruent</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>12.079</td>\n",
       "      <td>19.278</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>16.791</td>\n",
       "      <td>18.741</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>9.564</td>\n",
       "      <td>21.214</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>8.630</td>\n",
       "      <td>15.687</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>14.669</td>\n",
       "      <td>22.803</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   Congruent  Incongruent\n",
       "0     12.079       19.278\n",
       "1     16.791       18.741\n",
       "2      9.564       21.214\n",
       "3      8.630       15.687\n",
       "4     14.669       22.803"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Loading our dataset\n",
    "path = \"data/\"\n",
    "file = 'stroopdata.csv'\n",
    "df = pd.read_csv(path+file)\n",
    "df.head(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "<class 'pandas.core.frame.DataFrame'>\n",
      "RangeIndex: 24 entries, 0 to 23\n",
      "Data columns (total 2 columns):\n",
      "Congruent      24 non-null float64\n",
      "Incongruent    24 non-null float64\n",
      "dtypes: float64(2)\n",
      "memory usage: 464.0 bytes\n"
     ]
    }
   ],
   "source": [
    "# Display information of the dataset\n",
    "df.info()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This dataset seems to have been processed by our source. Thus, we will be ok to use it as it is. There is also something important on this dataset and that is the size of it. It seems a fairly small dataset of 24 samples per group (congruent and incongruent) to have a 100% accurate decision when determining if the data is normally distributed, but if the distributions get close to normal we can agree that it is close enough even with this little data.\n",
    "\n",
    "We can start by calculating the median and mean and see how far apart they are in each group"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Congruent Mean: 14.0511\n",
      "Congruent Median: 14.3565\n",
      "\n",
      "Incongruent Mean: 22.0159\n",
      "Incongruent Median: 21.0175\n"
     ]
    }
   ],
   "source": [
    "# Checking the mean and median for both groups\n",
    "print('Congruent Mean: {:.4f}\\nCongruent Median: {:.4f}'.\n",
    "      format(df['Congruent'].mean(),df['Congruent'].median()))\n",
    "\n",
    "print('\\nIncongruent Mean: {:.4f}\\nIncongruent Median: {:.4f}'.\n",
    "      format(df['Incongruent'].mean(),df['Incongruent'].median()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Right away we can notice the median and the mean for both groups is very close to each other. Thus, we can say our data is close to a normal distribution. It will be good to check the 25% and 75% percentiles, minimum and maximum values to understand a little bit better our data. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Congruent [25%,75%] percentiles: [11.8953,16.2007]\n",
      "Congruent (min,max): (8.6300,22.3280)\n",
      "\n",
      "Incongruent [25%,75%] percentiles: [18.7168,24.0515]\n",
      "Incongruent (min,max): (15.6870,35.2550)\n"
     ]
    }
   ],
   "source": [
    "# Checking the 25% and 75% percentiles, min and max values for both groups\n",
    "p25_congruent = np.percentile(df['Congruent'],25)\n",
    "p75_congruent = np.percentile(df['Congruent'],75)\n",
    "print('Congruent [25%,75%] percentiles: [{:.4f},{:.4f}]'.format(p25_congruent,p75_congruent))\n",
    "print('Congruent (min,max): ({:.4f},{:.4f})'.format(df['Congruent'].min(), df['Congruent'].max()))\n",
    "\n",
    "p25_incongruent = np.percentile(df['Incongruent'],25)\n",
    "p75_incongruent = np.percentile(df['Incongruent'],75)\n",
    "print('\\nIncongruent [25%,75%] percentiles: [{:.4f},{:.4f}]'.format(p25_incongruent,p75_incongruent))\n",
    "print('Incongruent (min,max): ({:.4f},{:.4f})'.format(df['Incongruent'].min(), df['Incongruent'].max()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This data corroborates our assumption that the **congruent** group's distribution is normal as we can clearly see now that the mean and median are sitting in between the 25% and 75% percentiles. However, there is something suspicious about our **incongruent** group's distribution, apparently the mean is leaning to the 75% percentile, while the median seems centered in the between the 25% and 75% percentile. Our _incongruent data seems skewed to the right_ as $mean > median$. This might be caused by outliers since the maximum value is further away form the 75% than the minimum value from the 25% percentile. We will explore this later in a graph to make this concept clearer, but as the variance and standard deviation of both groups can tell us their variability (spread)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Congruent Variance: 12.6690\n",
      "Congruent Std: 3.5594\n",
      "\n",
      "Incongruent Variance: 23.0118\n",
      "Incongruent Std: 4.7971\n"
     ]
    }
   ],
   "source": [
    "# Checking the var and STD for both groups\n",
    "print('Congruent Variance: {:.4f}\\nCongruent Std: {:.4f}'.\n",
    "      format(df['Congruent'].var(),df['Congruent'].std()))\n",
    "\n",
    "print('\\nIncongruent Variance: {:.4f}\\nIncongruent Std: {:.4f}'.\n",
    "      format(df['Incongruent'].var(),df['Incongruent'].std()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we expected, the **incongruent** group has a high variance and standard deviation as we believe it contains outliers that contribute to the increase in variability. The next step will be to perform an exploratory data analysis to visualize our believes."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id=\"eda\"></a>\n",
    "### Exploratory Data Analysis (EDA)\n",
    "\n",
    "The first thing we want to confirm is that the dataset has a normal distribution as ou descriptive statistics tells us we might have normal distributions, but also outliers in our incongruent group."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
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pK9DeCdBzEbE7QPHv8+1cviRJkqQK1t4J0N3AmGJ6DHBXO5cvaQvzt7/9jdNP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\n2kZLE6BXI+Kg2pmIOBh4vZn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ETI2IPxf/7lTOGFtbE+c8ISL+GhE1xevj5YyxNUXEByNi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\nYFUAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11d49b630>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Histogram and box plot of both groups\n",
    "plt.figure(figsize=(14,5))\n",
    "\n",
    "# Histogram\n",
    "plt.subplot(1,2,1)\n",
    "sns.distplot(df[\"Congruent\"], norm_hist=False,kde=False, bins=4, label = \"Congruent\")\n",
    "sns.distplot(df[\"Incongruent\"], norm_hist=False,kde=False,  bins=7, label = \"Incongruent\")\n",
    "plt.axvline(df['Congruent'].mean(),color='b',lw=3,label='Congruent mean')\n",
    "plt.axvline(df['Incongruent'].mean(),color='r',lw=3,label='Incongruent mean')\n",
    "plt.xlabel('Time')\n",
    "plt.ylabel('Frequency')\n",
    "plt.title('Histogram of Stroop Tests')\n",
    "plt.legend()\n",
    "\n",
    "# Box plot\n",
    "subplot_box =plt.subplot(1,2,2) # position of subplot\n",
    "df.plot(ax=subplot_box, kind='box')\n",
    "plt.title('Distributions of Stroop Tests')\n",
    "plt.ylabel('Time');\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The histogram and boxplot clearly show the incongruent group has two outliers that are shifting the mean to the right skewing the distribution to the right. The boxplot also shows that most of the samples in the incongruent group actually live between the 25% and 75% percentile, which is a good indication that the distribution is normal.\n",
    "The congruent group on the other hand has its mean centered on the highest frequency of the histogram which is as close as the mode and median it can get to indicate a normal distribution. On the box plot, however, we can notice the spread between the minimum and maximum values in the congruent group is larger than the incongruent group. This was not expected as our variance for the congruent group was smaller, but this might be an indicative that if we remove the ouliers from the incongruent group (consecuentively from the congruent group as well) we should find a smaller variance for the incongruent distribution.\n",
    "\n",
    "However, while removing outliers in one group, we might find there will be an outlier for the other group. This all will depend on our metric to remove outliers. In here, we will consider the boxplot out tool to determine of something is an outlier."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: left;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Congruent</th>\n",
       "      <th>Incongruent</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>18.200</td>\n",
       "      <td>35.255</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>12.369</td>\n",
       "      <td>34.288</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>22.328</td>\n",
       "      <td>24.524</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    Congruent  Incongruent\n",
       "14     18.200       35.255\n",
       "19     12.369       34.288\n",
       "10     22.328       24.524"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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ZZ0fEDcA3I+LjwP8Ci/pYBkmSJEnSBqJvCWxmXgO8qMPwmyn3w0qSJEmSNGJj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\n1Aj/H0Qo+KNhyK1WAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11d55c828>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Removing outliers\n",
    "plt.figure(figsize=(16,5))\n",
    "\n",
    "# creating a copy of our dataset in which outliers will be removed\n",
    "df_clean = df.copy()\n",
    "\n",
    "# Removing first outlier\n",
    "sub1 = plt.subplot(1,3,1)\n",
    "outlier1 = df_clean['Incongruent'].argmax()\n",
    "df_clean = df_clean.drop([outlier1])\n",
    "df_clean.plot(ax=sub1, kind='box')\n",
    "plt.ylabel('Time')\n",
    "plt.title('First Outlier Removal (index 14)')\n",
    "\n",
    "# Removing second outlier\n",
    "sub2 = plt.subplot(1,3,2)\n",
    "outlier2 = df_clean['Incongruent'].argmax()\n",
    "df_clean = df_clean.drop([outlier2])\n",
    "df_clean.plot(ax=sub2, kind='box')\n",
    "plt.title('Second Outlier Removal (index 19)')\n",
    "\n",
    "#Removing third outlier (surprisingly found in congruent)\n",
    "sub3 = plt.subplot(1,3,3)\n",
    "outlier3 = df_clean['Congruent'].argmax()\n",
    "df_clean = df_clean.drop([outlier3])\n",
    "df_clean.plot(ax=sub3, kind='box')\n",
    "plt.title('Third Outlier Removal (index 10)');\n",
    "\n",
    "# Displaying the outliers' values\n",
    "df.loc[[outlier1,outlier2,outlier3]]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From the boxplots we could see that when removing the first outlier, we were found that the congruent group had an outlier. Thus, the removal of two outliers became a task of removing three outliers. The process illustrated above left us with 21 samples. \n",
    "Moreover, we now can say that the incongruent group is actually closer to a normal distribution than the congruent group, but both are close to normal distributions. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Congruent Mean: 13.5395\n",
      "Congruent Median: 14.2330\n",
      "Congruent Std: 3.1412\n",
      "\n",
      "Incongruent Mean: 20.6817\n",
      "Incongruent Median: 20.7620\n",
      "Incongruent Std: 2.8259\n"
     ]
    }
   ],
   "source": [
    "# Checking the mean, median and standard deviation for both groups\n",
    "print('Congruent Mean: {:.4f}\\nCongruent Median: {:.4f}\\nCongruent Std: {:.4f}'.\n",
    "      format(df_clean['Congruent'].mean(),df_clean['Congruent'].median(), df_clean['Congruent'].std()))\n",
    "\n",
    "print('\\nIncongruent Mean: {:.4f}\\nIncongruent Median: {:.4f}\\nIncongruent Std: {:.4f}'.\n",
    "      format(df_clean['Incongruent'].mean(),df_clean['Incongruent'].median(), df_clean['Incongruent'].std()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see that the spread (standard deviation) decreased in both groups, which is a good indication that we effectively removed the outliers. The incongruent group in particular got a mean and median almost similar indicating a normal distribution. Moreover, the standard deviation is now smaller than the congruent group which is what we expected to see. However, the congruent group now has a median and mean that are not as close as before. Here we sacrificed the separation in median and mean to gain a tighter distribution by reducing our standard deviation when we removed the congruent outlier.\n",
    "\n",
    "All in all, we learned that our dataset has two distributions closed to normal and that removing the outliers is not as significant as we expected. When we removed them we proved the groups tend to be normally distributed and we can keep the outliers as our dataset is too small (24 samples) and we cannot afford to remove three samples, which is 1/8 of the sample size (20% of the data). Thus, we keep the outliers for our hypothesis test and we have done enough by identifying them, but we do not know if the actually are bad data.\n",
    "\n",
    ">**Note:** Since our dataset is small and this is a psychophysical test, it is expected to have a large variance. Thus, it is possible to do the analysis with the outliers."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id=\"stats_test\"></a>\n",
    "### Statistical Test\n",
    "\n",
    "For the purpose of the hypothesis testing, we have two options to do the test. The first one is :\n",
    "\n",
    "$$H_0: \\mu_0 \\geq \\mu_1$$\n",
    "\n",
    "$$H_1: \\mu_0 < \\mu_1$$\n",
    "\n",
    "in this option we use a **two-sample paired t-test** (compare two groups that are related, same participant took both tests: congruent and incongruent) . On the second option, we can use some math to transform our original hypotheis to its equivalent: \n",
    "\n",
    "$$H_0: \\mu_0 - \\mu_1 \\geq 0$$\n",
    "\n",
    "$$H_1: \\mu_0 - \\mu_1 < 0$$\n",
    "\n",
    "for this other expression of our hypothesis, we use **bootstrapping** to run a hypothesis test.\n",
    "\n",
    ">**Note:** For the second option of hypothesis, we can use a **one-sample t-test**. This test is embedded on the two-sample paired t-test, which would first substract the two means and perform an one-sample t-test, so if we want to do a one-sample test we need the substracted samples and the assumption that our mean is centered in 0.\n",
    "\n",
    "Given that we already know our distributions are fairly close to normal, we are ready to use the mean as our statistic to perform our hypothesis test, which due to the fact that our sample size is smaller than 30 we need a t-test.\n",
    "\n",
    "\n",
    "The first thing we do is define our desired confidence interval and as a result the type I error threshold ($\\alpha$). In this case, we want to have a high confidence interval as this a test to understand our human reaction time in this Stroop test. Thus, we choose an interval of 99%, resulting in $\\alpha = 0.01$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Confidence level is 99.0%\n",
      "Type I error threshold (alpha) is 0.0100\n"
     ]
    }
   ],
   "source": [
    "confidence_level = 0.99\n",
    "alpha = 1 - confidence_level\n",
    "print('Confidence level is {}%'.format(100.0*confidence_level))\n",
    "print('Type I error threshold (alpha) is {:.4f}'.format(alpha))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are options to perform the hypothesis test:\n",
    "* **Bootstrapping**. We simulate the null distribution/population and use our sample statistic to find the p-value (uses hypothesis $\\mu_0 - \\mu_1 \\geq 0$).\n",
    "* **Two-sample paired t-test**. We need a one-tail test (defined by our hypothesis $\\mu_0 \\geq \\mu_1$). For this test we will use our sample dataset, the t-test uses the sample to approximate population's predictions. This was one of the most used tests before computers as a small sample size could be used for bigger estimations.\n",
    "* **One-sample t-test**. Instead of using the data for each congruent and incongruent group, we will need to use the difference of congruent-incongruent, and assume our mean is 0 (uses hypothesis $\\mu_0 - \\mu_1 \\geq 0$).\n",
    "\n",
    "In this notebook, we perform the first two approaches to prove that they all work and result in similar conclusions.\n",
    "\n",
    "<a id='bootstrapping'></a>\n",
    "#### Bootstrapping\n",
    "\n",
    "The first thing we do for bootstrapping is to calculate our sample mean.\n",
    "\n",
    ">**Note:** Remember to use the dataset with outliers as we cannot afford to lose 20% of our data\n",
    "\n",
    "In this case, our sample mean is $\\mu_0 - \\mu_1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Dataset congruent_mean - incongruent_mean : -7.9648\n"
     ]
    }
   ],
   "source": [
    "# Calculating the mean difference from our dataset (21 samples) \n",
    "u_diff = df['Congruent'].mean() - df['Incongruent'].mean()\n",
    "print('Dataset congruent_mean - incongruent_mean : {:.4f}'.format(u_diff))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "After we have our sample mean, we need to generate all the samples 'congruent - incongruent' time to be able to sample from them and allow bootstrapping to generate the desired distribution.\n",
    "\n",
    ">**Note:** Since congruent and incongruent time are related variables, it is better for bootstrapping to sample from $\\mu_0 - \\mu_1$ than to sample $\\mu_0$ and $\\mu_1$ and compute their difference. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: left;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Congruent</th>\n",
       "      <th>Incongruent</th>\n",
       "      <th>Diff</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>12.079</td>\n",
       "      <td>19.278</td>\n",
       "      <td>-7.199</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>16.791</td>\n",
       "      <td>18.741</td>\n",
       "      <td>-1.950</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>9.564</td>\n",
       "      <td>21.214</td>\n",
       "      <td>-11.650</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>8.630</td>\n",
       "      <td>15.687</td>\n",
       "      <td>-7.057</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>14.669</td>\n",
       "      <td>22.803</td>\n",
       "      <td>-8.134</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   Congruent  Incongruent    Diff\n",
       "0     12.079       19.278  -7.199\n",
       "1     16.791       18.741  -1.950\n",
       "2      9.564       21.214 -11.650\n",
       "3      8.630       15.687  -7.057\n",
       "4     14.669       22.803  -8.134"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df['Diff'] = df['Congruent'] - df['Incongruent']\n",
    "df.head(5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then, we sample with replace our $\\mu_0 - \\mu_1$ dataset a large number of times. For every sample we do, we obtain the mean of each group and the difference of their means.\n",
    "\n",
    ">**Note:** Doing the sampling is a powerful concept as it intercepts with central limit theorem which says that the distribution of the sample mean is normal.\n",
    "\n",
    "In this case, we will use 10,000 sample means of size 3 each."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Diff mean: -7.9042 Diff median: -7.6137 Diff std: 2.7305\n",
      "\n",
      "Sample std: 4.8648\n"
     ]
    }
   ],
   "source": [
    "# Using bootstraping\n",
    "RESAMPLE = 10000\n",
    "\n",
    "diff = []\n",
    "\n",
    "# bootstrapping our difference in congruent-incongruent\n",
    "for _ in range(RESAMPLE):\n",
    "    diff.append(df['Diff'].sample(3,replace=True).mean())\n",
    "\n",
    "# converting our samples to numpy array    \n",
    "diff = np.array(diff)\n",
    "\n",
    "print('\\nDiff mean: {:.4f} Diff median: {:.4f} Diff std: {:.4f}'.\n",
    "      format(diff.mean(), np.median(diff), diff.std())) \n",
    "\n",
    "print('\\nSample std: {:.4f}'.format(df['Diff'].std()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "After the bootstrapping, we can see that the mean for the bootstrapped statistic is very close to the sample. \n",
    "\n",
    "The next step is to visualize our results from bootstrapping using a histogram."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
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mhBNOIPfsbu6Y77bbbvTr149zzz23zuMjSZJaB+9cN4GddtqJ1atX89FHH7Hddtvx8MMP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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11d8b7278>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Histograms of simulated bootstraped groups means difference\n",
    "plt.figure(figsize=(12,4))\n",
    "sns.distplot(diff, kde=False, color='pink', label='Samples Differences Mean Dist')\n",
    "plt.axvline(u_diff,color='m',lw=4,label='Dataset Difference in means')\n",
    "plt.ylabel('Frequency')\n",
    "plt.xlabel('Time')\n",
    "plt.title('Histograms of Bootstraped $x_0 - x_1$ (difference mean of samples)')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Once we visualized our bootstrapped distribution with the dataset mean displayed on it, we can say that the bootstrapped distribution is representative of our dataset. Using this idea, we are able to obtain the standard deviation of our bootstrapped difference samples' mean to simulate the null distribution centered in 0 and using the same number of samples the bootstrapped used.\n",
    "\n",
    "Then, we will plot the distribution of our null distribution and overlap the boostrapped difference in means and the dataset difference in means. The latter is the one we are ultimately interested."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Simulating the null distribution \n",
    "null_vals = np.random.normal(0, diff.std(), RESAMPLE)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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BEbFtRDwSES8Vz9sUdSMiroqIlyNiSkT0rmRskiRJUrlV+oLGnwK/zcweQBXw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\nHdXAeDYAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11d961f60>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Histogram of simulated normal null distribution\n",
    "plt.figure(figsize=(12,4))\n",
    "sns.distplot(null_vals,kde=False,norm_hist=False)\n",
    "plt.axvline(u_diff,color='b',lw=8,label='Dataset difference in means')\n",
    "plt.axvline(diff.mean(),color='r',lw=3,label='Bootstrapped difference in means')\n",
    "plt.title('Simulated Null Distribution of $\\mu_0 - \\mu_1$')\n",
    "plt.xlabel('Time Difference')\n",
    "plt.ylabel('Frequency')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From the plot, we can see our sample dataset difference in means $\\mu_0 - \\mu_1$ is far away to left on the distribution, which is already implying that such mean did not come from the null distribution with a high confidence. We will run a proper hypothesis test, but now we have an idea that most likely we expect to reject the null hypothesis.\n",
    "\n",
    "In order to do a hypothesis test using bootstrapping, we need:\n",
    "* **The null distribution** we simulated using the bootstrapping $\\mu_0 - \\mu_1$ standard deviation\n",
    "* **Sample dataset $\\mu_0 - \\mu_1$ **\n",
    "\n",
    "with the sample dataset $\\mu_0 - \\mu_1$ we can find the the p-value as a one-tailed test. We do it by finding the mean of all values in the null distribution greater or equal than the sample dataset $\\mu_0 - \\mu_1$.\n",
    "\n",
    "Then the confidence interval will be computed as 1 - p-value.\n",
    "\n",
    "Finally, we will determine if the given p-value is smaller than the type I error threshold and if we can reject the null hypothesis or we do not have enough information to do so."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Dataset 24 samples, Statistic (Diff mean): -7.9648\n",
      "Boostrapped 10000 samples, Statistic (Diff mean): -7.9042\n",
      "\n",
      "Hypothesis testing one-tail\n",
      "pval: 0.0012000000\n",
      "Computed confidence interval 99.88% > target CI 99.00% ? True \n",
      "\n",
      "Reject null hypothesis ? True , pval:0.0012000000 < alpha:0.01000\n"
     ]
    }
   ],
   "source": [
    "# Bootstrapping statistical test\n",
    "print('Dataset {} samples, Statistic (Diff mean): {:.4f}'.format(df.shape[0], u_diff))\n",
    "\n",
    "print('Boostrapped {} samples, Statistic (Diff mean): {:.4f}'.format(RESAMPLE, diff.mean()))\n",
    "\n",
    "print('\\nHypothesis testing one-tail')\n",
    "p_val = (u_diff >= null_vals).mean()\n",
    "confidence_interval = 1 - p_val\n",
    "print('pval: {:.10f}'.format(p_val))\n",
    "print('Computed confidence interval {:.2f}% > target CI {:.2f}% ? {} '.format(100.0*confidence_interval,\n",
    "                                                                     100.0*(1-alpha), p_val < alpha ))\n",
    "\n",
    "print('\\nReject null hypothesis ? {} , pval:{:.10f} < alpha:{:.5f}'.format(p_val<alpha,p_val,alpha))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "After running bootstrapping, we were able to reject the null hypothesis $H_0$ as the $pvalue < \\alpha = 0.01$ and our confidence interval is greater than 99%\n",
    "\n",
    "Another way to verify this, if we had the population distribution and parameters, is to use a Zscore test.\n",
    "Some people consider that the t-test should be used for sample sizes smaller than 30[4]. However, this is not mandatory, it all depends on your analysis and judgment. Since we got a sample size of 24 we wil not use the Z score."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='paired_t-test'></a>\n",
    "#### Two-sample Paired t-test\n",
    "\n",
    "Another way to do the hypothesis testing is through a T-test of one-tail using only our sample data.\n",
    "\n",
    ">**Note:** We used scipy ```ttest_rel()``` with input related samples congruent and incongruent.  We also have to be aware that this function is for two-tailed test, so we need to divide the p-value by 2.\n",
    "The ```ttest_rel()``` assumes our samples are normally distributed, but if we believed the distribution was not normal, we could use ```wilcoxon()``` which is a two-sample paired t-test for non-normal distributions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Scipy's two-sample paired t-test\n",
      " Statistic(Diff mean): -8.0207\n",
      " pvalue: 0.0000000205          \n",
      " Computed confidence Interval 99.999998% > target CI 99.00% ? True \n",
      "\n",
      "Reject null hypothesis ? True , pval:0.0000000205 < alpha:0.01000\n"
     ]
    }
   ],
   "source": [
    "# Scipy statistical test (a-b) - is only accounting for two tails, we need to do math on the p-value :/\n",
    "sp_tstat, sp_pval = stats.ttest_rel(a=df['Congruent'], b=df['Incongruent'])#, equal_var=False)\n",
    "\n",
    "sp_pval/=2 #divide pvalue by two because ttest_ind is a two-tailed test\n",
    "\n",
    "print('Scipy\\'s two-sample paired t-test\\n Statistic(Diff mean): {:.4f}\\n pvalue: {:.10f}\\\n",
    "          \\n Computed confidence Interval {:.6f}% > target CI {:.2f}% ? {} '.\n",
    "      format(sp_tstat, sp_pval, 100.0*(1-sp_pval),100.0*(1-alpha), sp_pval < alpha ))\n",
    "print('\\nReject null hypothesis ? {} , pval:{:.10f} < alpha:{:.5f}'.format(sp_pval<alpha,sp_pval,alpha))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this t-test we also found that we have a higer than 99% confidence interval and that our p-value is smaller than alpha. Thus, we have enough information to reject the null hypothesis. \n",
    "\n",
    ">**Note:** The p-value and the statistic we use have different values than the bootstrapping analysis. This is due to the fact that a t-test is using only the sample data (not any simulated distribution with tons of samples which is what bootstrapping does). Thus, there are different equations to calculate the t statistic that lead to a different p-value."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In both cases using bootstrapping and t-test, the pval < 0.01 which was our $\\alpha$ threshold for a Type I error. Thus we have enough information to reject the null hypothesis."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id=\"conclusions\"></a>\n",
    "## Conclusions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The null hypothesis ($H_0$) of our problem is that the reaction time of participants in the Stroop test was greater or equal for congruent words (words colored as their label, e.g. <font color='green'>GREEN</font>, <font color='blue'>BLUE</font>) than for incongruent words (words colored different than their label, e.g. <font color='red'>GREEN</font>, <font color='cyan'>PINK</font>). However, we challenged this idea proposing an alternative hypothesis saying that the reaction time for congruent words was faster than for incongruent words. During our hypothesis test, we decided to choose a confidence level of 99% resulting in a type I error threshold of $\\alpha = 0.01$. After our hypothesis test using bootstrapping, two-sample paired t-test and one-sample t-test one-tailed tests we got a $pvalue < \\alpha$. Since the pvalue was smaller than $\\alpha$ we got enough information to reject $H_0$ and conclude that based on our test we have more than 99% confidence that the participants have a faster reaction time when going through all the congruent words than when doing the same for incongruent words. This is in fact something we expected because we believed that having color not matching the words make us think more about what we see and what it means. We believe we will need to think more before speaking when looking at incongruent words. In other words, the Stroop effect indeed interferes people’s reaction time on a task when reading incogruent words."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id=\"further\"></a>\n",
    "## Further Investigation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A possible explanation for the Stroop effect is that the humand brain is trained to automatically process words rather than color when looking at text[6]. In other words, the human brain prioritizes some tasks and the way we shaped it is to read before deciding the color of words. As a result, when we are asked to only mention the color of a printed word, we need to step back and analyze the color instead of reading, which can cause a conflict in our automatic reply and give the entrance to our cognitive skill instead of the automatic reply we have developed over the time.\n",
    "\n",
    "There are some tests that branched from the Stroop effect to analyze humand cognition and sensory reactions. The one test that caught my attention is related to a test in which an arrow is drawn in up, down, left, right pointing positions and in every drawing the arrow has a word inside describing the direction it is pointing to. In the congruent case, the word matchs with the direction of the arrow. In the incongruent case, the arrow and the word would not match. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='appendix'></a>\n",
    "## Appendix - Additional Material\n",
    "\n",
    "<a id='one-sample_t-test'></a>\n",
    "### One-sample t-test\n",
    "\n",
    "As additional material to prove a two-sample paired t-test is at its core a one-sample t-test, we can run a one-sample test using ```ttest_1samp()``` function from scipy. The result should match the two-sample paired t-test as a two-sample paired t-test runs a one-sample t-test after it has the difference in means of both related groups.It also assumes their means centered at 0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Scipy's one-sample t-test\n",
      " Statistic(Diff mean): -8.0207\n",
      " pvalue: 0.0000000205        \n",
      " Computed confidence Interval 99.999998% > target CI 99.00% ? True \n",
      "\n",
      "Reject null hypothesis ? True , pval:0.0000000205 < alpha:0.01000\n"
     ]
    }
   ],
   "source": [
    "# One-sample t-test - is defined as a two-tailed test, so we must divide the resulting pvalue by 2\n",
    "onesamp_tstat, onesamp_pval = stats.ttest_1samp(df['Diff'], 0)\n",
    "\n",
    "onesamp_pval/=2 #divide pvalue by two because ttest_ind is a two-tailed test\n",
    "\n",
    "print('\\nScipy\\'s one-sample t-test\\n Statistic(Diff mean): {:.4f}\\n pvalue: {:.10f}\\\n",
    "        \\n Computed confidence Interval {:.6f}% > target CI {:.2f}% ? {} '.\n",
    "      format(onesamp_tstat, onesamp_pval, 100.0*(1-onesamp_pval),100.0*(1-alpha), onesamp_pval < alpha ))\n",
    "\n",
    "print('\\nReject null hypothesis ? {} , pval:{:.10f} < alpha:{:.5f}'.format(onesamp_pval<alpha,onesamp_pval,alpha))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='t-test_equation'></a>\n",
    "### t-test Statistic Equation\n",
    "\n",
    "\n",
    "The results from the two-sample paired t-test matches our result from a one-sample t-test, confirming that at the core of the first we run a one-sample t-test.\n",
    "\n",
    "In order to prove that our statistic is correct for the two-sample paired t-test and the one-sample t-test we can use the equation for t-test statistic:\n",
    "\n",
    "$$t = \\frac{\\bar{x_0}-\\bar{x_1}}{\\frac{s}{\\sqrt{n}}}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "T-test statistic: -8.0207\n"
     ]
    }
   ],
   "source": [
    "#Calculating our t-statistic from an equation\n",
    "n = df['Diff'].shape[0]\n",
    "t = (df['Congruent'].mean() - df['Incongruent'].mean())/(df['Diff'].std()/np.sqrt(n))\n",
    "print(\"T-test statistic: {:.4f}\".format(t))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "After implementing the equation above we can see it matches the two-sample paired and one-sample t-statistic we used to compute our p-value. This lead us to conclude that our results match and yield expected values. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## References\n",
    "\n",
    "[1] Statistics: The Science of Decisions Project Instructions.[https://docs.google.com/document/d/1-OkpZLjG_kX9J6LIQ5IltsqMzVWjh36QpnP2RYpVdPU/pub?embedded=True](https://docs.google.com/document/d/1-OkpZLjG_kX9J6LIQ5IltsqMzVWjh36QpnP2RYpVdPU/pub?embedded=True)\n",
    "\n",
    "[2] Kang,Eugine. \"Hypothesis Test\". [https://medium.com/@kangeugine/hypothesis-test-21795f788f7d](https://medium.com/@kangeugine/hypothesis-test-21795f788f7d) \n",
    "\n",
    "[3] Laerd Statistics.\"Dependent t-test for paired samples\".[https://statistics.laerd.com/statistical-guides/dependent-t-test-statistical-guide.php](https://statistics.laerd.com/statistical-guides/dependent-t-test-statistical-guide.php)\n",
    "\n",
    "[4] Statistics How To.\"T-Score vs. Z-Score: What’s the Difference?\"[http://www.statisticshowto.com/probability-and-statistics/hypothesis-testing/t-score-vs-z-score/](http://www.statisticshowto.com/probability-and-statistics/hypothesis-testing/t-score-vs-z-score/)\n",
    "\n",
    "[5] T-test table. [https://www2.palomar.edu/users/rmorrissette/Lectures/Stats/ttests/ttests.htm](https://www2.palomar.edu/users/rmorrissette/Lectures/Stats/ttests/ttests.htm)\n",
    "\n",
    "[6] Wikipedia. \"Stroop effect\". [https://en.wikipedia.org/wiki/Stroop_effect](https://en.wikipedia.org/wiki/Stroop_effect)\n",
    "\n",
    "[6]Macleod, M. Colin. [“The Stroop effect”](http://imbs.uci.edu/~kjameson/ECST/MacLeod_TheStroopEffect.pdf). Encyclopedia of Color Science and Technology,1,2015."
   ]
  }
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