From cd4f8bf8b78677070897d030e192cf1a14fabeea Mon Sep 17 00:00:00 2001
From: Jhonatan Mackalister <87587593+Jmackalister@users.noreply.github.com>
Date: Mon, 4 Sep 2023 21:23:26 -0500
Subject: [PATCH] Creado con Colaboratory

---
 .../04-stats-and-probability/assignment.ipynb | 337 +++++++++++++++---
 1 file changed, 278 insertions(+), 59 deletions(-)

diff --git a/1-Introduction/04-stats-and-probability/assignment.ipynb b/1-Introduction/04-stats-and-probability/assignment.ipynb
index 4bc3180b..45018741 100644
--- a/1-Introduction/04-stats-and-probability/assignment.ipynb
+++ b/1-Introduction/04-stats-and-probability/assignment.ipynb
@@ -1,15 +1,5 @@
 {
   "cells": [
-    {
-      "cell_type": "markdown",
-      "metadata": {
-        "id": "view-in-github",
-        "colab_type": "text"
-      },
-      "source": [
-        "<a href=\"https://colab.research.google.com/github/Jmackalister/Data-Science-For-Beginners/blob/main/1-Introduction/04-stats-and-probability/assignment.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
-      ]
-    },
     {
       "cell_type": "markdown",
       "source": [
@@ -24,7 +14,7 @@
     },
     {
       "cell_type": "code",
-      "execution_count": 29,
+      "execution_count": 37,
       "source": [
         "import pandas as pd\n",
         "import numpy as np\n",
@@ -32,23 +22,28 @@
         "import matplotlib.pyplot as plt\n",
         "\n",
         "df = pd.read_csv(\"https://raw.githubusercontent.com/Jmackalister/Data-Science-For-Beginners/main/data/diabetes.tsv\",sep='\\t')\n",
-        "df.head()"
+        "df.head(10)"
       ],
       "outputs": [
         {
           "output_type": "execute_result",
           "data": {
             "text/plain": [
-              "   AGE  SEX   BMI     BP   S1     S2    S3   S4      S5  S6    Y\n",
-              "0   59    2  32.1  101.0  157   93.2  38.0  4.0  4.8598  87  151\n",
-              "1   48    1  21.6   87.0  183  103.2  70.0  3.0  3.8918  69   75\n",
-              "2   72    2  30.5   93.0  156   93.6  41.0  4.0  4.6728  85  141\n",
-              "3   24    1  25.3   84.0  198  131.4  40.0  5.0  4.8903  89  206\n",
-              "4   50    1  23.0  101.0  192  125.4  52.0  4.0  4.2905  80  135"
+              "   AGE  SEX   BMI     BP   S1     S2    S3    S4      S5  S6    Y\n",
+              "0   59    2  32.1  101.0  157   93.2  38.0  4.00  4.8598  87  151\n",
+              "1   48    1  21.6   87.0  183  103.2  70.0  3.00  3.8918  69   75\n",
+              "2   72    2  30.5   93.0  156   93.6  41.0  4.00  4.6728  85  141\n",
+              "3   24    1  25.3   84.0  198  131.4  40.0  5.00  4.8903  89  206\n",
+              "4   50    1  23.0  101.0  192  125.4  52.0  4.00  4.2905  80  135\n",
+              "5   23    1  22.6   89.0  139   64.8  61.0  2.00  4.1897  68   97\n",
+              "6   36    2  22.0   90.0  160   99.6  50.0  3.00  3.9512  82  138\n",
+              "7   66    2  26.2  114.0  255  185.0  56.0  4.55  4.2485  92   63\n",
+              "8   60    2  32.1   83.0  179  119.4  42.0  4.00  4.4773  94  110\n",
+              "9   29    1  30.0   85.0  180   93.4  43.0  4.00  5.3845  88  310"
             ],
             "text/html": [
               "\n",
-              "  <div id=\"df-30c85aba-808e-438e-a0a9-81aba0543f8a\" class=\"colab-df-container\">\n",
+              "  <div id=\"df-45e91a8f-e634-4ee3-8ad4-be229a0fcbed\" class=\"colab-df-container\">\n",
               "    <div>\n",
               "<style scoped>\n",
               "    .dataframe tbody tr th:only-of-type {\n",
@@ -90,7 +85,7 @@
               "      <td>157</td>\n",
               "      <td>93.2</td>\n",
               "      <td>38.0</td>\n",
-              "      <td>4.0</td>\n",
+              "      <td>4.00</td>\n",
               "      <td>4.8598</td>\n",
               "      <td>87</td>\n",
               "      <td>151</td>\n",
@@ -104,7 +99,7 @@
               "      <td>183</td>\n",
               "      <td>103.2</td>\n",
               "      <td>70.0</td>\n",
-              "      <td>3.0</td>\n",
+              "      <td>3.00</td>\n",
               "      <td>3.8918</td>\n",
               "      <td>69</td>\n",
               "      <td>75</td>\n",
@@ -118,7 +113,7 @@
               "      <td>156</td>\n",
               "      <td>93.6</td>\n",
               "      <td>41.0</td>\n",
-              "      <td>4.0</td>\n",
+              "      <td>4.00</td>\n",
               "      <td>4.6728</td>\n",
               "      <td>85</td>\n",
               "      <td>141</td>\n",
@@ -132,7 +127,7 @@
               "      <td>198</td>\n",
               "      <td>131.4</td>\n",
               "      <td>40.0</td>\n",
-              "      <td>5.0</td>\n",
+              "      <td>5.00</td>\n",
               "      <td>4.8903</td>\n",
               "      <td>89</td>\n",
               "      <td>206</td>\n",
@@ -146,18 +141,88 @@
               "      <td>192</td>\n",
               "      <td>125.4</td>\n",
               "      <td>52.0</td>\n",
-              "      <td>4.0</td>\n",
+              "      <td>4.00</td>\n",
               "      <td>4.2905</td>\n",
               "      <td>80</td>\n",
               "      <td>135</td>\n",
               "    </tr>\n",
+              "    <tr>\n",
+              "      <th>5</th>\n",
+              "      <td>23</td>\n",
+              "      <td>1</td>\n",
+              "      <td>22.6</td>\n",
+              "      <td>89.0</td>\n",
+              "      <td>139</td>\n",
+              "      <td>64.8</td>\n",
+              "      <td>61.0</td>\n",
+              "      <td>2.00</td>\n",
+              "      <td>4.1897</td>\n",
+              "      <td>68</td>\n",
+              "      <td>97</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>6</th>\n",
+              "      <td>36</td>\n",
+              "      <td>2</td>\n",
+              "      <td>22.0</td>\n",
+              "      <td>90.0</td>\n",
+              "      <td>160</td>\n",
+              "      <td>99.6</td>\n",
+              "      <td>50.0</td>\n",
+              "      <td>3.00</td>\n",
+              "      <td>3.9512</td>\n",
+              "      <td>82</td>\n",
+              "      <td>138</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>7</th>\n",
+              "      <td>66</td>\n",
+              "      <td>2</td>\n",
+              "      <td>26.2</td>\n",
+              "      <td>114.0</td>\n",
+              "      <td>255</td>\n",
+              "      <td>185.0</td>\n",
+              "      <td>56.0</td>\n",
+              "      <td>4.55</td>\n",
+              "      <td>4.2485</td>\n",
+              "      <td>92</td>\n",
+              "      <td>63</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>8</th>\n",
+              "      <td>60</td>\n",
+              "      <td>2</td>\n",
+              "      <td>32.1</td>\n",
+              "      <td>83.0</td>\n",
+              "      <td>179</td>\n",
+              "      <td>119.4</td>\n",
+              "      <td>42.0</td>\n",
+              "      <td>4.00</td>\n",
+              "      <td>4.4773</td>\n",
+              "      <td>94</td>\n",
+              "      <td>110</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>9</th>\n",
+              "      <td>29</td>\n",
+              "      <td>1</td>\n",
+              "      <td>30.0</td>\n",
+              "      <td>85.0</td>\n",
+              "      <td>180</td>\n",
+              "      <td>93.4</td>\n",
+              "      <td>43.0</td>\n",
+              "      <td>4.00</td>\n",
+              "      <td>5.3845</td>\n",
+              "      <td>88</td>\n",
+              "      <td>310</td>\n",
+              "    </tr>\n",
               "  </tbody>\n",
               "</table>\n",
               "</div>\n",
               "    <div class=\"colab-df-buttons\">\n",
               "\n",
               "  <div class=\"colab-df-container\">\n",
-              "    <button class=\"colab-df-convert\" onclick=\"convertToInteractive('df-30c85aba-808e-438e-a0a9-81aba0543f8a')\"\n",
+              "    <button class=\"colab-df-convert\" onclick=\"convertToInteractive('df-45e91a8f-e634-4ee3-8ad4-be229a0fcbed')\"\n",
               "            title=\"Convert this dataframe to an interactive table.\"\n",
               "            style=\"display:none;\">\n",
               "\n",
@@ -209,12 +274,12 @@
               "\n",
               "    <script>\n",
               "      const buttonEl =\n",
-              "        document.querySelector('#df-30c85aba-808e-438e-a0a9-81aba0543f8a button.colab-df-convert');\n",
+              "        document.querySelector('#df-45e91a8f-e634-4ee3-8ad4-be229a0fcbed button.colab-df-convert');\n",
               "      buttonEl.style.display =\n",
               "        google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
               "\n",
               "      async function convertToInteractive(key) {\n",
-              "        const element = document.querySelector('#df-30c85aba-808e-438e-a0a9-81aba0543f8a');\n",
+              "        const element = document.querySelector('#df-45e91a8f-e634-4ee3-8ad4-be229a0fcbed');\n",
               "        const dataTable =\n",
               "          await google.colab.kernel.invokeFunction('convertToInteractive',\n",
               "                                                    [key], {});\n",
@@ -234,8 +299,8 @@
               "  </div>\n",
               "\n",
               "\n",
-              "<div id=\"df-52252daf-b4e1-41b4-b799-a0d24dffca46\">\n",
-              "  <button class=\"colab-df-quickchart\" onclick=\"quickchart('df-52252daf-b4e1-41b4-b799-a0d24dffca46')\"\n",
+              "<div id=\"df-0a8c3091-152e-4a01-8e2f-36e85b498fc1\">\n",
+              "  <button class=\"colab-df-quickchart\" onclick=\"quickchart('df-0a8c3091-152e-4a01-8e2f-36e85b498fc1')\"\n",
               "            title=\"Suggest charts.\"\n",
               "            style=\"display:none;\">\n",
               "\n",
@@ -354,7 +419,7 @@
               "    }\n",
               "    (() => {\n",
               "      let quickchartButtonEl =\n",
-              "        document.querySelector('#df-52252daf-b4e1-41b4-b799-a0d24dffca46 button');\n",
+              "        document.querySelector('#df-0a8c3091-152e-4a01-8e2f-36e85b498fc1 button');\n",
               "      quickchartButtonEl.style.display =\n",
               "        google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
               "    })();\n",
@@ -365,16 +430,16 @@
             ]
           },
           "metadata": {},
-          "execution_count": 29
+          "execution_count": 37
         }
       ],
       "metadata": {
-        "id": "HJs0xKfP4qCt",
-        "outputId": "12d0c557-36f5-4601-cb1a-df9506971412",
         "colab": {
           "base_uri": "https://localhost:8080/",
-          "height": 206
-        }
+          "height": 363
+        },
+        "id": "HJs0xKfP4qCt",
+        "outputId": "4d1b94d6-05d9-4fb0-b43e-c32923c5ddb2"
       }
     },
     {
@@ -432,11 +497,11 @@
         }
       ],
       "metadata": {
-        "id": "ZxVqeOXV4qC5",
-        "outputId": "4bf7a7c3-c22f-48f3-bbde-49f4c10f7005",
         "colab": {
           "base_uri": "https://localhost:8080/"
-        }
+        },
+        "id": "ZxVqeOXV4qC5",
+        "outputId": "4bf7a7c3-c22f-48f3-bbde-49f4c10f7005"
       }
     },
     {
@@ -450,14 +515,14 @@
     },
     {
       "cell_type": "code",
-      "execution_count": 15,
+      "execution_count": 38,
       "metadata": {
         "colab": {
           "base_uri": "https://localhost:8080/",
-          "height": 487
+          "height": 447
         },
         "id": "TFRfrvyKIUW8",
-        "outputId": "512eaa2a-63df-45cf-c437-efab647ecfb0"
+        "outputId": "440ae9a1-dbb6-4074-f0c3-1078da2c12bb"
       },
       "outputs": [
         {
@@ -466,15 +531,12 @@
             "text/plain": [
               "<Figure size 1500x500 with 3 Axes>"
             ],
-            "image/png": 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           },
           "metadata": {}
         }
       ],
       "source": [
-        "# Replace values 1 and 2 with \"men\" and \"women\"\n",
-        "df['SEX'] = df['SEX'].replace({1: 'men', 2: 'women'})\n",
-        "\n",
         "# Create a figure with three subplots\n",
         "fig, axes = plt.subplots(1, 3, figsize=(15, 5))\n",
         "\n",
@@ -536,12 +598,12 @@
         }
       ],
       "metadata": {
-        "id": "yKaqwzQb4qDD",
-        "outputId": "3027725f-b6ea-4802-a925-679ff84e6c8d",
         "colab": {
           "base_uri": "https://localhost:8080/",
           "height": 853
-        }
+        },
+        "id": "yKaqwzQb4qDD",
+        "outputId": "3027725f-b6ea-4802-a925-679ff84e6c8d"
       }
     },
     {
@@ -568,12 +630,12 @@
         "plt.show()"
       ],
       "metadata": {
-        "id": "-y55KMmPBtqx",
-        "outputId": "05353b4b-bdca-4019-beb0-68c962c7aafe",
         "colab": {
           "base_uri": "https://localhost:8080/",
           "height": 435
-        }
+        },
+        "id": "-y55KMmPBtqx",
+        "outputId": "05353b4b-bdca-4019-beb0-68c962c7aafe"
       },
       "execution_count": 31,
       "outputs": [
@@ -603,12 +665,12 @@
         "plt.show()"
       ],
       "metadata": {
-        "id": "gC5Fqp19FZo0",
-        "outputId": "4cc0e3ee-eb91-4fcc-81d5-3abebf656e39",
         "colab": {
           "base_uri": "https://localhost:8080/",
-          "height": 465
-        }
+          "height": 427
+        },
+        "id": "gC5Fqp19FZo0",
+        "outputId": "4cc0e3ee-eb91-4fcc-81d5-3abebf656e39"
       },
       "execution_count": 35,
       "outputs": [
@@ -653,12 +715,12 @@
         "df.groupby('SEX').agg({ 'Y' : 'mean'})"
       ],
       "metadata": {
-        "id": "9el94npsGWFw",
-        "outputId": "ec34d5d5-7ffb-4506-c279-54b01d456344",
         "colab": {
           "base_uri": "https://localhost:8080/",
           "height": 143
-        }
+        },
+        "id": "9el94npsGWFw",
+        "outputId": "ec34d5d5-7ffb-4506-c279-54b01d456344"
       },
       "execution_count": 36,
       "outputs": [
@@ -925,6 +987,163 @@
           "execution_count": 36
         }
       ]
+    },
+    {
+      "cell_type": "code",
+      "source": [
+        "import scipy.stats\n",
+        "\n",
+        "def mean_confidence_interval(data, confidence=0.95):\n",
+        "    a = 1.0 * np.array(data)\n",
+        "    n = len(a)\n",
+        "    m, se = np.mean(a), scipy.stats.sem(a)\n",
+        "    h = se * scipy.stats.t.ppf((1 + confidence) / 2., n-1)\n",
+        "    return m, h\n",
+        "\n",
+        "for p in [0.85, 0.9, 0.95]:\n",
+        "    m, h = mean_confidence_interval(df['Y'].fillna(method='pad'),p)\n",
+        "    print(f\"p={p:.2f}, mean = {m:.2f} ± {h:.2f}\")"
+      ],
+      "metadata": {
+        "id": "tZTMnAbkIkx-",
+        "outputId": "5616ad69-dfa0-4eb4-cdad-be89925b0c24",
+        "colab": {
+          "base_uri": "https://localhost:8080/"
+        }
+      },
+      "execution_count": 40,
+      "outputs": [
+        {
+          "output_type": "stream",
+          "name": "stdout",
+          "text": [
+            "p=0.85, mean = 152.13 ± 5.29\n",
+            "p=0.90, mean = 152.13 ± 6.04\n",
+            "p=0.95, mean = 152.13 ± 7.21\n"
+          ]
+        }
+      ]
+    },
+    {
+      "cell_type": "code",
+      "source": [
+        "for p in [0.85,0.9,0.95]:\n",
+        "    m1, h1 = mean_confidence_interval(df.loc[df['SEX']=='1',['Y']].fillna(method='pad'),p)\n",
+        "    m2, h2 = mean_confidence_interval(df.loc[df['SEX']=='2',['Y']].fillna(method='pad'),p)\n",
+        "    print(f'Conf={p:.2f}, 1: {m1-h1[0]:.2f}..{m1+h1[0]:.2f}, 2 : {m2-h2[0]:.2f}..{m2+h2[0]:.2f}')"
+      ],
+      "metadata": {
+        "id": "aOoz2G37HmaM",
+        "outputId": "4fbb7e19-442f-46ba-bead-9321838e3c50",
+        "colab": {
+          "base_uri": "https://localhost:8080/"
+        }
+      },
+      "execution_count": 46,
+      "outputs": [
+        {
+          "output_type": "stream",
+          "name": "stdout",
+          "text": [
+            "Conf=0.85, 1: nan..nan, 2 : nan..nan\n",
+            "Conf=0.90, 1: nan..nan, 2 : nan..nan\n",
+            "Conf=0.95, 1: nan..nan, 2 : nan..nan\n"
+          ]
+        },
+        {
+          "output_type": "stream",
+          "name": "stderr",
+          "text": [
+            "/usr/local/lib/python3.10/dist-packages/numpy/core/fromnumeric.py:3432: RuntimeWarning: Mean of empty slice.\n",
+            "  return _methods._mean(a, axis=axis, dtype=dtype,\n",
+            "/usr/local/lib/python3.10/dist-packages/numpy/core/_methods.py:190: RuntimeWarning: invalid value encountered in double_scalars\n",
+            "  ret = ret.dtype.type(ret / rcount)\n",
+            "/usr/local/lib/python3.10/dist-packages/numpy/core/_methods.py:265: RuntimeWarning: Degrees of freedom <= 0 for slice\n",
+            "  ret = _var(a, axis=axis, dtype=dtype, out=out, ddof=ddof,\n",
+            "/usr/local/lib/python3.10/dist-packages/numpy/core/_methods.py:223: RuntimeWarning: invalid value encountered in divide\n",
+            "  arrmean = um.true_divide(arrmean, div, out=arrmean, casting='unsafe',\n",
+            "/usr/local/lib/python3.10/dist-packages/numpy/core/_methods.py:254: RuntimeWarning: invalid value encountered in divide\n",
+            "  ret = um.true_divide(\n",
+            "/usr/local/lib/python3.10/dist-packages/numpy/core/fromnumeric.py:3432: RuntimeWarning: Mean of empty slice.\n",
+            "  return _methods._mean(a, axis=axis, dtype=dtype,\n",
+            "/usr/local/lib/python3.10/dist-packages/numpy/core/_methods.py:190: RuntimeWarning: invalid value encountered in double_scalars\n",
+            "  ret = ret.dtype.type(ret / rcount)\n",
+            "/usr/local/lib/python3.10/dist-packages/numpy/core/_methods.py:265: RuntimeWarning: Degrees of freedom <= 0 for slice\n",
+            "  ret = _var(a, axis=axis, dtype=dtype, out=out, ddof=ddof,\n",
+            "/usr/local/lib/python3.10/dist-packages/numpy/core/_methods.py:223: RuntimeWarning: invalid value encountered in divide\n",
+            "  arrmean = um.true_divide(arrmean, div, out=arrmean, casting='unsafe',\n",
+            "/usr/local/lib/python3.10/dist-packages/numpy/core/_methods.py:254: RuntimeWarning: invalid value encountered in divide\n",
+            "  ret = um.true_divide(\n",
+            "/usr/local/lib/python3.10/dist-packages/numpy/core/fromnumeric.py:3432: RuntimeWarning: Mean of empty slice.\n",
+            "  return _methods._mean(a, axis=axis, dtype=dtype,\n",
+            "/usr/local/lib/python3.10/dist-packages/numpy/core/_methods.py:190: RuntimeWarning: invalid value encountered in double_scalars\n",
+            "  ret = ret.dtype.type(ret / rcount)\n",
+            "/usr/local/lib/python3.10/dist-packages/numpy/core/_methods.py:265: RuntimeWarning: Degrees of freedom <= 0 for slice\n",
+            "  ret = _var(a, axis=axis, dtype=dtype, out=out, ddof=ddof,\n",
+            "/usr/local/lib/python3.10/dist-packages/numpy/core/_methods.py:223: RuntimeWarning: invalid value encountered in divide\n",
+            "  arrmean = um.true_divide(arrmean, div, out=arrmean, casting='unsafe',\n",
+            "/usr/local/lib/python3.10/dist-packages/numpy/core/_methods.py:254: RuntimeWarning: invalid value encountered in divide\n",
+            "  ret = um.true_divide(\n",
+            "/usr/local/lib/python3.10/dist-packages/numpy/core/fromnumeric.py:3432: RuntimeWarning: Mean of empty slice.\n",
+            "  return _methods._mean(a, axis=axis, dtype=dtype,\n",
+            "/usr/local/lib/python3.10/dist-packages/numpy/core/_methods.py:190: RuntimeWarning: invalid value encountered in double_scalars\n",
+            "  ret = ret.dtype.type(ret / rcount)\n",
+            "/usr/local/lib/python3.10/dist-packages/numpy/core/_methods.py:265: RuntimeWarning: Degrees of freedom <= 0 for slice\n",
+            "  ret = _var(a, axis=axis, dtype=dtype, out=out, ddof=ddof,\n",
+            "/usr/local/lib/python3.10/dist-packages/numpy/core/_methods.py:223: RuntimeWarning: invalid value encountered in divide\n",
+            "  arrmean = um.true_divide(arrmean, div, out=arrmean, casting='unsafe',\n",
+            "/usr/local/lib/python3.10/dist-packages/numpy/core/_methods.py:254: RuntimeWarning: invalid value encountered in divide\n",
+            "  ret = um.true_divide(\n",
+            "/usr/local/lib/python3.10/dist-packages/numpy/core/fromnumeric.py:3432: RuntimeWarning: Mean of empty slice.\n",
+            "  return _methods._mean(a, axis=axis, dtype=dtype,\n",
+            "/usr/local/lib/python3.10/dist-packages/numpy/core/_methods.py:190: RuntimeWarning: invalid value encountered in double_scalars\n",
+            "  ret = ret.dtype.type(ret / rcount)\n",
+            "/usr/local/lib/python3.10/dist-packages/numpy/core/_methods.py:265: RuntimeWarning: Degrees of freedom <= 0 for slice\n",
+            "  ret = _var(a, axis=axis, dtype=dtype, out=out, ddof=ddof,\n",
+            "/usr/local/lib/python3.10/dist-packages/numpy/core/_methods.py:223: RuntimeWarning: invalid value encountered in divide\n",
+            "  arrmean = um.true_divide(arrmean, div, out=arrmean, casting='unsafe',\n",
+            "/usr/local/lib/python3.10/dist-packages/numpy/core/_methods.py:254: RuntimeWarning: invalid value encountered in divide\n",
+            "  ret = um.true_divide(\n",
+            "/usr/local/lib/python3.10/dist-packages/numpy/core/fromnumeric.py:3432: RuntimeWarning: Mean of empty slice.\n",
+            "  return _methods._mean(a, axis=axis, dtype=dtype,\n",
+            "/usr/local/lib/python3.10/dist-packages/numpy/core/_methods.py:190: RuntimeWarning: invalid value encountered in double_scalars\n",
+            "  ret = ret.dtype.type(ret / rcount)\n",
+            "/usr/local/lib/python3.10/dist-packages/numpy/core/_methods.py:265: RuntimeWarning: Degrees of freedom <= 0 for slice\n",
+            "  ret = _var(a, axis=axis, dtype=dtype, out=out, ddof=ddof,\n",
+            "/usr/local/lib/python3.10/dist-packages/numpy/core/_methods.py:223: RuntimeWarning: invalid value encountered in divide\n",
+            "  arrmean = um.true_divide(arrmean, div, out=arrmean, casting='unsafe',\n",
+            "/usr/local/lib/python3.10/dist-packages/numpy/core/_methods.py:254: RuntimeWarning: invalid value encountered in divide\n",
+            "  ret = um.true_divide(\n"
+          ]
+        }
+      ]
+    },
+    {
+      "cell_type": "code",
+      "source": [
+        "from scipy.stats import ttest_ind\n",
+        "\n",
+        "tval, pval = ttest_ind(df.loc[df['SEX']=='1',['Y']], df.loc[df['SEX']=='2',['Y']],equal_var=False)\n",
+        "print(f\"T-value = {tval[0]:.2f}\\nP-value: {pval[0]}\")"
+      ],
+      "metadata": {
+        "id": "nUBcO-GJJKhQ",
+        "outputId": "765d6a56-cc37-4bff-e6ec-e5c2a9f009c8",
+        "colab": {
+          "base_uri": "https://localhost:8080/"
+        }
+      },
+      "execution_count": 44,
+      "outputs": [
+        {
+          "output_type": "stream",
+          "name": "stdout",
+          "text": [
+            "T-value = nan\n",
+            "P-value: nan\n"
+          ]
+        }
+      ]
     }
   ],
   "metadata": {
@@ -950,7 +1169,7 @@
     },
     "colab": {
       "provenance": [],
-      "include_colab_link": true
+      "toc_visible": true
     }
   },
   "nbformat": 4,