From 3508ee9fe9c20ef414c020a577731ef10ff13652 Mon Sep 17 00:00:00 2001
From: Eduardo Montealvo <montealvo@hotmail.com>
Date: Sun, 7 Sep 2025 12:01:10 +0200
Subject: [PATCH] Finish hypothesis testing lab

---
 lab-hypothesis-testing.ipynb | 1123 +++++++++++++++++++++++++++++++++-
 1 file changed, 1109 insertions(+), 14 deletions(-)

diff --git a/lab-hypothesis-testing.ipynb b/lab-hypothesis-testing.ipynb
index 0cc26d5..45d53b2 100644
--- a/lab-hypothesis-testing.ipynb
+++ b/lab-hypothesis-testing.ipynb
@@ -51,11 +51,782 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 3,
+   "execution_count": 2,
    "metadata": {},
    "outputs": [
     {
      "data": {
+      "application/vnd.microsoft.datawrangler.viewer.v0+json": {
+       "columns": [
+        {
+         "name": "index",
+         "rawType": "int64",
+         "type": "integer"
+        },
+        {
+         "name": "Name",
+         "rawType": "object",
+         "type": "unknown"
+        },
+        {
+         "name": "Type 1",
+         "rawType": "object",
+         "type": "string"
+        },
+        {
+         "name": "Type 2",
+         "rawType": "object",
+         "type": "unknown"
+        },
+        {
+         "name": "HP",
+         "rawType": "int64",
+         "type": "integer"
+        },
+        {
+         "name": "Attack",
+         "rawType": "int64",
+         "type": "integer"
+        },
+        {
+         "name": "Defense",
+         "rawType": "int64",
+         "type": "integer"
+        },
+        {
+         "name": "Sp. Atk",
+         "rawType": "int64",
+         "type": "integer"
+        },
+        {
+         "name": "Sp. Def",
+         "rawType": "int64",
+         "type": "integer"
+        },
+        {
+         "name": "Speed",
+         "rawType": "int64",
+         "type": "integer"
+        },
+        {
+         "name": "Generation",
+         "rawType": "int64",
+         "type": "integer"
+        },
+        {
+         "name": "Legendary",
+         "rawType": "bool",
+         "type": "boolean"
+        }
+       ],
+       "ref": "fa1a6123-1359-4524-88ac-200a125d4aad",
+       "rows": [
+        [
+         "0",
+         "Bulbasaur",
+         "Grass",
+         "Poison",
+         "45",
+         "49",
+         "49",
+         "65",
+         "65",
+         "45",
+         "1",
+         "False"
+        ],
+        [
+         "1",
+         "Ivysaur",
+         "Grass",
+         "Poison",
+         "60",
+         "62",
+         "63",
+         "80",
+         "80",
+         "60",
+         "1",
+         "False"
+        ],
+        [
+         "2",
+         "Venusaur",
+         "Grass",
+         "Poison",
+         "80",
+         "82",
+         "83",
+         "100",
+         "100",
+         "80",
+         "1",
+         "False"
+        ],
+        [
+         "3",
+         "Mega Venusaur",
+         "Grass",
+         "Poison",
+         "80",
+         "100",
+         "123",
+         "122",
+         "120",
+         "80",
+         "1",
+         "False"
+        ],
+        [
+         "4",
+         "Charmander",
+         "Fire",
+         null,
+         "39",
+         "52",
+         "43",
+         "60",
+         "50",
+         "65",
+         "1",
+         "False"
+        ],
+        [
+         "5",
+         "Charmeleon",
+         "Fire",
+         null,
+         "58",
+         "64",
+         "58",
+         "80",
+         "65",
+         "80",
+         "1",
+         "False"
+        ],
+        [
+         "6",
+         "Charizard",
+         "Fire",
+         "Flying",
+         "78",
+         "84",
+         "78",
+         "109",
+         "85",
+         "100",
+         "1",
+         "False"
+        ],
+        [
+         "7",
+         "Mega Charizard X",
+         "Fire",
+         "Dragon",
+         "78",
+         "130",
+         "111",
+         "130",
+         "85",
+         "100",
+         "1",
+         "False"
+        ],
+        [
+         "8",
+         "Mega Charizard Y",
+         "Fire",
+         "Flying",
+         "78",
+         "104",
+         "78",
+         "159",
+         "115",
+         "100",
+         "1",
+         "False"
+        ],
+        [
+         "9",
+         "Squirtle",
+         "Water",
+         null,
+         "44",
+         "48",
+         "65",
+         "50",
+         "64",
+         "43",
+         "1",
+         "False"
+        ],
+        [
+         "10",
+         "Wartortle",
+         "Water",
+         null,
+         "59",
+         "63",
+         "80",
+         "65",
+         "80",
+         "58",
+         "1",
+         "False"
+        ],
+        [
+         "11",
+         "Blastoise",
+         "Water",
+         null,
+         "79",
+         "83",
+         "100",
+         "85",
+         "105",
+         "78",
+         "1",
+         "False"
+        ],
+        [
+         "12",
+         "Mega Blastoise",
+         "Water",
+         null,
+         "79",
+         "103",
+         "120",
+         "135",
+         "115",
+         "78",
+         "1",
+         "False"
+        ],
+        [
+         "13",
+         "Caterpie",
+         "Bug",
+         null,
+         "45",
+         "30",
+         "35",
+         "20",
+         "20",
+         "45",
+         "1",
+         "False"
+        ],
+        [
+         "14",
+         "Metapod",
+         "Bug",
+         null,
+         "50",
+         "20",
+         "55",
+         "25",
+         "25",
+         "30",
+         "1",
+         "False"
+        ],
+        [
+         "15",
+         "Butterfree",
+         "Bug",
+         "Flying",
+         "60",
+         "45",
+         "50",
+         "90",
+         "80",
+         "70",
+         "1",
+         "False"
+        ],
+        [
+         "16",
+         "Weedle",
+         "Bug",
+         "Poison",
+         "40",
+         "35",
+         "30",
+         "20",
+         "20",
+         "50",
+         "1",
+         "False"
+        ],
+        [
+         "17",
+         "Kakuna",
+         "Bug",
+         "Poison",
+         "45",
+         "25",
+         "50",
+         "25",
+         "25",
+         "35",
+         "1",
+         "False"
+        ],
+        [
+         "18",
+         "Beedrill",
+         "Bug",
+         "Poison",
+         "65",
+         "90",
+         "40",
+         "45",
+         "80",
+         "75",
+         "1",
+         "False"
+        ],
+        [
+         "19",
+         "Mega Beedrill",
+         "Bug",
+         "Poison",
+         "65",
+         "150",
+         "40",
+         "15",
+         "80",
+         "145",
+         "1",
+         "False"
+        ],
+        [
+         "20",
+         "Pidgey",
+         "Normal",
+         "Flying",
+         "40",
+         "45",
+         "40",
+         "35",
+         "35",
+         "56",
+         "1",
+         "False"
+        ],
+        [
+         "21",
+         "Pidgeotto",
+         "Normal",
+         "Flying",
+         "63",
+         "60",
+         "55",
+         "50",
+         "50",
+         "71",
+         "1",
+         "False"
+        ],
+        [
+         "22",
+         "Pidgeot",
+         "Normal",
+         "Flying",
+         "83",
+         "80",
+         "75",
+         "70",
+         "70",
+         "101",
+         "1",
+         "False"
+        ],
+        [
+         "23",
+         "Mega Pidgeot",
+         "Normal",
+         "Flying",
+         "83",
+         "80",
+         "80",
+         "135",
+         "80",
+         "121",
+         "1",
+         "False"
+        ],
+        [
+         "24",
+         "Rattata",
+         "Normal",
+         null,
+         "30",
+         "56",
+         "35",
+         "25",
+         "35",
+         "72",
+         "1",
+         "False"
+        ],
+        [
+         "25",
+         "Raticate",
+         "Normal",
+         null,
+         "55",
+         "81",
+         "60",
+         "50",
+         "70",
+         "97",
+         "1",
+         "False"
+        ],
+        [
+         "26",
+         "Spearow",
+         "Normal",
+         "Flying",
+         "40",
+         "60",
+         "30",
+         "31",
+         "31",
+         "70",
+         "1",
+         "False"
+        ],
+        [
+         "27",
+         "Fearow",
+         "Normal",
+         "Flying",
+         "65",
+         "90",
+         "65",
+         "61",
+         "61",
+         "100",
+         "1",
+         "False"
+        ],
+        [
+         "28",
+         "Ekans",
+         "Poison",
+         null,
+         "35",
+         "60",
+         "44",
+         "40",
+         "54",
+         "55",
+         "1",
+         "False"
+        ],
+        [
+         "29",
+         "Arbok",
+         "Poison",
+         null,
+         "60",
+         "85",
+         "69",
+         "65",
+         "79",
+         "80",
+         "1",
+         "False"
+        ],
+        [
+         "30",
+         "Pikachu",
+         "Electric",
+         null,
+         "35",
+         "55",
+         "40",
+         "50",
+         "50",
+         "90",
+         "1",
+         "False"
+        ],
+        [
+         "31",
+         "Raichu",
+         "Electric",
+         null,
+         "60",
+         "90",
+         "55",
+         "90",
+         "80",
+         "110",
+         "1",
+         "False"
+        ],
+        [
+         "32",
+         "Sandshrew",
+         "Ground",
+         null,
+         "50",
+         "75",
+         "85",
+         "20",
+         "30",
+         "40",
+         "1",
+         "False"
+        ],
+        [
+         "33",
+         "Sandslash",
+         "Ground",
+         null,
+         "75",
+         "100",
+         "110",
+         "45",
+         "55",
+         "65",
+         "1",
+         "False"
+        ],
+        [
+         "34",
+         "Nidoran♀",
+         "Poison",
+         null,
+         "55",
+         "47",
+         "52",
+         "40",
+         "40",
+         "41",
+         "1",
+         "False"
+        ],
+        [
+         "35",
+         "Nidorina",
+         "Poison",
+         null,
+         "70",
+         "62",
+         "67",
+         "55",
+         "55",
+         "56",
+         "1",
+         "False"
+        ],
+        [
+         "36",
+         "Nidoqueen",
+         "Poison",
+         "Ground",
+         "90",
+         "92",
+         "87",
+         "75",
+         "85",
+         "76",
+         "1",
+         "False"
+        ],
+        [
+         "37",
+         "Nidoran♂",
+         "Poison",
+         null,
+         "46",
+         "57",
+         "40",
+         "40",
+         "40",
+         "50",
+         "1",
+         "False"
+        ],
+        [
+         "38",
+         "Nidorino",
+         "Poison",
+         null,
+         "61",
+         "72",
+         "57",
+         "55",
+         "55",
+         "65",
+         "1",
+         "False"
+        ],
+        [
+         "39",
+         "Nidoking",
+         "Poison",
+         "Ground",
+         "81",
+         "102",
+         "77",
+         "85",
+         "75",
+         "85",
+         "1",
+         "False"
+        ],
+        [
+         "40",
+         "Clefairy",
+         "Fairy",
+         null,
+         "70",
+         "45",
+         "48",
+         "60",
+         "65",
+         "35",
+         "1",
+         "False"
+        ],
+        [
+         "41",
+         "Clefable",
+         "Fairy",
+         null,
+         "95",
+         "70",
+         "73",
+         "95",
+         "90",
+         "60",
+         "1",
+         "False"
+        ],
+        [
+         "42",
+         "Vulpix",
+         "Fire",
+         null,
+         "38",
+         "41",
+         "40",
+         "50",
+         "65",
+         "65",
+         "1",
+         "False"
+        ],
+        [
+         "43",
+         "Ninetales",
+         "Fire",
+         null,
+         "73",
+         "76",
+         "75",
+         "81",
+         "100",
+         "100",
+         "1",
+         "False"
+        ],
+        [
+         "44",
+         "Jigglypuff",
+         "Normal",
+         "Fairy",
+         "115",
+         "45",
+         "20",
+         "45",
+         "25",
+         "20",
+         "1",
+         "False"
+        ],
+        [
+         "45",
+         "Wigglytuff",
+         "Normal",
+         "Fairy",
+         "140",
+         "70",
+         "45",
+         "85",
+         "50",
+         "45",
+         "1",
+         "False"
+        ],
+        [
+         "46",
+         "Zubat",
+         "Poison",
+         "Flying",
+         "40",
+         "45",
+         "35",
+         "30",
+         "40",
+         "55",
+         "1",
+         "False"
+        ],
+        [
+         "47",
+         "Golbat",
+         "Poison",
+         "Flying",
+         "75",
+         "80",
+         "70",
+         "65",
+         "75",
+         "90",
+         "1",
+         "False"
+        ],
+        [
+         "48",
+         "Oddish",
+         "Grass",
+         "Poison",
+         "45",
+         "50",
+         "55",
+         "75",
+         "65",
+         "30",
+         "1",
+         "False"
+        ],
+        [
+         "49",
+         "Gloom",
+         "Grass",
+         "Poison",
+         "60",
+         "65",
+         "70",
+         "85",
+         "75",
+         "40",
+         "1",
+         "False"
+        ]
+       ],
+       "shape": {
+        "columns": 11,
+        "rows": 800
+       }
+      },
       "text/html": [
        "<div>\n",
        "<style scoped>\n",
@@ -278,7 +1049,7 @@
        "[800 rows x 11 columns]"
       ]
      },
-     "execution_count": 3,
+     "execution_count": 2,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -288,6 +1059,58 @@
     "df"
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "0\n",
+      "0\n",
+      "386\n",
+      "int64\n"
+     ]
+    }
+   ],
+   "source": [
+    "# Check for missing values in the relevant columns\n",
+    "print(df[\"HP\"].isna().sum())\n",
+    "print(df[\"Type 1\"].isna().sum())\n",
+    "print(df[\"Type 2\"].isna().sum())\n",
+    "\n",
+    "# Check the data types\n",
+    "print(df[\"HP\"].dtype)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "n_dragon = 50, mean = 82.90\n",
+      "n_grass  = 95, mean = 66.05\n"
+     ]
+    }
+   ],
+   "source": [
+    "# Extract HP values for Dragon-type Pokémon\n",
+    "dragon_hp = df[(df[\"Type 1\"]==\"Dragon\") | (df[\"Type 2\"]==\"Dragon\")][\"HP\"].dropna()\n",
+    "\n",
+    "# Extract HP values for Grass-type Pokémon\n",
+    "grass_hp = df[(df[\"Type 1\"]==\"Grass\") | (df[\"Type 2\"]==\"Grass\")][\"HP\"].dropna()\n",
+    "\n",
+    "# Quick check: sample sizes and means\n",
+    "print(f\"n_dragon = {len(dragon_hp)}, mean = {dragon_hp.mean():.2f}\")\n",
+    "print(f\"n_grass  = {len(grass_hp)}, mean = {grass_hp.mean():.2f}\")"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -297,11 +1120,39 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 1,
+   "execution_count": 8,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "T-statistic = 4.098\n",
+      "P-value (two-sided) = 0.0001\n",
+      "P-value (one-sided, Dragon > Grass) = 0.0001\n"
+     ]
+    }
+   ],
+   "source": [
+    "# Perform Welch’s t-test (independent samples, unequal variances)\n",
+    "t_stat, p_two_sided = st.ttest_ind(dragon_hp, grass_hp, equal_var=False)\n",
+    "\n",
+    "# Convert to one-sided test: H1 = Dragon > Grass\n",
+    "p_one_sided = p_two_sided / 2 if t_stat > 0 else 1 - (p_two_sided / 2)\n",
+    "\n",
+    "print(f\"T-statistic = {t_stat:.3f}\")\n",
+    "print(f\"P-value (two-sided) = {p_two_sided:.4f}\")\n",
+    "print(f\"P-value (one-sided, Dragon > Grass) = {p_one_sided:.4f}\")\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
    "metadata": {},
    "outputs": [],
    "source": [
-    "#code here"
+    "# At 5% significance level, the p-value (0.0001) is much lower than 0.05.  \n",
+    "# Therefore, we reject H0 and conclude that Dragon-type Pokémon have, on average, higher HP than Grass-type Pokémon."
    ]
   },
   {
@@ -313,11 +1164,67 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 18,
+   "execution_count": 14,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Index(['Name', 'Type 1', 'Type 2', 'HP', 'Attack', 'Defense', 'Sp. Atk',\n",
+      "       'Sp. Def', 'Speed', 'Generation', 'Legendary'],\n",
+      "      dtype='object')\n"
+     ]
+    }
+   ],
+   "source": [
+    "# Clean column names: remove '#' and extra spaces\n",
+    "df.columns = df.columns.str.replace(\"#\", \"\").str.strip()\n",
+    "\n",
+    "print(df.columns)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 15,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "HP: t = 8.981, p = 0.0000 -> reject H0 (different means)\n",
+      "Attack: t = 10.438, p = 0.0000 -> reject H0 (different means)\n",
+      "Defense: t = 7.637, p = 0.0000 -> reject H0 (different means)\n",
+      "Sp. Atk: t = 13.417, p = 0.0000 -> reject H0 (different means)\n",
+      "Sp. Def: t = 10.016, p = 0.0000 -> reject H0 (different means)\n",
+      "Speed: t = 11.475, p = 0.0000 -> reject H0 (different means)\n"
+     ]
+    }
+   ],
+   "source": [
+    "# Independent samples t-test (Welch) for each stat: Legendary vs Non-Legendary\n",
+    "stats_cols = [\"HP\", \"Attack\", \"Defense\", \"Sp. Atk\", \"Sp. Def\", \"Speed\"]\n",
+    "alpha = 0.05  # significance level\n",
+    "\n",
+    "for col in stats_cols:\n",
+    "    leg = df.loc[df[\"Legendary\"] == True,  col].dropna()   # group 1\n",
+    "    non = df.loc[df[\"Legendary\"] == False, col].dropna()   # group 2\n",
+    "    \n",
+    "    t_stat, p_val = st.ttest_ind(leg, non, equal_var=False)  # two-sided by default\n",
+    "    decision = \"reject H0 (different means)\" if p_val < alpha else \"fail to reject H0\"\n",
+    "    \n",
+    "    print(f\"{col}: t = {t_stat:.3f}, p = {p_val:.4f} -> {decision}\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 16,
    "metadata": {},
    "outputs": [],
    "source": [
-    "#code here"
+    "# At 5% significance level, the p-values for all stats (HP, Attack, Defense, Sp. Atk, Sp. Def, Speed) are lower than 0.05.  \n",
+    "# Therefore, we reject H0 in all cases and conclude that Legendary Pokémon have significantly different stats compared to Non-Legendary Pokémon.\n"
    ]
   },
   {
@@ -337,11 +1244,132 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 5,
+   "execution_count": 19,
    "metadata": {},
    "outputs": [
     {
      "data": {
+      "application/vnd.microsoft.datawrangler.viewer.v0+json": {
+       "columns": [
+        {
+         "name": "index",
+         "rawType": "int64",
+         "type": "integer"
+        },
+        {
+         "name": "longitude",
+         "rawType": "float64",
+         "type": "float"
+        },
+        {
+         "name": "latitude",
+         "rawType": "float64",
+         "type": "float"
+        },
+        {
+         "name": "housing_median_age",
+         "rawType": "float64",
+         "type": "float"
+        },
+        {
+         "name": "total_rooms",
+         "rawType": "float64",
+         "type": "float"
+        },
+        {
+         "name": "total_bedrooms",
+         "rawType": "float64",
+         "type": "float"
+        },
+        {
+         "name": "population",
+         "rawType": "float64",
+         "type": "float"
+        },
+        {
+         "name": "households",
+         "rawType": "float64",
+         "type": "float"
+        },
+        {
+         "name": "median_income",
+         "rawType": "float64",
+         "type": "float"
+        },
+        {
+         "name": "median_house_value",
+         "rawType": "float64",
+         "type": "float"
+        }
+       ],
+       "ref": "58d9e304-eef4-48ad-8a71-6987b7448245",
+       "rows": [
+        [
+         "0",
+         "-114.31",
+         "34.19",
+         "15.0",
+         "5612.0",
+         "1283.0",
+         "1015.0",
+         "472.0",
+         "1.4936",
+         "66900.0"
+        ],
+        [
+         "1",
+         "-114.47",
+         "34.4",
+         "19.0",
+         "7650.0",
+         "1901.0",
+         "1129.0",
+         "463.0",
+         "1.82",
+         "80100.0"
+        ],
+        [
+         "2",
+         "-114.56",
+         "33.69",
+         "17.0",
+         "720.0",
+         "174.0",
+         "333.0",
+         "117.0",
+         "1.6509",
+         "85700.0"
+        ],
+        [
+         "3",
+         "-114.57",
+         "33.64",
+         "14.0",
+         "1501.0",
+         "337.0",
+         "515.0",
+         "226.0",
+         "3.1917",
+         "73400.0"
+        ],
+        [
+         "4",
+         "-114.57",
+         "33.57",
+         "20.0",
+         "1454.0",
+         "326.0",
+         "624.0",
+         "262.0",
+         "1.925",
+         "65500.0"
+        ]
+       ],
+       "shape": {
+        "columns": 9,
+        "rows": 5
+       }
+      },
       "text/html": [
        "<div>\n",
        "<style scoped>\n",
@@ -453,7 +1481,7 @@
        "4       624.0       262.0         1.9250             65500.0  "
       ]
      },
-     "execution_count": 5,
+     "execution_count": 19,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -483,22 +1511,89 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 18,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Index(['Name', 'Type 1', 'Type 2', 'HP', 'Attack', 'Defense', 'Sp. Atk',\n",
+      "       'Sp. Def', 'Speed', 'Generation', 'Legendary'],\n",
+      "      dtype='object')\n"
+     ]
+    }
+   ],
+   "source": [
+    "print(df.columns)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 20,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# School and hospital coordinates\n",
+    "school = (-118, 34)\n",
+    "hospital = (-122, 37)\n",
+    "\n",
+    "# Function to calculate Euclidean distance\n",
+    "def euclidean_distance(x1, y1, x2, y2):\n",
+    "    return np.sqrt((x1 - x2)**2 + (y1 - y2)**2)\n",
+    "\n",
+    "# Distances to school and hospital\n",
+    "df[\"dist_school\"] = euclidean_distance(df[\"longitude\"], df[\"latitude\"], school[0], school[1])\n",
+    "df[\"dist_hospital\"] = euclidean_distance(df[\"longitude\"], df[\"latitude\"], hospital[0], hospital[1])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 21,
    "metadata": {},
    "outputs": [],
-   "source": []
+   "source": [
+    "df[\"close\"] = ((df[\"dist_school\"] < 0.5) | (df[\"dist_hospital\"] < 0.5))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 22,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "T-statistic = 37.992\n",
+      "P-value = 0.0000\n"
+     ]
+    }
+   ],
+   "source": [
+    "near = df[df[\"close\"] == True][\"median_house_value\"]\n",
+    "far = df[df[\"close\"] == False][\"median_house_value\"]\n",
+    "\n",
+    "t_stat, p_val = st.ttest_ind(near, far, equal_var=False)\n",
+    "\n",
+    "print(f\"T-statistic = {t_stat:.3f}\")\n",
+    "print(f\"P-value = {p_val:.4f}\")"
+   ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 23,
    "metadata": {},
    "outputs": [],
-   "source": []
+   "source": [
+    "# At 5% significance level, the p-value is lower than 0.05.  \n",
+    "# Therefore, we reject H0 and conclude that houses close to either a school or a hospital are significantly more expensive than those farther away.\n"
+   ]
   }
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "Python 3",
+   "display_name": "data-analyst",
    "language": "python",
    "name": "python3"
   },
@@ -512,7 +1607,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.10.9"
+   "version": "3.13.5"
   }
  },
  "nbformat": 4,