diff --git a/notebooks/v2/intro_data_science/Regression_Evaluation_and_Interpretation.ipynb b/notebooks/v2/intro_data_science/Regression_Evaluation_and_Interpretation.ipynb
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+{
+  "cells": [
+    {
+      "cell_type": "markdown",
+      "metadata": {
+        "colab_type": "text",
+        "id": "view-in-github"
+      },
+      "source": [
+        "<a href=\"https://colab.research.google.com/github/datacommonsorg/api-python/blob/colabs/Regression_Evaluation_and_Interpretation.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "metadata": {
+        "id": "myiDKzkSIAUs"
+      },
+      "source": [
+        "Copyright 2025 Google LLC.\n",
+        "SPDX-License-Identifier: Apache-2.0"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "metadata": {
+        "id": "sbPgBMt01mSB"
+      },
+      "source": [
+        "# Regression: Evaluation and Interpretation\n",
+        "In the [previous notebook](https://github.com/datacommonsorg/api-python/blob/master/notebooks/v2/intro_data_science/Regression_Basics_and_Prediction.ipynb), we saw how powerful regression can be as a tool for prediction. In this Colab, we'll take that exploration one step further: what can regression models tell us about the statistical relationships between variables?\n",
+        "\n",
+        "In particular, this colab will take a more rigorous statistical approach to regressions. We'll look at how to evaluate and interpret our regression models using statistical methods.\n",
+        "\n",
+        "## Learning objectives:\n",
+        "* Hypothesis testing with regression\n",
+        "* Regression tables\n",
+        "* Pearson correlation coefficient, $r$\n",
+        "* $R^2$ and adjusted $R^2$\n",
+        "* Interpreting weights and intercepts\n",
+        "* How correlated variables affect models\n",
+        "---\n",
+        "**Need extra help?**\n",
+        "\n",
+        "If you're new to Google Colab, take a look at [this getting started tutorial](https://colab.research.google.com/notebooks/intro.ipynb).\n",
+        "\n",
+        "To build more familiarity with the Data Commons API, check out these [Data Commons tutorials](https://docs.datacommons.org/api/python/v2/tutorials.md).\n",
+        "\n",
+        "And for help with Pandas and manipulating data frames, take a look at the [Pandas documentation](https://pandas.pydata.org/docs/reference/index.html).\n",
+        "\n",
+        "We'll be using the scikit-learn library for implementing our models today. Documentation can be found [here](https://scikit-learn.org/stable/modules/classes.html).\n",
+        "\n",
+        "As usual, if you have any other questions, please reach out to your course staff!"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "metadata": {
+        "id": "gnoowEYUIQS-"
+      },
+      "source": [
+        "## Getting set up\n",
+        "\n",
+        "\n",
+        "Run the following code boxes to load the Python libraries and data we'll be using today."
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": null,
+      "metadata": {
+        "id": "YkuB0EIS59qX"
+      },
+      "outputs": [],
+      "source": [
+        "# Setup/Imports\n",
+        "!pip install \"datacommons-client[Pandas]\" --upgrade --quiet"
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": null,
+      "metadata": {
+        "id": "u2oFQ7-v8sxY"
+      },
+      "outputs": [],
+      "source": [
+        "# Data Commons Python and Pandas APIs\n",
+        "from datacommons_client.client import DataCommonsClient\n",
+        "client = DataCommonsClient(api_key=\"your API key\")\n",
+        "\n",
+        "# For manipulating data\n",
+        "import numpy as np\n",
+        "import pandas as pd\n",
+        "\n",
+        "# For implementing models and evaluation methods\n",
+        "from sklearn import linear_model\n",
+        "from sklearn.metrics import r2_score, mean_squared_error\n",
+        "from statsmodels import api as sm\n",
+        "\n",
+        "\n",
+        "# For plotting/printing\n",
+        "from matplotlib import pyplot as plt\n",
+        "import seaborn as sns"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "metadata": {
+        "id": "WQlAj-wYjI9L"
+      },
+      "source": [
+        "### The data\n",
+        "\n",
+        "In this assignment, we'll be returning to the scenario we started in the previous notebook. As a refresher, we'll be exploring how obesity rates vary with different health or societal factors across US cities.\n",
+        "\n",
+        "Our data science question: **What can we learn about the relationship of those health and lifestyle factors to obesity rates?**"
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": null,
+      "metadata": {
+        "colab": {
+          "base_uri": "https://localhost:8080/",
+          "height": 492
+        },
+        "id": "VuA3xgSQXhK6",
+        "outputId": "6c43f4e1-6424-4058-f681-6cf31b6fb05a"
+      },
+      "outputs": [
+        {
+          "data": {
+            "application/vnd.google.colaboratory.intrinsic+json": {
+              "summary": "{\n  \"name\": \"df\",\n  \"rows\": 498,\n  \"fields\": [\n    {\n      \"column\": \"place\",\n      \"properties\": {\n        \"dtype\": \"string\",\n        \"num_unique_values\": 498,\n        \"samples\": [\n          \"geoId/5363000\",\n          \"geoId/0639892\",\n          \"geoId/1714351\"\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"City Name\",\n      \"properties\": {\n        \"dtype\": \"string\",\n        \"num_unique_values\": 475,\n        \"samples\": [\n          \"Memphis\",\n          \"Plano\",\n          \"Avondale\"\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"Count_Person\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 471628.1194557695,\n        \"min\": 76212.0,\n        \"max\": 8258035.0,\n        \"num_unique_values\": 498,\n        \"samples\": [\n          755078.0,\n          78135.0,\n          81004.0\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"Percent_Person_Obesity\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 6.3675302626575006,\n        \"min\": 14.1,\n        \"max\": 48.9,\n        \"num_unique_values\": 220,\n        \"samples\": [\n          33.4,\n          41.6,\n          23.2\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"Percent_Person_PhysicalInactivity\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 5.79401923130349,\n        \"min\": 11.2,\n        \"max\": 41.8,\n        \"num_unique_values\": 209,\n        \"samples\": [\n          25.0,\n          39.5,\n          18.4\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"Percent_Person_SleepLessThan7Hours\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 4.3313227959895455,\n        \"min\": 24.9,\n        \"max\": 49.5,\n        \"num_unique_values\": 166,\n        \"samples\": [\n          42.5,\n          25.6,\n          28.4\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"Percent_Person_WithHighBloodPressure\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 4.56550336745893,\n        \"min\": 21.3,\n        \"max\": 45.7,\n        \"num_unique_values\": 170,\n        \"samples\": [\n          40.3,\n          25.6,\n          41.2\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"Percent_Person_WithHighCholesterol\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 2.089525781869934,\n        \"min\": 24.6,\n        \"max\": 35.6,\n        \"num_unique_values\": 95,\n        \"samples\": [\n          25.9,\n          34.1,\n          27.0\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"Percent_Person_WithMentalHealthNotGood\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 2.1019573952670365,\n        \"min\": 11.5,\n        \"max\": 23.3,\n        \"num_unique_values\": 103,\n        \"samples\": [\n          15.8,\n          16.3,\n          12.7\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    }\n  ]\n}",
+              "type": "dataframe",
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+              "\n",
+              "  <div id=\"df-c25564e5-a3d2-4ac5-9865-56f6ddbf1522\" class=\"colab-df-container\">\n",
+              "    <div>\n",
+              "<style scoped>\n",
+              "    .dataframe tbody tr th:only-of-type {\n",
+              "        vertical-align: middle;\n",
+              "    }\n",
+              "\n",
+              "    .dataframe tbody tr th {\n",
+              "        vertical-align: top;\n",
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+              "    .dataframe thead th {\n",
+              "        text-align: right;\n",
+              "    }\n",
+              "</style>\n",
+              "<table border=\"1\" class=\"dataframe\">\n",
+              "  <thead>\n",
+              "    <tr style=\"text-align: right;\">\n",
+              "      <th>variable</th>\n",
+              "      <th>City Name</th>\n",
+              "      <th>Count_Person</th>\n",
+              "      <th>Percent_Person_Obesity</th>\n",
+              "      <th>Percent_Person_PhysicalInactivity</th>\n",
+              "      <th>Percent_Person_SleepLessThan7Hours</th>\n",
+              "      <th>Percent_Person_WithHighBloodPressure</th>\n",
+              "      <th>Percent_Person_WithHighCholesterol</th>\n",
+              "      <th>Percent_Person_WithMentalHealthNotGood</th>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>place</th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "    </tr>\n",
+              "  </thead>\n",
+              "  <tbody>\n",
+              "    <tr>\n",
+              "      <th>geoId/0103076</th>\n",
+              "      <td>Auburn</td>\n",
+              "      <td>82025.0</td>\n",
+              "      <td>33.0</td>\n",
+              "      <td>23.6</td>\n",
+              "      <td>36.0</td>\n",
+              "      <td>34.3</td>\n",
+              "      <td>30.6</td>\n",
+              "      <td>17.8</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>geoId/0107000</th>\n",
+              "      <td>Birmingham</td>\n",
+              "      <td>196644.0</td>\n",
+              "      <td>44.9</td>\n",
+              "      <td>32.9</td>\n",
+              "      <td>42.9</td>\n",
+              "      <td>45.0</td>\n",
+              "      <td>31.6</td>\n",
+              "      <td>19.7</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>geoId/0135896</th>\n",
+              "      <td>Hoover</td>\n",
+              "      <td>92448.0</td>\n",
+              "      <td>32.5</td>\n",
+              "      <td>19.7</td>\n",
+              "      <td>33.6</td>\n",
+              "      <td>32.6</td>\n",
+              "      <td>31.0</td>\n",
+              "      <td>15.4</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>geoId/0137000</th>\n",
+              "      <td>Huntsville</td>\n",
+              "      <td>225564.0</td>\n",
+              "      <td>37.5</td>\n",
+              "      <td>24.0</td>\n",
+              "      <td>40.0</td>\n",
+              "      <td>36.5</td>\n",
+              "      <td>31.6</td>\n",
+              "      <td>18.0</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>geoId/0150000</th>\n",
+              "      <td>Mobile</td>\n",
+              "      <td>182595.0</td>\n",
+              "      <td>44.2</td>\n",
+              "      <td>28.7</td>\n",
+              "      <td>43.4</td>\n",
+              "      <td>39.8</td>\n",
+              "      <td>32.5</td>\n",
+              "      <td>19.9</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>...</th>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>geoId/5531000</th>\n",
+              "      <td>Green Bay</td>\n",
+              "      <td>105744.0</td>\n",
+              "      <td>38.9</td>\n",
+              "      <td>26.7</td>\n",
+              "      <td>33.1</td>\n",
+              "      <td>28.1</td>\n",
+              "      <td>30.7</td>\n",
+              "      <td>17.9</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>geoId/5539225</th>\n",
+              "      <td>Kenosha</td>\n",
+              "      <td>98211.0</td>\n",
+              "      <td>43.7</td>\n",
+              "      <td>23.8</td>\n",
+              "      <td>36.6</td>\n",
+              "      <td>29.9</td>\n",
+              "      <td>30.0</td>\n",
+              "      <td>18.6</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>geoId/5548000</th>\n",
+              "      <td>Madison</td>\n",
+              "      <td>280305.0</td>\n",
+              "      <td>32.1</td>\n",
+              "      <td>18.7</td>\n",
+              "      <td>29.9</td>\n",
+              "      <td>26.6</td>\n",
+              "      <td>28.5</td>\n",
+              "      <td>15.6</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>geoId/5553000</th>\n",
+              "      <td>Milwaukee</td>\n",
+              "      <td>561385.0</td>\n",
+              "      <td>43.4</td>\n",
+              "      <td>28.8</td>\n",
+              "      <td>40.0</td>\n",
+              "      <td>36.7</td>\n",
+              "      <td>30.1</td>\n",
+              "      <td>19.0</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>geoId/5566000</th>\n",
+              "      <td>Racine</td>\n",
+              "      <td>76602.0</td>\n",
+              "      <td>42.9</td>\n",
+              "      <td>28.2</td>\n",
+              "      <td>39.2</td>\n",
+              "      <td>32.3</td>\n",
+              "      <td>32.0</td>\n",
+              "      <td>18.4</td>\n",
+              "    </tr>\n",
+              "  </tbody>\n",
+              "</table>\n",
+              "<p>498 rows × 8 columns</p>\n",
+              "</div>\n",
+              "    <div class=\"colab-df-buttons\">\n",
+              "\n",
+              "  <div class=\"colab-df-container\">\n",
+              "    <button class=\"colab-df-convert\" onclick=\"convertToInteractive('df-c25564e5-a3d2-4ac5-9865-56f6ddbf1522')\"\n",
+              "            title=\"Convert this dataframe to an interactive table.\"\n",
+              "            style=\"display:none;\">\n",
+              "\n",
+              "  <svg xmlns=\"http://www.w3.org/2000/svg\" height=\"24px\" viewBox=\"0 -960 960 960\">\n",
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+              "  </svg>\n",
+              "    </button>\n",
+              "\n",
+              "  <style>\n",
+              "    .colab-df-container {\n",
+              "      display:flex;\n",
+              "      gap: 12px;\n",
+              "    }\n",
+              "\n",
+              "    .colab-df-convert {\n",
+              "      background-color: #E8F0FE;\n",
+              "      border: none;\n",
+              "      border-radius: 50%;\n",
+              "      cursor: pointer;\n",
+              "      display: none;\n",
+              "      fill: #1967D2;\n",
+              "      height: 32px;\n",
+              "      padding: 0 0 0 0;\n",
+              "      width: 32px;\n",
+              "    }\n",
+              "\n",
+              "    .colab-df-convert:hover {\n",
+              "      background-color: #E2EBFA;\n",
+              "      box-shadow: 0px 1px 2px rgba(60, 64, 67, 0.3), 0px 1px 3px 1px rgba(60, 64, 67, 0.15);\n",
+              "      fill: #174EA6;\n",
+              "    }\n",
+              "\n",
+              "    .colab-df-buttons div {\n",
+              "      margin-bottom: 4px;\n",
+              "    }\n",
+              "\n",
+              "    [theme=dark] .colab-df-convert {\n",
+              "      background-color: #3B4455;\n",
+              "      fill: #D2E3FC;\n",
+              "    }\n",
+              "\n",
+              "    [theme=dark] .colab-df-convert:hover {\n",
+              "      background-color: #434B5C;\n",
+              "      box-shadow: 0px 1px 3px 1px rgba(0, 0, 0, 0.15);\n",
+              "      filter: drop-shadow(0px 1px 2px rgba(0, 0, 0, 0.3));\n",
+              "      fill: #FFFFFF;\n",
+              "    }\n",
+              "  </style>\n",
+              "\n",
+              "    <script>\n",
+              "      const buttonEl =\n",
+              "        document.querySelector('#df-c25564e5-a3d2-4ac5-9865-56f6ddbf1522 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-c25564e5-a3d2-4ac5-9865-56f6ddbf1522');\n",
+              "        const dataTable =\n",
+              "          await google.colab.kernel.invokeFunction('convertToInteractive',\n",
+              "                                                    [key], {});\n",
+              "        if (!dataTable) return;\n",
+              "\n",
+              "        const docLinkHtml = 'Like what you see? Visit the ' +\n",
+              "          '<a target=\"_blank\" href=https://colab.research.google.com/notebooks/data_table.ipynb>data table notebook</a>'\n",
+              "          + ' to learn more about interactive tables.';\n",
+              "        element.innerHTML = '';\n",
+              "        dataTable['output_type'] = 'display_data';\n",
+              "        await google.colab.output.renderOutput(dataTable, element);\n",
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+              "        docLink.innerHTML = docLinkHtml;\n",
+              "        element.appendChild(docLink);\n",
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+              "      <button class=\"colab-df-quickchart\" onclick=\"quickchart('df-55b65718-94ff-4a23-8805-0585092f88a9')\"\n",
+              "                title=\"Suggest charts\"\n",
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+              "    </g>\n",
+              "</svg>\n",
+              "      </button>\n",
+              "\n",
+              "<style>\n",
+              "  .colab-df-quickchart {\n",
+              "      --bg-color: #E8F0FE;\n",
+              "      --fill-color: #1967D2;\n",
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+              "      --disabled-fill-color: #AAA;\n",
+              "      --disabled-bg-color: #DDD;\n",
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+              "\n",
+              "  [theme=dark] .colab-df-quickchart {\n",
+              "      --bg-color: #3B4455;\n",
+              "      --fill-color: #D2E3FC;\n",
+              "      --hover-bg-color: #434B5C;\n",
+              "      --hover-fill-color: #FFFFFF;\n",
+              "      --disabled-bg-color: #3B4455;\n",
+              "      --disabled-fill-color: #666;\n",
+              "  }\n",
+              "\n",
+              "  .colab-df-quickchart {\n",
+              "    background-color: var(--bg-color);\n",
+              "    border: none;\n",
+              "    border-radius: 50%;\n",
+              "    cursor: pointer;\n",
+              "    display: none;\n",
+              "    fill: var(--fill-color);\n",
+              "    height: 32px;\n",
+              "    padding: 0;\n",
+              "    width: 32px;\n",
+              "  }\n",
+              "\n",
+              "  .colab-df-quickchart:hover {\n",
+              "    background-color: var(--hover-bg-color);\n",
+              "    box-shadow: 0 1px 2px rgba(60, 64, 67, 0.3), 0 1px 3px 1px rgba(60, 64, 67, 0.15);\n",
+              "    fill: var(--button-hover-fill-color);\n",
+              "  }\n",
+              "\n",
+              "  .colab-df-quickchart-complete:disabled,\n",
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+              "          const quickchartButtonEl =\n",
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+              "          try {\n",
+              "            const charts = await google.colab.kernel.invokeFunction(\n",
+              "                'suggestCharts', [key], {});\n",
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+              "          quickchartButtonEl.classList.add('colab-df-quickchart-complete');\n",
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+              "    </style>\n",
+              "    <button class=\"colab-df-generate\" onclick=\"generateWithVariable('df')\"\n",
+              "            title=\"Generate code using this dataframe.\"\n",
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+              "        google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
+              "\n",
+              "      buttonEl.onclick = () => {\n",
+              "        google.colab.notebook.generateWithVariable('df');\n",
+              "      }\n",
+              "      })();\n",
+              "    </script>\n",
+              "  </div>\n",
+              "\n",
+              "    </div>\n",
+              "  </div>\n"
+            ],
+            "text/plain": [
+              "variable        City Name  Count_Person  Percent_Person_Obesity  \\\n",
+              "place                                                             \n",
+              "geoId/0103076      Auburn       82025.0                    33.0   \n",
+              "geoId/0107000  Birmingham      196644.0                    44.9   \n",
+              "geoId/0135896      Hoover       92448.0                    32.5   \n",
+              "geoId/0137000  Huntsville      225564.0                    37.5   \n",
+              "geoId/0150000      Mobile      182595.0                    44.2   \n",
+              "...                   ...           ...                     ...   \n",
+              "geoId/5531000   Green Bay      105744.0                    38.9   \n",
+              "geoId/5539225     Kenosha       98211.0                    43.7   \n",
+              "geoId/5548000     Madison      280305.0                    32.1   \n",
+              "geoId/5553000   Milwaukee      561385.0                    43.4   \n",
+              "geoId/5566000      Racine       76602.0                    42.9   \n",
+              "\n",
+              "variable       Percent_Person_PhysicalInactivity  \\\n",
+              "place                                              \n",
+              "geoId/0103076                               23.6   \n",
+              "geoId/0107000                               32.9   \n",
+              "geoId/0135896                               19.7   \n",
+              "geoId/0137000                               24.0   \n",
+              "geoId/0150000                               28.7   \n",
+              "...                                          ...   \n",
+              "geoId/5531000                               26.7   \n",
+              "geoId/5539225                               23.8   \n",
+              "geoId/5548000                               18.7   \n",
+              "geoId/5553000                               28.8   \n",
+              "geoId/5566000                               28.2   \n",
+              "\n",
+              "variable       Percent_Person_SleepLessThan7Hours  \\\n",
+              "place                                               \n",
+              "geoId/0103076                                36.0   \n",
+              "geoId/0107000                                42.9   \n",
+              "geoId/0135896                                33.6   \n",
+              "geoId/0137000                                40.0   \n",
+              "geoId/0150000                                43.4   \n",
+              "...                                           ...   \n",
+              "geoId/5531000                                33.1   \n",
+              "geoId/5539225                                36.6   \n",
+              "geoId/5548000                                29.9   \n",
+              "geoId/5553000                                40.0   \n",
+              "geoId/5566000                                39.2   \n",
+              "\n",
+              "variable       Percent_Person_WithHighBloodPressure  \\\n",
+              "place                                                 \n",
+              "geoId/0103076                                  34.3   \n",
+              "geoId/0107000                                  45.0   \n",
+              "geoId/0135896                                  32.6   \n",
+              "geoId/0137000                                  36.5   \n",
+              "geoId/0150000                                  39.8   \n",
+              "...                                             ...   \n",
+              "geoId/5531000                                  28.1   \n",
+              "geoId/5539225                                  29.9   \n",
+              "geoId/5548000                                  26.6   \n",
+              "geoId/5553000                                  36.7   \n",
+              "geoId/5566000                                  32.3   \n",
+              "\n",
+              "variable       Percent_Person_WithHighCholesterol  \\\n",
+              "place                                               \n",
+              "geoId/0103076                                30.6   \n",
+              "geoId/0107000                                31.6   \n",
+              "geoId/0135896                                31.0   \n",
+              "geoId/0137000                                31.6   \n",
+              "geoId/0150000                                32.5   \n",
+              "...                                           ...   \n",
+              "geoId/5531000                                30.7   \n",
+              "geoId/5539225                                30.0   \n",
+              "geoId/5548000                                28.5   \n",
+              "geoId/5553000                                30.1   \n",
+              "geoId/5566000                                32.0   \n",
+              "\n",
+              "variable       Percent_Person_WithMentalHealthNotGood  \n",
+              "place                                                  \n",
+              "geoId/0103076                                    17.8  \n",
+              "geoId/0107000                                    19.7  \n",
+              "geoId/0135896                                    15.4  \n",
+              "geoId/0137000                                    18.0  \n",
+              "geoId/0150000                                    19.9  \n",
+              "...                                               ...  \n",
+              "geoId/5531000                                    17.9  \n",
+              "geoId/5539225                                    18.6  \n",
+              "geoId/5548000                                    15.6  \n",
+              "geoId/5553000                                    19.0  \n",
+              "geoId/5566000                                    18.4  \n",
+              "\n",
+              "[498 rows x 8 columns]"
+            ]
+          },
+          "metadata": {},
+          "output_type": "display_data"
+        }
+      ],
+      "source": [
+        "# Load the data we'll be using\n",
+        "\n",
+        "# Fetch the population of the US cities\n",
+        "city_pop = client.observation.fetch_observations_by_entity_type(\n",
+        "    date=\"latest\",\n",
+        "    parent_entity=\"country/USA\",\n",
+        "    entity_type=\"City\",\n",
+        "    variable_dcids=\"Count_Person\",\n",
+        "    filter_facet_ids=\"2176550201\" # USCensusPEP_Annual_Population\n",
+        ").byVariable[\"Count_Person\"].byEntity\n",
+        "city_pop_dict = {\n",
+        "    city: data[\"orderedFacets\"][0].observations[0].value\n",
+        "    for city, data in city_pop.items()\n",
+        "    }\n",
+        "\n",
+        "# Filter to the top 500 cities\n",
+        "cities = [\n",
+        "    item[0]\n",
+        "    for item in sorted(\n",
+        "        city_pop_dict.items(),\n",
+        "        key=lambda item: item[1],\n",
+        "        reverse=True)[:500]\n",
+        "    ]\n",
+        "\n",
+        "# We've compiled a list of some nice Data Commons Statistical Variables\n",
+        "# to use as features for you\n",
+        "stat_vars_to_query = [\n",
+        "  \"Count_Person\",\n",
+        "  \"Percent_Person_PhysicalInactivity\",\n",
+        "  \"Percent_Person_SleepLessThan7Hours\",\n",
+        "  \"Percent_Person_WithHighBloodPressure\",\n",
+        "  \"Percent_Person_WithMentalHealthNotGood\",\n",
+        "  \"Percent_Person_WithHighCholesterol\",\n",
+        "  \"Percent_Person_Obesity\"\n",
+        "\n",
+        "]\n",
+        "\n",
+        "# Query Data Commons for the data\n",
+        "raw_features_df = client.observations_dataframe(\n",
+        "    variable_dcids=stat_vars_to_query,\n",
+        "    date=\"latest\",\n",
+        "    entity_dcids=cities)\n",
+        "\n",
+        "# Filter to highest ranked facet for each entity and variable\n",
+        "df = raw_features_df.copy(deep=True)\n",
+        "df = df.groupby([\"entity\", \"entity_name\", \"variable\"]).first().reset_index()\n",
+        "\n",
+        "# Select required columns and pivot by variable\n",
+        "df = df[[\"entity\", \"entity_name\", \"variable\", \"value\"]]\n",
+        "df = df.pivot(index=[\"entity\", \"entity_name\"], columns=\"variable\", values=\"value\")\n",
+        "df = df.dropna()\n",
+        "\n",
+        "# Rename columns and order alphabetically\n",
+        "df = df.reset_index()\n",
+        "df.rename(columns={\"entity\":\"place\", \"entity_name\": \"City Name\"}, inplace=True)\n",
+        "df.set_index(\"place\", inplace=True)\n",
+        "df = df.reindex(sorted(df.columns), axis=1)\n",
+        "\n",
+        "# Display results\n",
+        "display(df)"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "metadata": {
+        "id": "TbvPpqYDiGmY"
+      },
+      "source": [
+        "### The model\n",
+        "\n",
+        "Run the following code box to fit an [ordinary least squares](https://en.wikipedia.org/wiki/Ordinary_least_squares) regression model to our data."
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": null,
+      "metadata": {
+        "id": "V9x5v10LwYZG"
+      },
+      "outputs": [],
+      "source": [
+        "# Fit a regression model\n",
+        "dep_var = \"Percent_Person_Obesity\"\n",
+        "y = df[dep_var].to_numpy().reshape(-1, 1)\n",
+        "x = df.loc[:, ~df.columns.isin([dep_var, \"City Name\"])]\n",
+        "x = sm.add_constant(x)\n",
+        "\n",
+        "\n",
+        "model = sm.OLS(y, x)\n",
+        "results = model.fit()"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "metadata": {
+        "id": "_h32-nQChkE6"
+      },
+      "source": [
+        "## 0) Regression tables\n",
+        "\n",
+        "When performing regression analyses, statistical packages will usually provide a _**regression table**_, which summarizes the results of the analysis.\n",
+        "\n",
+        "Run the following codebox to display the regression table for our original model. In this Colab, we'll go over some of the statistics included in the table.\n"
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": null,
+      "metadata": {
+        "colab": {
+          "base_uri": "https://localhost:8080/"
+        },
+        "id": "KvCfUveghpcJ",
+        "outputId": "be61942c-174a-405d-b317-43439e1363b0"
+      },
+      "outputs": [
+        {
+          "name": "stdout",
+          "output_type": "stream",
+          "text": [
+            "                            OLS Regression Results                            \n",
+            "==============================================================================\n",
+            "Dep. Variable:                      y   R-squared:                       0.758\n",
+            "Model:                            OLS   Adj. R-squared:                  0.755\n",
+            "Method:                 Least Squares   F-statistic:                     256.0\n",
+            "Date:                Thu, 22 May 2025   Prob (F-statistic):          1.18e-147\n",
+            "Time:                        16:02:06   Log-Likelihood:                -1275.0\n",
+            "No. Observations:                 498   AIC:                             2564.\n",
+            "Df Residuals:                     491   BIC:                             2593.\n",
+            "Df Model:                           6                                         \n",
+            "Covariance Type:            nonrobust                                         \n",
+            "==========================================================================================================\n",
+            "                                             coef    std err          t      P>|t|      [0.025      0.975]\n",
+            "----------------------------------------------------------------------------------------------------------\n",
+            "const                                     -0.1937      2.656     -0.073      0.942      -5.412       5.024\n",
+            "Count_Person                           -7.183e-07   3.02e-07     -2.381      0.018   -1.31e-06   -1.25e-07\n",
+            "Percent_Person_PhysicalInactivity          0.3053      0.046      6.577      0.000       0.214       0.396\n",
+            "Percent_Person_SleepLessThan7Hours        -0.1246      0.057     -2.192      0.029      -0.236      -0.013\n",
+            "Percent_Person_WithHighBloodPressure       0.7572      0.054     14.062      0.000       0.651       0.863\n",
+            "Percent_Person_WithHighCholesterol        -0.1352      0.077     -1.756      0.080      -0.286       0.016\n",
+            "Percent_Person_WithMentalHealthNotGood     0.6901      0.103      6.704      0.000       0.488       0.892\n",
+            "==============================================================================\n",
+            "Omnibus:                        2.835   Durbin-Watson:                   1.396\n",
+            "Prob(Omnibus):                  0.242   Jarque-Bera (JB):                2.651\n",
+            "Skew:                           0.133   Prob(JB):                        0.266\n",
+            "Kurtosis:                       3.239   Cond. No.                     9.89e+06\n",
+            "==============================================================================\n",
+            "\n",
+            "Notes:\n",
+            "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
+            "[2] The condition number is large, 9.89e+06. This might indicate that there are\n",
+            "strong multicollinearity or other numerical problems.\n"
+          ]
+        }
+      ],
+      "source": [
+        "print(results.summary())"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "metadata": {
+        "id": "aFi0LWX0OlwA"
+      },
+      "source": [
+        "## 1) Hypothesis testing\n"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "metadata": {
+        "id": "DrYP7lZ9cyHL"
+      },
+      "source": [
+        "### 1.1) Null hypotheses\n",
+        "\n",
+        "When performing statistical analyses, one usually starts with a statement of the null hypothesis. Typically for regression models, these take the form of the coefficient for a variable equaling zero.\n",
+        "\n",
+        "**1.1)** Write out the null hypotheses for each of our independent variables."
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "metadata": {
+        "id": "wAOpaTGXeiQb"
+      },
+      "source": [
+        "### 1.2) T-test\n",
+        "\n",
+        "So how do we test our null hypotheses? We use the [T-test](https://en.wikipedia.org/wiki/Student%27s_t-test#Slope_of_a_regression_line).\n",
+        "\n",
+        "Take a look at the regression table above to answer the following questions\n",
+        "\n",
+        "**Q1.2A)** According to the t-test, which variables are statistically significant?\n",
+        "\n",
+        "**Q1.2B)** For variables that are not statistically significant, should we keep them in our model? Why or why not?"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "metadata": {
+        "id": "exV8u07Bek4z"
+      },
+      "source": [
+        "### 1.3) F-test\n",
+        "\n",
+        "Beyond testing the significance of our individual variables independently, we can also test the significance of our model overall using the [F-test](https://en.wikipedia.org/wiki/F-test#Regression_problems). In particular, the F-test compares our model to one without predictors (aka, just an intercept). In other words, can our model do statistically better than just predicting the mean?\n",
+        "\n",
+        "Again use the regression table above to answer the following questions:\n",
+        "\n",
+        "**1.3A)** What is the null hypothesis for the F-test?\n",
+        "\n",
+        "**1.3B)** Can we reject the null hypothesis for our model?"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "metadata": {
+        "id": "gpk0RO17VJXz"
+      },
+      "source": [
+        "## 2) Statistical measures"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "metadata": {
+        "id": "RS-U1bdrl-c2"
+      },
+      "source": [
+        "### 2.1) Correlation coefficient $r$\n",
+        "\n",
+        "We can quantify predictiveness of variables using a _correlation coefficient_, a number that represents the degree to which two variables have a statistical relationship. The most common correlation coefficient used is the [Pearson correlation coefficient](https://en.wikipedia.org/wiki/Pearson_correlation_coefficient), also known as _Pearson's r_, which measures the strength of linear relationships between variables.\n",
+        "\n",
+        "Mathematically, the correlation coefficient is defined as:\n",
+        "$$ r = \\frac{\\sum_i (x_i - \\bar{x})(y_i - \\bar{y})}{\\sqrt{\\sum_i (x_i - \\bar{x})^2}\\sqrt{\\sum_i (y_i - \\bar{y})^2}}\n",
+        "$$\n",
+        "\n",
+        "where $x$ and $y$ are the two variables.\n",
+        "\n",
+        "Those of you with a statistics background might recognize this as the ratio of covariance to the product of their standard deviations.\n",
+        "\n",
+        "**2.1A)** Either using the mathematical definition or by exploring with code, explain what the correlation coefficient would be in the following cases:\n",
+        "\n",
+        "A) $x = y$\n",
+        "\n",
+        "B) $x = -y$\n",
+        "\n",
+        "C) $x$ and $y$ are both normally distributed variables with mean 0 and variance 1, randomly sampled independently from each other."
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": null,
+      "metadata": {
+        "colab": {
+          "base_uri": "https://localhost:8080/",
+          "height": 35
+        },
+        "id": "FwEnQEWjMQv5",
+        "outputId": "9803eadf-dc00-4c69-b337-ef0b5ed92fad"
+      },
+      "outputs": [
+        {
+          "data": {
+            "application/vnd.google.colaboratory.intrinsic+json": {
+              "type": "string"
+            },
+            "text/plain": [
+              "'\\nOptional cell for 2.1A\\n'"
+            ]
+          },
+          "execution_count": 32,
+          "metadata": {},
+          "output_type": "execute_result"
+        }
+      ],
+      "source": [
+        "\"\"\"\n",
+        "Optional cell for 2.1A\n",
+        "\"\"\"\n",
+        "\n",
+        "# Hint: Try writing code to generate values for x and y, then either write or import\n",
+        "# a function to calculate the correlation coefficient\n",
+        "\n",
+        "# Your code here"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "metadata": {
+        "id": "mPxYc8tNMq52"
+      },
+      "source": [
+        "Now run the following code box to use the Pandas `.corr()`  function to calculate the correlation coefficient between our variables. Note that pandas outputs the results as a matrix."
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": null,
+      "metadata": {
+        "colab": {
+          "base_uri": "https://localhost:8080/",
+          "height": 320
+        },
+        "id": "TKrIjyt657ir",
+        "outputId": "80ea1cfc-18ec-49bd-8c21-5a0e33502bdf"
+      },
+      "outputs": [
+        {
+          "data": {
+            "application/vnd.google.colaboratory.intrinsic+json": {
+              "summary": "{\n  \"name\": \"df[stat_vars_to_query]\",\n  \"rows\": 7,\n  \"fields\": [\n    {\n      \"column\": \"variable\",\n      \"properties\": {\n        \"dtype\": \"string\",\n        \"num_unique_values\": 7,\n        \"samples\": [\n          \"Count_Person\",\n          \"Percent_Person_PhysicalInactivity\",\n          \"Percent_Person_WithHighCholesterol\"\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"Count_Person\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 0.3692768160533883,\n        \"min\": -0.032606435112257866,\n        \"max\": 1.0,\n        \"num_unique_values\": 7,\n        \"samples\": [\n          1.0,\n          0.05966842357472978,\n          0.04809963828980299\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"Percent_Person_PhysicalInactivity\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 0.3039891315555219,\n        \"min\": 0.05966842357472978,\n        \"max\": 1.0,\n        \"num_unique_values\": 7,\n        \"samples\": [\n          0.05966842357472978,\n          1.0,\n          0.4366429779497951\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"Percent_Person_SleepLessThan7Hours\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 0.3018122804213848,\n        \"min\": 0.07380691021769277,\n        \"max\": 1.0,\n        \"num_unique_values\": 7,\n        \"samples\": [\n          0.07380691021769277,\n          0.7788343765257514,\n          0.3694331027620301\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"Percent_Person_WithHighBloodPressure\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 0.32460607938114927,\n        \"min\": 0.025619158392611367,\n        \"max\": 1.0,\n        \"num_unique_values\": 7,\n        \"samples\": [\n          0.025619158392611367,\n          0.7446432625492557,\n          0.381625664582876\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"Percent_Person_WithMentalHealthNotGood\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 0.34286073197902867,\n        \"min\": -0.006579247299365092,\n        \"max\": 1.0,\n        \"num_unique_values\": 7,\n        \"samples\": [\n          -0.006579247299365092,\n          0.7007758800234068,\n          0.21400402260098858\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"Percent_Person_WithHighCholesterol\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 0.29745289241877604,\n        \"min\": 0.04809963828980299,\n        \"max\": 1.0,\n        \"num_unique_values\": 7,\n        \"samples\": [\n          0.04809963828980299,\n          0.4366429779497951,\n          1.0\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"Percent_Person_Obesity\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 0.3527430213065325,\n        \"min\": -0.032606435112257866,\n        \"max\": 1.0,\n        \"num_unique_values\": 7,\n        \"samples\": [\n          -0.032606435112257866,\n          0.7531559280354309,\n          0.29900147207085953\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    }\n  ]\n}",
+              "type": "dataframe"
+            },
+            "text/html": [
+              "\n",
+              "  <div id=\"df-f17c4d08-1545-4434-a6da-ab400f2da0cb\" class=\"colab-df-container\">\n",
+              "    <div>\n",
+              "<style scoped>\n",
+              "    .dataframe tbody tr th:only-of-type {\n",
+              "        vertical-align: middle;\n",
+              "    }\n",
+              "\n",
+              "    .dataframe tbody tr th {\n",
+              "        vertical-align: top;\n",
+              "    }\n",
+              "\n",
+              "    .dataframe thead th {\n",
+              "        text-align: right;\n",
+              "    }\n",
+              "</style>\n",
+              "<table border=\"1\" class=\"dataframe\">\n",
+              "  <thead>\n",
+              "    <tr style=\"text-align: right;\">\n",
+              "      <th>variable</th>\n",
+              "      <th>Count_Person</th>\n",
+              "      <th>Percent_Person_PhysicalInactivity</th>\n",
+              "      <th>Percent_Person_SleepLessThan7Hours</th>\n",
+              "      <th>Percent_Person_WithHighBloodPressure</th>\n",
+              "      <th>Percent_Person_WithMentalHealthNotGood</th>\n",
+              "      <th>Percent_Person_WithHighCholesterol</th>\n",
+              "      <th>Percent_Person_Obesity</th>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>variable</th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "    </tr>\n",
+              "  </thead>\n",
+              "  <tbody>\n",
+              "    <tr>\n",
+              "      <th>Count_Person</th>\n",
+              "      <td>1.000000</td>\n",
+              "      <td>0.059668</td>\n",
+              "      <td>0.073807</td>\n",
+              "      <td>0.025619</td>\n",
+              "      <td>-0.006579</td>\n",
+              "      <td>0.048100</td>\n",
+              "      <td>-0.032606</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>Percent_Person_PhysicalInactivity</th>\n",
+              "      <td>0.059668</td>\n",
+              "      <td>1.000000</td>\n",
+              "      <td>0.778834</td>\n",
+              "      <td>0.744643</td>\n",
+              "      <td>0.700776</td>\n",
+              "      <td>0.436643</td>\n",
+              "      <td>0.753156</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>Percent_Person_SleepLessThan7Hours</th>\n",
+              "      <td>0.073807</td>\n",
+              "      <td>0.778834</td>\n",
+              "      <td>1.000000</td>\n",
+              "      <td>0.745474</td>\n",
+              "      <td>0.619343</td>\n",
+              "      <td>0.369433</td>\n",
+              "      <td>0.657111</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>Percent_Person_WithHighBloodPressure</th>\n",
+              "      <td>0.025619</td>\n",
+              "      <td>0.744643</td>\n",
+              "      <td>0.745474</td>\n",
+              "      <td>1.000000</td>\n",
+              "      <td>0.690294</td>\n",
+              "      <td>0.381626</td>\n",
+              "      <td>0.825544</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>Percent_Person_WithMentalHealthNotGood</th>\n",
+              "      <td>-0.006579</td>\n",
+              "      <td>0.700776</td>\n",
+              "      <td>0.619343</td>\n",
+              "      <td>0.690294</td>\n",
+              "      <td>1.000000</td>\n",
+              "      <td>0.214004</td>\n",
+              "      <td>0.735612</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>Percent_Person_WithHighCholesterol</th>\n",
+              "      <td>0.048100</td>\n",
+              "      <td>0.436643</td>\n",
+              "      <td>0.369433</td>\n",
+              "      <td>0.381626</td>\n",
+              "      <td>0.214004</td>\n",
+              "      <td>1.000000</td>\n",
+              "      <td>0.299001</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>Percent_Person_Obesity</th>\n",
+              "      <td>-0.032606</td>\n",
+              "      <td>0.753156</td>\n",
+              "      <td>0.657111</td>\n",
+              "      <td>0.825544</td>\n",
+              "      <td>0.735612</td>\n",
+              "      <td>0.299001</td>\n",
+              "      <td>1.000000</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-f17c4d08-1545-4434-a6da-ab400f2da0cb')\"\n",
+              "            title=\"Convert this dataframe to an interactive table.\"\n",
+              "            style=\"display:none;\">\n",
+              "\n",
+              "  <svg xmlns=\"http://www.w3.org/2000/svg\" height=\"24px\" viewBox=\"0 -960 960 960\">\n",
+              "    <path d=\"M120-120v-720h720v720H120Zm60-500h600v-160H180v160Zm220 220h160v-160H400v160Zm0 220h160v-160H400v160ZM180-400h160v-160H180v160Zm440 0h160v-160H620v160ZM180-180h160v-160H180v160Zm440 0h160v-160H620v160Z\"/>\n",
+              "  </svg>\n",
+              "    </button>\n",
+              "\n",
+              "  <style>\n",
+              "    .colab-df-container {\n",
+              "      display:flex;\n",
+              "      gap: 12px;\n",
+              "    }\n",
+              "\n",
+              "    .colab-df-convert {\n",
+              "      background-color: #E8F0FE;\n",
+              "      border: none;\n",
+              "      border-radius: 50%;\n",
+              "      cursor: pointer;\n",
+              "      display: none;\n",
+              "      fill: #1967D2;\n",
+              "      height: 32px;\n",
+              "      padding: 0 0 0 0;\n",
+              "      width: 32px;\n",
+              "    }\n",
+              "\n",
+              "    .colab-df-convert:hover {\n",
+              "      background-color: #E2EBFA;\n",
+              "      box-shadow: 0px 1px 2px rgba(60, 64, 67, 0.3), 0px 1px 3px 1px rgba(60, 64, 67, 0.15);\n",
+              "      fill: #174EA6;\n",
+              "    }\n",
+              "\n",
+              "    .colab-df-buttons div {\n",
+              "      margin-bottom: 4px;\n",
+              "    }\n",
+              "\n",
+              "    [theme=dark] .colab-df-convert {\n",
+              "      background-color: #3B4455;\n",
+              "      fill: #D2E3FC;\n",
+              "    }\n",
+              "\n",
+              "    [theme=dark] .colab-df-convert:hover {\n",
+              "      background-color: #434B5C;\n",
+              "      box-shadow: 0px 1px 3px 1px rgba(0, 0, 0, 0.15);\n",
+              "      filter: drop-shadow(0px 1px 2px rgba(0, 0, 0, 0.3));\n",
+              "      fill: #FFFFFF;\n",
+              "    }\n",
+              "  </style>\n",
+              "\n",
+              "    <script>\n",
+              "      const buttonEl =\n",
+              "        document.querySelector('#df-f17c4d08-1545-4434-a6da-ab400f2da0cb 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-f17c4d08-1545-4434-a6da-ab400f2da0cb');\n",
+              "        const dataTable =\n",
+              "          await google.colab.kernel.invokeFunction('convertToInteractive',\n",
+              "                                                    [key], {});\n",
+              "        if (!dataTable) return;\n",
+              "\n",
+              "        const docLinkHtml = 'Like what you see? Visit the ' +\n",
+              "          '<a target=\"_blank\" href=https://colab.research.google.com/notebooks/data_table.ipynb>data table notebook</a>'\n",
+              "          + ' to learn more about interactive tables.';\n",
+              "        element.innerHTML = '';\n",
+              "        dataTable['output_type'] = 'display_data';\n",
+              "        await google.colab.output.renderOutput(dataTable, element);\n",
+              "        const docLink = document.createElement('div');\n",
+              "        docLink.innerHTML = docLinkHtml;\n",
+              "        element.appendChild(docLink);\n",
+              "      }\n",
+              "    </script>\n",
+              "  </div>\n",
+              "\n",
+              "\n",
+              "    <div id=\"df-9d80ec6f-6e5d-45e6-9251-550a8e397bee\">\n",
+              "      <button class=\"colab-df-quickchart\" onclick=\"quickchart('df-9d80ec6f-6e5d-45e6-9251-550a8e397bee')\"\n",
+              "                title=\"Suggest charts\"\n",
+              "                style=\"display:none;\">\n",
+              "\n",
+              "<svg xmlns=\"http://www.w3.org/2000/svg\" height=\"24px\"viewBox=\"0 0 24 24\"\n",
+              "     width=\"24px\">\n",
+              "    <g>\n",
+              "        <path d=\"M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2zM9 17H7v-7h2v7zm4 0h-2V7h2v10zm4 0h-2v-4h2v4z\"/>\n",
+              "    </g>\n",
+              "</svg>\n",
+              "      </button>\n",
+              "\n",
+              "<style>\n",
+              "  .colab-df-quickchart {\n",
+              "      --bg-color: #E8F0FE;\n",
+              "      --fill-color: #1967D2;\n",
+              "      --hover-bg-color: #E2EBFA;\n",
+              "      --hover-fill-color: #174EA6;\n",
+              "      --disabled-fill-color: #AAA;\n",
+              "      --disabled-bg-color: #DDD;\n",
+              "  }\n",
+              "\n",
+              "  [theme=dark] .colab-df-quickchart {\n",
+              "      --bg-color: #3B4455;\n",
+              "      --fill-color: #D2E3FC;\n",
+              "      --hover-bg-color: #434B5C;\n",
+              "      --hover-fill-color: #FFFFFF;\n",
+              "      --disabled-bg-color: #3B4455;\n",
+              "      --disabled-fill-color: #666;\n",
+              "  }\n",
+              "\n",
+              "  .colab-df-quickchart {\n",
+              "    background-color: var(--bg-color);\n",
+              "    border: none;\n",
+              "    border-radius: 50%;\n",
+              "    cursor: pointer;\n",
+              "    display: none;\n",
+              "    fill: var(--fill-color);\n",
+              "    height: 32px;\n",
+              "    padding: 0;\n",
+              "    width: 32px;\n",
+              "  }\n",
+              "\n",
+              "  .colab-df-quickchart:hover {\n",
+              "    background-color: var(--hover-bg-color);\n",
+              "    box-shadow: 0 1px 2px rgba(60, 64, 67, 0.3), 0 1px 3px 1px rgba(60, 64, 67, 0.15);\n",
+              "    fill: var(--button-hover-fill-color);\n",
+              "  }\n",
+              "\n",
+              "  .colab-df-quickchart-complete:disabled,\n",
+              "  .colab-df-quickchart-complete:disabled:hover {\n",
+              "    background-color: var(--disabled-bg-color);\n",
+              "    fill: var(--disabled-fill-color);\n",
+              "    box-shadow: none;\n",
+              "  }\n",
+              "\n",
+              "  .colab-df-spinner {\n",
+              "    border: 2px solid var(--fill-color);\n",
+              "    border-color: transparent;\n",
+              "    border-bottom-color: var(--fill-color);\n",
+              "    animation:\n",
+              "      spin 1s steps(1) infinite;\n",
+              "  }\n",
+              "\n",
+              "  @keyframes spin {\n",
+              "    0% {\n",
+              "      border-color: transparent;\n",
+              "      border-bottom-color: var(--fill-color);\n",
+              "      border-left-color: var(--fill-color);\n",
+              "    }\n",
+              "    20% {\n",
+              "      border-color: transparent;\n",
+              "      border-left-color: var(--fill-color);\n",
+              "      border-top-color: var(--fill-color);\n",
+              "    }\n",
+              "    30% {\n",
+              "      border-color: transparent;\n",
+              "      border-left-color: var(--fill-color);\n",
+              "      border-top-color: var(--fill-color);\n",
+              "      border-right-color: var(--fill-color);\n",
+              "    }\n",
+              "    40% {\n",
+              "      border-color: transparent;\n",
+              "      border-right-color: var(--fill-color);\n",
+              "      border-top-color: var(--fill-color);\n",
+              "    }\n",
+              "    60% {\n",
+              "      border-color: transparent;\n",
+              "      border-right-color: var(--fill-color);\n",
+              "    }\n",
+              "    80% {\n",
+              "      border-color: transparent;\n",
+              "      border-right-color: var(--fill-color);\n",
+              "      border-bottom-color: var(--fill-color);\n",
+              "    }\n",
+              "    90% {\n",
+              "      border-color: transparent;\n",
+              "      border-bottom-color: var(--fill-color);\n",
+              "    }\n",
+              "  }\n",
+              "</style>\n",
+              "\n",
+              "      <script>\n",
+              "        async function quickchart(key) {\n",
+              "          const quickchartButtonEl =\n",
+              "            document.querySelector('#' + key + ' button');\n",
+              "          quickchartButtonEl.disabled = true;  // To prevent multiple clicks.\n",
+              "          quickchartButtonEl.classList.add('colab-df-spinner');\n",
+              "          try {\n",
+              "            const charts = await google.colab.kernel.invokeFunction(\n",
+              "                'suggestCharts', [key], {});\n",
+              "          } catch (error) {\n",
+              "            console.error('Error during call to suggestCharts:', error);\n",
+              "          }\n",
+              "          quickchartButtonEl.classList.remove('colab-df-spinner');\n",
+              "          quickchartButtonEl.classList.add('colab-df-quickchart-complete');\n",
+              "        }\n",
+              "        (() => {\n",
+              "          let quickchartButtonEl =\n",
+              "            document.querySelector('#df-9d80ec6f-6e5d-45e6-9251-550a8e397bee button');\n",
+              "          quickchartButtonEl.style.display =\n",
+              "            google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
+              "        })();\n",
+              "      </script>\n",
+              "    </div>\n",
+              "\n",
+              "    </div>\n",
+              "  </div>\n"
+            ],
+            "text/plain": [
+              "variable                                Count_Person  \\\n",
+              "variable                                               \n",
+              "Count_Person                                1.000000   \n",
+              "Percent_Person_PhysicalInactivity           0.059668   \n",
+              "Percent_Person_SleepLessThan7Hours          0.073807   \n",
+              "Percent_Person_WithHighBloodPressure        0.025619   \n",
+              "Percent_Person_WithMentalHealthNotGood     -0.006579   \n",
+              "Percent_Person_WithHighCholesterol          0.048100   \n",
+              "Percent_Person_Obesity                     -0.032606   \n",
+              "\n",
+              "variable                                Percent_Person_PhysicalInactivity  \\\n",
+              "variable                                                                    \n",
+              "Count_Person                                                     0.059668   \n",
+              "Percent_Person_PhysicalInactivity                                1.000000   \n",
+              "Percent_Person_SleepLessThan7Hours                               0.778834   \n",
+              "Percent_Person_WithHighBloodPressure                             0.744643   \n",
+              "Percent_Person_WithMentalHealthNotGood                           0.700776   \n",
+              "Percent_Person_WithHighCholesterol                               0.436643   \n",
+              "Percent_Person_Obesity                                           0.753156   \n",
+              "\n",
+              "variable                                Percent_Person_SleepLessThan7Hours  \\\n",
+              "variable                                                                     \n",
+              "Count_Person                                                      0.073807   \n",
+              "Percent_Person_PhysicalInactivity                                 0.778834   \n",
+              "Percent_Person_SleepLessThan7Hours                                1.000000   \n",
+              "Percent_Person_WithHighBloodPressure                              0.745474   \n",
+              "Percent_Person_WithMentalHealthNotGood                            0.619343   \n",
+              "Percent_Person_WithHighCholesterol                                0.369433   \n",
+              "Percent_Person_Obesity                                            0.657111   \n",
+              "\n",
+              "variable                                Percent_Person_WithHighBloodPressure  \\\n",
+              "variable                                                                       \n",
+              "Count_Person                                                        0.025619   \n",
+              "Percent_Person_PhysicalInactivity                                   0.744643   \n",
+              "Percent_Person_SleepLessThan7Hours                                  0.745474   \n",
+              "Percent_Person_WithHighBloodPressure                                1.000000   \n",
+              "Percent_Person_WithMentalHealthNotGood                              0.690294   \n",
+              "Percent_Person_WithHighCholesterol                                  0.381626   \n",
+              "Percent_Person_Obesity                                              0.825544   \n",
+              "\n",
+              "variable                                Percent_Person_WithMentalHealthNotGood  \\\n",
+              "variable                                                                         \n",
+              "Count_Person                                                         -0.006579   \n",
+              "Percent_Person_PhysicalInactivity                                     0.700776   \n",
+              "Percent_Person_SleepLessThan7Hours                                    0.619343   \n",
+              "Percent_Person_WithHighBloodPressure                                  0.690294   \n",
+              "Percent_Person_WithMentalHealthNotGood                                1.000000   \n",
+              "Percent_Person_WithHighCholesterol                                    0.214004   \n",
+              "Percent_Person_Obesity                                                0.735612   \n",
+              "\n",
+              "variable                                Percent_Person_WithHighCholesterol  \\\n",
+              "variable                                                                     \n",
+              "Count_Person                                                      0.048100   \n",
+              "Percent_Person_PhysicalInactivity                                 0.436643   \n",
+              "Percent_Person_SleepLessThan7Hours                                0.369433   \n",
+              "Percent_Person_WithHighBloodPressure                              0.381626   \n",
+              "Percent_Person_WithMentalHealthNotGood                            0.214004   \n",
+              "Percent_Person_WithHighCholesterol                                1.000000   \n",
+              "Percent_Person_Obesity                                            0.299001   \n",
+              "\n",
+              "variable                                Percent_Person_Obesity  \n",
+              "variable                                                        \n",
+              "Count_Person                                         -0.032606  \n",
+              "Percent_Person_PhysicalInactivity                     0.753156  \n",
+              "Percent_Person_SleepLessThan7Hours                    0.657111  \n",
+              "Percent_Person_WithHighBloodPressure                  0.825544  \n",
+              "Percent_Person_WithMentalHealthNotGood                0.735612  \n",
+              "Percent_Person_WithHighCholesterol                    0.299001  \n",
+              "Percent_Person_Obesity                                1.000000  "
+            ]
+          },
+          "execution_count": 33,
+          "metadata": {},
+          "output_type": "execute_result"
+        }
+      ],
+      "source": [
+        "# calculate correlation\n",
+        "df[stat_vars_to_query].corr()"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "metadata": {
+        "id": "XHsUlBdXNC1X"
+      },
+      "source": [
+        "\n",
+        "**2.1B)** Explain why the diagonals of the matrix have the value 1.\n",
+        "\n",
+        "**2.1C)** What is the correlation coefficient between `Count_Person` and `Percent_Person_Obesity`? What does the correlation coefficient imply about the relationship between population and obesity rate?\n",
+        "\n",
+        "**2.1D)** What is the correlation coefficient between `Percent_Person_PhysicalInactivity` and `Percent_Person_Obesity`? What does the correlation coefficient imply about the relationship between physical inactivity and obesity rate?\n",
+        "\n",
+        "**2.1E)** In general, would you prefer to include features that correlate strongly with the dependent variable, or features with no correlation in a regression model?\n",
+        "\n",
+        "**2.1F)** You find a new feature with correlation coefficient $r=-0.97$ between it and obesity rates. Would it be a good idea to add this new feature to your model?\n"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "metadata": {
+        "id": "HXG9__t8YAqy"
+      },
+      "source": [
+        "### 2.2) $R^2$ score\n",
+        "\n",
+        "To quantify how predictive a linear regression model is overall, we can use the [coefficient of determination](https://en.wikipedia.org/wiki/Coefficient_of_determination), $R^2$ (pronounced \"R squared\").\n",
+        "\n",
+        "Mathematically, the $R^2$ score is defined as:\n",
+        "\n",
+        "$$S_{residuals} = \\sum_i{(y_i - f_i)^2} \\\\\n",
+        "S_{total} = \\sum_i{(y_i - \\bar{y})^2}\\\\\n",
+        "R^2 = 1 - \\frac{S_{residuals}}{S_{total}}$$\n",
+        "\n",
+        "where $y_i$s are the actual dependent variable values, $f_i$ are the predicted dependent variable values, and $\\bar{y}$ is the average of the $y_i$'s.\n",
+        "\n",
+        "Conceptually, the $R^2$ score is a measure of explained variance. If $R^2=0.75$, that means that 75% of the variance in the dependent variable has been accounted for by our model, while 25% of the remaining variability has not.\n",
+        "\n",
+        "**2.2A)** Based on the mathematic definition, what is the range of values possible for R^2?\n",
+        "\n",
+        "**2.2B)** Come up with a situation (e.g. what would the data look like) where:\n",
+        "\n",
+        "A) $R^2 = 1.0$\n",
+        "\n",
+        "B) $R^2 = 0.0$\n",
+        "\n",
+        "Let's now analyze what the $R^2$ value is for our model."
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": null,
+      "metadata": {
+        "colab": {
+          "base_uri": "https://localhost:8080/"
+        },
+        "id": "-rnvtExD5_U1",
+        "outputId": "51ee0f86-162f-4542-eeb6-809deb556b88"
+      },
+      "outputs": [
+        {
+          "name": "stdout",
+          "output_type": "stream",
+          "text": [
+            "Model R^2 = 0.7577718062114178\n"
+          ]
+        }
+      ],
+      "source": [
+        "# calculate R^2\n",
+        "print(\"Model R^2 =\", results.rsquared)"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "metadata": {
+        "id": "k_L-1iVbXKtc"
+      },
+      "source": [
+        "**2.2C)** Is the model's $R^2$ a \"good\" score?\n",
+        "\n",
+        "**2.2D)** Can you think of any ways we can change our model that would improve the $R^2$ score?"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "metadata": {
+        "id": "t_Eieuedizkv"
+      },
+      "source": [
+        "### 2.3) Adjusted $R^2$\n",
+        "\n",
+        "There's an issue with $R^2$ scores that one needs to be aware of when working with multiple independent variables: namely, that the number of independent variables used can affect the $R^2$ score.\n",
+        "\n",
+        "Let's see this in practice. Let's create a new dataframe with an extra 100 dummy variables (randomly sampled from a 0-mean 1-variance normal distribution) tacked on."
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": null,
+      "metadata": {
+        "colab": {
+          "base_uri": "https://localhost:8080/",
+          "height": 510
+        },
+        "id": "iF9B9dPJ1P8G",
+        "outputId": "66d7dae3-11cb-4b46-a15e-846812c9f5b7"
+      },
+      "outputs": [
+        {
+          "data": {
+            "application/vnd.google.colaboratory.intrinsic+json": {
+              "type": "dataframe",
+              "variable_name": "df_padded"
+            },
+            "text/html": [
+              "\n",
+              "  <div id=\"df-5110798e-d937-4e52-8199-1a9cdb2f375b\" class=\"colab-df-container\">\n",
+              "    <div>\n",
+              "<style scoped>\n",
+              "    .dataframe tbody tr th:only-of-type {\n",
+              "        vertical-align: middle;\n",
+              "    }\n",
+              "\n",
+              "    .dataframe tbody tr th {\n",
+              "        vertical-align: top;\n",
+              "    }\n",
+              "\n",
+              "    .dataframe thead th {\n",
+              "        text-align: right;\n",
+              "    }\n",
+              "</style>\n",
+              "<table border=\"1\" class=\"dataframe\">\n",
+              "  <thead>\n",
+              "    <tr style=\"text-align: right;\">\n",
+              "      <th></th>\n",
+              "      <th>City Name</th>\n",
+              "      <th>Count_Person</th>\n",
+              "      <th>Percent_Person_Obesity</th>\n",
+              "      <th>Percent_Person_PhysicalInactivity</th>\n",
+              "      <th>Percent_Person_SleepLessThan7Hours</th>\n",
+              "      <th>Percent_Person_WithHighBloodPressure</th>\n",
+              "      <th>Percent_Person_WithHighCholesterol</th>\n",
+              "      <th>Percent_Person_WithMentalHealthNotGood</th>\n",
+              "      <th>Random Variable 0</th>\n",
+              "      <th>Random Variable 1</th>\n",
+              "      <th>...</th>\n",
+              "      <th>Random Variable 90</th>\n",
+              "      <th>Random Variable 91</th>\n",
+              "      <th>Random Variable 92</th>\n",
+              "      <th>Random Variable 93</th>\n",
+              "      <th>Random Variable 94</th>\n",
+              "      <th>Random Variable 95</th>\n",
+              "      <th>Random Variable 96</th>\n",
+              "      <th>Random Variable 97</th>\n",
+              "      <th>Random Variable 98</th>\n",
+              "      <th>Random Variable 99</th>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>place</th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "    </tr>\n",
+              "  </thead>\n",
+              "  <tbody>\n",
+              "    <tr>\n",
+              "      <th>geoId/0103076</th>\n",
+              "      <td>Auburn</td>\n",
+              "      <td>82025.0</td>\n",
+              "      <td>33.0</td>\n",
+              "      <td>23.6</td>\n",
+              "      <td>36.0</td>\n",
+              "      <td>34.3</td>\n",
+              "      <td>30.6</td>\n",
+              "      <td>17.8</td>\n",
+              "      <td>-1.564312</td>\n",
+              "      <td>0.288515</td>\n",
+              "      <td>...</td>\n",
+              "      <td>0.762521</td>\n",
+              "      <td>0.251051</td>\n",
+              "      <td>-0.697129</td>\n",
+              "      <td>-1.697195</td>\n",
+              "      <td>0.399706</td>\n",
+              "      <td>-0.557155</td>\n",
+              "      <td>0.444760</td>\n",
+              "      <td>1.787642</td>\n",
+              "      <td>0.340410</td>\n",
+              "      <td>1.535658</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>geoId/0107000</th>\n",
+              "      <td>Birmingham</td>\n",
+              "      <td>196644.0</td>\n",
+              "      <td>44.9</td>\n",
+              "      <td>32.9</td>\n",
+              "      <td>42.9</td>\n",
+              "      <td>45.0</td>\n",
+              "      <td>31.6</td>\n",
+              "      <td>19.7</td>\n",
+              "      <td>-0.580159</td>\n",
+              "      <td>0.849181</td>\n",
+              "      <td>...</td>\n",
+              "      <td>0.215843</td>\n",
+              "      <td>1.553184</td>\n",
+              "      <td>-1.766115</td>\n",
+              "      <td>1.152941</td>\n",
+              "      <td>0.712426</td>\n",
+              "      <td>0.936660</td>\n",
+              "      <td>0.576485</td>\n",
+              "      <td>-0.127241</td>\n",
+              "      <td>-0.543845</td>\n",
+              "      <td>1.536037</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>geoId/0135896</th>\n",
+              "      <td>Hoover</td>\n",
+              "      <td>92448.0</td>\n",
+              "      <td>32.5</td>\n",
+              "      <td>19.7</td>\n",
+              "      <td>33.6</td>\n",
+              "      <td>32.6</td>\n",
+              "      <td>31.0</td>\n",
+              "      <td>15.4</td>\n",
+              "      <td>-0.322616</td>\n",
+              "      <td>-1.748737</td>\n",
+              "      <td>...</td>\n",
+              "      <td>2.036116</td>\n",
+              "      <td>0.993741</td>\n",
+              "      <td>-1.786077</td>\n",
+              "      <td>-0.264808</td>\n",
+              "      <td>-1.922278</td>\n",
+              "      <td>-1.227397</td>\n",
+              "      <td>-1.723762</td>\n",
+              "      <td>0.847944</td>\n",
+              "      <td>-0.446194</td>\n",
+              "      <td>-0.320127</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>geoId/0137000</th>\n",
+              "      <td>Huntsville</td>\n",
+              "      <td>225564.0</td>\n",
+              "      <td>37.5</td>\n",
+              "      <td>24.0</td>\n",
+              "      <td>40.0</td>\n",
+              "      <td>36.5</td>\n",
+              "      <td>31.6</td>\n",
+              "      <td>18.0</td>\n",
+              "      <td>0.768514</td>\n",
+              "      <td>-0.534476</td>\n",
+              "      <td>...</td>\n",
+              "      <td>0.950064</td>\n",
+              "      <td>0.730344</td>\n",
+              "      <td>0.007471</td>\n",
+              "      <td>3.514180</td>\n",
+              "      <td>0.145648</td>\n",
+              "      <td>-1.254448</td>\n",
+              "      <td>0.275048</td>\n",
+              "      <td>-1.241024</td>\n",
+              "      <td>-0.163577</td>\n",
+              "      <td>0.376057</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>geoId/0150000</th>\n",
+              "      <td>Mobile</td>\n",
+              "      <td>182595.0</td>\n",
+              "      <td>44.2</td>\n",
+              "      <td>28.7</td>\n",
+              "      <td>43.4</td>\n",
+              "      <td>39.8</td>\n",
+              "      <td>32.5</td>\n",
+              "      <td>19.9</td>\n",
+              "      <td>0.207217</td>\n",
+              "      <td>1.028760</td>\n",
+              "      <td>...</td>\n",
+              "      <td>-0.775507</td>\n",
+              "      <td>1.338210</td>\n",
+              "      <td>-0.395432</td>\n",
+              "      <td>-0.830337</td>\n",
+              "      <td>-0.558512</td>\n",
+              "      <td>-0.367606</td>\n",
+              "      <td>-1.049303</td>\n",
+              "      <td>-3.161325</td>\n",
+              "      <td>-0.586668</td>\n",
+              "      <td>0.934307</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>...</th>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>geoId/5531000</th>\n",
+              "      <td>Green Bay</td>\n",
+              "      <td>105744.0</td>\n",
+              "      <td>38.9</td>\n",
+              "      <td>26.7</td>\n",
+              "      <td>33.1</td>\n",
+              "      <td>28.1</td>\n",
+              "      <td>30.7</td>\n",
+              "      <td>17.9</td>\n",
+              "      <td>-0.104983</td>\n",
+              "      <td>-0.856795</td>\n",
+              "      <td>...</td>\n",
+              "      <td>-0.945322</td>\n",
+              "      <td>-0.219595</td>\n",
+              "      <td>-2.113165</td>\n",
+              "      <td>0.614379</td>\n",
+              "      <td>0.110795</td>\n",
+              "      <td>-0.250010</td>\n",
+              "      <td>0.926896</td>\n",
+              "      <td>-0.526254</td>\n",
+              "      <td>-0.359181</td>\n",
+              "      <td>-1.424956</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>geoId/5539225</th>\n",
+              "      <td>Kenosha</td>\n",
+              "      <td>98211.0</td>\n",
+              "      <td>43.7</td>\n",
+              "      <td>23.8</td>\n",
+              "      <td>36.6</td>\n",
+              "      <td>29.9</td>\n",
+              "      <td>30.0</td>\n",
+              "      <td>18.6</td>\n",
+              "      <td>-0.355349</td>\n",
+              "      <td>0.348573</td>\n",
+              "      <td>...</td>\n",
+              "      <td>-0.789575</td>\n",
+              "      <td>0.590118</td>\n",
+              "      <td>-0.193587</td>\n",
+              "      <td>0.502188</td>\n",
+              "      <td>0.124404</td>\n",
+              "      <td>-0.376209</td>\n",
+              "      <td>-0.331331</td>\n",
+              "      <td>0.697165</td>\n",
+              "      <td>1.029427</td>\n",
+              "      <td>-1.143744</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>geoId/5548000</th>\n",
+              "      <td>Madison</td>\n",
+              "      <td>280305.0</td>\n",
+              "      <td>32.1</td>\n",
+              "      <td>18.7</td>\n",
+              "      <td>29.9</td>\n",
+              "      <td>26.6</td>\n",
+              "      <td>28.5</td>\n",
+              "      <td>15.6</td>\n",
+              "      <td>-0.648119</td>\n",
+              "      <td>0.025662</td>\n",
+              "      <td>...</td>\n",
+              "      <td>-0.151965</td>\n",
+              "      <td>0.835380</td>\n",
+              "      <td>-1.381286</td>\n",
+              "      <td>0.303114</td>\n",
+              "      <td>0.540398</td>\n",
+              "      <td>-0.359988</td>\n",
+              "      <td>0.007904</td>\n",
+              "      <td>0.010788</td>\n",
+              "      <td>-0.276071</td>\n",
+              "      <td>0.979319</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>geoId/5553000</th>\n",
+              "      <td>Milwaukee</td>\n",
+              "      <td>561385.0</td>\n",
+              "      <td>43.4</td>\n",
+              "      <td>28.8</td>\n",
+              "      <td>40.0</td>\n",
+              "      <td>36.7</td>\n",
+              "      <td>30.1</td>\n",
+              "      <td>19.0</td>\n",
+              "      <td>-0.154089</td>\n",
+              "      <td>-0.339432</td>\n",
+              "      <td>...</td>\n",
+              "      <td>2.255458</td>\n",
+              "      <td>1.357828</td>\n",
+              "      <td>0.692794</td>\n",
+              "      <td>0.924034</td>\n",
+              "      <td>0.951688</td>\n",
+              "      <td>-0.071096</td>\n",
+              "      <td>0.097582</td>\n",
+              "      <td>0.952135</td>\n",
+              "      <td>-1.019633</td>\n",
+              "      <td>-0.778193</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>geoId/5566000</th>\n",
+              "      <td>Racine</td>\n",
+              "      <td>76602.0</td>\n",
+              "      <td>42.9</td>\n",
+              "      <td>28.2</td>\n",
+              "      <td>39.2</td>\n",
+              "      <td>32.3</td>\n",
+              "      <td>32.0</td>\n",
+              "      <td>18.4</td>\n",
+              "      <td>1.638922</td>\n",
+              "      <td>-0.543906</td>\n",
+              "      <td>...</td>\n",
+              "      <td>0.370553</td>\n",
+              "      <td>-0.606273</td>\n",
+              "      <td>1.066660</td>\n",
+              "      <td>0.022132</td>\n",
+              "      <td>0.039135</td>\n",
+              "      <td>1.102639</td>\n",
+              "      <td>-0.438601</td>\n",
+              "      <td>-1.744647</td>\n",
+              "      <td>1.245214</td>\n",
+              "      <td>2.216294</td>\n",
+              "    </tr>\n",
+              "  </tbody>\n",
+              "</table>\n",
+              "<p>498 rows × 108 columns</p>\n",
+              "</div>\n",
+              "    <div class=\"colab-df-buttons\">\n",
+              "\n",
+              "  <div class=\"colab-df-container\">\n",
+              "    <button class=\"colab-df-convert\" onclick=\"convertToInteractive('df-5110798e-d937-4e52-8199-1a9cdb2f375b')\"\n",
+              "            title=\"Convert this dataframe to an interactive table.\"\n",
+              "            style=\"display:none;\">\n",
+              "\n",
+              "  <svg xmlns=\"http://www.w3.org/2000/svg\" height=\"24px\" viewBox=\"0 -960 960 960\">\n",
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+              "  </svg>\n",
+              "    </button>\n",
+              "\n",
+              "  <style>\n",
+              "    .colab-df-container {\n",
+              "      display:flex;\n",
+              "      gap: 12px;\n",
+              "    }\n",
+              "\n",
+              "    .colab-df-convert {\n",
+              "      background-color: #E8F0FE;\n",
+              "      border: none;\n",
+              "      border-radius: 50%;\n",
+              "      cursor: pointer;\n",
+              "      display: none;\n",
+              "      fill: #1967D2;\n",
+              "      height: 32px;\n",
+              "      padding: 0 0 0 0;\n",
+              "      width: 32px;\n",
+              "    }\n",
+              "\n",
+              "    .colab-df-convert:hover {\n",
+              "      background-color: #E2EBFA;\n",
+              "      box-shadow: 0px 1px 2px rgba(60, 64, 67, 0.3), 0px 1px 3px 1px rgba(60, 64, 67, 0.15);\n",
+              "      fill: #174EA6;\n",
+              "    }\n",
+              "\n",
+              "    .colab-df-buttons div {\n",
+              "      margin-bottom: 4px;\n",
+              "    }\n",
+              "\n",
+              "    [theme=dark] .colab-df-convert {\n",
+              "      background-color: #3B4455;\n",
+              "      fill: #D2E3FC;\n",
+              "    }\n",
+              "\n",
+              "    [theme=dark] .colab-df-convert:hover {\n",
+              "      background-color: #434B5C;\n",
+              "      box-shadow: 0px 1px 3px 1px rgba(0, 0, 0, 0.15);\n",
+              "      filter: drop-shadow(0px 1px 2px rgba(0, 0, 0, 0.3));\n",
+              "      fill: #FFFFFF;\n",
+              "    }\n",
+              "  </style>\n",
+              "\n",
+              "    <script>\n",
+              "      const buttonEl =\n",
+              "        document.querySelector('#df-5110798e-d937-4e52-8199-1a9cdb2f375b 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-5110798e-d937-4e52-8199-1a9cdb2f375b');\n",
+              "        const dataTable =\n",
+              "          await google.colab.kernel.invokeFunction('convertToInteractive',\n",
+              "                                                    [key], {});\n",
+              "        if (!dataTable) return;\n",
+              "\n",
+              "        const docLinkHtml = 'Like what you see? Visit the ' +\n",
+              "          '<a target=\"_blank\" href=https://colab.research.google.com/notebooks/data_table.ipynb>data table notebook</a>'\n",
+              "          + ' to learn more about interactive tables.';\n",
+              "        element.innerHTML = '';\n",
+              "        dataTable['output_type'] = 'display_data';\n",
+              "        await google.colab.output.renderOutput(dataTable, element);\n",
+              "        const docLink = document.createElement('div');\n",
+              "        docLink.innerHTML = docLinkHtml;\n",
+              "        element.appendChild(docLink);\n",
+              "      }\n",
+              "    </script>\n",
+              "  </div>\n",
+              "\n",
+              "\n",
+              "    <div id=\"df-5fe34167-016f-4b30-8e93-e367dffc41ab\">\n",
+              "      <button class=\"colab-df-quickchart\" onclick=\"quickchart('df-5fe34167-016f-4b30-8e93-e367dffc41ab')\"\n",
+              "                title=\"Suggest charts\"\n",
+              "                style=\"display:none;\">\n",
+              "\n",
+              "<svg xmlns=\"http://www.w3.org/2000/svg\" height=\"24px\"viewBox=\"0 0 24 24\"\n",
+              "     width=\"24px\">\n",
+              "    <g>\n",
+              "        <path d=\"M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2zM9 17H7v-7h2v7zm4 0h-2V7h2v10zm4 0h-2v-4h2v4z\"/>\n",
+              "    </g>\n",
+              "</svg>\n",
+              "      </button>\n",
+              "\n",
+              "<style>\n",
+              "  .colab-df-quickchart {\n",
+              "      --bg-color: #E8F0FE;\n",
+              "      --fill-color: #1967D2;\n",
+              "      --hover-bg-color: #E2EBFA;\n",
+              "      --hover-fill-color: #174EA6;\n",
+              "      --disabled-fill-color: #AAA;\n",
+              "      --disabled-bg-color: #DDD;\n",
+              "  }\n",
+              "\n",
+              "  [theme=dark] .colab-df-quickchart {\n",
+              "      --bg-color: #3B4455;\n",
+              "      --fill-color: #D2E3FC;\n",
+              "      --hover-bg-color: #434B5C;\n",
+              "      --hover-fill-color: #FFFFFF;\n",
+              "      --disabled-bg-color: #3B4455;\n",
+              "      --disabled-fill-color: #666;\n",
+              "  }\n",
+              "\n",
+              "  .colab-df-quickchart {\n",
+              "    background-color: var(--bg-color);\n",
+              "    border: none;\n",
+              "    border-radius: 50%;\n",
+              "    cursor: pointer;\n",
+              "    display: none;\n",
+              "    fill: var(--fill-color);\n",
+              "    height: 32px;\n",
+              "    padding: 0;\n",
+              "    width: 32px;\n",
+              "  }\n",
+              "\n",
+              "  .colab-df-quickchart:hover {\n",
+              "    background-color: var(--hover-bg-color);\n",
+              "    box-shadow: 0 1px 2px rgba(60, 64, 67, 0.3), 0 1px 3px 1px rgba(60, 64, 67, 0.15);\n",
+              "    fill: var(--button-hover-fill-color);\n",
+              "  }\n",
+              "\n",
+              "  .colab-df-quickchart-complete:disabled,\n",
+              "  .colab-df-quickchart-complete:disabled:hover {\n",
+              "    background-color: var(--disabled-bg-color);\n",
+              "    fill: var(--disabled-fill-color);\n",
+              "    box-shadow: none;\n",
+              "  }\n",
+              "\n",
+              "  .colab-df-spinner {\n",
+              "    border: 2px solid var(--fill-color);\n",
+              "    border-color: transparent;\n",
+              "    border-bottom-color: var(--fill-color);\n",
+              "    animation:\n",
+              "      spin 1s steps(1) infinite;\n",
+              "  }\n",
+              "\n",
+              "  @keyframes spin {\n",
+              "    0% {\n",
+              "      border-color: transparent;\n",
+              "      border-bottom-color: var(--fill-color);\n",
+              "      border-left-color: var(--fill-color);\n",
+              "    }\n",
+              "    20% {\n",
+              "      border-color: transparent;\n",
+              "      border-left-color: var(--fill-color);\n",
+              "      border-top-color: var(--fill-color);\n",
+              "    }\n",
+              "    30% {\n",
+              "      border-color: transparent;\n",
+              "      border-left-color: var(--fill-color);\n",
+              "      border-top-color: var(--fill-color);\n",
+              "      border-right-color: var(--fill-color);\n",
+              "    }\n",
+              "    40% {\n",
+              "      border-color: transparent;\n",
+              "      border-right-color: var(--fill-color);\n",
+              "      border-top-color: var(--fill-color);\n",
+              "    }\n",
+              "    60% {\n",
+              "      border-color: transparent;\n",
+              "      border-right-color: var(--fill-color);\n",
+              "    }\n",
+              "    80% {\n",
+              "      border-color: transparent;\n",
+              "      border-right-color: var(--fill-color);\n",
+              "      border-bottom-color: var(--fill-color);\n",
+              "    }\n",
+              "    90% {\n",
+              "      border-color: transparent;\n",
+              "      border-bottom-color: var(--fill-color);\n",
+              "    }\n",
+              "  }\n",
+              "</style>\n",
+              "\n",
+              "      <script>\n",
+              "        async function quickchart(key) {\n",
+              "          const quickchartButtonEl =\n",
+              "            document.querySelector('#' + key + ' button');\n",
+              "          quickchartButtonEl.disabled = true;  // To prevent multiple clicks.\n",
+              "          quickchartButtonEl.classList.add('colab-df-spinner');\n",
+              "          try {\n",
+              "            const charts = await google.colab.kernel.invokeFunction(\n",
+              "                'suggestCharts', [key], {});\n",
+              "          } catch (error) {\n",
+              "            console.error('Error during call to suggestCharts:', error);\n",
+              "          }\n",
+              "          quickchartButtonEl.classList.remove('colab-df-spinner');\n",
+              "          quickchartButtonEl.classList.add('colab-df-quickchart-complete');\n",
+              "        }\n",
+              "        (() => {\n",
+              "          let quickchartButtonEl =\n",
+              "            document.querySelector('#df-5fe34167-016f-4b30-8e93-e367dffc41ab button');\n",
+              "          quickchartButtonEl.style.display =\n",
+              "            google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
+              "        })();\n",
+              "      </script>\n",
+              "    </div>\n",
+              "\n",
+              "  <div id=\"id_1f652a22-3cd6-40ad-baf6-fb40cc24f889\">\n",
+              "    <style>\n",
+              "      .colab-df-generate {\n",
+              "        background-color: #E8F0FE;\n",
+              "        border: none;\n",
+              "        border-radius: 50%;\n",
+              "        cursor: pointer;\n",
+              "        display: none;\n",
+              "        fill: #1967D2;\n",
+              "        height: 32px;\n",
+              "        padding: 0 0 0 0;\n",
+              "        width: 32px;\n",
+              "      }\n",
+              "\n",
+              "      .colab-df-generate:hover {\n",
+              "        background-color: #E2EBFA;\n",
+              "        box-shadow: 0px 1px 2px rgba(60, 64, 67, 0.3), 0px 1px 3px 1px rgba(60, 64, 67, 0.15);\n",
+              "        fill: #174EA6;\n",
+              "      }\n",
+              "\n",
+              "      [theme=dark] .colab-df-generate {\n",
+              "        background-color: #3B4455;\n",
+              "        fill: #D2E3FC;\n",
+              "      }\n",
+              "\n",
+              "      [theme=dark] .colab-df-generate:hover {\n",
+              "        background-color: #434B5C;\n",
+              "        box-shadow: 0px 1px 3px 1px rgba(0, 0, 0, 0.15);\n",
+              "        filter: drop-shadow(0px 1px 2px rgba(0, 0, 0, 0.3));\n",
+              "        fill: #FFFFFF;\n",
+              "      }\n",
+              "    </style>\n",
+              "    <button class=\"colab-df-generate\" onclick=\"generateWithVariable('df_padded')\"\n",
+              "            title=\"Generate code using this dataframe.\"\n",
+              "            style=\"display:none;\">\n",
+              "\n",
+              "  <svg xmlns=\"http://www.w3.org/2000/svg\" height=\"24px\"viewBox=\"0 0 24 24\"\n",
+              "       width=\"24px\">\n",
+              "    <path d=\"M7,19H8.4L18.45,9,17,7.55,7,17.6ZM5,21V16.75L18.45,3.32a2,2,0,0,1,2.83,0l1.4,1.43a1.91,1.91,0,0,1,.58,1.4,1.91,1.91,0,0,1-.58,1.4L9.25,21ZM18.45,9,17,7.55Zm-12,3A5.31,5.31,0,0,0,4.9,8.1,5.31,5.31,0,0,0,1,6.5,5.31,5.31,0,0,0,4.9,4.9,5.31,5.31,0,0,0,6.5,1,5.31,5.31,0,0,0,8.1,4.9,5.31,5.31,0,0,0,12,6.5,5.46,5.46,0,0,0,6.5,12Z\"/>\n",
+              "  </svg>\n",
+              "    </button>\n",
+              "    <script>\n",
+              "      (() => {\n",
+              "      const buttonEl =\n",
+              "        document.querySelector('#id_1f652a22-3cd6-40ad-baf6-fb40cc24f889 button.colab-df-generate');\n",
+              "      buttonEl.style.display =\n",
+              "        google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
+              "\n",
+              "      buttonEl.onclick = () => {\n",
+              "        google.colab.notebook.generateWithVariable('df_padded');\n",
+              "      }\n",
+              "      })();\n",
+              "    </script>\n",
+              "  </div>\n",
+              "\n",
+              "    </div>\n",
+              "  </div>\n"
+            ],
+            "text/plain": [
+              "                City Name  Count_Person  Percent_Person_Obesity  \\\n",
+              "place                                                             \n",
+              "geoId/0103076      Auburn       82025.0                    33.0   \n",
+              "geoId/0107000  Birmingham      196644.0                    44.9   \n",
+              "geoId/0135896      Hoover       92448.0                    32.5   \n",
+              "geoId/0137000  Huntsville      225564.0                    37.5   \n",
+              "geoId/0150000      Mobile      182595.0                    44.2   \n",
+              "...                   ...           ...                     ...   \n",
+              "geoId/5531000   Green Bay      105744.0                    38.9   \n",
+              "geoId/5539225     Kenosha       98211.0                    43.7   \n",
+              "geoId/5548000     Madison      280305.0                    32.1   \n",
+              "geoId/5553000   Milwaukee      561385.0                    43.4   \n",
+              "geoId/5566000      Racine       76602.0                    42.9   \n",
+              "\n",
+              "               Percent_Person_PhysicalInactivity  \\\n",
+              "place                                              \n",
+              "geoId/0103076                               23.6   \n",
+              "geoId/0107000                               32.9   \n",
+              "geoId/0135896                               19.7   \n",
+              "geoId/0137000                               24.0   \n",
+              "geoId/0150000                               28.7   \n",
+              "...                                          ...   \n",
+              "geoId/5531000                               26.7   \n",
+              "geoId/5539225                               23.8   \n",
+              "geoId/5548000                               18.7   \n",
+              "geoId/5553000                               28.8   \n",
+              "geoId/5566000                               28.2   \n",
+              "\n",
+              "               Percent_Person_SleepLessThan7Hours  \\\n",
+              "place                                               \n",
+              "geoId/0103076                                36.0   \n",
+              "geoId/0107000                                42.9   \n",
+              "geoId/0135896                                33.6   \n",
+              "geoId/0137000                                40.0   \n",
+              "geoId/0150000                                43.4   \n",
+              "...                                           ...   \n",
+              "geoId/5531000                                33.1   \n",
+              "geoId/5539225                                36.6   \n",
+              "geoId/5548000                                29.9   \n",
+              "geoId/5553000                                40.0   \n",
+              "geoId/5566000                                39.2   \n",
+              "\n",
+              "               Percent_Person_WithHighBloodPressure  \\\n",
+              "place                                                 \n",
+              "geoId/0103076                                  34.3   \n",
+              "geoId/0107000                                  45.0   \n",
+              "geoId/0135896                                  32.6   \n",
+              "geoId/0137000                                  36.5   \n",
+              "geoId/0150000                                  39.8   \n",
+              "...                                             ...   \n",
+              "geoId/5531000                                  28.1   \n",
+              "geoId/5539225                                  29.9   \n",
+              "geoId/5548000                                  26.6   \n",
+              "geoId/5553000                                  36.7   \n",
+              "geoId/5566000                                  32.3   \n",
+              "\n",
+              "               Percent_Person_WithHighCholesterol  \\\n",
+              "place                                               \n",
+              "geoId/0103076                                30.6   \n",
+              "geoId/0107000                                31.6   \n",
+              "geoId/0135896                                31.0   \n",
+              "geoId/0137000                                31.6   \n",
+              "geoId/0150000                                32.5   \n",
+              "...                                           ...   \n",
+              "geoId/5531000                                30.7   \n",
+              "geoId/5539225                                30.0   \n",
+              "geoId/5548000                                28.5   \n",
+              "geoId/5553000                                30.1   \n",
+              "geoId/5566000                                32.0   \n",
+              "\n",
+              "               Percent_Person_WithMentalHealthNotGood  Random Variable 0  \\\n",
+              "place                                                                      \n",
+              "geoId/0103076                                    17.8          -1.564312   \n",
+              "geoId/0107000                                    19.7          -0.580159   \n",
+              "geoId/0135896                                    15.4          -0.322616   \n",
+              "geoId/0137000                                    18.0           0.768514   \n",
+              "geoId/0150000                                    19.9           0.207217   \n",
+              "...                                               ...                ...   \n",
+              "geoId/5531000                                    17.9          -0.104983   \n",
+              "geoId/5539225                                    18.6          -0.355349   \n",
+              "geoId/5548000                                    15.6          -0.648119   \n",
+              "geoId/5553000                                    19.0          -0.154089   \n",
+              "geoId/5566000                                    18.4           1.638922   \n",
+              "\n",
+              "               Random Variable 1  ...  Random Variable 90  Random Variable 91  \\\n",
+              "place                             ...                                           \n",
+              "geoId/0103076           0.288515  ...            0.762521            0.251051   \n",
+              "geoId/0107000           0.849181  ...            0.215843            1.553184   \n",
+              "geoId/0135896          -1.748737  ...            2.036116            0.993741   \n",
+              "geoId/0137000          -0.534476  ...            0.950064            0.730344   \n",
+              "geoId/0150000           1.028760  ...           -0.775507            1.338210   \n",
+              "...                          ...  ...                 ...                 ...   \n",
+              "geoId/5531000          -0.856795  ...           -0.945322           -0.219595   \n",
+              "geoId/5539225           0.348573  ...           -0.789575            0.590118   \n",
+              "geoId/5548000           0.025662  ...           -0.151965            0.835380   \n",
+              "geoId/5553000          -0.339432  ...            2.255458            1.357828   \n",
+              "geoId/5566000          -0.543906  ...            0.370553           -0.606273   \n",
+              "\n",
+              "               Random Variable 92  Random Variable 93  Random Variable 94  \\\n",
+              "place                                                                       \n",
+              "geoId/0103076           -0.697129           -1.697195            0.399706   \n",
+              "geoId/0107000           -1.766115            1.152941            0.712426   \n",
+              "geoId/0135896           -1.786077           -0.264808           -1.922278   \n",
+              "geoId/0137000            0.007471            3.514180            0.145648   \n",
+              "geoId/0150000           -0.395432           -0.830337           -0.558512   \n",
+              "...                           ...                 ...                 ...   \n",
+              "geoId/5531000           -2.113165            0.614379            0.110795   \n",
+              "geoId/5539225           -0.193587            0.502188            0.124404   \n",
+              "geoId/5548000           -1.381286            0.303114            0.540398   \n",
+              "geoId/5553000            0.692794            0.924034            0.951688   \n",
+              "geoId/5566000            1.066660            0.022132            0.039135   \n",
+              "\n",
+              "               Random Variable 95  Random Variable 96  Random Variable 97  \\\n",
+              "place                                                                       \n",
+              "geoId/0103076           -0.557155            0.444760            1.787642   \n",
+              "geoId/0107000            0.936660            0.576485           -0.127241   \n",
+              "geoId/0135896           -1.227397           -1.723762            0.847944   \n",
+              "geoId/0137000           -1.254448            0.275048           -1.241024   \n",
+              "geoId/0150000           -0.367606           -1.049303           -3.161325   \n",
+              "...                           ...                 ...                 ...   \n",
+              "geoId/5531000           -0.250010            0.926896           -0.526254   \n",
+              "geoId/5539225           -0.376209           -0.331331            0.697165   \n",
+              "geoId/5548000           -0.359988            0.007904            0.010788   \n",
+              "geoId/5553000           -0.071096            0.097582            0.952135   \n",
+              "geoId/5566000            1.102639           -0.438601           -1.744647   \n",
+              "\n",
+              "               Random Variable 98  Random Variable 99  \n",
+              "place                                                  \n",
+              "geoId/0103076            0.340410            1.535658  \n",
+              "geoId/0107000           -0.543845            1.536037  \n",
+              "geoId/0135896           -0.446194           -0.320127  \n",
+              "geoId/0137000           -0.163577            0.376057  \n",
+              "geoId/0150000           -0.586668            0.934307  \n",
+              "...                           ...                 ...  \n",
+              "geoId/5531000           -0.359181           -1.424956  \n",
+              "geoId/5539225            1.029427           -1.143744  \n",
+              "geoId/5548000           -0.276071            0.979319  \n",
+              "geoId/5553000           -1.019633           -0.778193  \n",
+              "geoId/5566000            1.245214            2.216294  \n",
+              "\n",
+              "[498 rows x 108 columns]"
+            ]
+          },
+          "metadata": {},
+          "output_type": "display_data"
+        }
+      ],
+      "source": [
+        "# Pad our dataframe with more random variables\n",
+        "num_rows = len(df.index)\n",
+        "num_new_columns = 100\n",
+        "random_data = np.random.normal(loc=0, scale=1, size=(num_rows, num_new_columns))\n",
+        "new_column_names = [f\"Random Variable {i}\" for i in range(num_new_columns)]\n",
+        "random_data_df = pd.DataFrame(\n",
+        "    random_data,\n",
+        "    columns=new_column_names,\n",
+        "    index=df.index\n",
+        ")\n",
+        "df_padded = pd.concat([df, random_data_df], axis=1)\n",
+        "display(df_padded)"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "metadata": {
+        "id": "Q5f22xUmvoN_"
+      },
+      "source": [
+        "Now let's fit a new model to the data and compare R^2 scores."
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": null,
+      "metadata": {
+        "colab": {
+          "base_uri": "https://localhost:8080/"
+        },
+        "id": "rn57oEF82dju",
+        "outputId": "27dd8be5-fae5-45b4-a31f-5858a087d3d5"
+      },
+      "outputs": [
+        {
+          "name": "stdout",
+          "output_type": "stream",
+          "text": [
+            "Original Model R^2 =  0.7577718062114178\n",
+            "Padded Model R^2 = 0.7988444670439291\n"
+          ]
+        }
+      ],
+      "source": [
+        "# New R^2\n",
+        "y_padded = df_padded[dep_var].to_numpy().reshape(-1, 1)\n",
+        "x_padded = df_padded.loc[:, ~df_padded.columns.isin([dep_var, \"City Name\"])]\n",
+        "x_padded = sm.add_constant(x_padded)\n",
+        "\n",
+        "padded_model = sm.OLS(y_padded, x_padded)\n",
+        "padded_results = padded_model.fit()\n",
+        "\n",
+        "print(\"Original Model R^2 = \", results.rsquared)\n",
+        "print(\"Padded Model R^2 =\", padded_results.rsquared)\n"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "metadata": {
+        "id": "j-j4IbOtwFj8"
+      },
+      "source": [
+        "**2.3A)** Which model had a better $R^2$ score?\n",
+        "\n",
+        "**2.3B)** Think about the variables used in each model. Should one model be much more predictive than another?\n",
+        "\n",
+        "**2.3B)** In general, how would you expect $R^2$ to change as we increase the number of independent variables?\n",
+        "\n"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "metadata": {
+        "id": "2Ipg_orhxOF_"
+      },
+      "source": [
+        "So how do we fix this? We can adjust our $R^2$ metric to account for the number of variables. The most popular way to defined the _**adjusted $R^2$**_ score is as follows:\n",
+        "\n",
+        "$$R^{2}_{adj}=1-(1-R^{2}){n-1 \\over n-p-1}$$\n",
+        "\n",
+        "where $n$ is the number of data points and $p$ is the number of independent variables.\n",
+        "\n",
+        "Now let's compare the adjusted $R^2$ of our models."
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": null,
+      "metadata": {
+        "colab": {
+          "base_uri": "https://localhost:8080/"
+        },
+        "id": "7pZ9_NmZisGi",
+        "outputId": "bfa5cddd-dddf-45c9-8082-ab58dbe5c286"
+      },
+      "outputs": [
+        {
+          "name": "stdout",
+          "output_type": "stream",
+          "text": [
+            "Original Model Adjusted R^2 =  0.7548117875500502\n",
+            "Padded Model Adjusted R^2 = 0.7443112535059662\n"
+          ]
+        }
+      ],
+      "source": [
+        "# Adjusted R^2\n",
+        "print(\"Original Model Adjusted R^2 = \", results.rsquared_adj)\n",
+        "print(\"Padded Model Adjusted R^2 =\", padded_results.rsquared_adj)"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "metadata": {
+        "id": "qU9VwLsNHcKD"
+      },
+      "source": [
+        "**2.3D)** Which model had a better adjusted $R^2$ score?\n",
+        "\n",
+        "**2.3E)** When would you prefer to use adjusted R^2 over R^2 to evaluate model fit?"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "metadata": {
+        "id": "1tiopX7PWHiu"
+      },
+      "source": [
+        "## 3) Interpreting regression models\n"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "metadata": {
+        "id": "qC7aC0y-O3_D"
+      },
+      "source": [
+        "### 3.1) Analyzing weights and intercepts\n",
+        "The parameters of the regression model itself can also yield important insights.\n",
+        "\n",
+        "Run the following code box to display the weights and intercept of our original model."
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": null,
+      "metadata": {
+        "colab": {
+          "base_uri": "https://localhost:8080/",
+          "height": 304
+        },
+        "id": "_y0xeWysPIm6",
+        "outputId": "6ede53a7-bbc0-474e-bc4a-4567735c8d75"
+      },
+      "outputs": [
+        {
+          "data": {
+            "text/html": [
+              "<div>\n",
+              "<style scoped>\n",
+              "    .dataframe tbody tr th:only-of-type {\n",
+              "        vertical-align: middle;\n",
+              "    }\n",
+              "\n",
+              "    .dataframe tbody tr th {\n",
+              "        vertical-align: top;\n",
+              "    }\n",
+              "\n",
+              "    .dataframe thead th {\n",
+              "        text-align: right;\n",
+              "    }\n",
+              "</style>\n",
+              "<table border=\"1\" class=\"dataframe\">\n",
+              "  <thead>\n",
+              "    <tr style=\"text-align: right;\">\n",
+              "      <th></th>\n",
+              "      <th>0</th>\n",
+              "    </tr>\n",
+              "  </thead>\n",
+              "  <tbody>\n",
+              "    <tr>\n",
+              "      <th>const</th>\n",
+              "      <td>-0.19367</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>Count_Person</th>\n",
+              "      <td>-0.00000</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>Percent_Person_PhysicalInactivity</th>\n",
+              "      <td>0.30528</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>Percent_Person_SleepLessThan7Hours</th>\n",
+              "      <td>-0.12455</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>Percent_Person_WithHighBloodPressure</th>\n",
+              "      <td>0.75717</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>Percent_Person_WithHighCholesterol</th>\n",
+              "      <td>-0.13520</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>Percent_Person_WithMentalHealthNotGood</th>\n",
+              "      <td>0.69012</td>\n",
+              "    </tr>\n",
+              "  </tbody>\n",
+              "</table>\n",
+              "</div><br><label><b>dtype:</b> float64</label>"
+            ],
+            "text/plain": [
+              "const                                    -0.19367\n",
+              "Count_Person                             -0.00000\n",
+              "Percent_Person_PhysicalInactivity         0.30528\n",
+              "Percent_Person_SleepLessThan7Hours       -0.12455\n",
+              "Percent_Person_WithHighBloodPressure      0.75717\n",
+              "Percent_Person_WithHighCholesterol       -0.13520\n",
+              "Percent_Person_WithMentalHealthNotGood    0.69012\n",
+              "dtype: float64"
+            ]
+          },
+          "metadata": {},
+          "output_type": "display_data"
+        }
+      ],
+      "source": [
+        "# Display weights/coefficients\n",
+        "display(results.params.round(5))"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "metadata": {
+        "id": "dvpGBohWPymA"
+      },
+      "source": [
+        "**3.1A)** What is the intercept of our model? What are its units?\n",
+        "\n",
+        "**3.1B)** What are the units on each of the model weights (aka coefficients)?\n",
+        "\n",
+        "**3.1C)** Which variables matter most to our model?\n",
+        "\n",
+        "**3.1D)** In words, describe what a weight/coefficient in a linear regression means.\n",
+        "\n",
+        "**3.1E)** Our model is used to generate a predicted obesity rate for a fictional city named Dataopolis. If we increased `Percent_Person_WithMentalHealthNotGood` for Dataopolis by 1 unit, _while keeping the values for all remaining variables constant_, by how much would we expect our predicted obesity rate to change?"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "metadata": {
+        "id": "2ZoaATgWh-fR"
+      },
+      "source": [
+        "### 3.2) The effect of correlated variables\n",
+        "\n",
+        "When interpreting weights, one thing to look out for is if we have independent variables that are highly correlated with each other.\n",
+        "\n",
+        "Let's illustrate why this might be a problem, by adding a variable that is correlated with one of the existing variables"
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": null,
+      "metadata": {
+        "colab": {
+          "base_uri": "https://localhost:8080/",
+          "height": 845
+        },
+        "id": "uP4XtXkfLB1U",
+        "outputId": "f55a1573-1c38-49ad-ea26-9675b434e7aa"
+      },
+      "outputs": [
+        {
+          "name": "stdout",
+          "output_type": "stream",
+          "text": [
+            "New dataframe to fit:\n"
+          ]
+        },
+        {
+          "data": {
+            "application/vnd.google.colaboratory.intrinsic+json": {
+              "summary": "{\n  \"name\": \"correlated_df\",\n  \"rows\": 498,\n  \"fields\": [\n    {\n      \"column\": \"place\",\n      \"properties\": {\n        \"dtype\": \"string\",\n        \"num_unique_values\": 498,\n        \"samples\": [\n          \"geoId/5363000\",\n          \"geoId/0639892\",\n          \"geoId/1714351\"\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"City Name\",\n      \"properties\": {\n        \"dtype\": \"string\",\n        \"num_unique_values\": 475,\n        \"samples\": [\n          \"Memphis\",\n          \"Plano\",\n          \"Avondale\"\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"Count_Person\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 471628.1194557695,\n        \"min\": 76212.0,\n        \"max\": 8258035.0,\n        \"num_unique_values\": 498,\n        \"samples\": [\n          755078.0,\n          78135.0,\n          81004.0\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"Percent_Person_Obesity\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 6.3675302626575006,\n        \"min\": 14.1,\n        \"max\": 48.9,\n        \"num_unique_values\": 220,\n        \"samples\": [\n          33.4,\n          41.6,\n          23.2\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"Percent_Person_PhysicalInactivity\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 5.79401923130349,\n        \"min\": 11.2,\n        \"max\": 41.8,\n        \"num_unique_values\": 209,\n        \"samples\": [\n          25.0,\n          39.5,\n          18.4\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"Percent_Person_SleepLessThan7Hours\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 4.3313227959895455,\n        \"min\": 24.9,\n        \"max\": 49.5,\n        \"num_unique_values\": 166,\n        \"samples\": [\n          42.5,\n          25.6,\n          28.4\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"Percent_Person_WithHighBloodPressure\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 4.56550336745893,\n        \"min\": 21.3,\n        \"max\": 45.7,\n        \"num_unique_values\": 170,\n        \"samples\": [\n          40.3,\n          25.6,\n          41.2\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"Percent_Person_WithHighCholesterol\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 2.089525781869934,\n        \"min\": 24.6,\n        \"max\": 35.6,\n        \"num_unique_values\": 95,\n        \"samples\": [\n          25.9,\n          34.1,\n          27.0\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"Percent_Person_WithMentalHealthNotGood\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 2.1019573952670365,\n        \"min\": 11.5,\n        \"max\": 23.3,\n        \"num_unique_values\": 103,\n        \"samples\": [\n          15.8,\n          16.3,\n          12.7\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"Correlated Variable\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 2.2936580170071315,\n        \"min\": 9.748686615105605,\n        \"max\": 23.243226154344537,\n        \"num_unique_values\": 498,\n        \"samples\": [\n          16.440967097355323,\n          17.12077873386613,\n          17.01999484946955\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    }\n  ]\n}",
+              "type": "dataframe",
+              "variable_name": "correlated_df"
+            },
+            "text/html": [
+              "\n",
+              "  <div id=\"df-8b61f3cf-5ee3-47e5-9b76-59949bd902ae\" class=\"colab-df-container\">\n",
+              "    <div>\n",
+              "<style scoped>\n",
+              "    .dataframe tbody tr th:only-of-type {\n",
+              "        vertical-align: middle;\n",
+              "    }\n",
+              "\n",
+              "    .dataframe tbody tr th {\n",
+              "        vertical-align: top;\n",
+              "    }\n",
+              "\n",
+              "    .dataframe thead th {\n",
+              "        text-align: right;\n",
+              "    }\n",
+              "</style>\n",
+              "<table border=\"1\" class=\"dataframe\">\n",
+              "  <thead>\n",
+              "    <tr style=\"text-align: right;\">\n",
+              "      <th>variable</th>\n",
+              "      <th>City Name</th>\n",
+              "      <th>Count_Person</th>\n",
+              "      <th>Percent_Person_Obesity</th>\n",
+              "      <th>Percent_Person_PhysicalInactivity</th>\n",
+              "      <th>Percent_Person_SleepLessThan7Hours</th>\n",
+              "      <th>Percent_Person_WithHighBloodPressure</th>\n",
+              "      <th>Percent_Person_WithHighCholesterol</th>\n",
+              "      <th>Percent_Person_WithMentalHealthNotGood</th>\n",
+              "      <th>Correlated Variable</th>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>place</th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "      <th></th>\n",
+              "    </tr>\n",
+              "  </thead>\n",
+              "  <tbody>\n",
+              "    <tr>\n",
+              "      <th>geoId/0103076</th>\n",
+              "      <td>Auburn</td>\n",
+              "      <td>82025.0</td>\n",
+              "      <td>33.0</td>\n",
+              "      <td>23.6</td>\n",
+              "      <td>36.0</td>\n",
+              "      <td>34.3</td>\n",
+              "      <td>30.6</td>\n",
+              "      <td>17.8</td>\n",
+              "      <td>18.761300</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>geoId/0107000</th>\n",
+              "      <td>Birmingham</td>\n",
+              "      <td>196644.0</td>\n",
+              "      <td>44.9</td>\n",
+              "      <td>32.9</td>\n",
+              "      <td>42.9</td>\n",
+              "      <td>45.0</td>\n",
+              "      <td>31.6</td>\n",
+              "      <td>19.7</td>\n",
+              "      <td>17.655787</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>geoId/0135896</th>\n",
+              "      <td>Hoover</td>\n",
+              "      <td>92448.0</td>\n",
+              "      <td>32.5</td>\n",
+              "      <td>19.7</td>\n",
+              "      <td>33.6</td>\n",
+              "      <td>32.6</td>\n",
+              "      <td>31.0</td>\n",
+              "      <td>15.4</td>\n",
+              "      <td>14.736255</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>geoId/0137000</th>\n",
+              "      <td>Huntsville</td>\n",
+              "      <td>225564.0</td>\n",
+              "      <td>37.5</td>\n",
+              "      <td>24.0</td>\n",
+              "      <td>40.0</td>\n",
+              "      <td>36.5</td>\n",
+              "      <td>31.6</td>\n",
+              "      <td>18.0</td>\n",
+              "      <td>16.549451</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>geoId/0150000</th>\n",
+              "      <td>Mobile</td>\n",
+              "      <td>182595.0</td>\n",
+              "      <td>44.2</td>\n",
+              "      <td>28.7</td>\n",
+              "      <td>43.4</td>\n",
+              "      <td>39.8</td>\n",
+              "      <td>32.5</td>\n",
+              "      <td>19.9</td>\n",
+              "      <td>20.277958</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>...</th>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "      <td>...</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>geoId/5531000</th>\n",
+              "      <td>Green Bay</td>\n",
+              "      <td>105744.0</td>\n",
+              "      <td>38.9</td>\n",
+              "      <td>26.7</td>\n",
+              "      <td>33.1</td>\n",
+              "      <td>28.1</td>\n",
+              "      <td>30.7</td>\n",
+              "      <td>17.9</td>\n",
+              "      <td>18.645080</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>geoId/5539225</th>\n",
+              "      <td>Kenosha</td>\n",
+              "      <td>98211.0</td>\n",
+              "      <td>43.7</td>\n",
+              "      <td>23.8</td>\n",
+              "      <td>36.6</td>\n",
+              "      <td>29.9</td>\n",
+              "      <td>30.0</td>\n",
+              "      <td>18.6</td>\n",
+              "      <td>17.067335</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>geoId/5548000</th>\n",
+              "      <td>Madison</td>\n",
+              "      <td>280305.0</td>\n",
+              "      <td>32.1</td>\n",
+              "      <td>18.7</td>\n",
+              "      <td>29.9</td>\n",
+              "      <td>26.6</td>\n",
+              "      <td>28.5</td>\n",
+              "      <td>15.6</td>\n",
+              "      <td>15.665917</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>geoId/5553000</th>\n",
+              "      <td>Milwaukee</td>\n",
+              "      <td>561385.0</td>\n",
+              "      <td>43.4</td>\n",
+              "      <td>28.8</td>\n",
+              "      <td>40.0</td>\n",
+              "      <td>36.7</td>\n",
+              "      <td>30.1</td>\n",
+              "      <td>19.0</td>\n",
+              "      <td>19.073143</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>geoId/5566000</th>\n",
+              "      <td>Racine</td>\n",
+              "      <td>76602.0</td>\n",
+              "      <td>42.9</td>\n",
+              "      <td>28.2</td>\n",
+              "      <td>39.2</td>\n",
+              "      <td>32.3</td>\n",
+              "      <td>32.0</td>\n",
+              "      <td>18.4</td>\n",
+              "      <td>17.106196</td>\n",
+              "    </tr>\n",
+              "  </tbody>\n",
+              "</table>\n",
+              "<p>498 rows × 9 columns</p>\n",
+              "</div>\n",
+              "    <div class=\"colab-df-buttons\">\n",
+              "\n",
+              "  <div class=\"colab-df-container\">\n",
+              "    <button class=\"colab-df-convert\" onclick=\"convertToInteractive('df-8b61f3cf-5ee3-47e5-9b76-59949bd902ae')\"\n",
+              "            title=\"Convert this dataframe to an interactive table.\"\n",
+              "            style=\"display:none;\">\n",
+              "\n",
+              "  <svg xmlns=\"http://www.w3.org/2000/svg\" height=\"24px\" viewBox=\"0 -960 960 960\">\n",
+              "    <path d=\"M120-120v-720h720v720H120Zm60-500h600v-160H180v160Zm220 220h160v-160H400v160Zm0 220h160v-160H400v160ZM180-400h160v-160H180v160Zm440 0h160v-160H620v160ZM180-180h160v-160H180v160Zm440 0h160v-160H620v160Z\"/>\n",
+              "  </svg>\n",
+              "    </button>\n",
+              "\n",
+              "  <style>\n",
+              "    .colab-df-container {\n",
+              "      display:flex;\n",
+              "      gap: 12px;\n",
+              "    }\n",
+              "\n",
+              "    .colab-df-convert {\n",
+              "      background-color: #E8F0FE;\n",
+              "      border: none;\n",
+              "      border-radius: 50%;\n",
+              "      cursor: pointer;\n",
+              "      display: none;\n",
+              "      fill: #1967D2;\n",
+              "      height: 32px;\n",
+              "      padding: 0 0 0 0;\n",
+              "      width: 32px;\n",
+              "    }\n",
+              "\n",
+              "    .colab-df-convert:hover {\n",
+              "      background-color: #E2EBFA;\n",
+              "      box-shadow: 0px 1px 2px rgba(60, 64, 67, 0.3), 0px 1px 3px 1px rgba(60, 64, 67, 0.15);\n",
+              "      fill: #174EA6;\n",
+              "    }\n",
+              "\n",
+              "    .colab-df-buttons div {\n",
+              "      margin-bottom: 4px;\n",
+              "    }\n",
+              "\n",
+              "    [theme=dark] .colab-df-convert {\n",
+              "      background-color: #3B4455;\n",
+              "      fill: #D2E3FC;\n",
+              "    }\n",
+              "\n",
+              "    [theme=dark] .colab-df-convert:hover {\n",
+              "      background-color: #434B5C;\n",
+              "      box-shadow: 0px 1px 3px 1px rgba(0, 0, 0, 0.15);\n",
+              "      filter: drop-shadow(0px 1px 2px rgba(0, 0, 0, 0.3));\n",
+              "      fill: #FFFFFF;\n",
+              "    }\n",
+              "  </style>\n",
+              "\n",
+              "    <script>\n",
+              "      const buttonEl =\n",
+              "        document.querySelector('#df-8b61f3cf-5ee3-47e5-9b76-59949bd902ae 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-8b61f3cf-5ee3-47e5-9b76-59949bd902ae');\n",
+              "        const dataTable =\n",
+              "          await google.colab.kernel.invokeFunction('convertToInteractive',\n",
+              "                                                    [key], {});\n",
+              "        if (!dataTable) return;\n",
+              "\n",
+              "        const docLinkHtml = 'Like what you see? Visit the ' +\n",
+              "          '<a target=\"_blank\" href=https://colab.research.google.com/notebooks/data_table.ipynb>data table notebook</a>'\n",
+              "          + ' to learn more about interactive tables.';\n",
+              "        element.innerHTML = '';\n",
+              "        dataTable['output_type'] = 'display_data';\n",
+              "        await google.colab.output.renderOutput(dataTable, element);\n",
+              "        const docLink = document.createElement('div');\n",
+              "        docLink.innerHTML = docLinkHtml;\n",
+              "        element.appendChild(docLink);\n",
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+              "\n",
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+              "\n",
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+              "  .colab-df-quickchart {\n",
+              "    background-color: var(--bg-color);\n",
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+              "    height: 32px;\n",
+              "    padding: 0;\n",
+              "    width: 32px;\n",
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+              "    fill: var(--button-hover-fill-color);\n",
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+              "\n",
+              "  .colab-df-quickchart-complete:disabled,\n",
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+              "    background-color: var(--disabled-bg-color);\n",
+              "    fill: var(--disabled-fill-color);\n",
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+              "  }\n",
+              "\n",
+              "  .colab-df-spinner {\n",
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+              "    animation:\n",
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+              "  }\n",
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+              "    90% {\n",
+              "      border-color: transparent;\n",
+              "      border-bottom-color: var(--fill-color);\n",
+              "    }\n",
+              "  }\n",
+              "</style>\n",
+              "\n",
+              "      <script>\n",
+              "        async function quickchart(key) {\n",
+              "          const quickchartButtonEl =\n",
+              "            document.querySelector('#' + key + ' button');\n",
+              "          quickchartButtonEl.disabled = true;  // To prevent multiple clicks.\n",
+              "          quickchartButtonEl.classList.add('colab-df-spinner');\n",
+              "          try {\n",
+              "            const charts = await google.colab.kernel.invokeFunction(\n",
+              "                'suggestCharts', [key], {});\n",
+              "          } catch (error) {\n",
+              "            console.error('Error during call to suggestCharts:', error);\n",
+              "          }\n",
+              "          quickchartButtonEl.classList.remove('colab-df-spinner');\n",
+              "          quickchartButtonEl.classList.add('colab-df-quickchart-complete');\n",
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+              "        (() => {\n",
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+              "            document.querySelector('#df-bc921f8d-e327-427e-8077-e0f768859914 button');\n",
+              "          quickchartButtonEl.style.display =\n",
+              "            google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
+              "        })();\n",
+              "      </script>\n",
+              "    </div>\n",
+              "\n",
+              "  <div id=\"id_2edcbef2-0122-4e78-88e5-2a222f60df9b\">\n",
+              "    <style>\n",
+              "      .colab-df-generate {\n",
+              "        background-color: #E8F0FE;\n",
+              "        border: none;\n",
+              "        border-radius: 50%;\n",
+              "        cursor: pointer;\n",
+              "        display: none;\n",
+              "        fill: #1967D2;\n",
+              "        height: 32px;\n",
+              "        padding: 0 0 0 0;\n",
+              "        width: 32px;\n",
+              "      }\n",
+              "\n",
+              "      .colab-df-generate:hover {\n",
+              "        background-color: #E2EBFA;\n",
+              "        box-shadow: 0px 1px 2px rgba(60, 64, 67, 0.3), 0px 1px 3px 1px rgba(60, 64, 67, 0.15);\n",
+              "        fill: #174EA6;\n",
+              "      }\n",
+              "\n",
+              "      [theme=dark] .colab-df-generate {\n",
+              "        background-color: #3B4455;\n",
+              "        fill: #D2E3FC;\n",
+              "      }\n",
+              "\n",
+              "      [theme=dark] .colab-df-generate:hover {\n",
+              "        background-color: #434B5C;\n",
+              "        box-shadow: 0px 1px 3px 1px rgba(0, 0, 0, 0.15);\n",
+              "        filter: drop-shadow(0px 1px 2px rgba(0, 0, 0, 0.3));\n",
+              "        fill: #FFFFFF;\n",
+              "      }\n",
+              "    </style>\n",
+              "    <button class=\"colab-df-generate\" onclick=\"generateWithVariable('correlated_df')\"\n",
+              "            title=\"Generate code using this dataframe.\"\n",
+              "            style=\"display:none;\">\n",
+              "\n",
+              "  <svg xmlns=\"http://www.w3.org/2000/svg\" height=\"24px\"viewBox=\"0 0 24 24\"\n",
+              "       width=\"24px\">\n",
+              "    <path d=\"M7,19H8.4L18.45,9,17,7.55,7,17.6ZM5,21V16.75L18.45,3.32a2,2,0,0,1,2.83,0l1.4,1.43a1.91,1.91,0,0,1,.58,1.4,1.91,1.91,0,0,1-.58,1.4L9.25,21ZM18.45,9,17,7.55Zm-12,3A5.31,5.31,0,0,0,4.9,8.1,5.31,5.31,0,0,0,1,6.5,5.31,5.31,0,0,0,4.9,4.9,5.31,5.31,0,0,0,6.5,1,5.31,5.31,0,0,0,8.1,4.9,5.31,5.31,0,0,0,12,6.5,5.46,5.46,0,0,0,6.5,12Z\"/>\n",
+              "  </svg>\n",
+              "    </button>\n",
+              "    <script>\n",
+              "      (() => {\n",
+              "      const buttonEl =\n",
+              "        document.querySelector('#id_2edcbef2-0122-4e78-88e5-2a222f60df9b button.colab-df-generate');\n",
+              "      buttonEl.style.display =\n",
+              "        google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
+              "\n",
+              "      buttonEl.onclick = () => {\n",
+              "        google.colab.notebook.generateWithVariable('correlated_df');\n",
+              "      }\n",
+              "      })();\n",
+              "    </script>\n",
+              "  </div>\n",
+              "\n",
+              "    </div>\n",
+              "  </div>\n"
+            ],
+            "text/plain": [
+              "variable        City Name  Count_Person  Percent_Person_Obesity  \\\n",
+              "place                                                             \n",
+              "geoId/0103076      Auburn       82025.0                    33.0   \n",
+              "geoId/0107000  Birmingham      196644.0                    44.9   \n",
+              "geoId/0135896      Hoover       92448.0                    32.5   \n",
+              "geoId/0137000  Huntsville      225564.0                    37.5   \n",
+              "geoId/0150000      Mobile      182595.0                    44.2   \n",
+              "...                   ...           ...                     ...   \n",
+              "geoId/5531000   Green Bay      105744.0                    38.9   \n",
+              "geoId/5539225     Kenosha       98211.0                    43.7   \n",
+              "geoId/5548000     Madison      280305.0                    32.1   \n",
+              "geoId/5553000   Milwaukee      561385.0                    43.4   \n",
+              "geoId/5566000      Racine       76602.0                    42.9   \n",
+              "\n",
+              "variable       Percent_Person_PhysicalInactivity  \\\n",
+              "place                                              \n",
+              "geoId/0103076                               23.6   \n",
+              "geoId/0107000                               32.9   \n",
+              "geoId/0135896                               19.7   \n",
+              "geoId/0137000                               24.0   \n",
+              "geoId/0150000                               28.7   \n",
+              "...                                          ...   \n",
+              "geoId/5531000                               26.7   \n",
+              "geoId/5539225                               23.8   \n",
+              "geoId/5548000                               18.7   \n",
+              "geoId/5553000                               28.8   \n",
+              "geoId/5566000                               28.2   \n",
+              "\n",
+              "variable       Percent_Person_SleepLessThan7Hours  \\\n",
+              "place                                               \n",
+              "geoId/0103076                                36.0   \n",
+              "geoId/0107000                                42.9   \n",
+              "geoId/0135896                                33.6   \n",
+              "geoId/0137000                                40.0   \n",
+              "geoId/0150000                                43.4   \n",
+              "...                                           ...   \n",
+              "geoId/5531000                                33.1   \n",
+              "geoId/5539225                                36.6   \n",
+              "geoId/5548000                                29.9   \n",
+              "geoId/5553000                                40.0   \n",
+              "geoId/5566000                                39.2   \n",
+              "\n",
+              "variable       Percent_Person_WithHighBloodPressure  \\\n",
+              "place                                                 \n",
+              "geoId/0103076                                  34.3   \n",
+              "geoId/0107000                                  45.0   \n",
+              "geoId/0135896                                  32.6   \n",
+              "geoId/0137000                                  36.5   \n",
+              "geoId/0150000                                  39.8   \n",
+              "...                                             ...   \n",
+              "geoId/5531000                                  28.1   \n",
+              "geoId/5539225                                  29.9   \n",
+              "geoId/5548000                                  26.6   \n",
+              "geoId/5553000                                  36.7   \n",
+              "geoId/5566000                                  32.3   \n",
+              "\n",
+              "variable       Percent_Person_WithHighCholesterol  \\\n",
+              "place                                               \n",
+              "geoId/0103076                                30.6   \n",
+              "geoId/0107000                                31.6   \n",
+              "geoId/0135896                                31.0   \n",
+              "geoId/0137000                                31.6   \n",
+              "geoId/0150000                                32.5   \n",
+              "...                                           ...   \n",
+              "geoId/5531000                                30.7   \n",
+              "geoId/5539225                                30.0   \n",
+              "geoId/5548000                                28.5   \n",
+              "geoId/5553000                                30.1   \n",
+              "geoId/5566000                                32.0   \n",
+              "\n",
+              "variable       Percent_Person_WithMentalHealthNotGood  Correlated Variable  \n",
+              "place                                                                       \n",
+              "geoId/0103076                                    17.8            18.761300  \n",
+              "geoId/0107000                                    19.7            17.655787  \n",
+              "geoId/0135896                                    15.4            14.736255  \n",
+              "geoId/0137000                                    18.0            16.549451  \n",
+              "geoId/0150000                                    19.9            20.277958  \n",
+              "...                                               ...                  ...  \n",
+              "geoId/5531000                                    17.9            18.645080  \n",
+              "geoId/5539225                                    18.6            17.067335  \n",
+              "geoId/5548000                                    15.6            15.665917  \n",
+              "geoId/5553000                                    19.0            19.073143  \n",
+              "geoId/5566000                                    18.4            17.106196  \n",
+              "\n",
+              "[498 rows x 9 columns]"
+            ]
+          },
+          "metadata": {},
+          "output_type": "display_data"
+        },
+        {
+          "name": "stdout",
+          "output_type": "stream",
+          "text": [
+            "Correlated Model Weights and Intercept:\n"
+          ]
+        },
+        {
+          "data": {
+            "text/html": [
+              "<div>\n",
+              "<style scoped>\n",
+              "    .dataframe tbody tr th:only-of-type {\n",
+              "        vertical-align: middle;\n",
+              "    }\n",
+              "\n",
+              "    .dataframe tbody tr th {\n",
+              "        vertical-align: top;\n",
+              "    }\n",
+              "\n",
+              "    .dataframe thead th {\n",
+              "        text-align: right;\n",
+              "    }\n",
+              "</style>\n",
+              "<table border=\"1\" class=\"dataframe\">\n",
+              "  <thead>\n",
+              "    <tr style=\"text-align: right;\">\n",
+              "      <th></th>\n",
+              "      <th>0</th>\n",
+              "    </tr>\n",
+              "  </thead>\n",
+              "  <tbody>\n",
+              "    <tr>\n",
+              "      <th>const</th>\n",
+              "      <td>-0.28192</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>Count_Person</th>\n",
+              "      <td>-0.00000</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>Percent_Person_PhysicalInactivity</th>\n",
+              "      <td>0.30604</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>Percent_Person_SleepLessThan7Hours</th>\n",
+              "      <td>-0.12529</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>Percent_Person_WithHighBloodPressure</th>\n",
+              "      <td>0.75756</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>Percent_Person_WithHighCholesterol</th>\n",
+              "      <td>-0.13345</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>Percent_Person_WithMentalHealthNotGood</th>\n",
+              "      <td>0.55372</td>\n",
+              "    </tr>\n",
+              "    <tr>\n",
+              "      <th>Correlated Variable</th>\n",
+              "      <td>0.13921</td>\n",
+              "    </tr>\n",
+              "  </tbody>\n",
+              "</table>\n",
+              "</div><br><label><b>dtype:</b> float64</label>"
+            ],
+            "text/plain": [
+              "const                                    -0.28192\n",
+              "Count_Person                             -0.00000\n",
+              "Percent_Person_PhysicalInactivity         0.30604\n",
+              "Percent_Person_SleepLessThan7Hours       -0.12529\n",
+              "Percent_Person_WithHighBloodPressure      0.75756\n",
+              "Percent_Person_WithHighCholesterol       -0.13345\n",
+              "Percent_Person_WithMentalHealthNotGood    0.55372\n",
+              "Correlated Variable                       0.13921\n",
+              "dtype: float64"
+            ]
+          },
+          "metadata": {},
+          "output_type": "display_data"
+        }
+      ],
+      "source": [
+        "# New variable correlated with Percent_Person_WithMentalHealthNotGood\n",
+        "correlated_df = df.copy()\n",
+        "target_var = \"Percent_Person_WithMentalHealthNotGood\"\n",
+        "noise = np.random.normal(size=(len(correlated_df.index),))\n",
+        "correlated_df[\"Correlated Variable\"] = correlated_df[target_var] + noise\n",
+        "\n",
+        "# show new data frame\n",
+        "print(\"New dataframe to fit:\")\n",
+        "display(correlated_df)\n",
+        "\n",
+        "# Create a new model\n",
+        "y_corr = correlated_df[dep_var].to_numpy().reshape(-1, 1)\n",
+        "x_corr = correlated_df.loc[:, ~correlated_df.columns.isin([dep_var, \"City Name\"])]\n",
+        "x_corr = sm.add_constant(x_corr)\n",
+        "\n",
+        "correlated_model = sm.OLS(y_corr, x_corr)\n",
+        "correlated_results = correlated_model.fit()\n",
+        "\n",
+        "print(\"Correlated Model Weights and Intercept:\")\n",
+        "display(correlated_results.params.round(5))"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "metadata": {
+        "id": "HEHJWPxibiY3"
+      },
+      "source": [
+        "**3.2A)** Compare the new weights of the correlated model with the weights of our original model. What happened to the weights corresponding to `Percent_Person_WithMentalHealthNotGood`?\n",
+        "\n",
+        "**3.2B)** Thinking back to your answers for Q3.1C-E, how might correlated variables affect the interpretation of model weights?"
+      ]
+    }
+  ],
+  "metadata": {
+    "colab": {
+      "include_colab_link": true,
+      "provenance": [],
+      "toc_visible": true
+    },
+    "kernelspec": {
+      "display_name": "Python 3",
+      "name": "python3"
+    },
+    "language_info": {
+      "name": "python"
+    }
+  },
+  "nbformat": 4,
+  "nbformat_minor": 0
+}