{
 "cells": [
  {
   "cell_type": "markdown",
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   "source": [
    "# Homework\n",
    "\n",
    "- Use this assignment to practice the chapter's concepts and coding skills in runnable Python code.\n",
    "- Sign in from the account menu before you start working on the assignment.\n",
    "- Write and run your code directly in the coding cells.\n",
    "- Multiple submissions are allowed. The highest score is kept.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "::::{div} assignment-question-card\n",
    "\n",
    "**1. Correlation and Causation**\n",
    "\n",
    "Across 50 stores, the correlation between a product's price and units sold is -0.85. This shows that raising the price causes demand to fall.\n",
    "\n",
    "<div class=\"tf-options\" role=\"radiogroup\" aria-label=\"Question 1 answer\">\n",
    "  <label><input type=\"radio\" name=\"ch12-hw-q1\" value=\"true\"> True</label>\n",
    "  <label><input type=\"radio\" name=\"ch12-hw-q1\" value=\"false\"> False</label>\n",
    "</div>\n",
    "\n",
    "::::\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [],
   "source": [
    "### Solution\n",
    "# False\n",
    "# Correlation measures association, not causation. Other differences between stores (location, customers, competition) could explain the pattern.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "::::{div} assignment-question-card\n",
    "\n",
    "**2. Changing Units**\n",
    "\n",
    "Converting both variables to standard units, or measuring sales in dollars instead of thousands of dollars, does not change the correlation coefficient r.\n",
    "\n",
    "<div class=\"tf-options\" role=\"radiogroup\" aria-label=\"Question 2 answer\">\n",
    "  <label><input type=\"radio\" name=\"ch12-hw-q2\" value=\"true\"> True</label>\n",
    "  <label><input type=\"radio\" name=\"ch12-hw-q2\" value=\"false\"> False</label>\n",
    "</div>\n",
    "\n",
    "::::\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [],
   "source": [
    "### Solution\n",
    "# True\n",
    "# r is computed from standard units, so it has no units and does not change when either variable is rescaled.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "::::{div} assignment-question-card\n",
    "\n",
    "**3. The Regression Effect**\n",
    "\n",
    "The correlation between customers' monthly visits and their monthly spending is 0.6. A customer whose visits are 2 SDs above average is predicted to spend 1.2 SDs above average.\n",
    "\n",
    "<div class=\"tf-options\" role=\"radiogroup\" aria-label=\"Question 3 answer\">\n",
    "  <label><input type=\"radio\" name=\"ch12-hw-q3\" value=\"true\"> True</label>\n",
    "  <label><input type=\"radio\" name=\"ch12-hw-q3\" value=\"false\"> False</label>\n",
    "</div>\n",
    "\n",
    "::::\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [],
   "source": [
    "### Solution\n",
    "# True\n",
    "# In standard units, the regression prediction is r times x: 0.6 times 2 = 1.2. Predictions are closer to average than the x value is.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "::::{div} assignment-question-card\n",
    "\n",
    "**4. Least Squares**\n",
    "\n",
    "Among all straight lines that could be drawn through a scatter plot, the regression line has the smallest root mean squared error.\n",
    "\n",
    "<div class=\"tf-options\" role=\"radiogroup\" aria-label=\"Question 4 answer\">\n",
    "  <label><input type=\"radio\" name=\"ch12-hw-q4\" value=\"true\"> True</label>\n",
    "  <label><input type=\"radio\" name=\"ch12-hw-q4\" value=\"false\"> False</label>\n",
    "</div>\n",
    "\n",
    "::::\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [],
   "source": [
    "### Solution\n",
    "# True\n",
    "# That is why the regression line is also called the least squares line.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "::::{div} assignment-question-card\n",
    "\n",
    "**5. Fan-Shaped Residuals**\n",
    "\n",
    "In a residual plot, the residuals spread out more and more as the fitted values increase. The regression line is equally precise for small and large predictions.\n",
    "\n",
    "<div class=\"tf-options\" role=\"radiogroup\" aria-label=\"Question 5 answer\">\n",
    "  <label><input type=\"radio\" name=\"ch12-hw-q5\" value=\"true\"> True</label>\n",
    "  <label><input type=\"radio\" name=\"ch12-hw-q5\" value=\"false\"> False</label>\n",
    "</div>\n",
    "\n",
    "::::\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [],
   "source": [
    "### Solution\n",
    "# False\n",
    "# Uneven spread (heteroscedasticity) means predictions are less precise where the residuals are larger.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "tags": [
     "thebe-interactive"
    ]
   },
   "outputs": [],
   "source": [
    "### 6. Price and Units Sold\n",
    "# A store varied the price of a product over six weeks.\n",
    "#   1. Compute r between price and units, using standard units (np.mean, np.std).\n",
    "#   2. Print r rounded to 3 decimals.\n",
    "# Print exactly:\n",
    "# r: -0.995\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "\n",
    "demand = pd.DataFrame({\n",
    "    \"price\": [10, 12, 14, 16, 18, 20],\n",
    "    \"units\": [48, 45, 39, 36, 30, 27],\n",
    "})\n",
    "\n",
    "### Your code starts here.\n",
    "\n",
    "\n",
    "\n",
    "### Your code ends here.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "tags": [
     "hide-input",
     "homework-answer"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "r: -0.995\n"
     ]
    }
   ],
   "source": [
    "### Solution\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "\n",
    "demand = pd.DataFrame({\n",
    "    \"price\": [10, 12, 14, 16, 18, 20],\n",
    "    \"units\": [48, 45, 39, 36, 30, 27],\n",
    "})\n",
    "\n",
    "def standard_units(x):\n",
    "    return (x - np.mean(x)) / np.std(x)\n",
    "\n",
    "r = np.mean(standard_units(demand['price']) * standard_units(demand['units']))\n",
    "print(\"r:\", round(r, 3))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "tags": [
     "thebe-interactive"
    ]
   },
   "outputs": [],
   "source": [
    "### 7. Predicting Demand\n",
    "#   1. Compute the slope and intercept of the regression line of units on price.\n",
    "#   2. Predict units sold at a price of 15, rounded to 1 decimal\n",
    "#      (use the unrounded slope and intercept).\n",
    "# Print exactly:\n",
    "# Slope: -2.19\n",
    "# Intercept: 70.29\n",
    "# Predicted units at 15: 37.5\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "\n",
    "demand = pd.DataFrame({\n",
    "    \"price\": [10, 12, 14, 16, 18, 20],\n",
    "    \"units\": [48, 45, 39, 36, 30, 27],\n",
    "})\n",
    "\n",
    "### Your code starts here.\n",
    "\n",
    "\n",
    "\n",
    "### Your code ends here.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "tags": [
     "hide-input",
     "homework-answer"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Slope: -2.19\n",
      "Intercept: 70.29\n",
      "Predicted units at 15: 37.5\n"
     ]
    }
   ],
   "source": [
    "### Solution\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "\n",
    "demand = pd.DataFrame({\n",
    "    \"price\": [10, 12, 14, 16, 18, 20],\n",
    "    \"units\": [48, 45, 39, 36, 30, 27],\n",
    "})\n",
    "\n",
    "def standard_units(x):\n",
    "    return (x - np.mean(x)) / np.std(x)\n",
    "\n",
    "def correlation(x, y):\n",
    "    return np.mean(standard_units(x) * standard_units(y))\n",
    "\n",
    "def slope(x, y):\n",
    "    return correlation(x, y) * np.std(y) / np.std(x)\n",
    "\n",
    "def intercept(x, y):\n",
    "    return np.mean(y) - slope(x, y) * np.mean(x)\n",
    "\n",
    "a = slope(demand['price'], demand['units'])\n",
    "b = intercept(demand['price'], demand['units'])\n",
    "print(\"Slope:\", round(a, 2))\n",
    "print(\"Intercept:\", round(b, 2))\n",
    "print(\"Predicted units at 15:\", round(a * 15 + b, 1))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "tags": [
     "thebe-interactive"
    ]
   },
   "outputs": [],
   "source": [
    "### 8. Predicting in Standard Units\n",
    "# Monthly spending has mean $80 and SD $25; r between visits and spending is 0.6.\n",
    "# A customer's visits are 2.0 SDs above average.\n",
    "#   1. Predict the customer's spending in standard units: r * (visits in standard units).\n",
    "#   2. Convert the prediction to dollars: mean + (standard units) * SD.\n",
    "# Print exactly:\n",
    "# Predicted spend (standard units): 1.2\n",
    "# Predicted spend: 110.0\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "\n",
    "r = 0.6\n",
    "visits_su = 2.0\n",
    "spend_mean = 80\n",
    "spend_sd = 25\n",
    "\n",
    "### Your code starts here.\n",
    "\n",
    "\n",
    "\n",
    "### Your code ends here.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "tags": [
     "hide-input",
     "homework-answer"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Predicted spend (standard units): 1.2\n",
      "Predicted spend: 110.0\n"
     ]
    }
   ],
   "source": [
    "### Solution\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "\n",
    "r = 0.6\n",
    "visits_su = 2.0\n",
    "spend_mean = 80\n",
    "spend_sd = 25\n",
    "\n",
    "spend_su = r * visits_su\n",
    "print(\"Predicted spend (standard units):\", spend_su)\n",
    "print(\"Predicted spend:\", spend_mean + spend_su * spend_sd)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "tags": [
     "thebe-interactive"
    ]
   },
   "outputs": [],
   "source": [
    "### 9. How Far Off Are the Predictions?\n",
    "#   1. Compute the fitted values of the regression line of units on price.\n",
    "#   2. Compute the RMSE: sqrt(mean of (actual - fitted) ** 2), rounded to 2 decimals.\n",
    "# Print exactly:\n",
    "# RMSE: 0.72\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "\n",
    "demand = pd.DataFrame({\n",
    "    \"price\": [10, 12, 14, 16, 18, 20],\n",
    "    \"units\": [48, 45, 39, 36, 30, 27],\n",
    "})\n",
    "\n",
    "### Your code starts here.\n",
    "\n",
    "\n",
    "\n",
    "### Your code ends here.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "tags": [
     "hide-input",
     "homework-answer"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "RMSE: 0.72\n"
     ]
    }
   ],
   "source": [
    "### Solution\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "\n",
    "demand = pd.DataFrame({\n",
    "    \"price\": [10, 12, 14, 16, 18, 20],\n",
    "    \"units\": [48, 45, 39, 36, 30, 27],\n",
    "})\n",
    "\n",
    "def standard_units(x):\n",
    "    return (x - np.mean(x)) / np.std(x)\n",
    "\n",
    "def correlation(x, y):\n",
    "    return np.mean(standard_units(x) * standard_units(y))\n",
    "\n",
    "def slope(x, y):\n",
    "    return correlation(x, y) * np.std(y) / np.std(x)\n",
    "\n",
    "def intercept(x, y):\n",
    "    return np.mean(y) - slope(x, y) * np.mean(x)\n",
    "\n",
    "fitted = slope(demand['price'], demand['units']) * demand['price'] + intercept(demand['price'], demand['units'])\n",
    "print(\"RMSE:\", round(np.sqrt(np.mean((demand['units'] - fitted) ** 2)), 2))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "tags": [
     "thebe-interactive"
    ]
   },
   "outputs": [],
   "source": [
    "### 10. Does the Spread Grow?\n",
    "# Residuals from a regression of delivery cost on distance, sorted by fitted value.\n",
    "#   1. Compute the SD (np.std) of the residuals for the 5 smallest fitted values\n",
    "#      and for the 5 largest, each rounded to 2 decimals.\n",
    "#   2. Print whether the SD for the largest fitted values is more than twice the other.\n",
    "# Print exactly:\n",
    "# SD (small fitted): 1.47\n",
    "# SD (large fitted): 7.4\n",
    "# Spread grows: True\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "\n",
    "diag = pd.DataFrame({\n",
    "    \"fitted\": [20, 25, 30, 35, 40, 45, 50, 55, 60, 65],\n",
    "    \"residual\": [1, -1, 2, -2, 1, -6, 7, -8, 9, -7],\n",
    "})\n",
    "\n",
    "### Your code starts here.\n",
    "\n",
    "\n",
    "\n",
    "### Your code ends here.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "tags": [
     "hide-input",
     "homework-answer"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "SD (small fitted): 1.47\n",
      "SD (large fitted): 7.4\n",
      "Spread grows: True\n"
     ]
    }
   ],
   "source": [
    "### Solution\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "\n",
    "diag = pd.DataFrame({\n",
    "    \"fitted\": [20, 25, 30, 35, 40, 45, 50, 55, 60, 65],\n",
    "    \"residual\": [1, -1, 2, -2, 1, -6, 7, -8, 9, -7],\n",
    "})\n",
    "\n",
    "small = round(np.std(diag['residual'].iloc[:5]), 2)\n",
    "large = round(np.std(diag['residual'].iloc[5:]), 2)\n",
    "\n",
    "print(\"SD (small fitted):\", small)\n",
    "print(\"SD (large fitted):\", large)\n",
    "print(\"Spread grows:\", large > 2 * small)\n",
    "# Predictions for long distances are much less precise (heteroscedasticity).\n"
   ]
  },
  {
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    "\n",
    "  function applyFeedback(grader, feedback) {\n",
    "    var list = grader.querySelector('.ch12-hw-feedback-list');\n",
    "    list.hidden = false;\n",
    "    ['q1', 'q2', 'q3', 'q4', 'q5', 'q6', 'q7', 'q8', 'q9', 'q10'].forEach(function (question) {\n",
    "      var row = grader.querySelector('[data-question=' + question + ']');\n",
    "      var message = row.querySelector('span');\n",
    "      var item = feedback[question];\n",
    "      row.className = 'ch12-hw-feedback-row';\n",
    "      if (!item) return;\n",
    "      message.textContent = item.message + ' (' + item.score + ' / ' + item.max_score + ')';\n",
    "      row.classList.add(item.correct ? 'correct' : 'incorrect');\n",
    "    });\n",
    "  }\n",
    "\n",
    "  function clearFeedback(grader) {\n",
    "    var list = grader.querySelector('.ch12-hw-feedback-list');\n",
    "    list.hidden = true;\n",
    "    grader.querySelectorAll('.ch12-hw-feedback-row').forEach(function (row) {\n",
    "      row.className = 'ch12-hw-feedback-row';\n",
    "      var message = row.querySelector('span');\n",
    "      if (message) message.textContent = '';\n",
    "    });\n",
    "  }\n",
    "\n",
    "  function saveAttempt(payload) {\n",
    "    return fetch('/api/v1/homework-attempts.php', {\n",
    "      method: 'POST',\n",
    "      headers: { 'Content-Type': 'application/json' },\n",
    "      credentials: 'same-origin',\n",
    "      body: JSON.stringify(payload)\n",
    "    }).then(function (response) {\n",
    "      return response.text().then(function (text) {\n",
    "        var payload = text ? JSON.parse(text) : null;\n",
    "        if (!payload || !response.ok || !payload.ok) {\n",
    "          throw new Error((payload && (payload.message || payload.error)) || 'Submission API is unavailable.');\n",
    "        }\n",
    "        return payload;\n",
    "      });\n",
    "    });\n",
    "  }\n",
    "\n",
    "  function initHomeworkGrader() {\n",
    "    var grader = document.getElementById('ch12-hw-grader');\n",
    "    if (!grader || grader.dataset.ready === 'true') return;\n",
    "    grader.dataset.ready = 'true';\n",
    "\n",
    "    var identity = getParams();\n",
    "    var actions = grader.querySelector('.ch12-hw-actions');\n",
    "    var result = document.getElementById('ch12-hw-result');\n",
    "    var submit = document.getElementById('ch12-hw-submit');\n",
    "    var reset = document.getElementById('ch12-hw-reset');\n",
    "\n",
    "    loadSessionIdentity(identity, actions, submit, result);\n",
    "\n",
    "    submit.addEventListener('click', function () {\n",
    "      clearFeedback(grader);\n",
    "      submit.disabled = true;\n",
    "      result.textContent = 'Submitting homework.';\n",
    "\n",
    "      saveAttempt({\n",
    "        homework_id: homeworkId,\n",
    "        answers: collectAnswers(),\n",
    "        code: collectCodeCells(),\n",
    "        canvas_course_id: identity.canvas_course_id,\n",
    "        canvas_assignment_id: identity.canvas_assignment_id,\n",
    "        canvas_user_id: identity.canvas_user_id\n",
    "      })\n",
    "        .then(function (payload) {\n",
    "          var best = (payload.best_score === null || typeof payload.best_score === 'undefined') ? payload.score : payload.best_score;\n",
    "          var attempt = (payload.attempt_count === null || typeof payload.attempt_count === 'undefined') ? '' : ' Attempt ' + payload.attempt_count + '.';\n",
    "          applyFeedback(grader, payload.feedback || {});\n",
    "          result.textContent = 'Submission saved.' + attempt + ' Score: ' + payload.score + ' / ' + payload.max_score + '. Best score: ' + best + ' / ' + payload.max_score + '. Canvas sync: ' + payload.canvas_sync_status + '.';\n",
    "        })\n",
    "        .catch(function (error) {\n",
    "          result.textContent = 'Could not save this attempt: ' + error.message;\n",
    "        })\n",
    "        .finally(function () {\n",
    "          submit.disabled = false;\n",
    "        });\n",
    "    });\n",
    "\n",
    "    reset.addEventListener('click', function () {\n",
    "      clearFeedback(grader);\n",
    "      result.textContent = '';\n",
    "    });\n",
    "  }\n",
    "\n",
    "  if (document.readyState === 'loading') {\n",
    "    document.addEventListener('DOMContentLoaded', initHomeworkGrader);\n",
    "  } else {\n",
    "    initHomeworkGrader();\n",
    "  }\n",
    "})();\n",
    "</script>\n"
   ]
  }
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