{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"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. How Resampling Works**\n",
"\n",
"A bootstrap resample is drawn without replacement from the original sample and has the same size as the original sample.\n",
"\n",
"
\n",
" \n",
" \n",
"
\n",
"\n",
"::::\n"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"tags": [
"hide-input"
]
},
"outputs": [],
"source": [
"### Solution\n",
"# False\n",
"# A bootstrap resample is drawn WITH replacement. Without replacement, a resample of the same size would just be the original sample in a different order.\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"::::{div} assignment-question-card\n",
"\n",
"**2. Interval for a Mean**\n",
"\n",
"A 95% confidence interval for the mean order value is \\$42 to \\$48. This means that 95% of individual orders are between \\$42 and \\$48.\n",
"\n",
"
\n",
" \n",
" \n",
"
\n",
"\n",
"::::\n"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"tags": [
"hide-input"
]
},
"outputs": [],
"source": [
"### Solution\n",
"# False\n",
"# The interval estimates the average order value. Individual orders vary much more than their average does.\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"::::{div} assignment-question-card\n",
"\n",
"**3. What \"95% Confidence\" Means**\n",
"\n",
"For a reliable approximate procedure under suitable sampling conditions, if we repeated the whole process many times (draw a new random sample, bootstrap it, build a 95% interval), about 95% of those intervals would contain the true parameter.\n",
"\n",
"
\n",
" \n",
" \n",
"
\n",
"\n",
"::::\n"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"tags": [
"hide-input"
]
},
"outputs": [],
"source": [
"### Solution\n",
"# True\n",
"# The confidence level describes the process, not one particular interval.\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"::::{div} assignment-question-card\n",
"\n",
"**4. Lower Confidence**\n",
"\n",
"From the same bootstrap distribution, the 80% confidence interval is no wider than the 95% confidence interval.\n",
"\n",
"
\n",
" \n",
" \n",
"
\n",
"\n",
"::::\n"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"tags": [
"hide-input"
]
},
"outputs": [],
"source": [
"### Solution\n",
"# True\n",
"# The 80% interval runs from the 10th to the 90th percentile, which is inside the 2.5th to 97.5th percentile range.\n",
"# The widths can be equal when percentile endpoints tie.\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"::::{div} assignment-question-card\n",
"\n",
"**5. Estimating a Maximum**\n",
"\n",
"A bootstrap confidence interval works well for estimating the highest salary paid by a large company from a random sample of its employees.\n",
"\n",
"
\n",
" \n",
" \n",
"
\n",
"\n",
"::::\n"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"tags": [
"hide-input"
]
},
"outputs": [],
"source": [
"### Solution\n",
"# False\n",
"# The bootstrap works poorly for extremes. A resample can never contain a value larger than the sample's maximum, and the sample probably missed the highest salary.\n"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"tags": [
"thebe-interactive"
]
},
"outputs": [],
"source": [
"### 6. The Middle Half of Orders\n",
"# Describe individual order values (in dollars), not uncertainty in a mean.\n",
"# Use the chapter's nearest convention; this odd-size sample has the same\n",
"# ordinary median and nearest 50th percentile.\n",
"# 1. Compute the 25th, 50th, and 75th percentiles with method='nearest'.\n",
"# Print exactly:\n",
"# 25th percentile: 45\n",
"# Median: 49\n",
"# 75th percentile: 52\n",
"import numpy as np\n",
"import pandas as pd\n",
"\n",
"orders = pd.Series([38, 52, 45, 61, 47, 55, 40, 49, 58, 44, 51, 46, 63, 42, 50])\n",
"\n",
"### Your code starts here.\n",
"\n",
"\n",
"\n",
"### Your code ends here.\n"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"tags": [
"hide-input",
"homework-answer"
]
},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": "25th percentile: 45\nMedian: 49\n75th percentile: 52\n"
}
],
"source": [
"### Solution\n",
"import numpy as np\n",
"import pandas as pd\n",
"\n",
"orders = pd.Series([38, 52, 45, 61, 47, 55, 40, 49, 58, 44, 51, 46, 63, 42, 50])\n",
"\n",
"print(\"25th percentile:\", np.percentile(orders, 25, method='nearest'))\n",
"print(\"Median:\", np.percentile(orders, 50, method='nearest'))\n",
"print(\"75th percentile:\", np.percentile(orders, 75, method='nearest'))\n"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"tags": [
"thebe-interactive"
]
},
"outputs": [],
"source": [
"### 7. One Bootstrap Resample\n",
"# Draw one bootstrap resample of orders with .sample(): same size,\n",
"# replacement, and random_state=7. Then summarize that resample.\n",
"# 1. Compute the mean of the resample, rounded to 2 decimals.\n",
"# 2. Count the distinct values in the resample with nunique().\n",
"# Print exactly:\n",
"# Resample mean: 50.67\n",
"# Distinct values: 9 of 15\n",
"import numpy as np\n",
"import pandas as pd\n",
"\n",
"orders = pd.Series([38, 52, 45, 61, 47, 55, 40, 49, 58, 44, 51, 46, 63, 42, 50])\n",
"\n",
"### Your code starts here.\n",
"\n",
"\n",
"\n",
"### Your code ends here.\n"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"tags": [
"hide-input",
"homework-answer"
]
},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": "Resample mean: 50.67\nDistinct values: 9 of 15\n"
}
],
"source": [
"### Solution\n",
"import numpy as np\n",
"import pandas as pd\n",
"\n",
"orders = pd.Series([38, 52, 45, 61, 47, 55, 40, 49, 58, 44, 51, 46, 63, 42, 50])\n",
"resample = orders.sample(len(orders), replace=True, random_state=7)\n",
"\n",
"print(\"Resample mean:\", round(resample.mean(), 2))\n",
"print(\"Distinct values:\", resample.nunique(), \"of\", len(resample))\n",
"# Drawing with replacement repeats some orders and leaves others out.\n"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"tags": [
"thebe-interactive"
]
},
"outputs": [],
"source": [
"### 8. Interval for the Mean Order Value\n",
"# Generate 40 bootstrap means from orders: draw len(orders) values WITH\n",
"# replacement using rng.choice, and collect each mean in bootstrap_means.\n",
"# Set up the list before the loop. Keep the seed outside the loop.\n",
"# Then compute a central 95% interval with method='nearest'.\n",
"# Round only the displayed endpoints to 2 decimals.\n",
"# Forty repetitions make this a coarse workflow exercise; use thousands\n",
"# and check endpoint stability for a real management decision.\n",
"# Print exactly:\n",
"# 95% CI: 45.67 to 52.6\n",
"import numpy as np\n",
"import pandas as pd\n",
"\n",
"orders = pd.Series([38, 52, 45, 61, 47, 55, 40, 49, 58, 44, 51, 46, 63, 42, 50])\n",
"rng = np.random.default_rng(434)\n",
"\n",
"### Your code starts here.\n",
"\n",
"\n",
"### Your code ends here.\n"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"tags": [
"hide-input",
"homework-answer"
]
},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": "95% CI: 45.67 to 52.6\n"
}
],
"source": [
"### Solution\n",
"import numpy as np\n",
"import pandas as pd\n",
"\n",
"orders = pd.Series([38, 52, 45, 61, 47, 55, 40, 49, 58, 44, 51, 46, 63, 42, 50])\n",
"rng = np.random.default_rng(434)\n",
"bootstrap_means = []\n",
"for i in range(40):\n",
" resample = rng.choice(orders, size=len(orders), replace=True)\n",
" bootstrap_means.append(np.mean(resample))\n",
"\n",
"left = np.percentile(bootstrap_means, 2.5, method='nearest')\n",
"right = np.percentile(bootstrap_means, 97.5, method='nearest')\n",
"print(\"95% CI:\", round(left, 2), \"to\", round(right, 2))\n"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"tags": [
"thebe-interactive"
]
},
"outputs": [],
"source": [
"### 9. Sample Size and Interval Width\n",
"# Two surveys estimated the share of customers who use the mobile app.\n",
"# Survey A asked 100 customers; survey B asked 400.\n",
"# 1. Compute the width of each 95% interval, rounded to 3 decimals.\n",
"# 2. Compute how many times wider interval A is than interval B, rounded to 1 decimal.\n",
"# Print exactly:\n",
"# Width A: 0.18\n",
"# Width B: 0.09\n",
"# Ratio: 2.0\n",
"import numpy as np\n",
"import pandas as pd\n",
"\n",
"interval_a = [0.31, 0.49] # n = 100\n",
"interval_b = [0.355, 0.445] # n = 400\n",
"\n",
"### Your code starts here.\n",
"\n",
"\n",
"\n",
"### Your code ends here.\n"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"tags": [
"hide-input",
"homework-answer"
]
},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": "Width A: 0.18\nWidth B: 0.09\nRatio: 2.0\n"
}
],
"source": [
"### Solution\n",
"import numpy as np\n",
"import pandas as pd\n",
"\n",
"interval_a = [0.31, 0.49] # n = 100\n",
"interval_b = [0.355, 0.445] # n = 400\n",
"\n",
"width_a = interval_a[1] - interval_a[0]\n",
"width_b = interval_b[1] - interval_b[0]\n",
"\n",
"print(\"Width A:\", round(width_a, 3))\n",
"print(\"Width B:\", round(width_b, 3))\n",
"print(\"Ratio:\", round(width_a / width_b, 1))\n",
"# With comparable designs and variability, four times n gives roughly\n",
"# half the width. This is a planning approximation, not a guarantee.\n"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {
"tags": [
"thebe-interactive"
]
},
"outputs": [],
"source": [
"### 10. Checking a Supplier's Claim\n",
"# A supplier claims a 5% defect rate. In a random sample of 300 parts, 18 (0.06)\n",
"# were defective. The 40 rounded bootstrap proportions below are supplied\n",
"# for arithmetic practice, not a complete supplier-approval analysis.\n",
"# 1. Compute the 95% confidence interval (method='nearest').\n",
"# 2. Check whether the claimed rate of 0.05 is inside the interval.\n",
"# Print exactly:\n",
"# 95% CI: 0.033 to 0.083\n",
"# Claim inside interval: True\n",
"import numpy as np\n",
"import pandas as pd\n",
"\n",
"claimed = 0.05\n",
"bootstrap_props = pd.Series([0.043, 0.07, 0.033, 0.05, 0.093, 0.057, 0.063, 0.07, 0.067, 0.06,\n",
" 0.033, 0.067, 0.043, 0.05, 0.047, 0.083, 0.04, 0.067, 0.057, 0.083,\n",
" 0.057, 0.077, 0.083, 0.05, 0.057, 0.033, 0.06, 0.04, 0.06, 0.057,\n",
" 0.067, 0.07, 0.063, 0.05, 0.067, 0.073, 0.05, 0.057, 0.053, 0.073])\n",
"\n",
"### Your code starts here.\n",
"\n",
"\n",
"\n",
"### Your code ends here.\n"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {
"tags": [
"hide-input",
"homework-answer"
]
},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": "95% CI: 0.033 to 0.083\nClaim inside interval: True\n"
}
],
"source": [
"### Solution\n",
"import numpy as np\n",
"import pandas as pd\n",
"\n",
"claimed = 0.05\n",
"bootstrap_props = pd.Series([0.043, 0.07, 0.033, 0.05, 0.093, 0.057, 0.063, 0.07, 0.067, 0.06,\n",
" 0.033, 0.067, 0.043, 0.05, 0.047, 0.083, 0.04, 0.067, 0.057, 0.083,\n",
" 0.057, 0.077, 0.083, 0.05, 0.057, 0.033, 0.06, 0.04, 0.06, 0.057,\n",
" 0.067, 0.07, 0.063, 0.05, 0.067, 0.073, 0.05, 0.057, 0.053, 0.073])\n",
"\n",
"left = np.percentile(bootstrap_props, 2.5, method='nearest')\n",
"right = np.percentile(bootstrap_props, 97.5, method='nearest')\n",
"\n",
"print(\"95% CI:\", left, \"to\", right)\n",
"print(\"Claim inside interval:\", left <= claimed <= right)\n",
"# This practice interval does not rule out 5%; it does not prove the claim\n",
"# or show that the supplier meets the buyer's requirements.\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"