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```python?code_reference&code_event_index=2
\n" ); document.write( "import numpy as np
\n" ); document.write( "import pandas as pd
\n" ); document.write( "from collections import Counter\r
\n" ); document.write( "\n" ); document.write( "# Given data (18 four-digit numbers)
\n" ); document.write( "data = [
\n" ); document.write( " 3143, 4151, 9341, 3430, 8552, 4184,
\n" ); document.write( " 4045, 3120, 8132, 1660, 2821, 9139,
\n" ); document.write( " 1328, 4738, 8929, 4280, 4883, 3919
\n" ); document.write( "]
\n" ); document.write( "N = len(data)\r
\n" ); document.write( "\n" ); document.write( "# --- Task 1: Frequency Table of the Final Digit ---\r
\n" ); document.write( "\n" ); document.write( "final_digits = [n % 10 for n in data]
\n" ); document.write( "frequency_counts = Counter(final_digits)\r
\n" ); document.write( "\n" ); document.write( "# Create a frequency table dictionary ensuring all 10 digits are present, even if count is 0
\n" ); document.write( "frequency_table = {i: frequency_counts.get(i, 0) for i in range(10)}\r
\n" ); document.write( "\n" ); document.write( "# --- Task 2: Mean of the Sample Means of the Final Four Digits ---\r
\n" ); document.write( "\n" ); document.write( "sample_means = []\r
\n" ); document.write( "\n" ); document.write( "# Function to calculate the mean of the four digits of a number
\n" ); document.write( "def mean_of_digits(number):
\n" ); document.write( " s = str(number).zfill(4) # Ensure it's treated as 4 digits
\n" ); document.write( " digit_sum = sum(int(digit) for digit in s)
\n" ); document.write( " return digit_sum / 4\r
\n" ); document.write( "\n" ); document.write( "# Calculate the mean of the digits for each of the 18 numbers
\n" ); document.write( "for number in data:
\n" ); document.write( " sample_means.append(mean_of_digits(number))\r
\n" ); document.write( "\n" ); document.write( "# Compute the mean of these 18 sample means
\n" ); document.write( "mean_of_sample_means = np.mean(sample_means)\r
\n" ); document.write( "\n" ); document.write( "print(f\"Frequency Table (Final Digit): {frequency_table}\")
\n" ); document.write( "print(f\"Individual Sample Means: {sample_means}\")
\n" ); document.write( "print(f\"Mean of Sample Means: {mean_of_sample_means}\")\r
\n" ); document.write( "\n" ); document.write( "```\r
\n" ); document.write( "\n" ); document.write( "```text?code_stdout&code_event_index=2
\n" ); document.write( "Frequency Table (Final Digit): {0: 4, 1: 3, 2: 2, 3: 2, 4: 1, 5: 1, 6: 0, 7: 0, 8: 2, 9: 3}
\n" ); document.write( "Individual Sample Means: [2.75, 2.75, 4.25, 2.5, 5.0, 4.25, 3.25, 1.5, 3.5, 3.25, 3.25, 5.5, 3.5, 5.5, 7.0, 3.5, 5.75, 5.5]
\n" ); document.write( "Mean of Sample Means: 4.027777777777778\r
\n" ); document.write( "\n" ); document.write( "```\r
\n" ); document.write( "\n" ); document.write( "The data consists of 18 four-digit numbers:
\n" ); document.write( "$$3143, 4151, 9341, 3430, 8552, 4184, 4045, 3120, 8132, 1660, 2821, 9139, 1328, 4738, 8929, 4280, 4883, 3919$$\r
\n" ); document.write( "\n" ); document.write( "-----\r
\n" ); document.write( "\n" ); document.write( "## 1\. Frequency Table of the Final Digit\r
\n" ); document.write( "\n" ); document.write( "The final digits of the 18 numbers are:
\n" ); document.write( "$$3, 1, 1, 0, 2, 4, 5, 0, 2, 0, 1, 9, 8, 8, 9, 0, 3, 9$$\r
\n" ); document.write( "\n" ); document.write( "We count the frequency of each digit from 0 to 9:\r
\n" ); document.write( "\n" ); document.write( "| Final Digit | Frequency |
\n" ); document.write( "| :---: | :---: |
\n" ); document.write( "| 0 | 4 |
\n" ); document.write( "| 1 | 3 |
\n" ); document.write( "| 2 | 2 |
\n" ); document.write( "| 3 | 2 |
\n" ); document.write( "| 4 | 1 |
\n" ); document.write( "| 5 | 1 |
\n" ); document.write( "| 6 | 0 |
\n" ); document.write( "| 7 | 0 |
\n" ); document.write( "| 8 | 2 |
\n" ); document.write( "| 9 | 3 |
\n" ); document.write( "| **Total** | **18** |\r
\n" ); document.write( "\n" ); document.write( "-----\r
\n" ); document.write( "\n" ); document.write( "## 2\. Compute the Mean of the Sample Means\r
\n" ); document.write( "\n" ); document.write( "The \"sample mean of the final four digits\" is defined as the arithmetic mean of the four individual digits for each number (e.g., $9704 \to (9+7+0+4)/4 = 5$).\r
\n" ); document.write( "\n" ); document.write( "We calculate the mean of the digits for all 18 numbers:\r
\n" ); document.write( "\n" ); document.write( "| Number | Digit Sum | Sample Mean ($\bar{x}_i$) |
\n" ); document.write( "| :---: | :---: | :---: |
\n" ); document.write( "| 3143 | $3+1+4+3=11$ | $11/4 = 2.75$ |
\n" ); document.write( "| 4151 | $4+1+5+1=11$ | $11/4 = 2.75$ |
\n" ); document.write( "| 9341 | $9+3+4+1=17$ | $17/4 = 4.25$ |
\n" ); document.write( "| 3430 | $3+4+3+0=10$ | $10/4 = 2.50$ |
\n" ); document.write( "| 8552 | $8+5+5+2=20$ | $20/4 = 5.00$ |
\n" ); document.write( "| 4184 | $4+1+8+4=17$ | $17/4 = 4.25$ |
\n" ); document.write( "| 4045 | $4+0+4+5=13$ | $13/4 = 3.25$ |
\n" ); document.write( "| 3120 | $3+1+2+0=6$ | $6/4 = 1.50$ |
\n" ); document.write( "| 8132 | $8+1+3+2=14$ | $14/4 = 3.50$ |
\n" ); document.write( "| 1660 | $1+6+6+0=13$ | $13/4 = 3.25$ |
\n" ); document.write( "| 2821 | $2+8+2+1=13$ | $13/4 = 3.25$ |
\n" ); document.write( "| 9139 | $9+1+3+9=22$ | $22/4 = 5.50$ |
\n" ); document.write( "| 1328 | $1+3+2+8=14$ | $14/4 = 3.50$ |
\n" ); document.write( "| 4738 | $4+7+3+8=22$ | $22/4 = 5.50$ |
\n" ); document.write( "| 8929 | $8+9+2+9=28$ | $28/4 = 7.00$ |
\n" ); document.write( "| 4280 | $4+2+8+0=14$ | $14/4 = 3.50$ |
\n" ); document.write( "| 4883 | $4+8+8+3=23$ | $23/4 = 5.75$ |
\n" ); document.write( "| 3919 | $3+9+1+9=22$ | $22/4 = 5.50$ |
\n" ); document.write( "| **Sum** | **-** | **$72.50$** |\r
\n" ); document.write( "\n" ); document.write( "### Calculation:\r
\n" ); document.write( "\n" ); document.write( "The mean of the sample means ($\bar{\bar{x}}$) is the sum of the 18 sample means divided by 18.
\n" ); document.write( "$$\bar{\bar{x}} = \frac{\sum \bar{x}_i}{N} = \frac{72.50}{18}$$
\n" ); document.write( "$$\bar{\bar{x}} \approx 4.02777...$$\r
\n" ); document.write( "\n" ); document.write( "Rounding to 2 decimal places:
\n" ); document.write( "$$\mathbf{\text{Mean of the Sample Means} \approx 4.03}$$ \n" ); document.write( "