Question 1165219
```python?code_reference&code_event_index=2
import numpy as np
import pandas as pd
from collections import Counter

# Given data (18 four-digit numbers)
data = [
    3143, 4151, 9341, 3430, 8552, 4184,
    4045, 3120, 8132, 1660, 2821, 9139,
    1328, 4738, 8929, 4280, 4883, 3919
]
N = len(data)

# --- Task 1: Frequency Table of the Final Digit ---

final_digits = [n % 10 for n in data]
frequency_counts = Counter(final_digits)

# Create a frequency table dictionary ensuring all 10 digits are present, even if count is 0
frequency_table = {i: frequency_counts.get(i, 0) for i in range(10)}

# --- Task 2: Mean of the Sample Means of the Final Four Digits ---

sample_means = []

# Function to calculate the mean of the four digits of a number
def mean_of_digits(number):
    s = str(number).zfill(4) # Ensure it's treated as 4 digits
    digit_sum = sum(int(digit) for digit in s)
    return digit_sum / 4

# Calculate the mean of the digits for each of the 18 numbers
for number in data:
    sample_means.append(mean_of_digits(number))

# Compute the mean of these 18 sample means
mean_of_sample_means = np.mean(sample_means)

print(f"Frequency Table (Final Digit): {frequency_table}")
print(f"Individual Sample Means: {sample_means}")
print(f"Mean of Sample Means: {mean_of_sample_means}")

```

```text?code_stdout&code_event_index=2
Frequency Table (Final Digit): {0: 4, 1: 3, 2: 2, 3: 2, 4: 1, 5: 1, 6: 0, 7: 0, 8: 2, 9: 3}
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]
Mean of Sample Means: 4.027777777777778

```

The data consists of 18 four-digit numbers:
$$3143, 4151, 9341, 3430, 8552, 4184, 4045, 3120, 8132, 1660, 2821, 9139, 1328, 4738, 8929, 4280, 4883, 3919$$

-----

## 1\. Frequency Table of the Final Digit

The final digits of the 18 numbers are:
$$3, 1, 1, 0, 2, 4, 5, 0, 2, 0, 1, 9, 8, 8, 9, 0, 3, 9$$

We count the frequency of each digit from 0 to 9:

| Final Digit | Frequency |
| :---: | :---: |
| 0 | 4 |
| 1 | 3 |
| 2 | 2 |
| 3 | 2 |
| 4 | 1 |
| 5 | 1 |
| 6 | 0 |
| 7 | 0 |
| 8 | 2 |
| 9 | 3 |
| **Total** | **18** |

-----

## 2\. Compute the Mean of the Sample Means

The "sample mean of the final four digits" is defined as the arithmetic mean of the four individual digits for each number (e.g., $9704 \to (9+7+0+4)/4 = 5$).

We calculate the mean of the digits for all 18 numbers:

| Number | Digit Sum | Sample Mean ($\bar{x}_i$) |
| :---: | :---: | :---: |
| 3143 | $3+1+4+3=11$ | $11/4 = 2.75$ |
| 4151 | $4+1+5+1=11$ | $11/4 = 2.75$ |
| 9341 | $9+3+4+1=17$ | $17/4 = 4.25$ |
| 3430 | $3+4+3+0=10$ | $10/4 = 2.50$ |
| 8552 | $8+5+5+2=20$ | $20/4 = 5.00$ |
| 4184 | $4+1+8+4=17$ | $17/4 = 4.25$ |
| 4045 | $4+0+4+5=13$ | $13/4 = 3.25$ |
| 3120 | $3+1+2+0=6$ | $6/4 = 1.50$ |
| 8132 | $8+1+3+2=14$ | $14/4 = 3.50$ |
| 1660 | $1+6+6+0=13$ | $13/4 = 3.25$ |
| 2821 | $2+8+2+1=13$ | $13/4 = 3.25$ |
| 9139 | $9+1+3+9=22$ | $22/4 = 5.50$ |
| 1328 | $1+3+2+8=14$ | $14/4 = 3.50$ |
| 4738 | $4+7+3+8=22$ | $22/4 = 5.50$ |
| 8929 | $8+9+2+9=28$ | $28/4 = 7.00$ |
| 4280 | $4+2+8+0=14$ | $14/4 = 3.50$ |
| 4883 | $4+8+8+3=23$ | $23/4 = 5.75$ |
| 3919 | $3+9+1+9=22$ | $22/4 = 5.50$ |
| **Sum** | **-** | **$72.50$** |

### Calculation:

The mean of the sample means ($\bar{\bar{x}}$) is the sum of the 18 sample means divided by 18.
$$\bar{\bar{x}} = \frac{\sum \bar{x}_i}{N} = \frac{72.50}{18}$$
$$\bar{\bar{x}} \approx 4.02777...$$

Rounding to 2 decimal places:
$$\mathbf{\text{Mean of the Sample Means} \approx 4.03}$$