Question 1177421
**1. List of Possible Samples and Their Means**

Since we're sampling with replacement, the same person can be selected twice. Here's a list of all possible samples and their means:

| Sample | Ages  | Mean |
|---|---|---|
| 1 | 10, 10 | 10 |
| 2 | 10, 20 | 15 |
| 3 | 10, 30 | 20 |
| 4 | 10, 40 | 25 |
| 5 | 10, 50 | 30 |
| 6 | 20, 10 | 15 |
| 7 | 20, 20 | 20 |
| 8 | 20, 30 | 25 |
| 9 | 20, 40 | 30 |
| 10 | 20, 50 | 35 |
| 11 | 30, 10 | 20 |
| 12 | 30, 20 | 25 |
| 13 | 30, 30 | 30 |
| 14 | 30, 40 | 35 |
| 15 | 30, 50 | 40 |
| 16 | 40, 10 | 25 |
| 17 | 40, 20 | 30 |
| 18 | 40, 30 | 35 |
| 19 | 40, 40 | 40 |
| 20 | 40, 50 | 45 |
| 21 | 50, 10 | 30 |
| 22 | 50, 20 | 35 |
| 23 | 50, 30 | 40 |
| 24 | 50, 40 | 45 |
| 25 | 50, 50 | 50 |

**2. Sampling Distribution of the Mean**

Now, let's organize the sample means into a frequency distribution:

| Sample Mean | Frequency | Probability |
|---|---|---|
| 10 | 1 | 1/25 |
| 15 | 2 | 2/25 |
| 20 | 3 | 3/25 |
| 25 | 4 | 4/25 |
| 30 | 5 | 5/25 |
| 35 | 4 | 4/25 |
| 40 | 3 | 3/25 |
| 45 | 2 | 2/25 |
| 50 | 1 | 1/25 |

This table represents the sampling distribution of the mean. It shows the probability of obtaining each sample mean when drawing a random sample of size two with replacement from the population.

**3. Line Graph and Description**

Here's a line graph representing the sampling distribution:

[asy]
unitsize(1 cm);

int i;
real prob;

draw((0,0)--(10,0));
draw((0,0)--(0,0.25));
label("Sample Mean", (10,0), E);
label("Probability", (0,0.25), N);

for (i = 1; i <= 9; ++i) {
  prob = (i <= 5 ? i : 10 - i)/25;
  draw((i,prob)--(i + 1,prob));
  dot((i,prob));
}

label("$10$", (1,0), S);
label("$15$", (2,0), S);
label("$20$", (3,0), S);
label("$25$", (4,0), S);
label("$30$", (5,0), S);
label("$35$", (6,0), S);
label("$40$", (7,0), S);
label("$45$", (8,0), S);
label("$50$", (9,0), S);
[/asy]

**Description:**

The line graph shows a symmetric distribution centered around the mean of 30. The distribution is unimodal, with the highest probability at the mean. As you move away from the mean, the probabilities decrease. This shape is somewhat similar to a normal distribution, although it's more discrete and triangular in this case due to the small sample size and the nature of the original population.

**Key Observations:**

* The mean of the sampling distribution is equal to the population mean (30).
* The sampling distribution has less variability than the original population.
* Even though the original population is not normally distributed, the sampling distribution starts to exhibit some characteristics of a normal distribution. This is a hint of the Central Limit Theorem, which states that the sampling distribution of the mean approaches a normal distribution as the sample size increases.