Question 1180440
**1. Population Mean and Sample Means:**

*   **Population Mean:**
    The population mean (μ) is the average of all the ages:
    μ = (1 + 13 + 32 + 34 + 59) / 5 = 139 / 5 = 27.8

*   **Sample Means:**
    Since the sample size is 2, we need to find all possible combinations of 2 ages from the population. There are 10 such combinations:
    (1, 13), (1, 32), (1, 34), (1, 59), (13, 32), (13, 34), (13, 59), (32, 34), (32, 59), (34, 59)
    Now, calculate the mean of each sample:
    7, 16.5, 17.5, 30, 22.5, 23.5, 36, 33, 45.5, 46.5

*   **Probability Distribution of Sample Means:**
    Each of the 10 sample means has an equal probability of occurring (1/10). The probability distribution is:

    | Sample Mean | Probability |
    |---|---|
    | 7 | 1/10 |
    | 16.5 | 1/10 |
    | 17.5 | 1/10 |
    | 30 | 1/10 |
    | 22.5 | 1/10 |
    | 23.5 | 1/10 |
    | 36 | 1/10 |
    | 33 | 1/10 |
    | 45.5 | 1/10 |
    | 46.5 | 1/10 |

**2. Sampling Distribution of Sample Variances and Standard Deviations:**

*   **Sample Variances:**
    For each sample, calculate the variance using the formula:
    s² = Σ(xᵢ - x̄)² / (n - 1)
    where xᵢ are the ages in the sample, x̄ is the sample mean, and n is the sample size (2).
    The sample variances are:
    72, 480.5, 544.5, 1682, 180.5, 220.5, 1058, 2, 364.5, 312.5

*   **Sample Standard Deviations:**
    For each sample, take the square root of the variance:
    s = √s²
    The sample standard deviations are:
    8.49, 21.92, 23.33, 41.01, 13.44, 14.85, 32.53, 1.41, 19.09, 17.68

*   **Sampling Distribution:**
    Each sample variance and standard deviation has a probability of 1/10 of occurring. The sampling distributions are:

    **Sample Variances:**

    | Sample Variance | Probability |
    |---|---|
    | 72 | 1/10 |
    | 480.5 | 1/10 |
    | 544.5 | 1/10 |
    | 1682 | 1/10 |
    | 180.5 | 1/10 |
    | 220.5 | 1/10 |
    | 1058 | 1/10 |
    | 2 | 1/10 |
    | 364.5 | 1/10 |
    | 312.5 | 1/10 |

    **Sample Standard Deviations:**

    | Sample Standard Deviation | Probability |
    |---|---|
    | 8.49 | 1/10 |
    | 21.92 | 1/10 |
    | 23.33 | 1/10 |
    | 41.01 | 1/10 |
    | 13.44 | 1/10 |
    | 14.85 | 1/10 |
    | 32.53 | 1/10 |
    | 1.41 | 1/10 |
    | 19.09 | 1/10 |
    | 17.68 | 1/10 |