Question 1190594
Here's how to calculate the requested values for both scenarios (with and without replacement):

**Scenario 1: With Replacement**

1.  **Population Mean (μ):**
    μ = (3 + 8 + 10 + 15) / 4 = 36 / 4 = 9

2.  **Population Variance (σ²):**
    σ² = [(3-9)² + (8-9)² + (10-9)² + (15-9)²] / 4
    σ² = [36 + 1 + 1 + 36] / 4 = 74 / 4 = 18.5

3.  **Population Standard Deviation (σ):**
    σ = √σ² = √18.5 ≈ 4.3

4.  **Variance of the Sampling Distribution of Means:**
    When sampling with replacement, the variance of the sampling distribution of means (σₓ̄²) is given by:
    σₓ̄² = σ² / n  where n is the sample size.
    σₓ̄² = 18.5 / 2 = 9.25

5.  **Standard Deviation of the Sampling Distribution of Means (Standard Error):**
    This is the square root of the variance of the sampling distribution:
    σₓ̄ = √σₓ̄² = √9.25 ≈ 3.04

6.  **Mean of the Sampling Distribution of Means (μₓ̄):**
    When sampling with replacement, the mean of the sampling distribution of means is equal to the population mean:
    μₓ̄ = μ = 9

**Scenario 2: Without Replacement**

1.  **Population Mean (μ):**  (Same as before)
    μ = 9

2.  **Population Variance (σ²):** (Same as before)
    σ² = 18.5

3.  **Population Standard Deviation (σ):** (Same as before)
    σ ≈ 4.3

4.  **Variance of the Sampling Distribution of Means:**
    When sampling *without* replacement, the variance of the sampling distribution is adjusted by a finite population correction factor:
    σₓ̄² = (σ² / n) * [(N - n) / (N - 1)]
    Where N is the population size.
    σₓ̄² = (18.5 / 2) * [(4 - 2) / (4 - 1)]
    σₓ̄² = 9.25 * (2/3) = 6.17 (approximately)

5.  **Standard Deviation of the Sampling Distribution of Means (Standard Error):**
    σₓ̄ = √σₓ̄² = √6.17 ≈ 2.48

6.  **Mean of the Sampling Distribution of Means (μₓ̄):**
    Even when sampling without replacement, the mean of the sampling distribution of means is still equal to the population mean:
    μₓ̄ = μ = 9

**Summary Table:**

| Statistic | With Replacement | Without Replacement |
|---|---|---|
| Population Mean (μ) | 9 | 9 |
| Population Variance (σ²) | 18.5 | 18.5 |
| Population Standard Deviation (σ) | 4.3 | 4.3 |
| Variance of Sampling Distribution (σₓ̄²) | 9.25 | 6.17 |
| Standard Error of the Mean (σₓ̄) | 3.04 | 2.48 |
| Mean of Sampling Distribution (μₓ̄) | 9 | 9 |