Question 1190594: A population consists of (3,8,10,15). Consider all possible sample size 2 that can be drawn with replacement.
-what is the standard deviation of the sampling distribution of means? =
-what is the population mean? =
-what is the population standard deviation? =
-what is the population variance? =
-what is the variance of the sampling distribution of means? =
-what is the mean of the sampling distribution of means? =
A population consists of (3,8,10,15). Consider all possible sample size
2 that can be drawn without replacement.
-what is the variance of the sampling distribution of means? =
-what is the standard deviation of the sampling distribution of means? =
-what is the mean of the sampling distribution of means? =
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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 |
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