Question 1190599: A population consists of four number 2,5,8,and 1.
i.enumerate all possible samples of size 2 which can be drawn from this population with replacement.
ii. find mean and variance of the population.
iii. find the mean of sampling distribution of means and show that it is equal to the population mean.
iv. Find the variance of sampling distribution of mean
v. Find standard error of mean
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to work through this sampling problem:
**i. Possible Samples of Size 2 (with replacement):**
Since we're sampling with replacement, each draw is independent, and the same number can be selected twice. Here are all the possible samples of size 2:
(2,2), (2,5), (2,8), (2,1)
(5,2), (5,5), (5,8), (5,1)
(8,2), (8,5), (8,8), (8,1)
(1,2), (1,5), (1,8), (1,1)
There are 4 * 4 = 16 possible samples.
**ii. Population Mean and Variance:**
* **Population Mean (μ):**
μ = (2 + 5 + 8 + 1) / 4 = 16 / 4 = 4
* **Population Variance (σ²):**
σ² = Σ(xᵢ - μ)² / N
σ² = [(2-4)² + (5-4)² + (8-4)² + (1-4)²] / 4
σ² = [4 + 1 + 16 + 9] / 4
σ² = 30 / 4 = 7.5
**iii. Mean of the Sampling Distribution of the Means:**
1. **Calculate the mean of each sample:**
(2,2): 2; (2,5): 3.5; (2,8): 5; (2,1): 1.5
(5,2): 3.5; (5,5): 5; (5,8): 6.5; (5,1): 3
(8,2): 5; (8,5): 6.5; (8,8): 8; (8,1): 4.5
(1,2): 1.5; (1,5): 3; (1,8): 4.5; (1,1): 1
2. **Calculate the mean of these sample means:**
Mean of Sampling Distribution = (Sum of all sample means) / (Number of samples)
Mean of Sampling Distribution = (2+3.5+5+1.5+3.5+5+6.5+3+5+6.5+8+4.5+1.5+3+4.5+1)/16
Mean of Sampling Distribution = 64 / 16 = 4
As you can see, the mean of the sampling distribution of the means (4) is equal to the population mean (4).
**iv. Variance of the Sampling Distribution of the Mean:**
Variance of Sampling Distribution = Σ(Sample Mean - Population Mean)² / Number of Samples
Variance of Sampling Distribution = [(2-4)² + (3.5-4)² + (5-4)² + (1.5-4)² + (3.5-4)² + (5-4)² + (6.5-4)² + (3-4)² + (5-4)² + (6.5-4)² + (8-4)² + (4.5-4)² + (1.5-4)² + (3-4)² + (4.5-4)² + (1-4)²] / 16
Variance of Sampling Distribution = [4 + 0.25 + 1 + 6.25 + 0.25 + 1 + 6.25 + 1 + 1 + 6.25 + 16 + 0.25 + 6.25 + 1 + 0.25 + 9] / 16
Variance of Sampling Distribution = 52.5 / 16 = 3.28125
**v. Standard Error of the Mean:**
The standard error of the mean is the standard deviation of the sampling distribution of the means. It's the square root of the variance of the sampling distribution of the mean.
Standard Error = √Variance of Sampling Distribution
Standard Error = √3.28125 ≈ 1.81
|
|
|
