Answer:
To solve this problem, let's start by listing all possible samples of size 3 from the population \(\{2, 3, 5, 7\}\).
### Step 1: List all possible samples
We are drawing samples without replacement. The total number of possible samples of size 3 from a population of 4 elements is given by the combination formula \( \binom{4}{3} \), which is 4. The samples are:
1. \( \{2, 3, 5\} \)
2. \( \{2, 3, 7\} \)
3. \( \{2, 5, 7\} \)
4. \( \{3, 5, 7\} \)
### Step 2: Calculate the mean of each sample
- Mean of \( \{2, 3, 5\} \): \( \frac{2 + 3 + 5}{3} = 3.33 \)
- Mean of \( \{2, 3, 7\} \): \( \frac{2 + 3 + 7}{3} = 4.00 \)
- Mean of \( \{2, 5, 7\} \): \( \frac{2 + 5 + 7}{3} = 4.67 \)
- Mean of \( \{3, 5, 7\} \): \( \frac{3 + 5 + 7}{3} = 5.00 \)
### Step 3: Create the sampling probability distribution
Each sample has an equal probability of being chosen. Since there are 4 samples, the probability for each sample is \( \frac{1}{4} \).
| Sample | Sample Mean | Probability |
|--------------|-------------|-------------|
| \{2, 3, 5\} | 3.33 | 0.25 |
| \{2, 3, 7\} | 4.00 | 0.25 |
| \{2, 5, 7\} | 4.67 | 0.25 |
| \{3, 5, 7\} | 5.00 | 0.25 |
### Step 4: Find the mean of the sampling distribution
The mean of the sampling distribution (\( \mu_{\bar{X}} \)) is the expected value of the sample means:
\[ \mu_{\bar{X}} = \sum (\text{Sample Mean} \times \text{Probability}) \]
\[ \mu_{\bar{X}} = (3.33 \times 0.25) + (4.00 \times 0.25) + (4.67 \times 0.25) + (5.00 \times 0.25) \]
\[ \mu_{\bar{X}} = 0.8325 + 1.00 + 1.1675 + 1.25 \]
\[ \mu_{\bar{X}} = 4.25 \]
### Step 5: Find the variance of the sampling distribution
The variance of the sampling distribution (\( \sigma^2_{\bar{X}} \)) is given by:
\[ \sigma^2_{\bar{X}} = \sum ((\text{Sample Mean} - \mu_{\bar{X}})^2 \times \text{Probability}) \]
\[ \sigma^2_{\bar{X}} = ( (3.33 - 4.25)^2 \times 0.25 ) + ( (4.00 - 4.25)^2 \times 0.25 ) + ( (4.67 - 4.25)^2 \times 0.25 ) + ( (5.00 - 4.25)^2 \times 0.25 ) \]
\[ \sigma^2_{\bar{X}} = ( ( -0.92 )^2 \times 0.25 ) + ( ( -0.25 )^2 \times 0.25 ) + ( ( 0.42 )^2 \times 0.25 ) + ( ( 0.75 )^2 \times 0.25 ) \]
\[ \sigma^2_{\bar{X}} = ( 0.8464 \times 0.25 ) + ( 0.0625 \times 0.25 ) + ( 0.1764 \times 0.25 ) + ( 0.5625 \times 0.25 ) \]
\[ \sigma^2_{\bar{X}} = 0.2116 + 0.015625 + 0.0441 + 0.140625 \]
\[ \sigma^2_{\bar{X}} = 0.412 \]
### Step 6: Find the standard deviation of the sampling distribution
The standard deviation (\( \sigma_{\bar{X}} \)) is the square root of the variance:
\[ \sigma_{\bar{X}} = \sqrt{\sigma^2_{\bar{X}}} \]
\[ \sigma_{\bar{X}} = \sqrt{0.412} \]
\[ \sigma_{\bar{X}} \approx 0.64 \]
### Summary
- Mean of the sampling distribution: \( \mu_{\bar{X}} = 4.25 \)
- Variance of the sampling distribution: \( \sigma^2_{\bar{X}} = 0.412 \)
- Standard deviation of the sampling distribution: \( \sigma_{\bar{X}} \approx 0.64
