Certainly! Let's walk through the steps to answer the questions.
### Step 1: Identify all samples of size [tex]\( n = 2 \)[/tex]
Given the four pairs of shoes:
1. Black boots (B) cost [tex]$70
2. Red boots (R) cost $[/tex]60
3. White sneakers (W) cost [tex]$50
4. Green sandals (G) cost $[/tex]30
The samples of size 2, considering all possible pairs, are:
1. (B, R)
2. (B, W)
3. (B, G)
4. (R, W)
5. (R, G)
6. (W, G)
### Step 2: Calculate the mean price for each sample
To find the mean, add the costs of the two shoes in each sample and divide by 2:
1. (B, R): [tex]\(\frac{70 + 60}{2} = 65.0\)[/tex]
2. (B, W): [tex]\(\frac{70 + 50}{2} = 60.0\)[/tex]
3. (B, G): [tex]\(\frac{70 + 30}{2} = 50.0\)[/tex]
4. (R, W): [tex]\(\frac{60 + 50}{2} = 55.0\)[/tex]
5. (R, G): [tex]\(\frac{60 + 30}{2} = 45.0\)[/tex]
6. (W, G): [tex]\(\frac{50 + 30}{2} = 40.0\)[/tex]
### Step 3: Construct the sampling distribution of the sample means
The sample means found are:
1. 65.0
2. 60.0
3. 50.0
4. 55.0
5. 45.0
6. 40.0
These sample means are:
[tex]\[ [65.0, 60.0, 50.0, 55.0, 45.0, 40.0] \][/tex]
### Step 4: Calculate the sampling distribution
The sampling distribution shows the frequency of each mean:
[tex]\[ \{ 65.0: 1, 60.0: 1, 50.0: 1, 55.0: 1, 45.0: 1, 40.0: 1 \} \][/tex]
### Step 5: Summary
- The number of unique sample means is 6 dots.
- The unique sample means (dot positions) are: 65.0, 60.0, 50.0, 55.0, 45.0, and 40.0.
### Completing the table
| Sample | Cost (\[tex]$) | Mean (\$[/tex]) |
|--------|-----------|-----------|
| (B, R) | 70, 60 | 65.0 |
| (B, W) | 70, 50 | 60.0 |
| (B, G) | 70, 30 | 50.0 |
| (R, W) | 60, 50 | 55.0 |
| (R, G) | 60, 30 | 45.0 |
| (W, G) | 50, 30 | 40.0 |
### Question Answers
1. The mean for the first sample (B, R) is: \$65.0
2. The sampling distribution of the means of the samples can be described as: There will be 6 dots located at the unique means: 65.0, 60.0, 50.0, 55.0, 45.0, and 40.0.
Now we have a thorough, step-by-step breakdown of the problem and its results.
