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To solve this problem, we need to calculate the mean and standard deviation of the sampling distribution of the sample proportion, denoted as \(\hat{p}\), where:

1. **Given values**:
   - Population proportion (\(p\)) = 0.75 (75% of passengers show up).
   - Sample size (\(n\)) = 8.

### (a) Mean of the sampling distribution of \(\hat{p}\)

The mean of the sampling distribution of the sample proportion, \(\mu_{\hat{p}}\), is given by:
\[
\mu_{\hat{p}} = p
\]
So:
\[
\mu_{\hat{p}} = 0.75
\]

### (b) Standard deviation of the sampling distribution of \(\hat{p}\)

The standard deviation of the sampling distribution of the sample proportion, \(\sigma_{\hat{p}}\), is given by:
\[
\sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}}
\]
Substituting the values:
\[
\sigma_{\hat{p}} = \sqrt{\frac{0.75 \times (1 - 0.75)}{8}} = \sqrt{\frac{0.75 \times 0.25}{8}}
\]
\[
\sigma_{\hat{p}} = \sqrt{\frac{0.1875}{8}} = \sqrt{0.0234375} \approx 0.1531
\]

### Answers:

1. **(a)** \(\mu_{\hat{p}} = 0.75\)
2. **(b)** \(\sigma_{\hat{p}} \approx 0.15\)

Would you like further details on these calculations or have any questions?

#### Related Questions:
1. What conditions must be met to use the normal approximation for the sampling distribution of \(\hat{p}\)?
2. How would the mean and standard deviation change if the sample size were larger?
3. How is the sampling distribution of \(\hat{p}\) affected if \(p\) were closer to 0 or 1?
4. What does the standard deviation of \(\hat{p}\) indicate about the variability of sample proportions?
5. How would you interpret a sample proportion of 0.50 in this context?

**Tip:** When calculating proportions, double-check that the sample size meets requirements for normal approximation if used.

The large airline company, Absolute Airways, knows from experience that only 75% of the passengers booked for flights actually show up. Suppose that we will take a random sample of 8 Absolute Airways passengers. Let p represent the proportion of passengers from the sample who show up for their flight. Consider the sampling distribution of the sample proportion p̂. Complete the following:
(a) Find μₚ̂ (the mean of the sampling distribution of the sample proportion).
(b) Find σₚ̂ (the standard deviation of the sampling distribution of the sample proportion).

Learn how to find the mean and standard deviation of the sampling distribution of a sample proportion. This example explores a real-world scenario where 75% of airline passengers show up, with a sample size of 8. Suitable for Grades 10-12 students studying probability and statistics.

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