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.