To calculate the probability for part (c), we can use the following steps based on the normal approximation:

Given Data:

• Mean ($\mu_{\hat{p}}$) = 0.2735
• Standard deviation ($\sigma_{\hat{p}}$) = 0.01934
• We are finding $P(\hat{p} \leq 0.25)$.

Standardizing to the Z-Score:

The formula for the z-score is: $z = \frac{\hat{p} - \mu_{\hat{p}}}{\sigma_{\hat{p}}}$ Substitute the values: $z = \frac{0.25 - 0.2735}{0.01934} = \frac{-0.0235}{0.01934} \approx -1.215$

Finding the Probability:

Using standard normal distribution tables or a calculator, find $P(Z \leq -1.215)$: $P(Z \leq -1.215) \approx 0.1122$

Thus, the correct answer for $P(\hat{p} \leq 0.25)$ is approximately 0.1122.

Let me know if you want further clarification! Here are 5 questions that expand this context:

1. How do you interpret a z-score in the context of probabilities?
2. What is the significance of the standard deviation in these calculations?
3. How would the probability change if the sample size were larger?
4. How can you check if the normal approximation is appropriate for a problem like this?
5. How do these probabilities relate to confidence intervals in hypothesis testing?

Tip: Always confirm that the normal approximation is appropriate by checking if $np \geq 10$ and $n(1-p) \geq 10$. This ensures accuracy.