Let's go through the problem step by step:

Given Information:

• Historical records show that 80% of test subjects pass the test, so $p = 0.80$.
• Sample size: $n = 625$.

Let’s answer the questions:

(a) Find the mean of $\hat{p}$.

The mean of $\hat{p}$ (which is the sample proportion) in a binomial distribution is given by:

$\mu_{\hat{p}} = p$

Where $p$ is the population proportion.

$\mu_{\hat{p}} = 0.80$

So, the mean of $\hat{p}$ is 0.80.

(b) Find the standard deviation of $\hat{p}$.

The standard deviation of $\hat{p}$ is given by the formula:

$\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$

Substitute the values $p = 0.80$, $1 - p = 0.20$, and $n = 625$:

$\sigma_{\hat{p}} = \sqrt{\frac{0.80 \times 0.20}{625}} = \sqrt{\frac{0.16}{625}} = \sqrt{0.000256} = 0.016$

So, the standard deviation of $\hat{p}$ is 0.016.

(c) Compute an approximation for $P(\hat{p} > 0.76)$.

We are asked to find the probability that the proportion of people passing the test is greater than 76%, or $P(\hat{p} > 0.76)$.

To compute this, we need to standardize $\hat{p} = 0.76$ using a z-score:

$z = \frac{\hat{p} - \mu_{\hat{p}}}{\sigma_{\hat{p}}}$

Substitute the values $\hat{p} = 0.76$, $\mu_{\hat{p}} = 0.80$, and $\sigma_{\hat{p}} = 0.016$:

$z = \frac{0.76 - 0.80}{0.016} = \frac{-0.04}{0.016} = -2.5$

Now, use the z-score table or normal distribution calculator to find the probability corresponding to $z = -2.5$.

From standard z-tables, $P(z > -2.5)$ is approximately:

$P(z > -2.5) = 0.9938$

So, the probability that more than 76% of the people pass the test is approximately 0.9938.

Summary of Answers:

• (a) Mean of $\hat{p}$: 0.80.
• (b) Standard deviation of $\hat{p}$: 0.016.
• (c) Approximation for $P(\hat{p} > 0.76)$: 0.9938.

Would you like more details or explanations on any of these steps?

Here are some relative questions to expand on this:

1. How does the sample size affect the standard deviation of the sample proportion?
2. Can the mean $\hat{p}$ change if the population proportion remains the same?
3. How would the calculation differ if the sample size was smaller or larger?
4. What is the probability that less than 78% of the sample passes the test?
5. How can we compute confidence intervals for $\hat{p}$?

Tip: Understanding how to compute and use the z-score will help you analyze probabilities for normal distributions effectively.