Here are the answers:

a.

• The distribution of $Y$ is Exponential with a mean $\mu_Y = 4.3649$ minutes and a standard deviation $\sigma_Y = 4.3649$ minutes.
• The distribution of $\bar{Y}$ is Normal with a mean $\mu_{\bar{Y}} = 4.3649$ minutes and a standard deviation $\sigma_{\bar{Y}} = \frac{4.3649}{\sqrt{79}} = 0.49085$ minutes.

b.
The probability is 0.87352.

c.
The probability is 0.29469.

d.
The time is 4.54379 minutes.

Let me know if you have questions or want detailed explanations!

Here are 5 questions to explore next:

1. How is the exponential distribution related to the normal distribution of the sample mean?
2. Why does the standard deviation decrease for the sample mean?
3. How can you verify these calculations using statistical software or a calculator?
4. What are the implications of increasing the sample size from 79 to 200?
5. How does the probability change in part (b) if the range is shifted slightly?

Tip: Always double-check your calculations for probabilities using cumulative distribution functions (CDFs).