Let's go through the questions step-by-step.

Part (a)

The best point estimate for the population mean temperature, based on the sample, is simply the sample mean. Here, it is given as 58 degrees Fahrenheit.

Part (b)

Now, we’ll examine each scenario in the table to determine the appropriate distribution to use.

1. Sample size 16, normally distributed population, unknown standard deviation:

   • Since the sample size is small (n < 30) and the population standard deviation is unknown, use the t-distribution.

2. Sample size 95, non-normally distributed population:

   • With a large sample size (n > 30), the Central Limit Theorem applies, allowing us to use the Z-distribution, regardless of the population’s non-normality.

3. Sample size 20, normally distributed population, known standard deviation of 3.3:

   • Since the sample size is small, but the population standard deviation is known, we use the Z-distribution.

Let me know if you need further details or if you have any questions.

Here are 5 related questions to expand your understanding:

1. Why is the t-distribution preferred for small sample sizes with unknown population standard deviations?
2. How does the Central Limit Theorem justify using the Z-distribution for large samples?
3. What is the impact of sample size on the choice between Z and t distributions?
4. Why can’t we always use the Z-distribution, even for small samples?
5. How would you interpret a 95% confidence interval for this example?

Tip: For small sample sizes, the t-distribution accounts for more variability, which is crucial when the standard deviation is unknown.