Let's go through each part of the question and address the steps required to answer it.

Part (a)

The problem asks for the best point estimate, based on the sample, to use for the population mean temperature. The point estimate of the population mean, based on a sample, is the sample mean itself.

From the problem:

• The sample mean temperature is given as $63.5$ degrees Fahrenheit.

Thus, the best point estimate to use for the population mean is: 63.5 degrees.

Part (b)

For part (b), you are asked to choose the appropriate distribution to calculate the critical value for a 95% confidence interval in different sampling scenarios.

The scenario provided is:

• Sample size is 17
• The sample is from a normally distributed population
• The population has an unknown standard deviation

When the population standard deviation is unknown and the sample size is relatively small (less than 30), the t-distribution is generally used for constructing confidence intervals. This is because the t-distribution accounts for the extra variability due to estimating the population standard deviation from the sample.

Answer for (b):

• The appropriate choice in the table is t for this scenario.

Would you like further clarification on any part of this, or do you want details on calculating the confidence interval?

Here are five related questions that might expand your understanding:

1. How does the t-distribution differ from the z-distribution?
2. Why do we prefer the t-distribution over the z-distribution when the sample size is small?
3. What would change if the sample size were larger, say 50, in this problem?
4. How do you calculate a confidence interval using the t-distribution?
5. What role does the confidence level (like 95%) play in constructing confidence intervals?

Tip: When dealing with small samples and unknown population standard deviations, always check if the population is normally distributed; this often justifies the use of the t-distribution.