Let's go through the questions step by step.

Part (a)

Best Point Estimate:

The best point estimate for the population mean $\mu$ is the sample mean.

Given:

• Sample mean $\bar{x} = 481$

So, the best point estimate for the population mean is 481.

Part (b)

Determine the Distribution to Use:

To determine which distribution (Z or t) to use when calculating the critical value for the 90% confidence interval, we consider the sample size, whether the population standard deviation is known, and whether the population is normally distributed.

1. Sampling scenario:

• Sample size: 11
• Population distribution: Unknown distribution

Explanation:

• The sample size is small ($n < 30$).
• We know very little about the population distribution.
• When the sample size is small and the population distribution is unknown, we should use the t-distribution.

Answer: t

2. Sampling scenario:

• Sample size: 19
• Population distribution: Normally distributed population with an unknown standard deviation

Explanation:

• The sample size is small ($n < 30$).
• The population is normally distributed.
• The population standard deviation is unknown, so the t-distribution should be used.

Answer: t

3. Sampling scenario:

• Sample size: 95
• Population distribution: Non-normally distributed population with a known standard deviation of 75

Explanation:

• The sample size is large ($n \geq 30$), which typically allows the Central Limit Theorem to apply, meaning the sampling distribution of the sample mean will be approximately normal even if the population is not normal.
• The population standard deviation is known, so the Z-distribution can be used.

Answer: Z

Summary of the Answers:

• Part (a): Best point estimate = 481
• Part (b):
  1. t
  2. t
  3. Z

Would you like further details on any part? Here are some related questions you might be interested in:

1. How do you calculate the margin of error for a confidence interval?
2. What are the assumptions of the Central Limit Theorem?
3. How does sample size affect the choice of distribution for confidence intervals?
4. How do you interpret a confidence interval in context?
5. How does the confidence level affect the width of the confidence interval?

Tip: When dealing with small sample sizes, the t-distribution is often used because it accounts for the additional uncertainty due to the smaller sample.