(A) What is the best point estimate, based on the sample, to use for the population mean?

The best point estimate for the population mean is the sample mean. Given that the sample mean is 1.6 seconds, this is the best estimate.

• Answer: $\text{1.6 seconds}$

(B) Determine which distribution should be used to calculate the critical value for the 95% confidence interval for the population mean.

1. Sampling Scenario 1:

• Sample size: 85
• Population distribution: Non-normally distributed

Distribution to Use: Since the sample size is large (typically $n > 30$), the Central Limit Theorem applies, allowing us to use the z-distribution for the critical value, even though the population is non-normally distributed.

• Answer: $Z$

2. Sampling Scenario 2:

• Sample size: 18
• Population distribution: Normally distributed with a known standard deviation of 0.45

Distribution to Use: When the population is normally distributed and the population standard deviation is known, the z-distribution should be used, regardless of the sample size.

• Answer: $Z$

3. Sampling Scenario 3:

• Sample size: 11
• Population distribution: Normally distributed with an unknown standard deviation

Distribution to Use: When the sample size is small ($n \leq 30$), the population is normally distributed, and the population standard deviation is unknown, the t-distribution should be used.

• Answer: $t$

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Would you like more details or have any questions?

Here are some related questions:

1. How would the confidence interval change if the sample mean was 2.0 seconds instead of 1.6 seconds?
2. What is the formula to calculate the confidence interval for the population mean?
3. How does the sample size affect the width of the confidence interval?
4. What are the implications of using a t-distribution instead of a z-distribution?
5. Can you explain why the Central Limit Theorem allows us to use the z-distribution for large samples, even if the population distribution is not normal?

Tip: Always check the sample size and whether the population standard deviation is known to determine whether to use the z-distribution or t-distribution when constructing confidence intervals.