Part (a): Best Point Estimate

The best point estimate for the population mean, based on the sample, is the sample mean.

Thus, the best point estimate is: $\text{Sample Mean} = 40464 \, \text{dollars.}$

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Part (b): Distribution to Use for Critical Value

To determine which distribution to use for the critical value of the confidence interval, consider the size of the sample, the distribution of the population, and whether the population standard deviation is known.

1. Sample size = 85, non-normally distributed population

• Reasoning: Since the sample size is $n = 85$, which is sufficiently large ($n > 30$), the Central Limit Theorem (CLT) allows us to assume the sampling distribution of the mean is approximately normal even if the population is not. However, the population standard deviation is unknown, so we use the t-distribution.

• Answer: $t$.

2. Sample size = 95, normally distributed population with an unknown standard deviation

• Reasoning: The population is normally distributed, but the population standard deviation is unknown. In this case, the correct distribution is the t-distribution.

• Answer: $t$.

3. Sample size = 95, non-normally distributed population with a known standard deviation of 7756

• Reasoning: The sample size is $n = 95$, which is large ($n > 30$), and the population standard deviation is known. The CLT ensures the sampling distribution is approximately normal.

• Answer: $Z$.

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Final Answers

1. Sample size = 85, non-normally distributed population: $t$
2. Sample size = 95, normally distributed population, unknown standard deviation: $t$
3. Sample size = 95, non-normally distributed population, known standard deviation: $Z$

Would you like further clarification or help computing the confidence interval?

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Follow-up Questions:

1. What formula is used to calculate a confidence interval using the $t$-distribution?
2. How does the sample size affect the choice of $Z$- vs. $t$-distribution?
3. Why does the Central Limit Theorem allow the use of normal approximations for large sample sizes?
4. What are the degrees of freedom when using the $t$-distribution?
5. How do changes in confidence level affect the width of the confidence interval?

Tip:

The $t$-distribution is used when the population standard deviation is unknown, and its shape depends on the sample size (degrees of freedom).