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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.

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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.

You would like to construct a 95% confidence interval to estimate today's population mean temperature. Given that the mean of the measurements is 58 degrees Fahrenheit and their standard deviation is 3 degrees Fahrenheit, determine the best point estimate for the population mean and the appropriate distribution to use in different sampling scenarios.

Learn how to construct a 95% confidence interval to estimate population mean temperature. Determine the best point estimate and appropriate distribution (t or z) for different sampling scenarios. Given: sample mean of 58°F and standard deviation of 3°F.

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