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You would like to construct a 99% confidence interval to estimate today's population mean temperature, so you make a series of measurements (a random sample) throughout the day. The mean of these measurements is 64 degrees Fahrenheit, and their standard deviation is 3 degrees Fahrenheit.
(a) What is the best point estimate, based on the sample, to use for the population mean? (b) For each of the following sampling scenarios, determine which distribution should be used to calculate the critical value for the 99% confidence interval for the population mean. (In the table, Z refers to a standard normal distribution, and  refers to a *t* distribution.)

Let's address the problem step by step.

### (a) Best Point Estimate for the Population Mean
The best point estimate for the population mean based on the sample is the sample mean. Given that the mean of your measurements is 64 degrees Fahrenheit, this is the best point estimate for the population mean.

**Best Point Estimate**: \( \mu \approx 64 \) degrees Fahrenheit.

### (b) Determining the Appropriate Distribution for the Critical Value

To determine which distribution to use for the 99% confidence interval, we consider the information provided:

1. **Sample size (\(n\))**: Determines whether to use the *t*-distribution or the Z-distribution. Generally:
   - For small sample sizes (\(n < 30\)), the *t*-distribution is used.
   - For large sample sizes (\(n \geq 30\)), the Z-distribution can be used if the population standard deviation is known.
   
2. **Population distribution**: If the population is normally distributed, we use the *t*-distribution or Z-distribution based on the sample size and whether the population standard deviation is known.

3. **Population standard deviation (\(\sigma\))**: If the population standard deviation is known, the Z-distribution is used. If not, the *t*-distribution is used.

#### Analysis of the Scenarios from the Image

1. **Scenario 1**: Sample size is 14, population is normally distributed, known standard deviation.
   - Use the **Z-distribution** because the population standard deviation is known.

2. **Scenario 2**: Sample size is 14, population is normally distributed, unknown standard deviation.
   - Use the **t-distribution** because the population standard deviation is unknown and the sample size is small.

3. **Scenario 3**: Sample size is 105, population is not normally distributed.
   - Use the **Z-distribution** since the sample size is large (\(n \geq 30\)). The Central Limit Theorem allows us to assume that the sample mean is approximately normally distributed, regardless of the population distribution.

### Summary of the Table Based on the Image:
1. The first row should be marked under **Z**.
2. The second row should be marked under **t**.
3. The third row should be marked under **Z**.

Would you like further details or explanations on any part of this solution?

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Here are some related questions for further understanding:

1. What is the Central Limit Theorem, and how does it apply to large sample sizes?
2. How is the *t*-distribution different from the Z-distribution?
3. What are the degrees of freedom in the context of the *t*-distribution?
4. How does the confidence level affect the width of the confidence interval?
5. Why is it important to know if the population standard deviation is known or unknown?
6. What happens to the confidence interval if the sample size increases?
7. How do you calculate the critical value for a 99% confidence interval?
8. What are some common misconceptions when selecting between the *t*-distribution and the Z-distribution?

**Tip**: When dealing with small sample sizes and unknown population standard deviations, the *t*-distribution is generally preferred because it accounts for the additional uncertainty in the estimate.

Learn how to construct a 99% confidence interval for estimating today's population mean temperature using sample measurements. Understand the use of normal distribution and t-distribution based on sample size and population characteristics.

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