Use the standard normal distribution or the t-distribution to construct a 99% confidence interval for the population mean. Justify your decision. If neither distribution can be used, explain why. Interpret the results. In a random sample of 42 people, the mean body mass index (BMI) was 26.6 and the standard deviation was 6.12. Select the correct choice below and, if necessary, fill in any answer boxes to complete your choice. O A. The 99% confidence interval is ( ). (Round to two decimal places as needed.) O B. Neither distribution can be used to construct the confidence interval. Interpret the results. Choose the correct answer below. O A. It can be said that 99% of people have a BMI between the bounds of the confidence interval. B. If a large sample of people are taken approximately 99% of them will have a BMI between the bounds of the confidence interval. O C. With 99% confidence, it can be said that the population mean BMI is between the bounds of the confidence interval. O D. Neither distribution can be used to construct the confidence interval.

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**Constructing a 99% Confidence Interval for the Population Mean BMI**

To construct a 99% confidence interval for the population mean, utilize either the standard normal distribution or the t-distribution. The appropriate choice should be justified, and the results interpreted accordingly.

In a random sample of 42 people, the mean body mass index (BMI) was 26.6, and the standard deviation was 6.12.

### Question:
Select the correct choice below and, if necessary, fill in any answer boxes to complete your choice.

- **A.** The 99% confidence interval is [  ,  ]. (Round to two decimal places as needed.)
- **B.** Neither distribution can be used to construct the confidence interval.

### Interpretation of Results:
Choose the correct answer below.

- **A.** It can be said that 99% of people have a BMI between the bounds of the confidence interval.
- **B.** If a large sample of people are taken approximately 99% of them will have a BMI between the bounds of the confidence interval.
- **C.** With 99% confidence, it can be said that the population mean BMI is between the bounds of the confidence interval.
- **D.** Neither distribution can be used to construct the confidence interval.

---

**Explanation of Graphs or Diagrams:**

In this task, there are no graphs or diagrams presented. Instead, the focus is on constructing a confidence interval using statistical distributions and interpreting the results correctly.
Transcribed Image Text:--- **Constructing a 99% Confidence Interval for the Population Mean BMI** To construct a 99% confidence interval for the population mean, utilize either the standard normal distribution or the t-distribution. The appropriate choice should be justified, and the results interpreted accordingly. In a random sample of 42 people, the mean body mass index (BMI) was 26.6, and the standard deviation was 6.12. ### Question: Select the correct choice below and, if necessary, fill in any answer boxes to complete your choice. - **A.** The 99% confidence interval is [ , ]. (Round to two decimal places as needed.) - **B.** Neither distribution can be used to construct the confidence interval. ### Interpretation of Results: Choose the correct answer below. - **A.** It can be said that 99% of people have a BMI between the bounds of the confidence interval. - **B.** If a large sample of people are taken approximately 99% of them will have a BMI between the bounds of the confidence interval. - **C.** With 99% confidence, it can be said that the population mean BMI is between the bounds of the confidence interval. - **D.** Neither distribution can be used to construct the confidence interval. --- **Explanation of Graphs or Diagrams:** In this task, there are no graphs or diagrams presented. Instead, the focus is on constructing a confidence interval using statistical distributions and interpreting the results correctly.
---

### Constructing a 99% Confidence Interval for the Population Mean

**Problem Statement:**

Use the standard normal distribution or the t-distribution to construct a 99% confidence interval for the population mean. Justify your decision. If neither distribution can be used, explain why. Interpret the results.

In a random sample of 42 people, the mean body mass index (BMI) was 26.6, and the standard deviation was 6.12.

**Question:**

Which distribution should be used to construct the confidence interval? Choose the correct answer below.

**Options:**

- **A.** Use a normal distribution because the sample is random, \( n \geq 30 \), and \( \sigma \) is known.
- **B.** Use a t-distribution because the sample is random, the population is normal, and \( \sigma \) is unknown.
- **C.** Use a normal distribution because the sample is random, the population is normal, and \( \sigma \) is known.
- **D.** Use a t-distribution because the sample is random, \( n \geq 30 \), and \( \sigma \) is unknown.
- **E.** Neither a normal distribution nor a t-distribution can be used because either the sample is not random, or \( n < 30 \), and the population is not known to be normal.

**Solution:**

Select the correct choice below and, if necessary, fill in any answer boxes to complete your choice.

**Answer Box:**

- **A. The 99% confidence interval is (, ).**
  - *(Round to two decimal places as needed.)*

### Analysis:

Given the options, you must understand when to use the normal distribution or the t-distribution for constructing confidence intervals. Here are the essential points for deciding:

1. **Normal Distribution:** Generally used when the population standard deviation (\( \sigma \)) is known and the sample size is large (n ≥ 30).

2. **T-Distribution:** Used when the population standard deviation (\( \sigma \)) is unknown, the sample size is small (n < 30), or the population standard deviation is unknown regardless of the sample size.

In this case, the sample size is 42, which satisfies \( n \geq 30 \). The population standard deviation (\( \sigma \)) is unknown, hence the t-distribution should be used
Transcribed Image Text:--- ### Constructing a 99% Confidence Interval for the Population Mean **Problem Statement:** Use the standard normal distribution or the t-distribution to construct a 99% confidence interval for the population mean. Justify your decision. If neither distribution can be used, explain why. Interpret the results. In a random sample of 42 people, the mean body mass index (BMI) was 26.6, and the standard deviation was 6.12. **Question:** Which distribution should be used to construct the confidence interval? Choose the correct answer below. **Options:** - **A.** Use a normal distribution because the sample is random, \( n \geq 30 \), and \( \sigma \) is known. - **B.** Use a t-distribution because the sample is random, the population is normal, and \( \sigma \) is unknown. - **C.** Use a normal distribution because the sample is random, the population is normal, and \( \sigma \) is known. - **D.** Use a t-distribution because the sample is random, \( n \geq 30 \), and \( \sigma \) is unknown. - **E.** Neither a normal distribution nor a t-distribution can be used because either the sample is not random, or \( n < 30 \), and the population is not known to be normal. **Solution:** Select the correct choice below and, if necessary, fill in any answer boxes to complete your choice. **Answer Box:** - **A. The 99% confidence interval is (, ).** - *(Round to two decimal places as needed.)* ### Analysis: Given the options, you must understand when to use the normal distribution or the t-distribution for constructing confidence intervals. Here are the essential points for deciding: 1. **Normal Distribution:** Generally used when the population standard deviation (\( \sigma \)) is known and the sample size is large (n ≥ 30). 2. **T-Distribution:** Used when the population standard deviation (\( \sigma \)) is unknown, the sample size is small (n < 30), or the population standard deviation is unknown regardless of the sample size. In this case, the sample size is 42, which satisfies \( n \geq 30 \). The population standard deviation (\( \sigma \)) is unknown, hence the t-distribution should be used
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