Use the standard normal distribution or the t-distribution to construct a 95% 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 14 mortgage institutions, the mean interest rate was 3.41% and the standard deviation was 0.48%. Assume the interest rates are normally distributed. Which distribution should be used to construct the confidence interval? O A. Use a t-distribution because it is a random sample, o is unknown, and the interest rates are normally distributed. O B. Use a t-distribution because the interest rates are normally distributed and o is known. OC. Use a normal distribution because the interest rates are normally distributed and o is known. O D. Use a normal distribution because n< 30 and the interest rates are normally distributed. O E. Cannot use the standard normal distribution or the t-distribution because o is unknown, n< 30, and the interest rates are not normally distributed. Select the correct choice below and, if necessary, fill in any answer boxes to complete your choice. O A. The 95% 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 95% of institutions have an interest rate between the bounds of the confidence interval. O B. With 95% confidence, it can be said that the population mean interest rate is between the bounds of the confidence interval. OC. Ifa large sample of institutions are taken approximately 95% of them will have an interest rate between the bounds of the confidence interval. O D. Neither distribution can be used to construct the confidence interval.
Use the standard normal distribution or the t-distribution to construct a 95% 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 14 mortgage institutions, the mean interest rate was 3.41% and the standard deviation was 0.48%. Assume the interest rates are normally distributed. Which distribution should be used to construct the confidence interval? O A. Use a t-distribution because it is a random sample, o is unknown, and the interest rates are normally distributed. O B. Use a t-distribution because the interest rates are normally distributed and o is known. OC. Use a normal distribution because the interest rates are normally distributed and o is known. O D. Use a normal distribution because n< 30 and the interest rates are normally distributed. O E. Cannot use the standard normal distribution or the t-distribution because o is unknown, n< 30, and the interest rates are not normally distributed. Select the correct choice below and, if necessary, fill in any answer boxes to complete your choice. O A. The 95% 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 95% of institutions have an interest rate between the bounds of the confidence interval. O B. With 95% confidence, it can be said that the population mean interest rate is between the bounds of the confidence interval. OC. Ifa large sample of institutions are taken approximately 95% of them will have an interest rate between the bounds of the confidence interval. O D. Neither distribution can be used to construct the confidence interval.
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Question
![**Constructing a 95% Confidence Interval for the Population Mean**
**Scenario Overview:**
In a random sample of 14 mortgage institutions, the mean interest rate was 3.41%, with a standard deviation of 0.48%. We assume the interest rates follow a normal distribution.
**Choosing the Appropriate Distribution:**
Which distribution should be used to construct the confidence interval?
- **A.** Use a t-distribution because it is a random sample, σ is unknown, and the interest rates are normally distributed.
- **B.** Use a t-distribution because the interest rates are normally distributed and σ is known.
- **C.** Use a normal distribution because the interest rates are normally distributed and σ is known.
- **D.** Use a normal distribution because n < 30 and the interest rates are normally distributed.
- **E.** Cannot use the standard normal distribution or the t-distribution because σ is unknown, n < 30, and the interest rates are not normally distributed.
**Calculating the Confidence Interval:**
Select the correct choice below and, if necessary, fill in any answer boxes to complete your choice.
- **A.** The 95% confidence interval is ( [ ], [ ] ).
(Round to two decimal places as needed.)
- **B.** Neither distribution can be used to construct the confidence interval.
**Interpreting the Results:**
Choose the correct answer below.
- **A.** It can be said that 95% of institutions have an interest rate between the bounds of the confidence interval.
- **B.** With 95% confidence, it can be said that the population mean interest rate is between the bounds of the confidence interval.
- **C.** If a large sample of institutions are taken, approximately 95% of them will have an interest rate between the bounds of the confidence interval.
- **D.** Neither distribution can be used to construct the confidence interval.
**Explanation of Diagrams (if any):**
There are no graphs or diagrams in this content. All information is presented in text format for instructional purposes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F30ed8941-ce9c-4a61-953f-230a29ff6a7f%2F1e75d6ac-1ace-4bb8-873a-3856e7f6f201%2F16vmowg_processed.png&w=3840&q=75)
Transcribed Image Text:**Constructing a 95% Confidence Interval for the Population Mean**
**Scenario Overview:**
In a random sample of 14 mortgage institutions, the mean interest rate was 3.41%, with a standard deviation of 0.48%. We assume the interest rates follow a normal distribution.
**Choosing the Appropriate Distribution:**
Which distribution should be used to construct the confidence interval?
- **A.** Use a t-distribution because it is a random sample, σ is unknown, and the interest rates are normally distributed.
- **B.** Use a t-distribution because the interest rates are normally distributed and σ is known.
- **C.** Use a normal distribution because the interest rates are normally distributed and σ is known.
- **D.** Use a normal distribution because n < 30 and the interest rates are normally distributed.
- **E.** Cannot use the standard normal distribution or the t-distribution because σ is unknown, n < 30, and the interest rates are not normally distributed.
**Calculating the Confidence Interval:**
Select the correct choice below and, if necessary, fill in any answer boxes to complete your choice.
- **A.** The 95% confidence interval is ( [ ], [ ] ).
(Round to two decimal places as needed.)
- **B.** Neither distribution can be used to construct the confidence interval.
**Interpreting the Results:**
Choose the correct answer below.
- **A.** It can be said that 95% of institutions have an interest rate between the bounds of the confidence interval.
- **B.** With 95% confidence, it can be said that the population mean interest rate is between the bounds of the confidence interval.
- **C.** If a large sample of institutions are taken, approximately 95% of them will have an interest rate between the bounds of the confidence interval.
- **D.** Neither distribution can be used to construct the confidence interval.
**Explanation of Diagrams (if any):**
There are no graphs or diagrams in this content. All information is presented in text format for instructional purposes.
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