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 50 people, the mean body mass index (BMI) was 26.7 and the standard deviation was 6.06.
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 50 people, the mean body mass index (BMI) was 26.7 and the standard deviation was 6.06.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![**Title: Constructing a Confidence Interval for the Population Mean**
**Objective:**
Learn how to use either the standard normal distribution or the t-distribution to construct a 95% confidence interval for the population mean. Determine which distribution to use based on the given data, and justify your choice.
**Context:**
A random sample of 50 people was taken, revealing a mean body mass index (BMI) of 26.7 and a standard deviation of 6.06.
**Question:**
Which distribution should be used to construct the confidence interval?
**Options:**
- **A.** Use a normal distribution because the sample is random, the population is normal, and the population standard deviation (σ) is known.
- **B.** Use a t-distribution because the sample is random, the sample size (n) is greater than or equal to 30, and σ is unknown. *(Correct Answer)*
- **C.** Use a normal distribution because the sample is random, n ≥ 30, and σ is known.
- **D.** Use a t-distribution because the sample is random, the population is normal, and σ 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.
**Result:**
Select option **B** because the population standard deviation (σ) is unknown, and the sample size (n) is 50, which is greater than 30. This justifies the use of the t-distribution.
**Next Step:**
Choose the appropriate choice for computing the confidence interval:
- **A.** The 95% confidence interval is [ , ]. (Round to two decimal places as needed.)
- **B.** Neither distribution can be used to construct the confidence interval.
For this scenario, proceed with option **A** to calculate the 95% confidence interval using the t-distribution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F35b6c70d-8de6-4ced-ba13-39b6b43ecdee%2F5c6f1b3f-f9c7-478d-b333-7d6c6ee69318%2Flofo1gn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Constructing a Confidence Interval for the Population Mean**
**Objective:**
Learn how to use either the standard normal distribution or the t-distribution to construct a 95% confidence interval for the population mean. Determine which distribution to use based on the given data, and justify your choice.
**Context:**
A random sample of 50 people was taken, revealing a mean body mass index (BMI) of 26.7 and a standard deviation of 6.06.
**Question:**
Which distribution should be used to construct the confidence interval?
**Options:**
- **A.** Use a normal distribution because the sample is random, the population is normal, and the population standard deviation (σ) is known.
- **B.** Use a t-distribution because the sample is random, the sample size (n) is greater than or equal to 30, and σ is unknown. *(Correct Answer)*
- **C.** Use a normal distribution because the sample is random, n ≥ 30, and σ is known.
- **D.** Use a t-distribution because the sample is random, the population is normal, and σ 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.
**Result:**
Select option **B** because the population standard deviation (σ) is unknown, and the sample size (n) is 50, which is greater than 30. This justifies the use of the t-distribution.
**Next Step:**
Choose the appropriate choice for computing the confidence interval:
- **A.** The 95% confidence interval is [ , ]. (Round to two decimal places as needed.)
- **B.** Neither distribution can be used to construct the confidence interval.
For this scenario, proceed with option **A** to calculate the 95% confidence interval using the t-distribution.
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