A simple random sample of size n is drawn from a population that is normally distributed. The sample mean, X, is found to be 108, and the sample standard deviation, s, is found to be 10. (a) Construct a 96% confidence interval about µ if the sample size, n, is 13. (b) Construct a 96% confidence interval about μ if the sample size, n, is 29. (c) Construct a 99% confidence interval about μ if the sample size, n, is 13. (d) Could we have computed the confidence intervals in parts (a)-(c) if the population had not been normally distributed? Click the icon to view the table of areas under the t-distribution. Compare the results to those obtained in part (a). How does increasing the level of confidence affect the size of the margin of error, E? A. As the level of confidence increases, the size of the interval increases. OB. As the level of confidence increases, the size of the interval decreases. OC. As the level of confidence increases, the size of the interval stays the same. (d) Could we have computed the confidence intervals in parts (a)-(c) if the population had not been normally distributed? O A. Yes, the population does not need to be normally distributed. O B. Yes, the population needs to be normally distributed. OC. No, the population does not need to be normally distributed. O D. No, the population needs to be normally distributed.

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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(d) Could we have computed the confidence intervals in parts (a)-(c) if the population had not been normally distributed?

Yes, the population does not need to be normally distributed.

Yes, the population needs to be normally distributed.

No, the population does not need to be normally distributed.

No, the population needs to be normally distributed.

**Transcription for Educational Website**

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A simple random sample of size \( n \) is drawn from a population that is normally distributed. The sample mean, \(\bar{x}\), is found to be 108, and the sample standard deviation, \( s \), is found to be 10.

(a) Construct a 96% confidence interval about \( \mu \) if the sample size, \( n \), is 13.
(b) Construct a 96% confidence interval about \( \mu \) if the sample size, \( n \), is 29.
(c) Construct a 99% confidence interval about \( \mu \) if the sample size, \( n \), is 13.
(d) Could we have computed the confidence intervals in parts (a)-(c) if the population had not been normally distributed?

\[ \text{Click the icon to view the table of areas under the t-distribution.} \]

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**Comparison and Analysis:**

Compare the results to those obtained in part (a). How does increasing the level of confidence affect the size of the margin of error, \( E \)?

- **A.** ✅ As the level of confidence increases, the size of the interval increases.
- **B.** As the level of confidence increases, the size of the interval decreases.
- **C.** As the level of confidence increases, the size of the interval stays the same.

(d) Could we have computed the confidence intervals in parts (a)-(c) if the population had not been normally distributed?

- **A.** Yes, the population does not need to be normally distributed.
- **B.** Yes, the population needs to be normally distributed.
- **C.** ✅ No, the population does not need to be normally distributed.
- **D.** No, the population needs to be normally distributed.

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Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

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