Question

a simple random sample of size n is drawn from a population that is normally distributed. the sample mean, x, is found to be 106, and the sample standard deviation, s, is found to be 10.
(a) construct a 98% confidence interval about μ if the sample size, n, is 23
(b) construct a 98% confidence interval about μ if the sample size, n, is 19
(c) construct a 96% confidence interval about μ if the sample size, n, is 23
(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 decreasing the level of confidence affect the size of the margin of error, e?
a. as the level of confidence decreases, the size of the interval increases
b. as the level of confidence decreases, the size of the interval stays the same
c. as the level of confidence decreases, the size of the interval decreases
(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. no, the population does not need to be normally distributed
c. yes, the population needs to be normally distributed
d. no, the population needs to be normally distributed

Explanation:

When constructing confidence intervals for the population mean \(\mu\) using the \(t\) - distribution (since the population standard deviation \(\sigma\) is unknown), one of the key assumptions is that the population from which the sample is drawn is normally distributed. If the population is not normally distributed, and the sample size \(n\) is small (\(n<30\)), the \(t\) - distribution based confidence interval may not be valid. In parts (a) - (c), the sample sizes \(n = 23\) and \(n=19\) are small. The Central Limit Theorem (which allows us to use normal - based or \(t\) - based intervals for non - normal populations when \(n\geq30\)) does not apply here. So, the population needs to be normally distributed for the confidence intervals (constructed using the \(t\) - distribution) in parts (a) - (c) to be valid.

Answer:

D. No, the population needs to be normally distributed