Statistics 1

In-Class Exercise 11

Yu-You Liou

Shih Chien University

2026-09-16

Question 1

Confidence Interval When \sigma Is Known

A sample of 40 delivery times has a mean of 23.7 minutes. The population standard deviation is known to be 4.2 minutes.

1. Find z_{\alpha/2} for a 95% confidence level with qnorm().
2. Compute the margin of error E=z_{\alpha/2}\cdot\sigma/\sqrt{n}.
3. Report the 95% confidence interval for \mu, rounded to one decimal place.

Question 2

Confidence Interval When \sigma Is Unknown

The following are the weekly study hours reported by 15 students.

hours <- c(12, 9, 15, 7, 11, 14, 10, 8, 13, 16, 9, 12, 11, 6, 14)

1. Compute mean(hours) and sd(hours).
2. Find t_{\alpha/2} for a 99% confidence level with qt() and d.f.=n-1.
3. Report the 99% confidence interval for \mu.

Question 3

Width of the Interval

Using the data in Question 2:

1. Compute the 90%, 95%, and 99% confidence intervals, and report the width of each.
2. What happens to the width as the confidence level rises?
3. Explain in one or two sentences what “95% confident” means. Does it mean that the probability that \mu lies in your interval is 0.95?