Question

the waiting times (in minutes) of a random sample of 20 people at a bank have a sample standard deviation of 3.2 minutes. construct a confidence interval for the population variance σ² and the population standard deviation σ. use a 95% level of confidence. assume the sample is from a normally distributed population.
what is the confidence interval for the population variance σ²?
(5.9, 21.8) (round to one decimal place as needed.)
interpret the results. select the correct choice below and fill in the answer box(es) to complete your choice. (round to one decimal place as needed.)
○ a. with 5% confidence, you can say that the population variance is greater than.
○ b. with 95% confidence, you can say that the population variance is between 5.9 and 21.8.
○ c. with 5% confidence, you can say that the population variance is between and.
○ d. with 95% confidence, you can say that the population variance is less than.
what is the confidence interval for the population standard deviation σ?
(, ) (round to one decimal place as needed.)

Explanation:

Step1: Recall the relationship between variance and standard deviation

The standard deviation \(\sigma\) is the square - root of the variance \(\sigma^{2}\). If the confidence interval for \(\sigma^{2}\) is \((a,b)\), then the confidence interval for \(\sigma\) is \((\sqrt{a},\sqrt{b})\)

Step2: Calculate the square - roots of the endpoints of the variance confidence interval

Given the confidence interval for \(\sigma^{2}\) is \((5.9,21.8)\)
For the lower bound: \(\sqrt{5.9}\approx2.4\)
For the upper bound: \(\sqrt{21.8}\approx4.7\)

Answer:

\((2.4,4.7)\)