Question

the diameters (in inches) of 17 randomly selected bolts produced by a machine are listed. use a 90% level of confidence to construct a confidence interval for (a) the population variance $sigma^{2}$ and (b) the population standard deviation $sigma$. interpret the results.
4.472 4.424 4.027 4.311 4.007 3.789
3.816 3.771 4.239 3.957 4.148 4.528
3.935 3.763 3.883 3.809 4.449
interpret the results. select the correct choice below and fill in the answer box(es) to complete your choice. (round to three decimal places as needed.)
a. with 10% confidence, it can be said that the population variance is between and.
b. with 10% confidence, it can be said that the population variance is less than.
c. with 90% confidence, it can be said that the population variance is greater than.
d. with 90% confidence, it can be said that the population variance is between 0.046 and 0.151.
(b) the confidence interval for the population standard deviation is (). (round to three decimal places 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 the variance is \((a,b)\), then the confidence interval for the standard deviation is \((\sqrt{a},\sqrt{b})\)

Step2: Calculate the square - roots

Given the confidence interval for the variance \(\sigma^{2}\) is \((0.046,0.151)\)
For the lower bound of the standard deviation: \(\sqrt{0.046}\approx0.214\)
For the upper bound of the standard deviation: \(\sqrt{0.151}\approx0.389\)

Answer:

The confidence interval for the population standard deviation is \((0.214,0.389)\)