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
(a) the confidence interval for the population variance is (0.046, 0.151).
(round to three decimal places as needed.)
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
b. with 10% confidence, it can be said that the
population variance is between and.
population variance is less than.
c. with 90% confidence, it can be said that the
d. with 90% confidence, it can be said that the
population variance is greater than.
population variance is between and.

Explanation:

Step1: Recall confidence interval interpretation

A confidence interval for the population variance at a \(90\%\) confidence level means that if we were to construct many such intervals from different samples, about \(90\%\) of them would contain the true population variance.

Step2: Analyze each option

• Option A: Incorrect. The confidence level is \(90\%\), not \(10\%\).
• Option B: Incorrect. The confidence level is \(90\%\), not \(10\%\), and it's an interval, not just an upper - bound.
• Option C: Incorrect. It's an interval estimate, not just a lower - bound.
• Option D: Correct. Since we constructed a \(90\%\) confidence interval for the population variance \(\sigma^{2}\) which is \((0.046,0.151)\), with \(90\%\) confidence, it can be said that the population variance is between \(0.046\) and \(0.151\).

Answer:

D. With \(90\%\) confidence, it can be said that the population variance is between \(0.046\) and \(0.151\).