Question

the number of hours of reserve capacity of 10 randomly selected automotive batteries is shown to the right. 1.78 1.85 1.53 1.67 1.77 1.91 1.38 1.53 1.49 2.08 assume the sample is taken from a normally distributed population. construct 98% confidence intervals for (a) the population variance \\( \sigma^{2} \\) and (b) the population standard deviation \\( \sigma \\). (a) the confidence interval for the population variance is (0.020, 0.205). (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 2% confidence, it can be said that the population variance is greater than. b. with 98% confidence, it can be said that the population variance is between 0.020 and 0.205. c. with 2% confidence, it can be said that the population variance is between and. d. with 98% confidence, it can be said that the population variance is less than. (b) the confidence interval for the population standard deviation is (, ). (round to three decimal places as needed.)

Explanation:

Step1: Recall the formula for confidence interval of standard deviation

The confidence interval for the population standard deviation \(\sigma\) when the population is normally distributed is given by \(\sqrt{\frac{(n - 1)s^{2}}{\chi_{\alpha/2}^{2}}}<\sigma<\sqrt{\frac{(n - 1)s^{2}}{\chi_{1-\alpha/2}^{2}}}\), where \(n\) is the sample size, \(s^{2}\) is the sample variance, and \(\chi_{\alpha/2}^{2}\) and \(\chi_{1-\alpha/2}^{2}\) are the critical values from the chi - square distribution.

We know from part (a) that the confidence interval for the variance \(\sigma^{2}\) is \((0.020,0.205)\) (since \(\frac{(n - 1)s^{2}}{\chi_{\alpha/2}^{2}}=0.020\) and \(\frac{(n - 1)s^{2}}{\chi_{1-\alpha/2}^{2}}=0.205\) with \(n = 10\), \(\alpha=1 - 0.98=0.02\), \(\alpha/2=0.01\), \(1-\alpha/2 = 0.99\))

Step2: Calculate the confidence interval for the standard deviation

Take the square - root of the endpoints of the variance confidence interval.
For the lower bound: \(\sqrt{0.020}\approx0.141\)
For the upper bound: \(\sqrt{0.205}\approx0.453\)

Answer:

The confidence interval for the population standard deviation \(\sigma\) is \((0.141,0.453)\)