Questions: Question 6, 6.4.20 Part 1 of 4 Points: 0.75 of 1 As part of a survey, a marketing representative asks a random sample of 30 business owners how much they would be willing to pay for a website for their company. She finds that the sample standard deviation is 3548. Assume the sample is taken from a normally distributed population. Construct 99% confidence intervals for (a) the population variance σ^2 and (b) the population standard deviation σ. Interpret the results. (a) The confidence interval for the population variance is (Round to the nearest integer as needed.)

Solution Steps

The sample standard deviation \( s \) is given as \( 3548 \). The sample variance \( s^2 \) is calculated as follows:

\[ s^2 = 3548^2 = 12588304 \]

To construct the confidence interval for the population variance \( \sigma^2 \), we use the formula:

\[ \left( \frac{(n - 1)s^2}{\chi^2_{\alpha/2}}, \frac{(n - 1)s^2}{\chi^2_{1 - \alpha/2}} \right) \]

where \( n = 30 \) is the sample size and \( \alpha = 0.01 \) for a \( 99\% \) confidence level. The calculated confidence interval is:

\[ CI = \left( 6975380.0, 27822321.0 \right) \]

The confidence interval for the population standard deviation \( \sigma \) is obtained by taking the square root of the endpoints of the variance confidence interval:

\[ \left( \sqrt{6975380.0}, \sqrt{27822321.0} \right) \approx (2641, 5275) \]

Final Answer