Use technology to construct the confidence intervals for the population variance $\sigma^2$ and the population standard deviation $\sigma$ . Assume the sample is taken from a normally distributed population. $\newline$ $c=0.99$ , $s=33$ , $n=16$ $\newline$ The confidence interval for the population variance is $\newline$ $\square$ , $\newline$ $\square$ . (Round to two decimal places as needed.)

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Q. Use technology to construct the confidence intervals for the population variance

\sigma^2

and the population standard deviation

\sigma

. Assume the sample is taken from a normally distributed population.

\newline

c=0.99

s=33

n=16

\newline

The confidence interval for the population variance is

\newline

\square

\newline

\square

. (Round to two decimal places as needed.)

Calculate degrees of freedom: Use the chi-square distribution to find the critical values for the confidence interval of the population variance. The degrees of freedom (df) is $n - 1$ . $df = n - 1 = 16 - 1 = 15$
Find chi-square critical values: Find the chi-square critical values using a chi-square table or technology for $df = 15$ and $c = 0.99$ . The lower critical value (chi-square left) corresponds to the upper tail and the upper critical value (chi-square right) corresponds to the lower tail because the chi-square distribution is not symmetric. $\newline$ $\chi^2(0.005, 15)$ $\newline$ $\chi^2(0.995, 15)$
Use formula for confidence interval: Plug the critical values and the sample standard deviation into the formula for the confidence interval of the population variance. $\newline$ Confidence interval for variance = $[\left(n - 1\right) \cdot s^2] / \chi^2_{\text{right}}, [\left(n - 1\right) \cdot s^2] / \chi^2_{\text{left}}$
Calculate confidence interval: Calculate the confidence interval using the sample standard deviation $s = 33$ and the sample size $n = 16$ . Confidence interval for variance = $[15 \times (33)^2] / \chi^2_{\text{right}}, [15 \times (33)^2] / \chi^2_{\text{left}}$
Find exact chi-square critical values: Use technology to find the exact chi-square critical values. $\newline$ $\chi^2(0.005, 15) \approx 5.23$ $\newline$ $\chi^2(0.995, 15) \approx 30.58$
Substitute critical values: Substitute the critical values into the confidence interval formula. $\newline$ Confidence interval for variance = $[\left(15\right) \times \left(33\right)^{2}] / 30.58$ , $[\left(15\right) \times \left(33\right)^{2}] / 5.23$
Perform calculations: Perform the calculations to find the confidence interval for the population variance. $\newline$ Confidence interval for variance = $[\frac{(15) \times (1089)}{30.58}, \frac{(15) \times (1089)}{5.23}]$ $\newline$ Confidence interval for variance = $[\frac{16335}{30.58}, \frac{16335}{5.23}]$ $\newline$ Confidence interval for variance $\approx 534.16, 3124.28$
Round variance interval: Round the confidence interval for the population variance to two decimal places. $\newline$ Confidence interval for variance $\approx 534.16, 3124.28$
Calculate square roots: To find the confidence interval for the population standard deviation, take the square root of the variance interval endpoints. $\newline$ Confidence interval for standard deviation = $\sqrt{534.16}$ , $\sqrt{3124.28}$
Round standard deviation interval: Calculate the square roots. $\newline$ Confidence interval for standard deviation $\approx 23.11, 55.89$
Round standard deviation interval: Calculate the square roots. $\newline$ Confidence interval for standard deviation $\approx 23.11, 55.89$ Round the confidence interval for the population standard deviation to two decimal places. $\newline$ Confidence interval for standard deviation $\approx 23.11, 55.89$