Assume the sample is taken from a normally distributed population and construct the indicated confidence interval.
-Construct the indicated confidence intervals for the population variance $\sigma ^ { 2 }$ and the population standard deviation ?. Assume the sample is from a normally distributed population $\mathrm { c } = 0.95 , \mathrm {~s} ^ { 2 } = 15.3 , \mathrm { n } = 25$
A)(9.33, 29.61); (3.05, 5.44)
B)(13.9, 26.52); (3.73, 5.15)
C)(9.03, 27.99); (3, 5.29)
D)(9.75, 25.13); (3.12, 5.01)

Question

Quizplus · Accepted Answer

To construct the confidence intervals for the population variance $\sigma^2$ and the population standard deviation $\sigma$, we use the chi-square distribution because the sample is from a normally distributed population. The formula for the confidence interval for the variance is given by:$$\left( \frac{(n-1)s^2}{\chi^2_{\alpha/2, n-1}}, \frac{(n-1)s^2}{\chi^2_{1-\alpha/2, n-1}} \right)$$where $n$ is the sample size, $s^2$ is the sample variance, and $\chi^2_{\alpha/2, n-1}$ and $\chi^2_{1-\alpha/2, n-1}$ are the critical values from the chi-square distribution for $n-1$ degrees of freedom.Given: $c = 0.95$, $s^2 = 15.3$, $n = 25$, we find the critical values for $\alpha = 0.05$ (since $c = 1 - \alpha$) and $n-1 = 24$ degrees of freedom. Using chi-square distribution tables or a calculator, we find the critical values and plug them into the formula to calculate the confidence interval for the variance. Then, to find the confidence interval for the standard deviation, we simply take the square root of the variance interval endpoints.The correct answer, (9.33, 29.61) for the variance and (3.05, 5.44) for the standard deviation, matches the calculations based on the chi-square distribution and the given sample statistics.

Assume the Sample Is Taken from a Normally Distributed Population σ2\sigma ^ { 2 }σ2

Assume the Sample Is Taken from a Normally Distributed Population $\sigma ^ { 2 }$