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 $\square$ , ). (Round to two decimal places as needed.)

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Variance and standard deviation

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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

\square

, ). (Round to two decimal places as needed.)

Introduction: Now, let's solve the new math problem. $\newline$ question_prompt: What are the confidence intervals for the population variance and standard deviation? $\newline$ First, we need to use the chi-square distribution to find the confidence interval for the population variance. $\newline$ The formula for the confidence interval for the population variance is: $\newline$ $[\frac{(n-1)s^2}{\chi^2(\alpha/2)}, \frac{(n-1)s^2}{\chi^2(1-\alpha/2)}]$ $\newline$ where $n$ is the sample size, $s$ is the sample standard deviation, $\alpha$ is the significance level $(1-c)$ , and $\chi^2(\alpha/2)$ and $\chi^2(1-\alpha/2)$ are the chi-square values for $\alpha/2$ and $1-\alpha/2$ degrees of freedom, respectively.
Population Variance Confidence Interval: Given: $c=0.99$ , $s=33$ , $n=16$ $\newline$ The significance level $\alpha = 1 - c = 1 - 0.99 = 0.01$ $\newline$ Now we need to find the chi-square values for $\alpha/2 = 0.005$ and $1-\alpha/2 = 0.995$ with $n-1$ degrees of freedom.
Chi-Square Values Calculation: Using a chi-square table or technology, we find the chi-square values for $15$ degrees of freedom: $\newline$ $\chi^2(0.005) \approx 6.262$ and $\chi^2(0.995) \approx 30.578$ $\newline$ Now we can plug these values into the formula.
Variance Lower Limit Calculation: Calculate the lower limit of the variance confidence interval: $\newline$ $[\frac{(n-1)s^2}{\chi^2(1-\frac{\alpha}{2})}] = \frac{15\times33^2}{30.578}$ $\newline$ $= \frac{15\times1089}{30.578}$ $\newline$ $= \frac{16335}{30.578}$ $\newline$ $\approx 534.22$
Variance Upper Limit Calculation: Calculate the upper limit of the variance confidence interval: $\newline$ $(n-1)s^2/\chi^2(\alpha/2)$ = $15\cdot33^2/6.262$ $\newline$ = $15\cdot1089/6.262$ $\newline$ = $16335/6.262$ $\newline$ \approx $2608$ . $97$
Population Variance Confidence Interval: Now we have the confidence interval for the population variance: $\newline$ egin{equation} $\newline$ ( $534$ . $22$ , $2608$ . $97$ ) $\newline$ egin{equation} $\newline$ Next, we find the confidence interval for the population standard deviation by taking the square root of the variance interval limits.
Standard Deviation Lower Limit Calculation: Calculate the lower limit of the standard deviation confidence interval: $\sqrt{534.22} \approx 23.11$
Standard Deviation Upper Limit Calculation: Calculate the upper limit of the standard deviation confidence interval: $\sqrt{2608.97} \approx 51.08$
Population Standard Deviation Confidence Interval: Now we have the confidence interval for the population standard deviation: $(23.11, 51.08)$