Question

In Exercises 9–12, construct the indicated confidence intervals for (a) the population variance and (b) the population standard deviation . Assume the sample is from a normally distributed population.
c = 0.98, s^2 = 278.1, n =41

Accepted Answer

Step 1: Understand the problem. We are tasked with constructing confidence intervals for (a) the population variance (σ²) and (b) the population standard deviation (σ) using the given sample variance (s² = 278.1), sample size (n = 41), and confidence level (c = 0.98). The population is assumed to be normally distributed, which allows us to use the Chi-Square distribution for this calculation. Step 2: Identify the formula for the confidence interval of the population variance. The confidence interval for the population variance is given by: $$ \frac{(n-1)s^2}{\chi^2_{\alpha/2}} \leq \sigma^2 \leq \frac{(n-1)s^2}{\chi^2_{1-\alpha/2}} $$, where $$ \chi^2_{\alpha/2} $$ and $$ \chi^2_{1-\alpha/2} $$ are the critical values of the Chi-Square distribution with $$ df = n-1 $$ degrees of freedom, and $$ \alpha = 1 - c . $$Step 3: Calculate the degrees of freedom and the significance level. The degrees of freedom (df) is $$ n-1 = 41-1 = 40 . $$The significance level $$ \alpha  is$$ $$ 1 - c = 1 - 0.98 = 0.02 . $$Divide $$ \alpha  by 2 to $$find $$ \alpha/2 = 0.01 . $$Step 4: Look up the critical values of the Chi-Square distribution. Using a Chi-Square table or statistical software, find $$ \chi^2_{\alpha/2} $$ and $$ \chi^2_{1-\alpha/2} $$ for $$ df = 40 . $$These values correspond to the 0.01 and 0.99 percentiles of the Chi-Square distribution, respectively. Step 5: Plug the values into the formula for the confidence interval of the population variance. Substitute $$ n-1 = 40 $$, $$ s^2 = 278.1 $$, and the critical values $$ \chi^2_{\alpha/2} $$ and $$ \chi^2_{1-\alpha/2} $$ into the formula to calculate the lower and upper bounds of the confidence interval for $$ \sigma^2 . To $$find the confidence interval for the population standard deviation $$ \sigma $$, take the square root of the lower and upper bounds of the variance confidence interval.

In Exercises 9–12, construct the indicated confidence intervals for (a) the population variance and (b) the population standard deviation . Assume the sample is from a normally distributed population.
c = 0.98, s^2 = 278.1, n =41

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Confidence Interval

Chi-Squared Distribution

Sample Variance and Standard Deviation

In Exercises 9–12, construct the indicated confidence intervals for (a) the population variance and (b) the population standard deviation . Assume the sample is from a normally distributed population.c = 0.98, s^2 = 278.1, n =41

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Confidence Interval

Chi-Squared Distribution

Sample Variance and Standard Deviation

In Exercises 9–12, construct the indicated confidence intervals for (a) the population variance and (b) the population standard deviation . Assume the sample is from a normally distributed population.
c = 0.98, s^2 = 278.1, n =41