Exercise 1102 A sample of 20 items provides a sample standar

{Exercise 11.02}

A sample of 20 items provides a sample standard deviation of 4.

a. Compute the 90% confidence interval estimate of the population variance (to 2 decimals).

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b. Compute the 95% confidence interval estimate of the population variance (to 2 decimals).

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c. Compute the 95% confidence interval estimate of the population standard deviation (to 1 decimal).

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Solution

For a sample size of n=20, we will have df=n1=19 degrees of freedom. For a 90% confidence interval, we have =0.1, which gives 5% of the area at each end of the chi-square distribution. We find values of (X2)(chi-square) 0.95=10.117 and (2)0.05=30.1435. Evaluating (n1)s2/2, we obtain 10.0850 and 30.0484 . This leads to the inequalities 10.0850 230.0484 for the variance.

a) 10.08 2 30.05

For a sample size of n=20, we will have df=n1=19 degrees of freedom. For a 95% confidence interval, we have =0.05, which gives 2.5% of the area at each end of the chi-square distribution. We find values of (X2)(chi-square) 0.975=8.9065 and (2)0.025=32.8523. Evaluating (n1)s2/2, we obtain 9.2535 and 34.132 . This leads to the inequalities 9.2535234.132 for the variance, and taking square roots, 3.04 5.84 for the standard deviation.

b)   9.25 2 34.13

c)     3.0 5.8