Question

Use technology to construct the confidence intervals for the population variance o? and the population standard deviations. Assume the sample is taken from a normally distributed population C+0.95, 82 = 7.29, n = 27 The confidence interval for the population variance is (Round to two decimal places as needed.) The confidence interval for the population standard deviation is (Round to two decimal places as needed.)

Answer 1

Solution :

Given that,

c = 0.95

s2 = 7.29

n = 27

At 95% confidence level the $\chi ^{2}$ is ,

$\alpha$ = 1 - 95% = 1 - 0.95 = 0.05

$\alpha$ / 2 = 0.05 / 2 = 0.025

$\chi ^{2}$$\alpha$ /2,df = $\chi ^{2}$ 0.025,18 = 41.92

and

$\chi ^{2}$1- $\alpha$ /2,df = $\chi ^{2}$ 0.975,18 = 13.84

Point estimate = s2 = 7.29

$\chi$ 2L = $\chi$ 2$\alpha$/2,df = 41.92

$\chi$ 2R = $\chi$ 21 - $\alpha$ /2,df = 13.84

The 95% confidence interval for $\sigma$ 2 is,

(n - 1)s2 / $\chi$ 2$\alpha$/2 < $\sigma$ 2 < (n - 1)s2 / $\chi$ 21 - $\alpha$ /2

( 26 *7.29) / 34.81 < $\sigma$ 2 < ( 26 * 7.29) / 13.84

4.52 < $\sigma$ 2 < 13.69

(4.52 , 13.69)

The 95% confidence interval for $\sigma$ is,

s $\sqrt{}$ (n-1) / $\chi ^{2}$ $\alpha$ /2,df < $\sigma$ < s $\sqrt{}$ (n-1) / $\chi ^{2}$ 1- $\alpha$ /2,df

2.7 $\sqrt{}$ ( 27 - 1 ) / 41.92 < $\sigma$ < 2.7 $\sqrt{}$ ( 27 - 1 ) / 13.84

2.13 < $\sigma$ < 3.70

( 2.13 , 3.70)