Question

The proper operation of typical home appliances requires voltage levels that do not vary much. A sample of voltage levels for 10 days was recorded and it was found that the standard deviation of the sample was 0.23 volt. Construct a 90% confidence interval estimate of the standard deviation of all voltage levels. Write a sentence summarizing your results.

Answer 1

Solution :

Given that,

s = 0.23

s² = 0.0529

n = 10

Degrees of freedom = df = n - 1 = 10 - 1 = 9

At 90% confidence level the $\chi$ ² value is ,

$\alpha$ = 1 - 90% = 1 - 0.90 = 0.10

$\alpha$ / 2 = 0.10 / 2 = 0.05

1 -  $\alpha$ / 2 = 1 - 0.05 = 0.95

$\chi$²_L = $\chi$ ²_{$\alpha$/2,df} = 16.919

$\chi$²_R = $\chi$ ²_{1 - $\alpha$/2,df} = 3.325

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

$\sqrt$(n - 1)s² / $\chi$ ²_$\alpha$/2 < $\sigma$ < $\sqrt$ (n - 1)s² / $\chi$ ²_{1 - $\alpha$/2}

$\sqrt$9 * 0.0529 / 16.919 < $\sigma$ < $\sqrt$ 9 * 0.0529 / 3.325

0.17 < $\sigma$ < 0.38

( 0.17, 0.38)

The 95% confidence interval for $\sigma$ is, ( 0.17, 0.38) .