Question

(21) A sample of size 20 is drawn from a normal distribution with unknown variance and mean. The sample variance s2 = 0.012. Find a 95% two-sided confidence interval for the standard deviation o of the population. A. [0.0833,0.1600] B. (0.010,0.0130] C. (0.0069,0.0256] D. None of the above

Answer 1

Solution :

Given that,

s2 = 0.012

n = 20

Degrees of freedom = df = n - 1 = 20 - 1 = 19

At 95% confidence level the $\chi$ 2 value is ,

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

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

1 - $\alpha$ / 2 = 1 - 0.025 = 0.975

$\chi$2L = $\chi$ 2$\alpha$/2,df = 32.853

$\chi$2R = $\chi$ 21 - $\alpha$ /2,df = 8.907

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

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

$\sqrt$(19)(0.012) / 32.853 < $\sigma$ < $\sqrt$ (19)(0.012) / 8.907

0.0833 < $\sigma$ < 0.1600

(0.0833 , 0.1600)

option A. is correct