Question

A sample of 51 elements is selected to estimate a 95% confidence interval for the variance of the population. The chi-square values to be used for this interval estimation are
a. -1.96 and 1.96
b. 32.357 and 71.420
c, 34.764 and 67.505
d. 12.8786 and 46.9630

Answer 1

Answer:

Correct option: (b) 32.357 and 71.420

Explanation:

The confidence interval for population variance σ² is:

$\frac{(n-1)s^{2}}{\chi^{2}_{\alpha/2, (n-1) }}\leq \sigma^{2}\leq \frac{(n-1)s^{2}}{\chi^{2}_{(1-\alpha/2), (n-1) }}$

Given:

$n=51\\\alpha =1-0.95=0.05$

Compute the critical values of chi-square as follows:

$\chi^{2}_{\alpha/2, (n-1)}=\chi^{2}_{0.025,50}=71.42$

$\chi^{2}_{(1-\alpha/2), (n-1)}=\chi^{2}_{0.975,50}=32.36$

Use the chi-square table for the critical value.

Thus, the critical values are 32.36 and 71.42.