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

Correct option: (b) 32.357 and 71.420

Step-by-step explanation:

The confidence interval for population variance σ² is:

`((n-1)s^(2))/(\chi^(2)_(\alpha/2, (n-1) ))\leq \sigma^(2)\leq ((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.