Question

The number of hours of reserve capacity of 1010 randomly selected automotive batteries is shown to the right.

1.791.79 1.861.86 1.581.58 1.691.69 1.731.73 1.951.95 1.371.37 1.551.55 1.421.42 2.052.05

Assume the sample is taken from a normally distributed population. Construct

9595​%

confidence intervals for​ (a) the population variance

sigmaσsquared2

and​ (b) the population standard deviation

sigmaσ.

​(a) The confidence interval for the population variance is

​(nothing​,nothing​).

​(Round to three decimal places as​ needed.)

Answer 1

Solution:

Given: The number of hours of reserve capacity of 10 randomly selected automotive batteries is shown in table below.

1.79	1.86	1.58	1.69	1.73
1.95	1.37	1.55	1.42	2.05

the sample is taken from a normally distributed population.

(a) Construct 95​% confidence intervals for​ the population variance

where

and Chi-square critical values for right and left tail respectively.

Find s² .

x	x^2
1.79	3.2041
1.86	3.4596
1.58	2.4964
1.69	2.8561
1.73	2.9929
1.95	3.8025
1.37	1.8769
1.55	2.4025
1.42	2.0164
2.05	4.2025

Thus

Find Chi-square critical values:

df = n - 1 = 10 - 1 = 9

c = confidence level = 95% = 0.95

then

Thus right tail area =

For left tail critical value , we use Area =

Thus look in Chi-square table for df = 9 and Area = 0.975 and for 0.025

and find chi-square critical values.

Thus and  

Thus

Thus a 95% confidence interval for the population variance is :

Part b)  the population standard deviation

Thus a 95% confidence interval for the population standard deviation :