Question

A random sample of 27 observations is used to estimate the population variance. The sample mean and sample standard deviation are calculated as 44 and 4.5, respectively. Assume that the population is normally distributed. (You may find it useful to reference the appropriate table: chi-square table or F table)

a. Construct the 95% interval estimate for the population variance. (Round intermediate calculations to at least 4 decimal places and final answers to 2 decimal places.)

b. Construct the 99% interval estimate for the population variance. (Round intermediate calculations to at least 4 decimal places and final answers to 2 decimal places.)

Answer 1

Solution :

Given that,

s = 4.5

Point estimate = s² = 20.25

(a)

²_L = ²_/2,df = 41.923

²_R = ²_{1 - /2,df} = 13.844

The 95% confidence interval for ² is,

(n - 1)s² / ²_/2 < ² < (n - 1)s² / ²_{1 - /2}

26 * 20.25 / 41.923 < ² < 28 * 20.25 / 13.844

12.56 < ² < 38.03

(12.56 , 38.03)

b)

²_L = ²_/2,df = 48.290

²_R = ²_{1 - /2,df} = 11.160

The 99% confidence interval for ² is,

(n - 1)s² / ²_/2 < ² < (n - 1)s² / ²_{1 - /2}

26 * 20.25 / 48.290 < ² < 26*20.25 / 11.160

10.90 < ² < 47.18

(10.90 , 47.18 )