Question

The following are heights (in inches) of randomly selected women: 63.7, 61.2, 66.0, 62.5, 65.7, 64.4, 63.0, 61.9

a) Find the best point estimate of the population variance σ².

b) Construct a 95% confidence interval estimate of the population standard deviation σ.

c) Does the confidence interval contain the standard deviation value of 2.5 inches? (Women’s heights are known to have a standard deviation of 2.5 inches).

PLEASE WRITE CLEARLY

Answer 1

From the given sample data,

Sample mean = X / n = 508.4 / 8 = 63.55 ,

Sample variance = S² = (X² - n  ² ) / n -1

= 32329.84 - 8 * 63.55² / 7

= 3.0029

a)

Sample variance S² is best point estimate of population variance ² .

= 3.0029

b)

df = n - 1 = 8 - 1 = 7

From the chi square critical value table,

Critical values for 7 df with 0.05 significance level are 16.013 , 1.690

95% confidence interval for is

Sqrt [ (n-1) S² / /2 ] < < Sqrt [ (n-1) S² / 1-/2 ]

Sqrt [ (8-1) * 3.0029 / 16.013 ] < < Sqrt [ (8-1) * 3.0029 / 1.690]

Sqrt ( 7 * 3.0029 / 16.013 ) < < sqrt( 7 * 3.0029 / 1.690 )

Sqrt( 1.3127) < < sqrt( 12.4380)

1.1457 < < 3.5268

95% CI for is ( 1.1457 , 3.5268 )

c)

Confidence interval for   is from 1.1457 to 3.5268, So it contains 2.5.

Yes, the confidence interval contain the standard deviation value of 2.5 inches.