Question

The diameter of holes for a cable harness is known to have a normal distribution with a standard deviation of 0. 01 inch. A random sample of size 10 yields an average diameter of 1. 5045 inch. Find the 99% and 95% two-tailed/two-sided confidence intervals on the mean diameter and explain the difference of the interval between the two confidence levels.

Answer 1

For the given standard deviation and sample mean the two sided confidence interval on mean diameter foe 95% and 99% is given by        ( 1.4983 , 1.5107 ) and ( 1.49635 , 1.51265 ) respectively.

As given in the question,

Confidence interval for population mean is given by:

$\bar{x}$ ± z × ( σ / √n)

Where,

σ = population standard deviation.

n= sample size

$\bar{x}$ = Sample mean

Given values from the random sample size are :

σ = 0.01 inches

n= 10

$\bar{x}$ = 1.5045

Critical value confidence interval (using x-value table)

z  for 99% = 2.576 and Significance level = 0.01

z for 95% = 1.960 and Significance level = 0.05

99% two-sided confidence interval on the mean hole diameter is equal to :

$\bar{x}$ ± z × ( σ / √n)

= 1.5045 ± 2.576 × ( 0.01 / √10)

= 1.5045 ± 0.00815 (approximately)

= 1.5045 - 0.00815 , 1.5045 + 0.00815

= ( 1.49635 , 1.51265 )

95% two-sided confidence interval on the mean hole diameter is equal to :

$\bar{x}$ ± z × ( σ / √n)

= 1.5045 ± 1.960 × ( 0.01 / √10)

= 1.5045 ± 0.00620 (approximately)

= 1.5045 - 0.00620 , 1.5045 + 0.00620

= ( 1.4983 , 1.5107 )

Therefore, for the given standard deviation and sample mean the two sided confidence interval on mean diameter foe 95% and 99% is given by ( 1.4983 , 1.5107 ) and  ( 1.49635 , 1.51265 ) respectively.

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