Applied Statistics and Probability for Engineers

10-2 INFERENCE FOR A DIFFERENCE IN MEANS OF TWO NORMAL DISTRIBUTIONS, VARIANCES KNOWN 335

EXAMPLE 10-4 Tensile strength tests were performed on two different grades of aluminum spars used in

manufacturing the wing of a commercial transport aircraft. From past experience with the spar

manufacturing process and the testing procedure, the standard deviations of tensile strengths

are assumed to be known. The data obtained are as follows: n 1 10, 87.6,  1 1,

n 2 12, 74.5, and  2 1.5. If  1 and  2 denote the true mean tensile strengths for the

two grades of spars, we may find a 90% confidence interval on the difference in mean strength

 1  2 as follows:

Therefore, the 90% confidence interval on the difference in mean tensile strength (in kilo-

grams per square millimeter) is

(in kilograms per square millimeter)

Notice that the confidence interval does not include zero, implying that the mean

strength of aluminum grade 1 ( 1 ) exceeds the mean strength of aluminum grade 2 ( 2 ). In

fact, we can state that we are 90% confident that the mean tensile strength of aluminum

grade 1 exceeds that of aluminum grade 2 by between 12.22 and 13.98 kilograms per

square millimeter.

Choice of Sample Size

If the standard deviations  1 and  2 are known (at least approximately) and the two sample

sizes n 1 and n 2 are equal (n 1  n 2  n, say), we can determine the sample size required so that

the error in estimating  1   2 by will be less than Eat 100(1  )% confidence. The

required sample size from each population is

x 1 x 2

12.22 1  2 13.98

87.674.51.645 ̨

B

1122

10

1 1.5 23

12

 1  2 87.674.5 1.645

B

1122

10

1 1.5 22

12

x 1 x 2 z 

2 ̨

B

^21

n 1

^22

n 2 ^1 ^2 x^1 x^2 z^2 ̨B

^21

n 1

^22

n 2

x 2

x 1

na (10-8)

z 

2

E

b

2

1 ^21 ^222

Remember to round up if nis not an integer. This will ensure that the level of confidence does

not drop below 100(1)%.

One-Sided Confidence Bounds

One-sided confidence bounds on  1  2 may also be obtained. A 100(1)% upper-

confidence bound on  1  2 is

 1  2 x 1 x 2 z (10-9)

B

^21

n 1

^22

n 2

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