Page 391 - Applied Statistics And Probability For Engineers

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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 10, x 1 87.6, 1,
1
1
n 12, x 2 74.5, and 1.5. If and denote the true mean tensile strengths for the
2
1
2
2
two grades of spars, we may find a 90% confidence interval on the difference in mean strength
as follows:
1
2
2 2 2 2
1 2 1 2
x x z
2 ˛ x x z
2 ˛
1
2
1
2
2
1
n
n
B 1 n 2 B 1 n 2
2
112 2 11.52 3 11 2 11.52 2
87.6 74.5 1.645
B 10 12 B 10 12
87.6 74.5 1.645 ˛ 1 2
Therefore, the 90% confidence interval on the difference in mean tensile strength (in kilo-
grams per square millimeter) is
12.22 13.98 (in kilograms per square millimeter)
2
1
Notice that the confidence interval does not include zero, implying that the mean
strength of aluminum grade 1 ( ) exceeds the mean strength of aluminum grade 2 ( 2 ). In
1
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 x x 2 will be less than E at 100(1 )% confidence. The
1
required sample size from each population is
2
z
2 2 2
n a b 1 1 2 2 (10-8)
E

Remember to round up if n is 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 may also be obtained. A 100(1 )% upper-
1
2
confidence bound on is
1
2
2 2
1 2
1
2 x 1 x 2 z n n (10-9)
B 1 2