Applied Statistics and Probability for Engineers

334 CHAPTER 10 STATISTICAL INFERENCE FOR TWO SAMPLES

cause and effect here has proven enormously difficult. Many individuals had decided to

smoke long before the start of the research studies, and many factors other than smoking could

have a role in contracting lung cancer.

10-2.4 Confidence Interval on a Difference in Means,

Variances Known

The 100(1)% confidence interval on the difference in two means  1  2 when the vari-

ances are known can be found directly from results given previously in this section. Recall

thatX 11 , X 12 ,p, is a random sample ofn 1 observations from the first population and X 21 ,

X 22 ,p, is a random sample ofn 2 observations from the second population. The difference

in sample means is a point estimator of  1   2 , and

has a standard normal distribution if the two populations are normal or is approximately stan-

dard normal if the conditions of the central limit theorem apply, respectively. This implies that

, or

This can be rearranged as

Therefore, the 100(1)% confidence interval for  1  2 is defined as follows.

P aX 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 b^1 

P ≥z 

2 

X 1 X 2  1  1  22

B

^21

n 1

^22

n 2

z 

2 ¥ 1 

P 1 z 

2 Zz 

22  1 

Z

X 1 X 2  1  1  22

B

^21

n 1

^22

n 2

X 1 X 2

X 2 n 2

X 1 n 1

If and are the means of independent random samples of sizes n 1 and n 2 from

two independent normal populations with known variances ^21 and  22 , respectively,

a100(1)% confidence interval for  1  2 is

(10-7)

where z  2 is the upper 2 percentage point of the standard normal distribution.

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 1 x 2

Definition

The confidence level 1is exact when the populations are normal. For nonnormal popu-

lations, the confidence level is approximately valid for large sample sizes.

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