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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
that X 11 , X 12 , p , X is a random sample of n 1 observations from the first population and X 21 ,
1n 1
X 22 , p , X is a random sample ofn 2 observations from the second population. The difference
2n 2
in sample means X X 2 is a point estimator of 1 2 , and
1
X X 2 1 1 2 2
1
Z
2 2
1 2
n
B 1 n 2
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
P1 z
2 Z z
2 2 1 , or

X 1 X 2 1 1 2 2
P ≥ z
2 2 2 z
2 ¥ 1
1 2
n n
B 1 2

This can be rearranged as

2 2 2 2
1 2 1 2
P aX X z
2 X X z
2 b 1
n
2
1
1
2
2
1
n
B 1 n 2 B 1 n 2
Therefore, the 100(1 )% confidence interval for 1 2 is defined as follows.
Definition
If x 1 and x 2 are the means of independent random samples of sizes n 1 and n 2 from
2
2
two independent normal populations with known variances and , respectively,
2
1
a 100(1 )% confidence interval for 1 2 is
2 2 2 2
1 2 1 2
x z x x z (10-7)
x 1 2
2 n n 1 2 1 2
2 n n
B 1 2 B 1 2
where z 2 is the upper 2 percentage point of the standard normal distribution.
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.