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10-2 INFERENCE FOR A DIFFERENCE IN MEANS OF TWO NORMAL DISTRIBUTIONS, VARIANCES KNOWN 329

Assumptions
1. X , X , p , X 1n 1 is a random sample from population 1.
12
11
2. X , X , p , X 2n 2 is a random sample from population 2.
21
22
3. The two populations represented by X and X are independent.
2
1
4. Both populations are normal.
A logical point estimator of is the difference in sample means X X . Based
2
1
2
1
on the properties of expected values
E1X X 2 E1X 2 E1X 2 2
1
1
1
2
2
is
and the variance of X 1 X 2
2 2
1 2
V1X X 2 V1X 2 V1X 2 n 1 n 2
2
2
1
1
Based on the assumptions and the preceding results, we may state the following.
The quantity
1
2
X 1 X 1 2
2
Z (10-1)
2 2
1 2

B n 1 n 2
has a N(0, 1) distribution.

This result will be used to form tests of hypotheses and confidence intervals on .
1
2
ˆ
Essentially, we may think of as a parameter , and its estimator is X X 2
2
1
1
2 2 2
with variance ˆ 1
n 2
n . If is the null hypothesis value specified for , the test
1
2

0
ˆ
statistic will be 1 2
ˆ . Notice how similar this is to the test statistic for a single mean

0
used in Equation 9-8 of Chapter 9.
10-2.1 Hypothesis Tests for a Difference in Means, Variances Known
We now consider hypothesis testing on the difference in the means of two normal
1
2
populations. Suppose that we are interested in testing that the difference in means is
1
2
equal to a specified value . Thus, the null hypothesis will be stated as H : .
2
1
0
0
0
Obviously, in many cases, we will specify 0 so that we are testing the equality of two
0
means (i.e., H : ). The appropriate test statistic would be found by replacing 2
0
1
2
1
in Equation 10-1 by , and this test statistic would have a standard normal distribution under
0
. That is, the standard normal distribution is the reference distribution for the test statistic.
H 0
Suppose that the alternative hypothesis is H : . Now, a sample value of x x 2
1
2
0
1
1
that is considerably different from is evidence that H is true. Because Z has the N(0, 1)
0
0
1