Applied Statistics and Probability for Engineers

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

This result will be used to form tests of hypotheses and confidence intervals on  1  2.

Essentially, we may think of  1  2 as a parameter , and its estimator is

with variance If  0 is the null hypothesis value specified for , the test

statistic will be Notice how similar this is to the test statistic for a single mean

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  1  2 of two normal

populations. Suppose that we are interested in testing that the difference in means  1  2 is

equal to a specified value  0. Thus, the null hypothesis will be stated as H 0 :  1  2  0.

Obviously, in many cases, we will specify  0 0 so that we are testing the equality of two

means (i.e., H 0 :  1  2 ). The appropriate test statistic would be found by replacing  1  2

in Equation 10-1 by  0 , and this test statistic would have a standard normal distribution under

H 0. That is, the standard normal distribution is the reference distributionfor the test statistic.

Suppose that the alternative hypothesis is H 1 :  1  2  0. Now, a sample value of

that is considerably different from  0 is evidence that H 1 is true. Because Z 0 has the N(0, 1)

x 1 x 2

1 ˆ 02
 ˆ.

 ˆ^2  12
n 1 ^22 n 2.

ˆ X 1 X 2

A logical point estimator of  1  2 is the difference in sample means Based

on the properties of expected values

and the variance of is

Based on the assumptions and the preceding results, we may state the following.

V 1 X 1 X 22 V 1 X 12 V 1 X 22 

^21

n 1

^22

n 2

X 1 X 2

E 1 X 1 X 22 E 1 X 12 E 1 X 22  1  2

X 1 X 2.

1. X 11 , X 12 ,p, is a random sample from population 1.


2. X 21 , X 22 ,p, is a random sample from population 2.


3. The two populations represented by X 1 and X 2 are independent.


4. Both populations are normal.

X 2 n 2

X 1 n 1

Assumptions

The quantity

(10-1)

has a N(0, 1) distribution.

Z

X 1 X 2  1  1  22

B

^21

n 1

^22

n 2

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