Applied Statistics and Probability for Engineers

338 CHAPTER 10 STATISTICAL INFERENCE FOR TWO SAMPLES

develop the test procedure, but moderate departures from normality do not adversely affect

the procedure. Two different situations must be treated. In the first case, we assume that the

variances of the two normal distributions are unknown but equal; that is, ^2. In the

second, we assume that and are unknown and not necessarily equal.

Case 1: 12  22 ^2

Suppose we have two independent normal populations with unknown means  1 and  2 , and

unknown but equal variances, ^2. We wish to test

(10-11)

Let X 11 , X 12 , p, be a random sample of n 1 observations from the first population and

X 21 ,X 22 ,p, be a random sample of n 2 observations from the second population.

Let , ,S^21 , and S^22 be the sample means and sample variances, respectively. Now the ex-

pected value of the difference in sample means is  1  2 , so

is an unbiased estimator of the difference in means. The variance of is

It seems reasonable to combine the two sample variances and to form an estimator

of 2. The pooled estimatorof 2 is defined as follows.

S^21 S^22

V 1 X 1 X 22 

2

n 1

2

n 2 

(^2) a^1
n 1
1
n 2 b
X 1 X 2 X 1 X 2
X 1 X 2 E 1 X 1 X 22
X 1 X 2
X 2 n 2
X 1 n 1
H 1 :  1  2
 0
H 0 :  1  2  0
21 22
21 22
(^2122)
The pooled estimatorof 2 , denoted by S^2 p, is defined by
Sp^2  (10-12)
1 n 1  12 S^21
1 n 2  12 S^22
n 1
n 2  2
It is easy to see that the pooled estimator can be written as
where 0 w1. Thus Sp^2 is a weighted averageof the two sample variances S 12 and S 22 ,
where the weights wand 1wdepend on the two sample sizes n 1 and n 2. Obviously, if n 1 
n 2 n, w0.5 and Sp^2 is just the arithmetic average of S 12 and S 22. If n 1 10 and n 2  20
(say), w0.32 and 1w0.68. The first sample contributes n 1 1 degrees of freedom
toSp^2 and the second sample contributes n 2 1 degrees of freedom. Therefore, Sp^2 has
n 1
n 2 2 degrees of freedom.
Now we know that
has a N(0, 1) distribution. Replacing by Spgives the following.
Z
X 1 X 2  1  1  22
B
1
n 1
1
n 2
S^2 p
n 1  1
n 1
n 2  2
S^21
n 2  1
n 1
n 2  2
S^22 wS^21
11 w 2 S^22
S^2 p
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