Applied Statistics and Probability for Engineers

and have approximate normal distributions. We are interested in

testing the hypotheses

The statistic

H 1 : p 1 p 2

H 0 : p 1 p 2

Pˆ 1 X 1
n 1 Pˆ 2 X 2
n 2

362 CHAPTER 10 STATISTICAL INFERENCE FOR TWO SAMPLES

Z (10-32)

Pˆ 1 Pˆ 2  1 p 1 p 22

B

p 111 p 12

n 1

p 211 p 22

n 2

is distributed approximately as standard normal and is the basis of a test for H 0 : p 1 p 2.

Specifically, if the null hypothesis H 0 : p 1 p 2 is true, using the fact that p 1 p 2 p, the

random variable

is distributed approximately N(0, 1). An estimator of the common parameter pis

The test statisticfor H 0 : p 1 p 2 is then

This leads to the test procedures described below.

Z 0 

Pˆ 1 Pˆ 2

B

Pˆ 11 Pˆ 2 a

1

n 1

1

n 2 b

Pˆ

X 1 X 2

n 1 n 2

Z

Pˆ 1 Pˆ 2

B

p 11 p 2 a

1

n 1

1

n 2 b

Null hypothesis: H 0 : p 1 p 2

Test statistic: (10-33)

Alternative Hypotheses Rejection Criterion

H 1 : p 1

 p 2 z 0

 z

H 1 : p 1 p 2 z 0 z

H 1 : p 1 p 2 z 0

 z 

2 or z 0 z 

2

Z 0 

Pˆ 1 Pˆ 2

B

Pˆ 11 Pˆ 2 a

1

n 1

1

n 2 b

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