Applied Statistics and Probability for Engineers

10-3

which is the probability density function in the theorem on page 10-1.

10-6.2 Small-Sample Test for H 0 : p 1 p 2 (CD Only)

Many problems involving the comparison of proportions p 1 and p 2 have relatively large sam-

ple sizes, so the procedure based on the normal approximation to the binomial is widely used

in practice. However, occasionally, a small-sample-size problem is encountered. In such

cases, the Z-tests are inappropriate and an alternative procedure is required. In this section we

describe a procedure based on the hypergeometric distribution.

Suppose that X 1 and X 2 are the number of successes in two random samples of size n 1 and

n 2 , respectively. The test procedure requires that we view the total number of successes as

fixed at the value X 1 X 2 Y. Now consider the hypotheses

Given that X 1 X 2 Y, large values of X 1 support H 1 , and small moderate values of X 1 sup-

port H 0. Therefore, we will reject H 0 whenever X 1 is sufficiently large.

Since the combined sample of n 1 n 2 observations contains X 1 X 2 Ytotal suc-

cesses, if H 0 : p 1 p 2 the successes are no more likely to be concentrated in the first sample

than in the second. That is, all the ways in which the n 1 n 2 responses can be divided into one

sample of n 1 responses and a second sample of n 2 responses are equally likely. The number of

ways of selecting X 1 successes for the first sample leaving YX 1 successes for the second is

Because outcomes are equally likely, the probability of exactly X 1 successes in sample 1 is the

ratio of the number of sample 1 outcomes having X 1 successes to the total number of outcomes, or

(S10-1)

given that H 0 : p 1 p 2 is true. We recognize Equation S10-1 as a hypergeometric distribution.

To use Equation S10-1 for hypothesis testing, we would compute the probability of find-

ing a value of X 1 at least as extreme as the observed value of X 1. Note that this probability is a

P 1 X 1 x 1 | Y success in n 1 n 2 responses 2 

a

Y

x 1

b a

n 1 n 2 Y

n 1 x 1

b

a

n 1 n 2

n 1

b

a

Y

X 1

b a

n 1 n 2 Y

n 1 X 1

b

H 1 : p 1 p 2

H 0 : p 1 p 2



 a

 1  2

2

b a

 1

 2 b

 1 2

x^1    2 ^1

 a

 1

2

b  a

 2

2

b a

 1

 2 x^1 b

1  1  22 2 ,^0 x^



a

 1

 2 b

 1 2

x^1    2 ^1

 a

 1

2

b  a

 2

2

b a

 1

 2 x^1 b

1  1  22 2 

0

z^1 ^1 ^22   2 ^1 ez dz

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