Applied Statistics and Probability for Engineers

312 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE

Another form of the test statistic Z 0 in Equation 9-32 is occasionally encountered. Note

that if Xis the number of observations in a random sample of size nthat belongs to a class of

interest, then is the sample proportion that belongs to that class. Now divide both

numerator and denominator of Z 0 in Equation 9-32 by n, giving

or

(9-33)

This presents the test statistic in terms of the sample proportion instead of the number of items

Xin the sample that belongs to the class of interest.

Statistical software packages usually provide the one sample Z-test for a proportion. The

Minitab output for Example 9-10 follows.

Z 0 

Pˆ p 0

1 p 011 p 02

 n

Z 0 

X n p 0

1 p 011 p 02

 n

PˆX n

Notice that both the test statistic (and accompanying P-value) and the 95% one-sided upper

confidence bound are displayed. The 95% upper confidence bound is 0.036283, which is less

than 0.05. This is consistent with rejection of the null hypothesis Ho: p0.05.

9-5.2 Small-Sample Tests on a Proportion (CD Only)

9-5.3 Type II Error and Choice of Sample Size

It is possible to obtain closed-form equations for the approximate -error for the tests in

Section 9-5.1. Suppose that pis the true value of the population proportion. The approximate

-error for the two-sided alternative H 1 :p p 0 is

(9-34)

If the alternative is H 1 : pp 0 ,

 1  a (9-35)

p 0 p z 1 p 011 p 02

 n

1 p 11 p 2

 n

b

 a

p 0 pz 

21 p 011 p 02

 n

1 p 11 p 2

 n

b a

p 0 p z 

21 p 011 p 02

 n

1 p 11 p 2

 n

b

Test and CI for One Proportion

Test of p0.05 vs p0.05

Sample X N Sample p 95.0% Upper Bound Z-Value P-Value

1 4 200 0.020000 0.036283 1.95 0.026

* NOTE * The normal approximation may be inaccurate for small samples.

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