Applied Statistics and Probability for Engineers

9-5 TESTS ON A POPULATION PROPORTION 311

An approximate test based on the normal approximation to the binomial will be given. As

noted above, this approximate procedure will be valid as long as pis not extremely close to

zero or one, and if the sample size is relatively large. Let Xbe the number of observations in

a random sample of size nthat belongs to the class associated with p. Then, if the null

hypothesis H 0 : pp 0 is true, we have XN[np 0 , np 0 (1 p 0 )], approximately. To test

H 0 :pp 0 , calculate the test statistic

(9-32)

Z 0 

X np 0

1 np 011 p 02

and reject H 0 : pp 0 if

Note that the standard normal distribution is the reference distributionfor this test statistic.

Critical regions for the one-sided alternative hypotheses would be constructed in the usual manner.

EXAMPLE 9-10 A semiconductor manufacturer produces controllers used in automobile engine applications.

The customer requires that the process fallout or fraction defective at a critical manufacturing

step not exceed 0.05 and that the manufacturer demonstrate process capability at this level of

quality using 0.05. The semiconductor manufacturer takes a random sample of 200

devices and finds that four of them are defective. Can the manufacturer demonstrate process

capability for the customer?

We may solve this problem using the eight-step hypothesis-testing procedure as follows:

1. The parameter of interest is the process fraction defective p.


2. H 0 : p0.05


3. H 1 : p0.05
   This formulation of the problem will allow the manufacturer to make a strong claim
   about process capability if the null hypothesis H 0 : p0.05 is rejected.


4. 0.05


5. The test statistic is (from Equation 9-32)

where x4, n200, and p 0 0.05.

6. Reject H 0 : p0.05 if z 0 z0.051.645


7. Computations: The test statistic is


8. Conclusions: Since z 0 1.95z0.05 1.645, we reject H 0 and conclude that the
   process fraction defective pis less than 0.05. The P-value for this value of the test statistic
   z 0 is P0.0256, which is less than 0.05. We conclude that the process is capable.

z 0 

42001 0.05 2

12001 0.05 21 0.95 2

1.95

z 0 

x np 0

1 np 011 p 02

z 0 z

2 or z 0 z
2

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