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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 p is not extremely close to
zero or one, and if the sample size is relatively large. Let X be the number of observations in
a random sample of size n that belongs to the class associated with p. Then, if the null
hypothesis H 0 : p p 0 is true, we have X N[np 0 , np 0 (1 p 0 )], approximately. To test
H 0 : p p 0 , calculate the test statistic
X np 0
Z
0
1np 11 p 2 (9-32)
0
0
and reject H : p p if
0
0
z z
2 or z z
2
0
0
Note that the standard normal distribution is the reference distribution for 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 : p 0.05
0
3. H : p 0.05
1
This formulation of the problem will allow the manufacturer to make a strong claim
about process capability if the null hypothesis H : p 0.05 is rejected.
0
4. 0.05
5. The test statistic is (from Equation 9-32)
x np 0
z
0
1np 11 p 2
0
0
0.05.
where x 4, n 200, and p 0
6. Reject H : p 0.05 if z z 0.05 1.645
0
0
7. Computations: The test statistic is
4 20010.052
1.95
z 0
120010.05210.952
8. Conclusions: Since z 0 1.95 z 0.05 1.645, we reject H and conclude that the
0
process fraction defective p is less than 0.05. The P-value for this value of the test statistic
z is P 0.0256, which is less than 0.05. We conclude that the process is capable.
0