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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
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