Question

A semiconductor manufacturer produces controllers used in automobile engine applications. The customer required that the process fallout or fraction defective at a critical manufacturing step not exceed 0.05 and that the manufacturer demonstrate process capacity at this level of quality using α = 0.05. The semiconductor manufacturer takes a random sample of 200 devices and finds that four of them are defective. Can tha manufaturer demonstrate process capability for the customer?

Answer 1

 We need to test $H_0:p=0.05$ against $H_1:p<0.05.$ The test statistic is 

$$z={X-np\over \sqrt{np(1-p)}}$$

Given $n=200, X=4$

$$z={4-200(0.05)\over \sqrt{200(0.05)(1-0.05)}}\approx-1.946657$$

The critical value is $z_c=z_{1-\alpha}=-1.645.$ Rhe rejection region is $z<-1.645.$

Since $z=-1.946657<-1.645,$ we reject $H_0,$ and we can conclude that there is enough evidence to support the fact that the process fallout does not exceed.

Note that p-value $P(Z\leq-1.946657)=0.025788.$  Since p − value $=0.025788<0.05=\alpha,$ we reject $H_0,$ and we can conclude that there is enough evidence to support the fact that the process fallout does not exceed.