When a thread-cutting machine is operating properly, only 2% of the units produc

Question

When a thread-cutting machine is operating properly, only 2% of the units produced are defective. Since the machine was bumped by a forklift truck, however, the quality-control manager is concerned that it may require expensive downtime and a re-adjustment. The manager would like to set up a right-tailed test at the 0.01 level of significance to identify the proportion of defectives using a sample of 400 units produced. (a) State the null and alternative hypotheses. (b) If 12 of the 400 units produced are defective, what conclusion would the quality-control manager make? (c) Find the probability of type II error if the actual population proportion of defectives is (i) 0.02? (ii) 0.03? (iii) 0.04? (iv) 0.05? (v) 0.06? (d) When the probability of type II error (vertical axis) is plotted against a range of population parameter values for which the null hypothesis would be false (horizontal axis), and all the points are connected by a smooth curve, the result is known as the operating characteristic curve, or simply OC curve. Operating characteristic curves are widely used in industrial applications to provide a visual display of the merits of the test criterion. Making use of the probabilities calculated in part (c), plot the operating characteristic curve for the test.

Explanation / Answer

a) Ho:p=0.02

Ha:p is not equal to 0.02

b) Normal approximation to the Binomial distribution:

Mean=np=400*0.02=8

Standard deviation=sqrt(Var)=sqrt(np*(1-p))=sqrt(8*0.98)=sqrt(7.84)=2.8

z=(12-8)/2.8=1.4286

Two tailed p value=0.1531( at z=1.4286)

p-value of 0.1531>0.01,hence we can not reject the null hypothesis that p=0.02 or we accept the null hypotheis and we reject the alternate hypothesis(that p is not equal to 0.02).

Here,we know that null hypothesis is false,but we fail to reject it at significance level of 0.01,so we make type 2 error .

c) The type 2 error with significance level 0.01 will be equal to

=1-0.000069=0.999931

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