A shipment of 20 projection screen television sets is at the receiving dock of a large department store. The department store has a policy of not accepting shipments that contain more than 10 percent defective merchandise. The receiving clerk is instructed to have a quality inspection done on two sets that are randomly drawn, without replacement, from the shipment. Letting \(k\) represent the unknown number of defective sets in the shipment of 20 sets, the null hypothesis \(H_{0}: k \leq 2\) will be rejected iff both sets that are inspected are found to be defective.

(a) Calculate the probabilities of committing type I errors when \(k=0,1\), or 2 .

(b) Calculate the probabilities of committing type II errors when \(k \geq 3\).

(c) Plot the power function of this testing procedure. Interpret the implications of the power function from the standpoint of both the department store and the television manufacturer.