A distributor receives a large shipment of components. The distributor would like to accept the shipment if \(10 \%\) or fewer of the components are defective and to return it if more than \(10 \%\) of the components are defective. The distributor decides to sample 10 components and will return the shipment if more than 1 of the 10 is defective.

a. If the proportion of defectives in the batch is in fact \(10 \%\), what is the probability that the shipment will be returned?

b. If the proportion of defectives in the batch is \(20 \%\), what is the probability that the shipment will be returned?

c. If the proportion of defectives in the batch is \(2 \%\), what is the probability that the shipment will be returned?

d. The distributor decides that the shipment will be accepted only if none of the sampled items are defective. What is the minimum number of items that should be sampled in order to have a probability no greater than 0.01 of accepting the shipment if \(20 \%\) of the components in the shipment are defective?