In Exercise 34.9 we met an apple-grower who treated his apples to reduce the skin disorder called scald. His apples developed scald with probability 1/5 if treated, but with probability 3/5 if untreated. Treatment could not be observed directly, but we were able to observe the number X of apples that developed scald in a batch of n. We found that X ∼ Binomial 

n, 3 − 2p 5



,where p was the proportion of apples that were successfully treated. The maximum likelihood estimator of p based on X was, pb=

3n − 5X 2n

.

Suppose now that a chemical test has become available that can determine directly for each apple whether it has been successfully treated or not. We apply this direct test to the same n apples. Let Y be the number of these n apples that the test determines were successfully treated.

a. What is the distribution of Y, in terms of the unknown parameter p?

b. Define bs to be the maximum likelihood estimator of p based on the results Y of the direct screening test. What is bs?

c. Find the expectation of each estimator, pband bs. Are both estimators unbiased for p?

d. Show that the variance of the original, indirect estimator, pb, is Var( pb) =

(3 − 2p)(1 + p)

2n

.

e. What is the variance of the direct screening estimator, Var(bs)?

f. By comparing Var( pb) with Var(bs), decide which of pband bs is the better estimator. Explain all your reasoning.

g. The original observations on 200 apples showed that 94 of them had the scald disorder. The new direct screening test determines that 52 apples were successfully treated.

Find pbandbs. Is thebs estimate necessarily closer to the true value of p? Explain your answer.