In this exercise we will investigate some more properties of binomial confidence sets and the Sterne (1954) construction in particular. As in Example 9.2.11, we will again consider the binomial(3,p) distribution.€ƒ
a. Draw, as a function of p, a graph of the four probability functions Pp(X = x), x = 0,..., 3. Identify the maxima of Pp(X = 1) and Pp(X = 2).
b. Show that for small ˆˆ, Pp(X = 0) > Pp(X = 2) for p = 1/3 + ˆˆ.
c. Show that the most probable construction is to blame for the difficulties with the Sterne sets by showing that the following acceptance regions can be inverted to obtain a 1 - α = .442 confidence interval.

(This is essentially Crow's 1956 modification of Sterne's construction; see Miscellanea 9.5.2.)

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