Question: Let N have the binomial distribution based on 10 trials with success probability p. Given N = n, let X1, , Xn be i.i.d.

Let N have the binomial distribution based on 10 trials with success probability p. Given N = n, let X1, ··· , Xn be i.i.d. normal with mean θ and variance one. The data consists of (N, X1, ··· , X N ).

(i). If p has a known value p0, show there does not exist a UMP test of θ = 0 versus

θ > 0. [In fact, a UMPU test does not exist either.]

(ii). If p is unknown (taking values in (0,1)), find a UMPU test of θ = 0 versus θ > 0.

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