Let X1,..., Xn be independently distributed with density (2)1 ex/2, x 0, and let Y1

Let X1,..., Xn be independently distributed with density (2θ)−1 e−x/2θ, x ≥ 0, and let Y1 ≤···≤ Yn be the ordered X’s. Assume that Y1 becomes available first, then Y2, and so on, and that observation is continued until Yr has been observed. On the basis of Y1,..., Yr it is desired to test H : θ ≥ θ0 = 1000 at level

α = 0.05 against θ < θ0.

(i) Determine the rejection region when r = 4, and find the power of the test against

θ1 = 500.

(ii) Find the value of r required to get power β ≥ 0.95 against the alternative.

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