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Q40Section B (Section B)40 of 60

Values of the sample total at which the UMP test rejects

Let X1,X2,X3,X4X_1,X_2,X_3,X_4 be a random sample from an N(θ,1)N(\theta,1) distribution, where θ∈(−∞,∞)\theta\in(-\infty,\infty). Suppose the null hypothesis H0:θ=1H_0:\theta=1 is to be tested against the hypothesis H1:θ<1H_1:\theta<1 at α=0.05\alpha=0.05 level of significance. For what observed values of ∑i=14Xi\sum_{i=1}^4X_i, the uniformly most powerful test would reject H0H_0?

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