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Statistics Question on Testing of Hypotheses

Let X1,X2,…,XnX_1, X_2, \ldots, X_n be a random sample from an Exp(λ)\text{Exp}(\lambda) distribution, where λ∈1,2\lambda \in \\{1, 2\\}. For testing H0:λ=1H_0: \lambda = 1 against H1:λ=2H_1: \lambda = 2, the most powerful test of size α\alpha, α∈(0,1)\alpha \in (0,1), will reject H0H_0 if and only if

A

∑i=1nXi≤12χ2n,1−α2\sum_{i=1}^{n} X_i \leq \frac{1}{2} \chi^2_{2n,1-\alpha}

B

∑i=1nXi≥2χ2n,1−α2\sum_{i=1}^{n} X_i \geq 2 \chi^2_{2n,1-\alpha}

C

∑i=1nXi≤12χn,1−α2\sum_{i=1}^{n} X_i \leq \frac{1}{2} \chi^2_{n,1-\alpha}

D

∑i=1nXi≥2χn,1−α2\sum_{i=1}^{n} X_i \geq 2 \chi^2_{n,1-\alpha}

Answer

∑i=1nXi≤12χ2n,1−α2\sum_{i=1}^{n} X_i \leq \frac{1}{2} \chi^2_{2n,1-\alpha}

Explanation

Solution

The correct option is (A): ∑i=1nXi≤12χ2n,1−α2\sum_{i=1}^{n} X_i \leq \frac{1}{2} \chi^2_{2n,1-\alpha}