Theory and Methods of Statistics

6.7.3 Examples

Example 6.7.1

Let $\mathit{X}=\left(X_1,\dots ,X_n\right)$ be a random sample from $N\left(\mu ,{\sigma }^2\right)$ with σ known. The goal is to find UMP unbiased level α tests for (a) H₀: μ = μ₀ vs H₁: μ≠μ₀, and (b) H₀: μ₁ ≤ μ ≤ μ₂ vs $H_1:\mu \notin \left[{\mu }_1,{\mu }_2\right].$

Solution

(a) Denote $T=\sqrt{n}(\stackrel{-}{X}-{\mu }_0)/\sigma$. Then the UMP unbiased level α test is

$\begin{array}{l}\hfill \phi (\mathbf{x})=\left\{\begin{array}{ll}0& \text{if}|T|\le {\Phi }^{-1}(1-\alpha /2)\\ 1& \text{if}|T|>{\Phi }^{-1}(1-\alpha /2),\end{array}\right.\end{array}$

because $\stackrel{-}{X}$ is sufficient, ...