Question: Let \(X\) have a Poisson distribution with mean \(\theta\). Consider the simple hypothesis \(H_{0}: \theta=\frac{1}{2}\) and the alternative hypothesis \(H_{A}: \theta<\frac{1}{2} .\) Thus \(\Omega=\left\{\theta: 0<\theta \leq \frac{1}{2}\right\}\). Let

\(X_{1}, X_{2}, \ldots, X_{12}\) denote a random sample of size 12 from this distribution. We reject \(H_{0}\) if and only if the observed value of \(Y=X_{1}+X_{2}+\ldots+X_{12} \leq 2\). If \(\beta(\theta)\) is the power function of the test, find the powers \(\beta\left(\frac{1}{2}\right), \beta\left(\frac{1}{3}\right), \beta\left(\frac{1}{4}\right), \beta\left(\frac{1}{6}\right)\), and \(\beta\left(\frac{1}{12}\right)\). Sketch the graph of \(\beta(\theta)\). What is the significance level of the test?

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