Math 6010 • Linear Models

Let \(X\) be discrete taking values in \(\{0,1\}\) and have the pdf \(f(x;\theta )=\theta ^x(1-\theta )^{1-x}\) for \(x\in \{0,1\}\). We test the simple hypothesis \(H_0:\theta =1/4\) agains the alternative composite hypothesis \(H_A:\theta <1/4\) by taking a random sample of size 10 and rejecting \(H_0\) if and only if the observed values \(x_1,\ldots ,x_{10}\) satisfy \[ \sum _{i=1}^{10}x_i\leq 1. \] Find the power function \(K(\theta )\) of this test for \(0\leq \theta \leq 1/4\).