Solution
Power of a One-Sided Uniform Test
Show the problem again
Let $X_1,\dots,X_4$ be independent and uniform on $[0,\theta]$. To test $H_0:\theta=1$ against $H_1:\theta<1$, reject $H_0$ when $\max_i X_i\le 1/2$. The size of this test is $1/16$. What is the power of the test when the true value is $\theta=3/4$? Give a reduced fraction.
Worked solution
Under $\theta$, $P(\max\le t)=(t/\theta)^4$ for $0\le t\le\theta$. Under $H_0$ and $t=1/2$ this is $1/16$, confirming the size. At $\theta=3/4$ it is $\left(\frac{1/2}{3/4}\right)^4=(2/3)^4=16/81\approx0.198$. The type II error is therefore $65/81$: even a 25% shrinkage of the range is detected only about one time in five.
Source: Original problem written for AxiomIQ.