Question

Let X have a p.d.f. of the form:
f(x,θ) = { 1/θ e^-x/θ ,0<x<∞,θ>0
0 ,elsewhere.

To test H0 : θ = 2 against H1:θ =1,use the random sample x1,x2 of size 2 and define a critical region:W = {(x1,x2): 9.5≤x1+x2}.
Find i) Power of the test. ii) Significance level of the test.

Answer 1

Suppose $x_1, x_2$ is our sample of size 2.

The joint p. d. f. of $x_1 \text { and } x_2$ is $f(x_1,x_2,\theta)=f(x_1,\theta)f(x_2,\theta),\text{ that is }(\frac{1}{\theta})^2e^{-\frac{x_1+x_2} {\theta}} \text{ if } 0<x_1,x_2<\infty \text{ and }0 \text{ otherwise}.\\ \alpha=P(W;\theta=2) \text{ --- significance level of the test (Type I error).}\\ \text{W is the critical region}, H_0: \theta=2.$

$\gamma_w(2)=P(x_1+x_2\geq 9.5;2)=1-P(x_1+x_2\leq 9.5;2)=\\ =\int_0^{9.5}\int_0^{9.5-x_2}\frac{1}{4}e^{-\frac{x_1+x_2} {2}}dx_1dx_2=0.05.\\ ii) \alpha=0.05 - \text{significance level of the test}.$

$i) 1-\beta=\int_0^{9.5}\int_0^{9.5-x_2}e^{-(x_1+x_2)}dx_1dx_2=1-10.5e^{-9.5}=0.99 - \text{power of the test }(\beta\text{ is Type II error}).$