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Suppose that a single observation $X$ is to be taken from uniform distribution $[-\theta,\theta]$, it is desired to test the following hypotheses: $H_0:\theta=3,H_1:\theta=4$. Rejects $H_0$, if $|X|\geq c$. The level of significant $\alpha=0.2$ . Find the probability of type II error.

Whether I should use c.d.f to calculate the value of $c$ first? As $\dfrac{\alpha}{2}=1-\dfrac{c+3}{6},c=2.4$, and just calculate $\beta=\dfrac{2.4+4}8-\dfrac{-2.4+4}8=0.6$? Is it correct?

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Note that the critical region is $W=\{{x: |x| \geq c}\}$.

So the power function of the test is $$\begin{align}\beta(\theta) &=P[|X| \geq c|\theta] \\ &=1-\frac{c}{\theta}\end{align}$$

So Probability of Type I Error $=P[X\in W|\theta=3]=1-\frac{c}{3}$

So Probability of Type II Error $=P[X\in W'|\theta=4]=1-P[X\in W|\theta=4]=\frac{c}{4}$

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  • $\begingroup$ thank you.your expression is formal and clear logic. $\endgroup$ – nina li Jan 24 '13 at 13:41
  • $\begingroup$ @ninali : WELCOME $\endgroup$ – A.D Jan 24 '13 at 16:25

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