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# Power Functions

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We say a critical region C is of size &#61537; if &#61537;= max P&#61553;{(X1,....,Xn) &#61646; C], &#61553;&#61646;&#61559;0
We define the power function of a critical region to be &#61543;c(&#61553;)= P&#61553;[(X1,...Xn) &#61646;C], &#61553;&#61646;&#61559;1

1)Let X have a pdf of the from f(x; &#61553;)= &#61553;x^(&#61553;-1), 0<x<1, 0 elsewhere, where &#61553;&#61646; {&#61553;:&#61553; =1,2}. To test the simple hypothesis H0: &#61553;=1 against the alternative simple hypothesis H1: &#61553;=2, use a random sample X1, X2 of size n=2 and define the critical region to be C= {(x1,x2): Â¾ <= x1x2}. Find the power function of the test.

https://brainmass.com/statistics/hypothesis-testing/power-functions-32779

#### Solution Summary

We say a critical region C is of size a if = max P0{(X1,....,Xn) E C], &#61553;&#61646;&#61559;0
We define the power function of a critical region to be &#61543;c(&#61553;)= P&#61553;[(X1,...Xn) &#61646;C], &#61553;&#61646;&#61559;1

1)Let X have a pdf of the from f(x; &#61553;)= &#61553;x^(&#61553;-1), 0<x<1, 0 elsewhere, where &#61553;&#61646; {&#61553;:&#61553; =1,2}. To test the simple hypothesis H0: &#61553;=1 against the alternative simple hypothesis H1: &#61553;=2, use a random sample X1, X2 of size n=2 and define the critical region to be C= {(x1,x2): Â¾ <= x1x2}. Find the power function of the test.

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