Suppose that we have 9 independent observations from a normal distribution with

Question

Suppose that we have 9 independent observations from a normal distribution with standard deviation 10. We wish to test H_0: mu = 150 vs. H_A: mu notequalto 150 The LRT with level alpha = 0.05 has the rejection region {x^OverBar 156.5}. Calculate the power of this test against the alternative mu = 151. Calculate the power of this test against the alternative mu = 123. Alternatively, consider the test with another rejection region {x^OverBar 160.9}. Calculate the level of this second test. Calculate the power against the alternatives mu = 151. Calculate the power of this test against the alternative mu =123. Which of the two tests do you think is better? Why?

Explanation / Answer

a) Calculate the power against, the alternatives µ = 151

=10

n=9

P( <143.5 or >156.6) = Area on left of 143.5 or on right of 156.5

       Z= (Xbat -µ) /( /sqrt(n) )

Z1=(143.5-151)/(10/sqrt(9) )

    =-2.250

Z2=(156.5-151) /(10/ sqrt(9))

    =1.650

P( <143.5 or >156.6) =P(Z<-2.250 or Z>1.650)

                                              =P(Z<-2.250)+P(Z>1.650)

                                                =P(Z<-2.250)+1- P(Z>1.650)

                                               =0 .0122 +1-0 .9505

                                              =0.0617

                                               =6.17%

  

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