# 12PowerFTest - POWER OF THE F-TEST Suppose C is a q p...

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POWER OF THE F-TEST Suppose C is a q × p matrix such that C β is testable. We have established that F = ( C ˆ β - d ) 0 [ C ( X 0 X ) - C 0 ] - 1 ( C ˆ β - d ) / q ˆ σ 2 F q , n - r ( δ 2 ) where δ 2 = ( C β - d ) 0 [ C ( X 0 X ) - C 0 ] - 1 ( C β - d ) σ 2 . Copyright c 2012 (Iowa State University) Statistics 511 1 / 10

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Let F q , n - r , 1 - α denote the 1 - α quantile of the central F distribution with q and n - r d.f. The power of the significance level α test of H 0 : C β = d is given by P ( F F q , n - r , 1 - α ) . The power is an increasing function of the noncentrality parameter δ 2 . Copyright c 2012 (Iowa State University) Statistics 511 2 / 10

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x=seq(0,15,by=.01) y1=df(x,3,26) y2=df(x,3,26,ncp=10) plot(c(x,x),c(y1,y2),pch=" ", xlab="x",ylab="Density", main="A Comparison of Central and Non-Central F Densities") lines(x,y1,col="blue",lwd=2) lines(x,y2,lty=2,col="red",lwd=2) legend(6,.6, c(expression(paste(F[list(3,26)]," density")), expression(paste(F[list(3,26)](10)," density"))), lty=1:2, col=c("blue","red"), lwd=2)
qf(.95,3,26) 2.975154 1-pf(qf(.95,3,26),3,26,10) 0.6895487

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plot(c(x,x),c(y1,y2),pch=" ", xlab="x",ylab="Density", main="A Comparison of Central and Non-Central F Densities")

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