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Some have argued that the F test for equality of variances should be used to decide whether to use the pooled t-test or the non-pooled t-test. Others have argued that the variances should never be assumed to be exactly equal, and the non-pooled t-test should always be used. You will explore this here via simulation. We will consider testing the following hypotheses (using either a pooled or a non-pooled t-test) at the 5% level. H0: µ1 - µ2 = 0 H1: µ1 - µ2 ¹ 0 Assume that we take independent simple random samples of size 20 from each of two populations. The populations are N(0,12) and a N(µ,52). Use a simulation-based method to plot the power curves for the pooled and non-pooled t-tests (on the same graph) as a function of a sequence of µ values from -5 to 5. mu.vals <- seq(-5,5,by=.25) nsim <- 10000 reject.mat.pooled <- matrix(NA,length(mu.vals),nsim) reject.mat.nonpooled <- matrix(NA,length(mu.vals),nsim) for (i in 1:length(mu.vals)){ sim.mat1 <- matrix(rnorm(nsim*20,mean=0,sd=1),nrow=nsim,ncol=20) sim.mat2 <- matrix(rnorm(nsim*20,mean=mu.vals[i],sd=5),nrow=nsim,ncol=20) reject.mat.pooled[i,] <- unlist(lapply(1:nsim,FUN=function(x){ pval1<-t.test(sim.mat1[x,],sim.mat2[x,],alt="two.sided",var.equal=TRUE)$p.value;ifelse(pval1<.05,1,0)})) reject.mat.nonpooled[i,] <- unlist(lapply(1:nsim,FUN=function(x){ pval1<-t.test(sim.mat1[x,],sim.mat2[x,],alt="two.sided",var.equal=FALSE)$p.value;ifelse(pval1<.05,1,0)})) } power.vals.pooled <- rowMeans(reject.mat.pooled) power.vals.nonpooled <- rowMeans(reject.mat.nonpooled) plot(c(-5,5),c(0,1),type="n",xlab="True Difference in Means",ylab="Approx Prob of Rejecting") lines(mu.vals,power.vals.pooled) lines(mu.vals,power.vals.nonpooled,col="blue",lty=2) abline(h=.05,lty=2,col="red") 1. Which test would you recommend using, the pooled or non-pooled test? Why? Base your answer on the power curves plotted above? 2. Now, assume that the two populations are N(0,12 ) and a N(µ,12). a) Use a simulation-based method to plot the power curves for the pooled and non-pooled t-tests (on the same graph) as a function of a sequence of µ values from -5 to 5. You may want to use portions of the above R code. b) When the populations are N(0,12 ) and a N(µ,12 ), which test would you recommend using, the pooled or non-pooled test? Why? Base your answer on the power curves plotted above? 3. Now, assume that the two populations are N(0,12) and a N(µ,102). a) Use a simulation-based method to plot the power curves for the pooled and non-pooled t-tests (on the same graph) as a function of a sequence of µ values from -5 to 5. You may want to use portions of the above R code. b) When the populations are N(0,12 ) and a N(µ,102 ), which test would you recommend using, the pooled or non-pooled test? Why? Base your answer on the power curves plotted above? 4. If you have no idea whether the variances are equal or not, which test do you recommend based on your answers for 1-3? Explain your answer.

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