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...

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:m₁-m₂= 0

H₁:m₁-m₂¹0

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(m,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 ofmvalues from -5 to 5.

Use the following R code to plot the power functions (based on simulation) for this test when conditioning on the true difference in means.

mu.vals

nsim

reject.mat.pooled

reject.mat.nonpooled

for (i in 1:length(mu.vals)){

sim.mat1

sim.mat2

reject.mat.pooled[i,]

pval1

reject.mat.nonpooled[i,]

pval1

}

power.vals.pooled

power.vals.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")

2b) In this scenario, which test would you recommend using, the pooled or non-pooled test? Justify your answer and base your answer on the power curves plotted above?

2c) Now, assume that the two populations are N(0,1²) and a N(m,1²). 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 ofmvalues from -5 to 5. You will want to modify the above R code.

2d) When the populations are N(0,1²) and a N(m,1²), which test would you recommend using, the pooled or non-pooledP test? Why? Base your answer on the power curves plotted above?