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?



Oct 05, 2021
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