Question

Random sample of size n=19 is taken from a Normal population, sample mean is 11.5895, standard deviation is 1.0883.

1. At the 2% level, test whether it is reasonable to believe that the true population variance is larger than 1.

Using the scenario from above, do the following

2. Derive the power function. Show your work.
3. Using R, graph the power function for 0.5 < sigma^2 < 3.5.
4. Pretend that the sample size was actually 56. Plot this power function on the same graph.
5. Based on the power functions graphed in c, why is the test described in part c "better" than the original test? Explain your answer using the power functions.

Answer 1

4.

R code:

sigma=seq(0.5,3.5,by=0.01)
n1=19
n2=56
p1=1-pchisq(qchisq(0.98,n1-1)/sigma,n1-1)
p2=1-pchisq(qchisq(0.98,n2-1)/sigma,n2-1)
plot(sigma,p1,lwd=2,type="l",xlab=expression(sigma^2),ylab=expression(beta(sigma^2)),main="Power Curve",ylim=c(0,1.1))
lines(sigma,p2,lwd=2,type="l",col=2)
text(2.0,0.49,expression(n==19))
text(2.0,0.88,expression(n==56))

5. Since Power for n=56 is larger than power for n=19, hence the test based on n=56 is better than the test based on n=19.