Self-Check Questions (ungraded)

We use the t-test for H0: \(\mu\)= -2 versus HA: \(\mu\)< -2 at \(\alpha\)=0.05 level (reject when p-value<0.05). The sample size is n=10. We check the real Type I error rate of this test, when data comes from N(mean= -2, sd=3) distribution, using 10,000 Monte Carlo simulation runs. Write the R commands for the simulation below.

x.sim<-matrix(rnorm(10000*10, mean=-2,sd=3), ncol=10)
p.sim<-apply(x.sim, 1, function(x) t.test(x, mu= -2, alternative="less")$p.value)
mean(p.sim<0.05)

We use the t-test for H0: \(\mu\)= -2 versus HA: \(\mu\)> -2 at \(\alpha\)=0.05 level. We wish to know the power of this test when data comes from N(mean=1, sd=3) distribution with a sample size n=10. Write the R command that calculate this power.

power.t.test(n=10,delta=(1-(-2))/3,sig.level=0.05,  type='one.sample', alternative ='one.sided')