## Power function for Example 2, Section 2.4 notes:

# Decision rule is:  Reject H0 if T <= 6

possible.ps <- seq(0,1,by=.01)
power.for.p <- pbinom(6, size=20, prob=possible.ps)

plot(possible.ps, power.for.p, type='l', xlab='p', ylab='power', main="Power Function, Example 2")

# Significance level:

power.for.p[possible.ps==.5]

#########################################################

## Power function for Example 4(a), Section 2.4 notes:

# Decision rule is:  Reject H0 if  (xbar-5)/sqrt(1/100) > 1.645
# that is, Reject H0 if  xbar > 5.1645

possible.mus <- seq(4,6,by=.02)
power.for.mu <- pnorm(5.1645, mean=possible.mus, sd=sqrt(1/100), lower=F)

plot(possible.mus, power.for.mu, type='l', xlab='mu', ylab='power', main="Power Function, Example 4(a)")

# Significance level:

power.for.mu[possible.mus==5]


#########################################################

## Power function for Example 4(b), Section 2.4 notes:

# Decision rule is:  Reject H0 if  (xbar-5)/sqrt(1/100) > 1.96 or (xbar-5)/sqrt(1/100) < -1.96
# that is, Reject H0 if  xbar > 5.196 or xbar < 4.804

possible.mus <- seq(4,6,by=.02)
power.for.mu <- pnorm(5.196, mean=possible.mus, sd=sqrt(1/100), lower=F) + pnorm(4.804, mean=possible.mus, sd=sqrt(1/100), lower=T)

plot(possible.mus, power.for.mu, type='l', xlab='mu', ylab='power', main="Power Function, Example 4(b)")

# Significance level:

power.for.mu[possible.mus==5]

#########################################################

## Power functions for Example 4(a), Section 2.4 notes, FOR SEVERAL VALUES OF n:

# Decision rule is:  Reject H0 if  (xbar-5)/sqrt(1/n) > 1.645
# that is, Reject H0 if  xbar > 1.645*sqrt(1/n) + 5

possible.mus <- seq(4.5,5.5,by=.02)
plot(possible.mus, seq(0,1,length=length(possible.mus)), xlab='mu', ylab='power', main="Power Function, Example 4(a)", type='n')

for (n in c(5,10,20,50,100)){
power.for.mu <- pnorm(1.645*sqrt(1/n) + 5, mean=possible.mus, sd=sqrt(1/n), lower=F)
lines(possible.mus, power.for.mu, lwd=n/10)
}

# Lines are thicker for the larger values of n