##### Calculating Power Functions:

## Example 1 from Section 10.4:  Test Ho: mu = 500 vs. Ha: mu > 500


# Rejection rule was:  Reject H0 if  (ybar-500)/(97.4/sqrt(80)) > 1.645
# that is, Reject H0 if  ybar > 517.91

possible.mus <- seq(475,575,by=.2)
power.for.mu <- pnorm(517.91, mean=possible.mus, sd=97.4/sqrt(80), lower=F)

plot(possible.mus, power.for.mu, type='l', xlab='mu', ylab='power', main="Power Function, Example 1")

# Significance level:

power.for.mu[possible.mus==500]

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

## Example 1 (altered) from Section 10.4:  Test Ho: mu = 500 vs. Ha: mu =/= 500

# Rejection rule is:  Reject H0 if  (ybar-500)/(97.4/sqrt(80)) > 1.96 or (ybar-500)/(97.4/sqrt(80)) < -1.96
# that is, Reject H0 if  xbar > 521.34 or xbar < 478.66

possible.mus <- seq(440,560,by=.2)
power.for.mu <- pnorm(521.34, mean=possible.mus, sd=97.4/sqrt(80), lower=F) + pnorm(478.66, mean=possible.mus, sd=97.4/sqrt(80), 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==500]

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

# Note the effect of the sample size on the power functions:

## Power functions for an example z-test, FOR SEVERAL VALUES OF n:

# Suppose we 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)), type='n', xlab='mu', ylab='power', main="Power Functions for varying 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)
}
abline(h=.05,lty=3)

# Lines are thicker for the larger values of n