# 2.b
n <- 3
a.sample <- sample( exercise2data$height , size=n ) 
likelihood.normal <- function( x ) sapply(x , function(z)
prod( dnorm( a.sample , mean=z , sd=sd(a.sample) ) )) 
curve( likelihood.normal(x) , from=60 , to=75 ) 
lines( c( mean(a.sample) , mean(a.sample) ) , c(0,1) , col="red" ) 
points( a.sample , rep(0,n) )

# 2.d
n <- 10 
sample.means <- sapply( rep(n,10000) , function(x)
mean( sample( exercise2data$height , size=x ) ) ) 
hist( sample.means , breaks=50 ) 
lines( c( mean(exercise2data$height) , mean(exercise2data$height) ) , c(0,5000) ,
col="red" )

# 3.a

# sample 
sample.means <- {} 
sample.ci.hi <- {} 
sample.ci.lo <- {} 
n <- 100 
for ( i in 1:n ) {
    s <- sample( exercise2data$h , size=5 ) 
    sample.means[i] <- mean(s) 
    sample.ci <- confint.default( lm( s ~ 1 ) , level=0.95 ) 
    sample.ci.hi[i] <- sample.ci[2] 
    sample.ci.lo[i] <- sample.ci[1]
} 
# count instances of true mean being inside intervals 
is.inside <- function( true.mean , lo , hi ) {
    ifelse( true.mean>lo & true.mean<hi , TRUE , FALSE )
} 
count.inside <- 0 
for ( i in 1:n ) {
    if ( is.inside( mean(h) , sample.ci.lo[i] , sample.ci.hi[i] ) ) { 
        count.inside <- count.inside + 1
    }
} 
p.inside <- count.inside / n 
# plot sample means and confidence intervals 
plot( 1:n ~ sample.means , xlim=c( min( sample.ci.lo ) , max( sample.ci.hi ) ) ,
ylab="sample number" , xlab="sample mean" ,
main=paste(p.inside,"of confidence intervals contain true mean") ) 
for ( i in 1:n ) {
    lines( c( sample.ci.hi[i] , sample.ci.lo[i] ) , c( i , i ) , col=ifelse( is.inside( mean(h) , sample.ci.lo[i] , sample.ci.hi[i] ) , "black" , "red" ) )
} 
# plot true mean 
lines( c( mean(h) , mean(h) ) , c(1,n) , col="red" )