### Class examples from Week 5: Probability and the Normal Curve
### Ken Benoit Feb 2010


## Tossing two coins examples
rbinom(100,2,.5)    # this randomly generates 100 tosses of 2 coins
                    # where Pr(outcome=1) is 0.5

# Table of proportions of 10 cosses of 2 coins with Pr=0.5
table(rbinom(10,2,.5))/10 

# Table of proportions of 1000 cosses of 2 coins with Pr=0.5
table(rbinom(1000,2,.5))/1000 

# Histogram of the outcomes of tosses, N=50
barplot(table(rbinom(50,2,.5))/50)

# Histogram of the outcomes of tosses, N=1000
barplot(table(rbinom(1000,2,.5))/1000)

# Means of frequency distributions of 2 coin tosses
mean(rbinom(10,2,.5))
mean(rbinom(100,2,.5))
mean(rbinom(1000,2,.5))
mean(rbinom(10000,2,.5))

# compute some areas under the normal curve
1-pnorm(1.4) 
round((1-pnorm(1.4))*100, 2) 
.50-pnorm(1.33) 
1-pnorm(1.33)
pnorm(1.33) - pnorm(-1.33)

# plot a normal curve
x <- seq(-5, 5, 0.1)         # numeric sequence every .1 from 5 to 5
plot(x, dnorm(x), type="l")  # plot (standard) normal function for x

# plot a normal curve with shaded areas -1.33 and +1.33
x <- seq(-4,4,.1)
plot(x, dnorm(x), type="n", xaxs="i", yaxs="i", ylim=c(0,.4), bty="l",
     xaxt="n")
polysection <- function(a,b,dist=dnorm,col="blue",n=11) {
  dx <- seq(a,b,length.out=n)
  polygon(c(a,dx,b),c(0,dist(dx),0),border=NA,col=col)
}
polysection(-5,-1.33, col="gray")
polysection(1.33, 5, col="gray")
axis(1, c(-5,-1.33, 0, 1.33,5))
lines(x,dnorm(x))

# simple example of how to standardize scores
x <- c(1,4, 5, 7, 14, 0, 21)
(stdx <- (x - mean(x)) / sd(x))  # the long way
scale(x)                         # short-cut way

# check normality with a Q-Q plot
load("dail2002.Rdata")
attach(dail2002)
qqnorm(spend_total)
qqline(spend_total)
detach(dail)