## R Lab Session 2, 10/3/08 ##

## Review: Boxplots

some.data <- c(13,10,7,7,12,13,12.5,7,11,11,13,13,12,13,10,12,13,8,11.5,12,11.5,14,5.5)
boxplot(some.data, ylab="Homework Scores", main="Boxplot of Homework Scores")

## Review: Normal probabilities
# ACT Test; Mean = 21.0, SD = 4.7 [range: 1-36]

# Probability of scoring between 25 and 32?
pnorm(25, mean=21, sd=4.7) - pnorm(32, mean=21, sd=4.7, lower.tail=F)
pnorm(25, mean=21, sd=4.7) - pnorm(-((32-21)/4.7), mean=0, sd=1)

# Probability of scoring over 30?
pnorm(30, mean=21, sd=4.7, lower.tail=F)
pnorm(-((30-21)/4.7), mean=0, sd=1)

# Probability of scoring either below 17 or above 33?
pnorm(17, mean=21, sd=4.7) + pnorm(33, mean=21, sd=4.7, lower.tail=F)
pnorm(17, mean=21, sd=4.7) + pnorm(-((33-21)/4.7), mean=0, sd=1)

## Independent Events

# Coin Toss, 3 heads in a row?

0.5 * 0.5 * 0.5

## Box Models

# SAT => +1 point for getting a question right; -1/4 point for getting a question wrong (or fail to answer*).  Multiple choice questions have 4 answers, one of which is right.  100 questions. EV?  SE of EV?  Pr(>= 10 questions right)?

box.input <- c(1, rep(-0.25, 3))
draws <- 100

G <- 10000
holder <- rep(NA, G)

for(g in 1:G){
	holder[g] <- sum(sample(box.input, draws, replace=T))
}

mean(holder)
sd(holder)
sum(holder >= 10)/G

(1 + (-0.25*3))/4 
= 0.0625

0.0625 * 100 
= 6.25

sqrt((1^2 + 3*((-0.25)^2))/4) * sqrt(100)
= 5.45