# S-PLUS hints for assignment due on March 25
#
# Problem 1, Exercise 9.2.10

# (a) The menu options Statistics:Compare Samples:One Sample:t test
#     opens a dialog box to fill in to do a t test.

# (b) To do a sign test for variable conc in data frame prob1
#     with the null hypothesis that mu = 2.6
#     versus the alternative that mu > 2.6, you want to count the number
#     of observations greater than 2.6 and find the sum of binomial probabilities
#     with n equal to the sample size and p = 0.5 greater than or equal to this
#     count.  A sequence of S-PLUS commands that do this as follows.

# this allows you to refer to variables in prob1 by name
attach(prob1)

# create a vector of signs
x <- sign(conc - 2.6)

# count the positive signs
count <- sum(x == 1)

# find the sum of the binomial probabilities
n <- length(x)
sum(dbinom(count:n,n,0.5))
# or, alternatively
1 - pbinom(count-1,n,0.5)

# A short command to do all of this in one command is
1 - pbinom(sum(sign(prob1$conc-2.6)==1)-1,length(prob1$conc),0.5)

# (c) The menu options Statistics:Compare Samples:One Sample:Wilcoxon signed rank test
#     opens a dialog box to fill in to do a nonparametric Wilcoxon test.

# You could mimic this in steps by doing this.

attach(prob1)
s <- sign(conc - 2.6)
r <- rank(abs(conc - 2.6))
w <- sum(s*r)

wilcox.test(conc,mu=2.6,alternative="greater",exact=T,correct=T)

# Problem 2, data from Exercise 9.4.6

# (a) The menu options Statistics:Compare Samples:Two Samples:t test
#     opens a dialog box to fill in to do a t test.

# For this data, there is a grouping variable (esp).

# (c) The menu options Statistics:Compare Samples:Two Samples:Wilcoxon rank test
#     opens a dialog box to fill in to do a nonparametric Wilcoxon test.

# You could compute the test statistic directly like this.

attach(prob2)
# CORRECTED ON MARCH 24!
#x <- score[esp="high"] is wrong
#y <- score[esp="low"] is wrong
x <- score[esp=="high"]
y <- score[esp=="low"]
w <- sum(rank(c(x,y))[seq(along=x)])

# Problem 3, data from Exercise 9.4.12

# (a) The menu options Statistics:Compare Samples:Two Samples:t test
#     opens a dialog box to fill in to do a t test.

# In this case, game1 and game2 are the two variables, and it is a paired t-test.

# (b) To do a permutation test, you need to find the distribution of the test
#     statistic when the labels are randomly assigned to each pair.  This code
#     will estimate a p-value by simulation.

attach(prob3)
d <- game1 - game2

# create a matrix of size 5000 by length(d) with i.i.d. equally probable +/- 1's.
# 
# rs <- matrix(sample(c(-1,1),5000*length(d),replace=T)) THIS WAS WRONG!
#
# CORRECTED ON MARCH 24, 1999
rs <- matrix(sample(c(-1,1),5000*length(d),replace=T),5000,length(d))

# create an array of simulated test statistics by taking the dot product
# of each row of the random signs with the array d, via matrix multiplication
x <- as.vector(rs %*% d)

# find a p-value for the one-sided test that mu(d) < 0
p <- sum(x <= sum(d))/5000