Question

Write a simulation in R that shows the distribution of the t-­‐test statistic for a two-­‐sample t test when the null hypothesis is true (i.e. when H0: µ1 -­‐ µ2 = 0 is true). To do this, you should use a for loop that repeatedly performs t-­‐tests comparing sample means of data that come from distributions with the same population mean and standard deviation. Use rnorm() to take samples, t.test() to perform the t-­‐ tests, and use “$statistic” to extract the t-­‐test statistic from the t.test() procedure (e.g. t.test(x,y)$statistic). Make a histogram of the test statistics.

Answer 1

R code with comments (all statements starting with # are comments and can be removed)

#set the random seed
set.seed(123)

#set the mean of the 2 populations
mu<-10
#set the standard deviation of 2 populations
sigma<-5
#set the number of simulations
R<-1000
#set the sample size
n<-20
#initialize the variable to hold the test statistics
tstat<-numeric(R)

#loop R times
for (i in 1:R){
#draw a sample of size n
s1<-rnorm(n,mu,sigma)
#draw a second sample of size n
s2<-rnorm(n,mu,sigma)
#store the statistics part of the t test
tstat[i]<-t.test(s1,s2)$statistic
}
#Make a histogram
hist(tstat,main="Histogram of t-­test statistic",xlab="t stat")

# get this