demo/clt2.R

In IPSUR: Introduction to Probability and Statistics Using R

##################################################
# The Central Limit Theorem                       

clt2 <- function(population = "runif",
                 a = 0,
                 b = 10,
                 sample.size = 2,
                 N.iter = 100000){

#################################################################
# initialize variables
population <- get(population, mode = "function")
xbar <- rep(0, N.iter)
graphics.off()

f <- function(x) dunif(x, min = a, max = b )
curve( f,
       xlim = c(a-1,b+1), ylim = c(0, 1.3/(b-a)),
       xlab = "Support Set",
       ylab = "Density",
       lwd = 2,
       main = "The Population Distribution \n (while we're waiting)" )

abline( h = 0 , col = "grey" )


########################################
# Label the plot with mu
text(   (a+b)/2, 
        0.9/(b-a), 
        bquote( mu ==.((a+b)/2) ),
        cex = 1.5, 
        pos = 1 )              

# Label the plot with sigma^2
text(   (a+b)/2, 
        0.8/(b-a), 
        bquote( sigma^2 ==.( (b-a)^2/12 ) ),
        cex = 1.5, 
        pos = 1 )
        

#############################################
# simulate xbar's
xbar <-rowMeans(matrix(population(sample.size * N.iter, min = a, max = b),
                       nrow = N.iter)
                )

# Find mean and variance of xbar
xbar.mean <- round( mean( xbar ), 4)
xbar.var <- round( var( xbar ), 4)

# window for graph
low <- xbar.mean - 3*sqrt(xbar.var)
up <- xbar.mean + 3*sqrt(xbar.var)

dev.new()
dev.set(3)
# Draw histogram of simulated x-bars
hist(   xbar,
        breaks = 80,
        xlim = c(low,up),
        xlab = "",
        prob = TRUE,
        main = "Sampling Distribution of X-bar",
        sub = "Click to see Limiting Normal Density (in red)")

########################################
# Label the histogram with mean(xbar)
text(   up, 
        dnorm(xbar.mean, mean = xbar.mean, sd = sd(xbar)), 
        bquote( mean(xbar)==.(xbar.mean) ),
        cex = 1, 
        pos = 2 )              

# Label the histogram with var(xbar)
text(   up, 
        dnorm(xbar.mean, mean = xbar.mean, sd = sd(xbar))*0.9, 
        bquote( var(xbar)==.(xbar.var) ),
        cex = 1, 
        pos = 2 )
        
# Label the histogram with n
text(   up, 
        dnorm(xbar.mean, mean = xbar.mean, sd = sd(xbar))*0.8, 
        bquote( n ==.(sample.size) ),
        cex = 1.85, 
        pos = 2 )              
        
######################################
# Draw limiting Normal curve
z <- locator( n = 1 )   
g <- function(x) dnorm(x, mean = xbar.mean, sd = sd(xbar))
curve(  g, 
        lwd = 2,
        col = "red",
        add = TRUE )
}