Finding distribution of sample mean by central limit theorem - r

Question

let X1,...,X25 be a random sampe from normal distribution with mean=37 and sd=45.
Let xbar be the sample mean. How is xbar distributed? I have to verify it by central limit theorem.
Also compute P(xbar>43.1)
my attempt
for(i in 1:1000){
x=rnorm(25,mean=37,sd=45)
xbar=mean(x)
z=(xbar-37)/(45/sqrt(25))
}
z
But i couldn't find the distribution of xbar.

Answer 1

Change your for loop and use replicate instead
set.seed(1)
X <- replicate(1000, rnorm(25,mean=37,sd=45))
X_bar <- colMeans(X)
hist(X_bar) # this is how the distribution of X_bar looks like

Answer 2

xbar=c()
for(i in 1:1000){
x=rnorm(25,mean=37,sd=45)
xbar=c(xbar,mean(x)) #save every time the value of xbar
}
hist(xbar) #plot the hist of xbar
#compute the probability to b e bigger thant 43.1
prob=which(xbar>43.1)/length(xbar)

Answer 3

Just to expand in this a little bit.
The Central Limit Theorem states the distribution of the mean is asymptotically N[mu, sd/sqrt(n)]. Where mu and sd are the mean and standard deviation of the underlying distribution, and n is the sample size used in calculating the mean. So, in the example below data is a dataset of size 2500 drawn from N[37,45], arbitrarily segmented into 100 groups of 25. means is a dataset of the means of each group. Note that both the data and the means are (aprox.) normally distributed, but the distribution of the means is much tighter (lower sigma). From the CLT we expect sd(mean) ~ sd(data)/sqrt(25), which it is.
data <- data.frame(sample=rep(1:100,each=25),x = rnorm(2500,mean=37,sd=45))
means <- aggregate(data$x,by=list(data$sample),mean)
#plot histoggrams
par(mfrow=c(1,2))
hist(data$x,main="",sub="Histogram of Underlying Data",xlim=c(-150,200))
hist(means$x,main="",sub="Histogram of Means", xlim=c(-150,200))
mtext("Underlying Data ~ N[37,45]",outer=T,line=-3)
c(sd.data=sd(data$x), sd.means=sd(means$x))
sd.data sd.means
43.548570 7.184518
But the real power of the CLT is that it shows that the distribution of the means is asymptotically normal, regardless of the distribution of the underlying data. This is shown here, where the underlying data is sampled from a uniform distribution. Again, sd(mean) ~ sd(data)/sqrt(25).
data <- data.frame(sample=rep(1:100,each=25),x = runif(2500,min=-150, max=200))
means <- aggregate(data$x,by=list(data$sample),mean)
#plot histoggrams
par(mfrow=c(1,2))
hist(data$x,main="",sub="Histogram of Underlying Data",xlim=c(-150,200))
hist(means$x,main="",sub="Histogram of Means", xlim=c(-150,200))
mtext("Underlying Data ~ U[-150,200]",outer=T,line=-3)
c(sd.data=sd(data$x), sd.means=sd(means$x))
sd.data sd.means
99.7800 18.8176