Please answer this question with RStudio.
R code with comments (all statements starting with # are comments)
#set the parameters of gamma distribution
k<-2
theta<-3
#part a)
#set the values of x
x<-seq(0,20,length=100)
#get the density of x
p<-dgamma(x,shape=k,scale=theta)
#plot
plot(x,p,type="l",ylab="density",main=bquote("Density of gamma
distribution with k=2,"*theta*"=3"))
#get this plot
b) The expected value (true mean) of X is
the variance of X is
#part b)
#expected value is
mu<-k*theta
sigma2<-k*theta^2
sprintf('The true mean of X is %.4f',mu)
sprintf('The true variance of X is %.4f',sigma2)
#get this poutput
c) Each sample of size 12 is a draw from Gamma distribution and they are different.
Since s are random variables, the sum
is also a random variable.
If we divide this sum by n, the average that we get of these s is also random.
Hence is a random variable.
d) the true mean of is
The true variance of is
The true standard deviation of is
R code
#part d)
#set the sample size
n<-12
#get the true mean of sample mean
muxbar<-mu
#get the true variance of sample mean
sigma2xbar<-sigma2/n
#get the true standard deviation of sample mean
sigmaxbar<-sqrt(sigma2xbar)
sprintf('The true mean of sample mean is %.4f',muxbar)
sprintf('The true variance of sample mean is
%.4f',sigma2xbar)
sprintf('The true standard deviation of sample mean is
%.4f',sigmaxbar)
#get this output
e) R code
#part e)
#set the random seed
set.seed(123)
#set the sample size
n<-12
#set the number of repetition
r<-10000
#intialize the variable to store the z values
z<-numeric(r)
for (i in 1:r) {
x<-rgamma(n,shape=k,scale=theta)
#get the sample mean m
m<-mean(x)
#calculate the test statistics
z[i]<-(m-6)/sigmaxbar
}
#plot the histogram of z
hist(z,breaks=50,freq=FALSE,xlab="Test statistics
(z)",main="Histogram of the test statistics")
#add dnorm
curve(dnorm(x),from=min(z),to=max(z),add=TRUE,col="red")
#get this plot
The distribution of the test statistics is reasonably well approximated by a normal distribution.
f) Using sample standard deviation
#part f)
#set the random seed
set.seed(123)
#set the sample size
n<-12
#set the number of repeatition
r<-10000
#intialize the variable to store the z values
t<-numeric(r)
for (i in 1:r) {
x<-rgamma(n,shape=k,scale=theta)
#get the sample mean m
m<-mean(x)
#get the sample standard deviation
s<-sd(x)
#calculate the test statistics
t[i]<-(m-6)/(s/sqrt(n))
}
#plot the histogram of t
hist(t,breaks=50,freq=FALSE,xlab="Test statistics
(t)",main="Histogram of the test statistics",ylim=c(0,0.4))
#add dnorm
curve(dnorm(x),from=min(t),to=max(t),add=TRUE,col="red")
#add dt
curve(dt(x,df=n-1),from=min(t),to=max(t),add=TRUE,col="blue")
legend("topleft",c("test statistics","standard normal","standard
t"),lty=1,col=c("black","red","blue"))
#get this plot
t distribution with its thicket tail seems to be a better fit than standard normal, particularly for the left tail.
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