Question

you can do it by hand

2 Distribution of sample mean 1, write a script to obtain the sample mean values with n = 4 for a total of 10000 experiments, where each sample is drawn normally from N(0,1). This results in 10000 sample mean values. (10pt) variance? (5pt) (5pt) 2. Consider the sample mean as a random variable X, what are its mean and 3. Does your observation from (a) match with your calculation from (b)?

Answer 1

Let X be a random variable having a standard normal distribution with mean
μ=0
and variance
σ2 = 1

Let be the sample mean of any sample of size n=4 drawn from the above standard normal distribution.

2. We know that has normal distribution with mean

$mu_{ar{X}}=mu=0$

and variance

σ2 1 σχ = 2 0.25 7n

Now we will check using a script if the above is true

1. R script for doing this, with comments (all statements starting with # are comments)

#set a random seed
set.seed(123)

#part 1)
#set the sample size
n<-4
#set the number of experiments
r<-10000
#draw n*r samples from standard normal N(0,1)
x<-rnorm(n*r,0,1)
#convert this into a matrix of n columns, each with r values
x<-matrix(x,ncol=n,nrow=r)
#get r sample means
xbar<-apply(x,1,mean)

#part 3)
#print the mean and variance of xbar
sprintf('Mean of sample mean is %.4f',mean(xbar))
sprintf('Variance of sample mean is %.4f',var(xbar))

# get this output

> #part 3) > #print the mean and variance of Xbar > sprintf ('Mean of sample mean is % . 4f",mean(xbar)) 1] "Mean of sample mean is -0.0060" > sprint f ('Variance of sample mean is % .4f',var (xbar)) 1] "Variance of sample mean is 0.2474"

The mean of a sample means of size 4 has a value of -0.0060 and it is close enough to the theoretical mean of 0 given in part 2)

The variance of a sample means of size 4 has a value of 0.2474 and it is close enough to the theoretical variance of 0.25 given in part 2)