Question

For expert using R , I appreciate everything can be helpful..

Thank you

Answer 1

(a)

R code:

x=rnorm(100,0,1)
p=length(subset(x,x<1.645))/100
p

pnorm(1.645)

$Output:\\ \hat{F}(1.645)=0.93,~F(1.645)=0.95$

(b)

R code:

x=matrix(0,800,1000)
p=1:800*0
for(i in 1:800)
{
x[i,]=rnorm(1000,0,1)
p[i]=length(subset(x[i,],x[i,]<1.645))/1000
}
mean(p)
var(p)

Output:

> mean(p)
[1] 0.9499062
> var(p)
[1] 4.545178e-05
Yes mean(p) is very closed to 0.95.