Question

R Simulation:

For n = 1000, simulate a random sample of size n from N(0,1). Use the generated data to give an approximation to the critical values when α = 0.01,0.05,0.1, and compare them with the theoretical values zα/2. Repeat with n = 10,000 to get a better approximation.

Please use Commands quantile( ) or qnorm( ) or both.

Answer 1

code

x<- rnorm(1000)
z0.01 <- quantile(x,0.995)
z0.05 <- quantile(x,0.975)
z0.1 <- quantile(x,0.95)

c0.01 <- qnorm(0.995)
c0.05 <- qnorm(0.975)
c0.1 <- qnorm(0.95)


z0.01
c0.01

z0.05
c0.05

z0.1
c0.1

y <- rnorm(10000)
z0.01 <- quantile(y,0.995)
z0.05 <- quantile(y,0.975)
z0.1 <- quantile(y,0.95)

c0.01 <- qnorm(0.995)
c0.05 <- qnorm(0.975)
c0.1 <- qnorm(0.95)

z0.01
c0.01

z0.05
c0.05

z0.1
c0.1

#running the code

> x<- rnorm(1000)
> z0.01 <- quantile(x,0.995)
> z0.05 <- quantile(x,0.975)
> z0.1 <- quantile(x,0.95)
> 
> c0.01 <-  qnorm(0.995)
>   c0.05 <- qnorm(0.975)
>   c0.1 <- qnorm(0.95)
>   
>   
> z0.01
   99.5% 
2.614842 
> c0.01
[1] 2.575829
> 
> z0.05
   97.5% 
1.818711 
> c0.05
[1] 1.959964
> 
> z0.1
     95% 
1.567166 
> c0.1
[1] 1.644854
> 
> 
> y <- rnorm(10000)
> z0.01 <- quantile(y,0.995)
> z0.05 <- quantile(y,0.975)
> z0.1 <- quantile(y,0.95)
> 
> c0.01 <-  qnorm(0.995)
> c0.05 <- qnorm(0.975)
> c0.1 <- qnorm(0.95)
> 
> 
> z0.01
   99.5% 
2.561284 
> c0.01
[1] 2.575829
> 
> z0.05
   97.5% 
1.942748 
> c0.05
[1] 1.959964
> 
> z0.1
     95% 
1.649045 
> c0.1
[1] 1.644854

we see that when n = 10000

both are close to each other