Question about sampledistribution

#1
Hi, I have a question I was hoping someone could answer me.

It goes as follows:
The correlation r between two variables x_1 and x_2 equals 0.33. For testing the significance of the correlation, H_0 : p = 0, the sampledistribution of the correlationcoefficient has to be determined. Wat goes for the sampledistribution of the correlation whilst testing the 0-hypothesis? The sample distribution whilst testing this hypothesis is:
a, skewed to the left
b, skewed to the right
c, symmetric around 0.33
d, symmetric around 0

(I hope I translated everything in a way that makes sense..)

I don't really understand the question so I'm hoping someone could answer it and explain why that is the answer.
 

Englund

TS Contributor
#2
I'd say that the distribution of the sample correlation coefficient is independent of the null hypothesis. Can you specify exactly what confuses you?

Here's a couple of simulations that might help you.


jpg images

click image upload
Code:
samplingdist <- function(M,n,r) {
  corr <- numeric(M);
  for (i in 1:M) {
    x <- mvrnorm(n, rep(1, 2), matrix(c(1,r,r,1),2,2))
    corr[i] <- cor(x[,1],x[,2])
  }
  hist(corr,freq=F,breaks=80); list(median(corr),mean(corr))
}
samplingdist(20000,100,0.33)
Code:
samplingdist <- function(M,k,n) {
  corr <- numeric(M); meandiff <- numeric(k); r <- numeric(k)
  for (j in 0:99) {
    for (i in 1:M) {
      x <- mvrnorm(n, rep(1, 2), matrix(c(1,j/100,j/100,1),2,2))
      corr[i] <- cor(x[,1],x[,2])
    }
    r[j] <- j/100
    meandiff[j] <- median(corr)-mean(corr)
  }
plot(r,meandiff,ylim=c(-0.0035,0.0035)); abline(mean(meandiff),0)
}
samplingdist(1000,100,500)
 

Dason

Ambassador to the humans
#3
I'm thinking the question is asking about the sampling distribution under the null hypothesis (so assuming the null is true what would we expect to see).
 

Englund

TS Contributor
#4
I'm thinking the question is asking about the sampling distribution under the null hypothesis (so assuming the null is true what would we expect to see).
Ah, of course.

The distribution of the sample correlation coefficient under the null is always symmetrical around zero when the errors are normally distributed. When the errors are not normally distributed the sampling distribution is normal and symmetric in large samples. The sampling distribution in small samples when the errors aren't normal depends on the distribution.

Have a look on a proof of why the beta coefficient is normal and you'll find your answer of why this is the case.