Question

I don't understand where the sqrt(n) from the z-score goes when doing the CLT. Could someone help, please?

Answer 1

Let X and Y be the two independent random variables , then

Var(X+Y)=Var(X)+Var(Y) and Var(a⋅X)=a2⋅Var(X). So the variance of ∑Xi is ∑σ^2=nσ^2, and the variance of Xbar=1/n∑Xi is nσ^2/n^2=σ^2/n.

This is for the variance. To standardize a random variable, you divide it by its standard deviation. And we know that, the expected value of X bar is μ, so the variable

has expected value 0 and variance 1. So if it tends to a Gaussian, it has to be the standard Gaussian N(0,1). Your formulation in the first equation is equivalent. By multiplying the left hand side by σ you set the variance to σ^2.

So the sqrt (n) in z score comes from variance property