Let X1,X2,…,Xn be i.i.d. random variables with mean μ and variance σ2 . Denote the sample mean by X¯¯¯¯n=∑ni=1Xin .

Question

Assume that n is large enough that the central limit theorem (clt) holds. Find a random variable Z with approximate distribution N(0,1) , in terms of X¯¯¯¯n , n , μ and σ . (Note that μ and σ2 refers to the mean and variance of Xi , not X¯¯¯¯n .)

Answer 1

continuation of the above question:
Z∼N(0,1) for Z=

Answer 2

It is the sample mean minus the mean mu divided by the sample variance

(barX_n-mu)/(sigma/sqrt(n))

Answer 3

Correction: divided by sample standard deviation