As shown in the first image we first assume that the distribution of heights is follows a normal distribution.

Then as shown in second image we say that the sampling distribution of sample means is also normally distributed with μ_sample = μ_population

and σ_sample = (σ_population / √n).

I want to ask is this because we assumed that the heights are normally distributed then as we learned in the lecture of Sum of Gaussians (Week-2) what we’re doing is:

1: E[(X1 + X2 + … + Xn) / n] = (μ1 + μ2 + … μn) / n = μ

(since μ1=μ2=…=μn=μ)

2: Var[(X1 + X2 + … + Xn) / n] = (1/n^2) * Var[(X1 + X2 + … + Xn)]

= (1/n^2) * [Var(X1) + Var(X2) + … + Var(Xn)], (X1, X2…Xn are iid)

= (1/n^2) * [n * Var(X)]

= Var(X) / n

Am i Correct or is there another reason behind the assumption that the mean and variance of the sampling distribution of the sample means has the same mean as population mean and deviation = population_dev / n