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Q: If X is a random variable with X normally distributed. Let x bar be a random sample of size n and z = square root n times xbar - mean / standard deviation, which of the following is true due to CLT?

Let X1, X2,..., XnX1, X2,..., Xn be a random sample from X∼N(μ,σ2). X∼N(μ,σ2).
Now let X¯=1n∑ni=1XiX¯=1n∑i=1nXi,
and Z=n−−√X¯−μσ.Z=nX¯−μσ.

A linear combination of normally distributed variables is also normally distributed.
X¯∼N(E[X¯], V[X¯])X¯∼N(E[X¯], V[X¯]),
where
E[X¯]=...=μE[X¯]=...=μ
V[X¯]=...=σ2/nV[X¯]=...=σ2/n.

ZZ is a linear combination of normally distributed variables and is thus normally distributed itself. Z∼N(E[Z],V[Z])Z∼N(E[Z],V[Z]),
where
E[Z]=...=0E[Z]=...=0
V[Z]=...=1.V[Z]=...=1.

What about CLT, then?

N(0,σ2)N(0,σ2) :

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Sample Math homework statistics questions from our experts

Answer

What is the distribution of X¯X¯?

What is the distribution of ZZ?

Which holds for any sample size nn.

Above, the stochastic variable XX is normally distributed, and any linear combination of normally distributed variables is also normally distributed. However, what happens if XX is not normally distributed?
Suppose that X1,...,XnX1,...,Xn is a random sample (i.i.d) from an unknown distribution with mean E[X]=μE[X]=μ and variance V[X]=σ2<∞V[X]=σ2<∞. Then as nn approaches infinity, n−−√(X¯−μ)n(X¯−μ) converges in distribution to a normal

Z=n−−√X¯−μσ−→d.N(0,1)Z=nX¯−μσ→d.N(0,1),
as n→∞n→∞.

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