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Do My Statistics Homework
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Types of Statistics Academic Papers We Can Write
Statistics term papers
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Sample Math homework statistics questions from our experts
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?
Answer
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.
What is the distribution of X¯X¯?
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.
What is the distribution of ZZ?
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.
Which holds for any sample size nn.
What about CLT, then?
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
N(0,σ2)N(0,σ2) :
Z=n−−√X¯−μσ−→d.N(0,1)Z=nX¯−μσ→d.N(0,1),
as n→∞n→∞.
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