Question

Let X1,X2,…,X25 be random sample ~ normal(0,16) and Y1, Y2,…,Y25 be random sample ~ normal(1,9), assume the two samples are independent ,comput P(X>Y) where X=Sum(Xi)/25 and Y=Sum(Yi)/25

Answer 1

We have two samples, n1=n2=25
both are normally distributed, and are independent.
Under these given conditions,
E(X̄1-X̄2)=E(X̄1)-E(X̄2)=μ1-μ2
VAR(X̄1-X̄2)=V(X̄1)+V(X̄2)=σ1²/n1 + σ2²/n2

Z=(X̄1-X̄2)-(μ1-μ2) / sqrt(σ1²/n1 + σ2²/n2)
should have a N(0,1) distribution.
Thus looking up the normal distribution table will give the required probability if we consider (X̄1-X̄2)=0

Answer 2

Correction to missing parentheses for above expression for Z


Z=[(X̄1-X̄2)-(μ1-μ2)] / sqrt(σ1²/n1 + σ2²/n2)