Example: Suppose we have two populations, A and B,
and we know that the distribution of scores in these two populations
on an identical examination are normally distributed with the following
parameters:
X ~ N(625, 100) in population A
Y ~ N(600, 150) in population B
Suppose we take a random sample of two people from
population A and a random sample of three people from population
B. What is the probability that the average score of the two
people from population A will be higher than the average score
of the three people from population B. That is:
_ _ _ _
P(X_{2} > Y_{3}) = P(X_{2} - Y_{3} > 0)
_
Clearly X_{2} ~ N(625, 50)
_
and Y_{3} ~ N(600, 50)
_ _
so that X_{2} - Y_{3} ~ N(25, 100)
_ _
Hence: P[(X_{2} - Y_{3} - 25)/10 > (0 - 25)/10] = P(Z > -2.5) =
1 - F(-2.5) = 1 - .0062 = .9938