Advanced High-School Mathematics

374 CHAPTER 6 Inferential Statistics

x 1 ,x 2 ,...,xnof this random variable. Associated with this sample are

Thesample mean: this is defined by setting

x =

x 1 +x 2 +···+xn

n

.

The basic reason for considering the sample mean is the follow-

ing. Suppose that we have taken the samplesx 1 , x 2 , ..., xnfrom

a population whose mean is μ. Would we expect that x ≈ μ?

Fortunately, the answer is in agreement with our intuition; that

we really do expect that the sample mean to approximate the

theoretical (or population) mean. The reason, simply is that if we

form the random variable

X=

X 1 +X 2 ···+Xn

n

,

then it is clear that E(X) = μ. (Indeed, we already noted this

fact back on page 324.) That is to say, when we take nindepen-

dent samples from a population, then we “expect” to get back the

theoretical meanμ. Another way to state this is to say thatxis

anunbiased estimate of the population meanμ.

Next, notice that sinceX 1 , X 2 ,···, Xnare independent, we have

that

Var(X) = Var

(X

1 +X 2 +···+Xn

n

)

=

1

n^2

Var(X 1 +X 2 +···+Xn)

=

1

n^2

(Var(X 1 ) + Var(X 2 ) +···+ Var(Xn))

=

σ^2

n

.