Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

242 Some Elementary Statistical Inferences

LetS 12 =(n 1 −1)−^1

∑n 1

i=1(Xi−X)

(^2) andS 2
2 =(n 2 −1)
− 1 ∑n 2
i=1(Yi−Y)
(^2) be the
sample variances. Then estimating the variances by the sample variances, consider
the random variable
Z=
Δ̂−Δ
√
S^21
n 1 +
S^22
n 2

. (4.2.8)

By the independence of the samples and Theorem 4.2.1, this pivot variable has
an approximateN(0,1) distribution. This leads to the approximate (1−α)100%
confidence interval for Δ =μ 1 −μ 2 given by

⎛

⎝(x−y)−zα/ 2

√

s^21

n 1

+

s^22

n 2

,(x−y)+zα/ 2

√

s^21

n 1

+

s^22

n 2

⎞

⎠, (4.2.9)

where

√
(s^21 /n 1 )+(s^22 /n 2 ) is the standard error ofX−Y. Thisisalargesample
(1−α)100% confidence interval forμ 1 −μ 2.

The above confidence interval is approximate. In this situation we can obtain
exact confidence intervals if we assume that the distributions ofXandYare normal
with the same variance; i.e.,σ 12 =σ^22. Thus the distributions can differ only in
location, i.e., alocation model. Assume then thatXis distributedN(μ 1 ,σ^2 )
andY is distributedN(μ 2 ,σ^2 ), whereσ^2 is the common variance ofX andY.
As above, letX 1 ,...,Xn 1 be a random sample from the distribution ofX,let
Y 1 ,...,Yn 2 be a random sample from the distribution ofY, and assume that the
samples are independent of one another. Letn=n 1 +n 2 be the total sample size.
Our estimator of Δ isX−Y. Our goal is to show that a pivot random variable,
defined below, has at-distribution, which is defined in Section 3.6.

BecauseXis distributedN(μ 1 ,σ^2 /n 1 ),Y is distributedN(μ 2 ,σ^2 /n 2 ), andX
andYare independent, we have the result

(X−Y)−(μ 1 −μ 2 )

σ

q 1

n 1 +n^12

has aN(0,1) distribution. (4.2.10)

This serves as the numerator of ourT-statistic.

Let

Sp^2 =

(n 1 −1)S 12 +(n 2 −1)S 22

n 1 +n 2 − 2

. (4.2.11)

Note thatSp^2 is a weighted average ofS 12 andS 22. It is easy to see thatSp^2 is
an unbiased estimator ofσ^2. It is called thepooled estimatorofσ^2 .Also,
because (n 1 −1)S 12 /σ^2 has aχ^2 (n 1 −1) distribution, (n 2 −1)S 22 /σ^2 has aχ^2 (n 2 −1)
distribution, andS^21 andS 22 are independent, we have that (n−2)S^2 p/σ^2 has a
χ^2 (n−2) distribution; see Corollary 3.3.1. Finally, becauseS 12 is independent of
XandS^22 is independent ofY, and the random samples are independent of each
other, it follows thatSp^2 is independent of expression (4.2.10). Therefore, from the