Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

4.2. Confidence Intervals 243

result of Section 3.6.1 concerning Student’st-distribution, we have that

T =

[(X−Y)−(μ 1 −μ 2 )]/σ

√

n− 11 +n− 21

√

(n−2)S^2 p/(n−2)σ^2

=

(X−Y)−(μ 1 −μ 2 )

Sp

√

1

n 1 +

1

n 2

(4.2.12)

has at-distribution withn−2 degrees of freedom. From this last result, it is easy
to see that the following interval is an exact (1−α)100% confidence interval for
Δ=μ 1 −μ 2 :

(

(x−y)−t(α/ 2 ,n−2)sp

√

1

n 1

+

1

n 2

,(x−y)+t(α/ 2 ,n−2)sp

√

1

n 1

+

1

n 2

)

. (4.2.13)

A consideration of the difficulty encountered when the unknown variances of the
two normal distributions are not equal is assigned to one of the exercises.

Example 4.2.4.To illustrate the pooledt-confidence interval, consider the baseball
data presented in Hettmansperger and McKean (2011). It consists of 6 variables
recorded on 59 professional baseball players of which 33 are hitters and 26 are pitch-
ers. The data can be found in the filebb.rdalocated at the site listed in Chapter

1. The height in inches of a player is one of these measurements and in this exam-
   ple we consider the difference in heights between pitchers and hitters. Denote the
   true mean heights of the pitchers and hitters byμpandμh, respectively, and let
   Δ=μp−μh. The sample averages of the heights are 75.19 and 72.67 inches for
   the pitchers and hitters, respectively. Hence, our point estimate of Δ is 2.53 inches.
   Assuming the filebb.rdahas been loaded in R, the following R segment computes
   the 95% confidence interval for Δ:
   hitht=height[hitpitind==1]; pitht=height[hitpitind==0]
   t.test(pitht,hitht,var.equal=T)
   The confidence interval computes to (1. 42 , 3 .63). Note that all values in the confi-
   dence interval are positive, indicating that on the average pitchers are taller than
   hitters.

Remark 4.2.1.SupposeX andY are not normally distributed but that their

distributions differ only in location. As we show in Chapter 5, the above interval,

(4.2.13), is then approximate and not exact.

4.2.2 Confidence Interval for Difference in Proportions

LetX andY be two independent random variables with Bernoulli distributions
b(1,p 1 )andb(1,p 2 ), respectively. Let us now turn to the problem of finding a confi-
dence interval for the differencep 1 −p 2 .LetX 1 ,...,Xn 1 be a random sample from
the distribution ofXand letY 1 ,...,Yn 2 be a random sample from the distribution
ofY. As above, assume that the samples are independent of one another and let