Matched Pairs (Dependent Samples)
t-distibutions: confidence intervals and hypothesis tests
Matched Pairs are when two samples are taken from two populations such that the samples are
dependent on each other and can be matched up. This is not a two-sample distribution, rather, we treat
the differences of the two samples as one sample. We therefore apply t-procedures to the one
population of differences.
1. Imagine that you want to compare two sets of golf club brands: Titleist and Calloway. Suppose
that we want to show that Titleist hits farther than Calloway. We do this by comparing 10 random
pairs of golf clubs where each pair consists of one Titleist club and one Calloway club of the same
type. Each pair is hit by a different randomly chosen professional golfer (this makes the pairs
dependent since one pair can be matched up through the one golfer hitting those two clubs).
Measurements in the number of yards hit are recorded below:
Golfer Titleist Calloway Difference
1 188 194 -6
2 201 197 4
3 195 195 0
4 195 190 5
5 202 195 7
6 175 178 -3
7 182 172 10
8 188 184 4
9 193 201 -8
10 194 192 2
Is this sample of differences sufficient evidence to claim that Titleist hits farther than Calloway?
Choose 05.0=
α
.
ANSWER:
First note that the first population consists of the number of yards a Titleist club is hit, and the
second population consists of the number of yards a Calloway club is hit. Therefore, denote
T
μ
= the average number of yards a Titleist club hits
C
μ
=the average number of yards a Calloway club hits
We will assume CT
H
μ
μ
=
:
0
and we want to show CTa
H
μ
μ
>:
It is important to note that we always subtracted Calloway from Titleist in our sample. If we subtract
off C
μ
then it follows that
We will assume 0:
0
=
−
CT
H
μ
μ
and we want to show 0: >
−
CTa
H
μ
μ
Define the average of the population of differences as CTd
μ
μ
μ
−
=
. Then
We will assume 0:
0
=
d
H
μ
and we want to show 0: >
da
H
μ
1
