Stat 312: Lecture 14
Two-sample tests
Moo K. Chung
March 11, 2003
Concepts
1. Let X1,···, Xnand Y1,···, Ymbe two in-
dependent samples with mean µX,µYand
variance σ2
X,σ2
Yrespectively.
E(¯
X−¯
Y) = µX−µY.
Var(¯
X−¯
Y) = σ2
X/n +σ2
Y/m.
2. Let X1,···, Xnand Y1,···, Ymbe two inde-
pendent samples with Xi∼N(µX, σ2
X) and
Yj∼N(µY, σ2
Y). Then
Z=¯
X−¯
Y
qσ2
X/n +σ2
Y/m ∼N(0,1).
3. Assume nand mto be large. The test
statistic for testing
H0:µX=µYvs. H1:µX6=µY
Z=¯
X−¯
Y
qσ2
X/n +σ2
Y/m ∼N(0,1).
In-class problems
Example 1. A study was conducted to compare
the reaction times of men and women to a stim-
ulus. 12 men and 9 women were employed in
the experiment. Let Xibe the reaction time
of the i-th man and Yjbe the reaction time of
the j-th woman. There is a reason to believe
that Xi∼N(µX,0.18) and Yj∼N(µY,0.14).
We want to test if the population means are the
same. The following measurements were taken:
¯x= 3.6 seconds, ¯y= 3.8 seconds. Is there any
difference between the reaction times of men and
women?
> z=(3.6-3.8)/sqrt(0.18/12+0.14/9)
> z
[1] -1.144155
Example 2. A study was conducted to compare
the reaction times of men and women to a stim-
ulus. 50 men and 65 women were employed in
the experiment. Let Xibe the reaction time
of the i-th man and Yjbe the reaction time of
the j-th woman. The following measurements
were taken: ¯x= 3.6 seconds, ¯y= 3.8 seconds,
s2
X= 0.18, s2
Y= 0.14. Is there any difference
between the reaction times of men and women?
> z=(3.6-3.8)/sqrt(0.18/50+0.14/65)
> z
[1] -2.63664
> qnorm(0.025)
[1] -1.959964
What is the main reason you are getting the
different results in Example 1 and 2?
Self-study problems
Determine the P-value in Example 1 and 2.