difference
between population
means
Two-sided:
Question
just asks about any
difference
H
0
:
1
=
2
H
0
:
1
–
2
= 0
All are contradicted by the same null
hypothesis

0
:
H
0
wsu
uw
0
:
H
a
wsu
uw
Set up a test assuming that the null (H
O
) is true and see whether the
facts of our sample contradict that assumption.
Key question: How likely is it that we would observe a sample
difference (14.5-12.5 = 2 hrs) if the null (H
O
) were true?
If very unlikely, we will reject H
O
and support H
a
If not very unlikely, we will fail to
reject H
O
Two-sample hypothesis test
We want to know if UW students differ from WSU students in terms of average study
time.
We draw a random sample of 100 UW students and find an average study time of
14.5 hours per week with a standard deviation of 8.25.
In contrast, our sample of 81
WSU students has an average study time of 12.5 hours per week with a standard
deviation of 7.0.
Use a .05 alpha level to test the statistical significance of the observed
difference between means.
Plan: State the HYPOTHESES

0
:
H
0
wsu
uw
0
:
H
a
wsu
uw
Set up a test assuming that the null (H
O
) is true and see whether the
facts of our sample contradict that assumption.
Key question: How likely is it that we would observe a sample
difference (14.5-12.5 = 2 hrs) if the null (H
O
) were true?
If very unlikely, we will reject H
O
and support H
a
If not very unlikely, we will fail to
reject H
O
Have to think about
probability of receiving our
sample result (difference)
if H
0
were true
Have to think about the
SAMPLING DISTRIBUTION of all
possible sample results
Plan: State the HYPOTHESES
Two-sample hypothesis test

Sampling theory:
Hypothesis test for difference between means
Sampling Distribution for the
difference
between means
•
A theoretical probability distribution that would be obtained by
calculating all of the possible mean differences (
) for all possible
pairs
of random, independent samples of size n
1
and n
2
drawn from two
populations.
2
1
X
X

Characteristics of the Sampling Distribution for Difference
between Two Means:
•
IF the random samples are independently drawn
AND the
sample
sizes are large
(n
1
+ n
2
> 100) then:
•
The sampling distribution of differences between means would be
NORMAL
[but use the t-distribution when population standard deviations are unknown]
•
The
MEAN
of this distribution would equal the real difference between the
population means
•
The
STANDARD ERROR
of the sampling distribution of differences would be
equal to:
2
2
2
1
2
1
2
1
n
n
X
X
2
1
2
1
X
X
2
2
2
1
2
1
2
1
n
s
n
s
s
X
X
ESTIMATE
Sampling theory:
Hypothesis test for difference between means