1
Psych 100A
Winter 2010
Lecture 12: Statistical Inference
1. Inference on two population means
2. Estimates of independent groups’
1

2
a. Known variances
b. Unknown but equal variance
3. Hypothesis testing for
1

2
4. Inference on two population means:
Dependent samples
1
1. Inference on two population means
•
To find the effect of a new treatment, we wish
to compare at least two groups:
•
One with treatment
•
One sans treatment (or treatment #2)
•
Experiment can be designed in two ways:
1. Independent (unpaired) samples.
Objects selected from population #1
have no bearing on objects selected
from population #2
2. Dependent (paired) samples. Each
object selected from population #1 is
naturally or forced paired with an
object selected from population #2
2
2. Point estimates of
1

2
, independent
groups
Population
of
units
If
1

2
,
treatment 1
“better” than treatment 2
If
1

2
=0,
treatment 1
equals than treatment 2
Inference
Independent
selection
If
1

2
,
treatment 2
“better” than treatment 1
Point Estimate:
2
1
2
1
x
Treatment groups
Sample #2
2
2
,
n
SRS
Sample #1
1
1
,
3
2. Interval estimates of
1

2
, independent
groups
•
Require a model for the chance variation of
1

2
•
Assume that each sample came from a normal
population, or that n
1
and n
2
are both sufficiently
large (eg, >25 each) such that the CLT holds, then
2
D
•
Can show that
4
2
2
2
1
2
1
2
1
2
1
,
~
N
2
2
2
2
2
1
1
1
1
,
~
,
~
Theorem: confidence interval for the
population mean difference,
2
’s known
If x
11
, x
12
, …, x
1n
1
and x
21
, x
22
, …, x
2n
2
are
independent SRSs from normal populations (or
large independent SRSs) with unknown means (
1
and
2
) and known variances (
1
2
and
2
2
), then
is a (1
)100% confidence interval for
1

2
(CI for
1

2
).
5
2
1
2
2
2
1
2
1
2
2
1
n
n
z
x
x
•
Example. Suppose we wish to compare two diets to
determine the difference between their average
weight losses.
•
Based on the following SRSs, obtain a point estimate
of
1

2
and 95% CI for
1

2
.
•
Data: diet loss in gm
A. 448, 229, 316, 105, 516, 496, 130, 242, 470,
195, 389,
97, 458, 347, 340, 212
B. 232, 200, 184, 180, 265, 125, 193, 322, 211
•
Assumptions:
1. Independent SRS of dieters A and B.
2. Weight losses are normally distributed
3.
2
s are known (say,
1
2
=
2
2
= 14,000 gm
2
)
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