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Psych 100A
Winter 2010
Lecture 8: Statistical Inference
1. Interval estimate (recap)
2. Hypothesis testing: tests of
significance
Example
•
•
Notations and definitions
3. Test for
,
known
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1. Interval estimates
•
How accurate are the point estimates?
Example: Suppose a city manager in Los Angeles
wants to know the mean income (
) of 25,000
families living in South Los Angeles.
She
interviews 900 (n) families “at random” and
finds that the sample mean = $14 400
Based
finds that the sample mean = $14,400.
on past study, she knows that the population SD
(
) of incomes equals $9,000.
Problem: Put a “give or take” number on the
point estimator (ie, the sample mean).
x
?
2
Solution: Although the exact distribution of
incomes for this community is unknown,
)
,
(
~
2
n
N
1. Sampling distribution of mean incomes is
normally distributed with mean
and
variance
2
/n due to central limit theorem.
Or,
2. Magnitude of chance errors in sample mean
when repeatedly drawing a random sample of
size 900 from all 25,000 families is
300
$
900
000
,
9
$
3. Chance error in sample mean is called
standard error.
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Solution (cont.):
%
95
96
.
1
96
.
1
Pr
4. Because
is normally distributed, the
probability that
is within 1.96 standard
errors of
equals:
5. “1.96” comes from … Table A.
0
25
50
-4 -3 -2 -1 0
1
2
3
4
Units of z
% Density
95%
2.5%
2.5%
-1.96
1.96
96
.
1
96
.
1
Units of
4
Solution (cont.):
equals
96
.
1
96
.
1
96
.
1
96
.
1
6. Algebraic manipulation yields
or
96
.
1
96
.
1
7. On repeated random sampling, the true
population mean,
will be captured by the
interval
, 95% of the
time.
96
.
1
5
Solution (cont.):
300
$
400
,
14
$
8. Thus,
300
$
96
.
1
400
,
14
$
equals
take"
or
give
"
the
So,
And, we can reasonably expect that the
interval ($13,812, $14,988) is near the true
population mean,
of incomes in South Los
Angeles.
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