1
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
1
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
x
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
$
n
3. Chance error in sample mean is called
standard error.
3
Solution (cont.):
%
95
96
.
1
96
.
1
Pr
n
x
n
x
4. Because
is normally distributed, the
probability that
is within 1.96 standard
errors of
equals:
x
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
n
96
.
1
n
96
.
1
Units of
x
4
Solution (cont.):
x
equals
n
x
n
96
.
1
96
.
1
96
.
1
96
.
1
6. Algebraic manipulation yields
n
x
n
x
or
n
n
96
.
1
96
.
1
7. On repeated random sampling, the true
population mean,
will be captured by the
interval
, 95% of the
time.
n
x
96
.
1
5
Solution (cont.):
300
$
400
,
14
$
n
x
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.
6
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2
Theorem: confidence interval for the
population mean,
2
known
If x
1
, x
2
, …, x
n
is a random sample from a
normal population or a large random sample from
any population with unknown mean
and known
variance
2
then
n
z
x
2
, then
is a (1
)100% confidence interval for
(CI for
).
7
Comments: CI for
,
2
known
•
1
is called the confidence coefficient and
reflects the desired level of accuracy.
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 Winter '10
 FIRSTENBERG,I.
 Statistics, Normal Distribution, Statistical hypothesis testing

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