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You already learned about
the standard error for the sampling distribution of means,
s.emean = 
My lecture notes for
yesterday gave the formula for computing the standard error
for proportions, which is simply a mean computed for data
scored 1 (for p) or 0 (for q). It so happens that the
variance for data in proportions is simply
- variance =
pq
- So the
standard deviation =
In case you don't believe
this, here is a computed example for these data inspired by
the CBS/New York Times poll reported on October 29,
2001.
Sixty-one
percent think the war in Afghanistan would be worth it
even if it meant several thousand American troops would
lose their lives; 27 percent say the war there would not
be worth that cost.
Let's round off the 61% to
60% for easier computation and consider only a sub-sample of
ten cases:
|
Case
|
Worth
It?
|
Score
(X)
|
Mean
|
(X-mean)
|
(X-mean)2
|
|
1
|
yes
|
1
|
0.6
|
0.4
|
0.16
|
|
2
|
no
|
0
|
0.6
|
-0.6
|
0.36
|
|
3
|
no
|
0
|
0.6
|
-0.6
|
0.36
|
|
4
|
yes
|
1
|
0.6
|
0.4
|
0.16
|
|
5
|
yes
|
1
|
0.6
|
0.4
|
0.16
|
|
6
|
yes
|
1
|
0.6
|
0.4
|
0.16
|
|
7
|
yes
|
1
|
0.6
|
0.4
|
0.16
|
|
8
|
no
|
0
|
0.6
|
-0.6
|
0.36
|
|
9
|
yes
|
1
|
0.6
|
0.4
|
0.16
|
|
10
|
no
|
0
|
0.6
|
-0.6
|
0.36
|
|
|
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6/10 =.6
(mean of proportion)
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|
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· = 2.4
(sum of squares)
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- Given a sum of squares
of 2.4 for ten cases, the variance is .24.
- Now let's multiply
the p (.6) by the q (.4): .6 * .4 = .24
-- so pq = variance.
- We can compute the s.e.
of the proportion for the CBS/New York Times poll of
1,024 respondents, using yesterday's formula:
- This result is one
standard error of a proportion; we multiply by 100 to
make it a percentage: 1.5%
- But remember we need to
double the 1.5% to produce an estimate of +/- 3%--such
that it will embrace 95% of the possible
samples.
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