Plan ahead for desired CI Margin of Error (pp. 545-6)
(Large n) CI:
| Hand in:
p. 549 sample size: 8.26, 8.27 Sample size Note, no surprise I hope, that your graph is a parabola. Two-sample, p. 566 8.31 choose which. 8.41, 42 downloading music 8.48 drunken cyclists 8.47, and p.574, 8.66, 8.67 gender bias in 10 textbooks This is an example of how the same data set can be re-analyzed in many ways. If you get tired of doing these by hand, at least read and understand the solutions. (The text gives no HW problems where the two-sample plus-four technique needs to be implemented. Sorry about that!) |
Read, discuss
8.33 what's wrong? p.570, 8.54 presenting results |
Optional |
Ch. 8, Inference for proportions.
Plan ahead for desired CI Margin of Error
(pp.
545-6)
(Large n) CI:
Decide on desired Margin of error m, and C
(thus
z*). Guesstimate a true p, p*. (p*=1/2 requires the
largest sample size--safest. Remember you showed that p(1-p)
had its maximum at p = .5).
Solve margin-of-error equation for n. 
(Some results, p. 546. If you wonder why so many polls
feature a 3% margin of error and a sample size of 1060 or so, this tells
you why. )
Sec. 8.2, Comparing 2 proportions (from independent samples)
Comparing means from an experiment with two treatments (usually
control and "treatment").
/--- Group 1, n1---- Treatment 1---\
/
\
Random asst.
Compare results --"proportions"
\
/
\--- Group 2, n2---- Treatment 2---/
To examine the difference of the two proportions, p1
-p2:
We use the difference of the two sample proportions, D =
, and assume it's approximately Normal.
We find the standard deviation of D, 
and
make estimates of the p's as before.
Large sample CI: (90%-99% C, all successes
and failures > 10) : D + z* SED,where
(Plus four (p. 559): Add 2 to each n, one to each of
the successes and the failures. Good down to n's >5)
Test: H0 : p1 =
p2
As usual, find D =
; the mean of D is 0 under the null hypothesis, so all that remains to
do the test is to divide by the standard deviation of D to get a z-value,
and find a P-value from the normal table.
What should we use for the standard deviation of D, under the null
hypothesis? We aren't assuming that we know either p1
or p2 now, only that they are the same.
We could use SED, as in the CI.
BUT since we're assuming the two p's are equal, we can use a "pooled"
technique, which gives each observation equal weight.
Assuming p1=p2 =
p, the common value, 
We still need to estimate the common p.
Do it by throwing both set of data into the same pot, so we have a
total of
n1 + n2 observations, and
we have X1 + X2 total "successes",
so our pooled estimate is 
,
and we use this to build a "pooled" SE,
.
Note how this development parallels the development for the pooled two-sample
t.
Another approach to comparison of two proportions--Relative risk
(pp. 563-4)
Looks at the ratio of the two proportions, p1/
p2. For instance,
if the proportion of people who die from disease A under treatment 1 is
.30, and the proportion who die from disease 2 is .60, then the relative
risk of treatment 1 to treatment 2 is .30/.60 = .5; the risk of treatment
1 is half that of treatment 2.
CI's for relative risk can be built based on sample proportions; they
are not of the form estimate + m, and aren't symmetrical
around the estimate. I delved into SPSS to find out where they calculated
these, and it's buried very deep, inside "log linear" analyses.
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