Workshop Statistics: Discovery with Data, Second
Edition
Topic 24: Comparing Two Proportions
Activity 24-1: Friendly Observers (cont.)
(a) A
= .250; B
= .727
(b) yes
(c) yes
Students' answers to (d)-(g) may differ since
the data is chosen randomly. These are meant to be sample answers.
(d) 6, no
(e)
Repetition #
1
2
3
4
5
"successes" assigned to group A
6
5
5
4
5
as extreme as actual study?
no
no
no
no
no
(f)
<----------------------------- number of successes --------------------->
(g) 5-7 are the most common values. This makes sense because there
are 23 cards, 11 of which are "successes." So, usually we'd expect
the successes to split roughly in half between the treatment groups.
(h) The values of the dotplot would be centered around zero since the
proportions should be about the same.
Students' answers to (i) may differ since the
data is chosen randomly. It is meant to be a sample answer.
(i) 100 repetitions were performed by the class as a whole. 6
of them gave a result at least as extreme as the actual sample. This
is a proportion of .06.
(k) The data provide some evidence in support of the researchers' conjecture.
Our simulation has shown that it is rare that such an extreme value would
occur by randomization alone providing evidence against the hypothesis
that there was no difference between the two groups.
Activity 24-2: Pregnancy, AZT, and HIV (cont.)
(a) Ho: qplac = qAZT
(the proportions are the same in the two groups)
Ha: qplac > qAZT
(those taking AZT are less likely to have HIV-positive babies)
(b) AZT
= .079; plac
= .250
(c) 324 women were subjects in this study. 53 had HIV-positive
babies. c
= .163
(d) z = 4.17
(e) p-value < .001
(f) There is a almost no chance that we would get a difference in sample
proportions this large by randomization alone.
(g) The sample data support the researchers' conjecture at the .05
level (very signifiant evidence).
(h) (-.249, -.092)
(i) The AZT group has a smaller proportion of HIV positive babies,
from 25% to 9% fewer.
Activity 24-3: Perceptions of Self-Attractiveness
(a) We need to know how many men and women were surveyed.
(b) small sample sizes
(c) large sample sizes
(d)
sample size
satisfied women
satisfied men
test stat
p-value
alpha = .10?
alpha = .05?
alpha = .01?
100
71
81
1.65
.0495
yes
barely
no
200
142
162
2.34
.0096
yes
yes
barely
500
355
405
3.70
< .0002
yes
yes
yes
(e) All else being equal, as sample size increases, a difference between
two sample proportions becomes more statistically significant.
(a) Ho: qmen = qwomen
(the proportion of men accepted is equal to the proportion of women accepted)
Ha: qmen > qwomen
(men are accepted at a higher rate)
z = 9.56
p-value < .0002
The sample results are statistically significant at commonly used levels
of significance, such as .10, .05, and .01. Thisis very strong evidence
of a difference in the acceptance rates for me and women.
(b) Even though these data are statistically significant, one cannot
draw conclusions about causation from an observational study. Therefore,
we cannot regard this statistically significant difference as evidence
of discrimination.