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)-(k) may differ since the data is chosen randomly. This 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.
(j) Our outcome is pretty unlikely to occur if there is no difference between the two groups.
(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) H_o: q_plac = q_AZT (the proportions are the same in the two groups)
H_a: q_plac > q_AZT (those taking AZT are less likely to have HIV-positive babies)
(b)

_AZT = .079;

_plac = .250

_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) Since the p-value is small, the sample data support the researchers' conjecture at the .05 level (very significant evidence).
(h) (-.249, -.092)
Note: The answer to (i) below is the answer to (j) in the Calculator version.
(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	-1.66	.049	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.

Activity 24-4: Graduate Admissions Discrimination (cont.)

(a) H_o: q_men = q_women (the proportion of men accepted is equal to the proportion of women accepted)
H_a: q_men > q_women (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.