Section 6.9
Test for a Difference
in Proportions
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Outline
 Pooled proportion
 Test for a difference in proportions
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Split or Steal?
http://www.youtube.com/watch?v=p3Uos2fzIJ0
• Both choose “split”: split the jackpot
• Both choose “steal”: both get nothing
• One “steal,” one “split: stealer gets everything
What would you do???
(a) Split
(b) Steal
Van den Assem, M., Van Dolder, D., and Thaler, R., “Split or Steal? Cooperative
Behavior When the Stakes Are Large,” 2/19/11.
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Split or Steal?
SPLIT
STEAL
Total
MALE
140
129
269
FEMALE
163
142
305
Total
303
271
574
Are males or females more cooperative?
pM = proportion of males who split
pF = proportion of females who split
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Split or Steal?
Are males or females more cooperative?
pM = proportion of males who split
pF = proportion of females who split
The relevant hypotheses are:
(a) H0: pM = pF , Ha: pM > pF
(b) H0: pM = pF , Ha: pM < pF
(c) H0: pM = pF , Ha: pM ≠ pF
(d) H0: pM ≠ pF , Ha: pM = pF
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Split or Steal?
SPLIT
STEAL
Total
MALE
140
129
269
FEMALE
163
142
305
Total
303
271
574
pM = proportion of males who split
pF = proportion of females who split
Calculate the sample statistic, 𝑝𝑀 − 𝑝𝐹
140/269 – 163/305
= 0.520 – 0.534 = -0.014
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Hypothesis Testing
For hypothesis testing, we want the
distribution of the sample proportion
assuming the null hypothesis is true
H0: p1 = p2
SE =
𝑝1 (1−𝑝1 )
𝑛1
+
𝑝2 (1−𝑝2 )
𝑛2
What to use for p1 and p2?
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Pooled Proportion
 We assume the proportions are the same
between the two groups, and want to use one
proportion that is our best guess for what they
would be, if they were equal
 Combine both groups into one big group, and
use the overall proportion, called the pooled
proportion, 𝒑
 Hint: the pooled proportion will always be
somewhere in between 𝑝1 and 𝑝2
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Split or Steal?
MALE FEMALE Total
SPLIT 140
163
303
STEAL 129
142
271
Total 269
305
574
Pooled proportion = overall proportion who split
140 + 163 303
𝑝=
=
= 0.528
269 + 305 574
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Test for a Difference in Proportions
𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐 − 𝑛𝑢𝑙𝑙
𝑧=
𝑆𝐸
𝑧=
𝑝1 −𝑝2
𝑝(1 − 𝑝) 𝑝(1 − 𝑝)
+
𝑛1
𝑛2
• If 𝑛𝑝 ≥ 10 and 𝑛(1 − 𝑝) ≥ 10 for both sample
sizes, then the p-value can be computed as
the area in the tail(s) of a standard normal
beyond z.
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Split or Steal?
SPLIT
STEAL
Total
MALE
140
129
269
FEMALE
163
142
305
Total
303
271
574
Based on these data, can we conclude whether
males or females are significantly more
cooperative when playing Golden Balls?
(a) Yes
(b) No
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Split or Steal?
Counts are greater than 10 in each category
H0: pM = pF , Ha: pM ≠ pF
140 + 163 303
𝑝=
=
= 0.528
269 + 305 574
𝑝1 −𝑝2
𝑧 = 𝑝(1−𝑝) 𝑝 1−𝑝
𝑛1
=
+ 𝑛
2
p-value = 2(0.371)
0.528(1 − 0.528) 0.528(1 − 0.528) = 0.742
+
0.520 − 0.534
269
−0.014
=
= −0.33
0.042
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Based on these data, we cannot
conclude whether males or females
are more cooperative.
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Split or Steal
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Difference in Proportions
 Come up with your own categorical variable
that you would like to analyze by gender, using
this class.
 This should be a categorical variable with
possible answers: yes or no
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Difference in Proportions
MALES ONLY ANSWER:
Your question here.
(a) Yes
(b) No
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Difference in Proportions
FEMALES ONLY ANSWER:
Your question here.
(a) Yes
(b) No
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Difference in Proportions
Is there a significant difference in the
proportion answering “yes” between males
and females?
(a) Yes
(b) No
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