Testing a single proportion
Researchers tested whether Underdogs in American football tend to “cover the spread” in a majority of games. For bet to
“pay-off” in long-run, the Underdog would have to win the bet with Probability > .5238. Based on Arena Football
League:
H
0
:
 
0.5238
H
A
:
 
.5238 Sample: n

992 games y

534 underdog wins bet
Comparing 2 Proportions – Independent / Large Samples
Researchers compared use of tanning beds among students who do and do not watch reality beauty shows
H
0
:

1
 
2
H
A
:

1
 
2
Group 1: Watch Reality Beauty Shows: n
1

224 y
1

Group 1: Do not Watch Reality Beauty Shows: n
2

352
29 use tanning beds y
2

13 use tanning beds
Comparing 2 Proportions – Independent / Small Samples
John Lister Data on Deaths from upper limb amputations: H
0
:

Post
 
Pre
H
A
:

Post
 
Pre
Pre-Antiseptic
Post-Antiseptic
Total
Death
1
7
6
NoDeath Total
6
11
17
12
12
24
Comparing 2 Proportions – McNemar’s Test
A sample of 150 people asked to choose between a Sure Thing and a Risky alternative in terms of a Gain, and in terms of a loss. Let

be the probability a person chooses the Risky alternative. H
0
:

G
 
L
H
A
:

G
 
L
Gain\Loss Sure
Sure
Risky
16
4
Risky
110
20
Goodness-of-Fit for Multinomial Distribution
120 Subjects were given warm beverage in cups of 4 colors (Blue, Green, Yellow, Red) and asked to rate which was warmest (all were of same temperature). Researchers were testing whether there are inherent differences of choices from random selection.
H
0
:

B
 
G
 
Y
 
R

0.25 Data: n
B

16, n
G

24, n
Y

34, n
R

46

Chi-Square Test
People were assigned to either Simultaneous choosing items for 3 future product uses, or Sequential choice of items for a single usage on 3 occasions. Subjects were classified as having selected either High, Medium, or Low variety based on their purchase choices. H
0
: No association between choice condition and Variety Seeking. H
A
: Association exists.
Observed H
Sim
Seq
Total
141
91
232
M
67
85
152
L
22
62
84
Total
230
238
468
Ordinal Association
People were told a beer was either: low, medium, or high price; and then after tasting it, rating it as: undrinkable (0), poor
(1), fair (2), good (3), or very favorable 4. Compute the sample value of

, the measure of ordinal association.
Price
Quality
2
3
0
1
4
1
4
20
23
9
4
2
1
21
22
12
4
26
15
7
3
4
8
Large-Sample Z-test for Proportions and Chi-Square Test in 2x2 Table
Prospect Theory: (Independent Samples)
Problem 11: In addition to what you own, you have been given 1000. (Gain Condition)
Choice A: 50% Chance of Win 1000, 50% Chance of Win 0 Choice B: 100% Chance of Win 500 (n
1
= 70, 58 take B)
Problem 12: In addition to what you own, you have been given 2000. (Loss Condition)
Choice C: 50% Chance of Lose 1000, 50% Chance of Lose 0 Choice D: 100% Chance of Lose 500 (n
2
= 68, 21 take D)
Goal: Test whether true proportions choosing sure bet are equal for 2 Conditions. H
0
:

G
 
L
Sample proportions in Each Condition (and overall) choosing sure Bet and Z-test:




1

1
70 68




0.084046

^
G

59

70
0.842857
obs

^

L
0.084046

21

68
0.308824
^
 
59 21


80
70 68 138

0.579710


6.354056 RR: Z obs
 z
.025

1.96

Chi-Square Test: H
0
: No association between Gain/Loss perspective and Choice obs
Gain
Loss
Sure
59
21
80
Risk
11
47
58
70
68
138 exp
Gain
Loss
Sure Risk
40.57971 29.42029
39.42029 28.57971
80 58
70
68
138
Chi-squareSure
Gain 8.361496
Risk
11.5331
Loss 8.607422 11.87231
40.37432
Observed Counts: n i ij

1, 2; j

1, 2 Expected Counts: E ij
 n n i
  j n

2 obs
 i
2 2 

1 j

1
 n ij

E ij

2
E ij

40.37432
2 obs
 
2
.05

3.841
2
Note: Z obs
 2
X obs
2
and Z
.025
 
2
.05
i

1, 2; j

1, 2
Odds Ratio
Study compared Olestra and triglyceride potato chips for gastrointestinal symptoms.
Olestra: n
O
= 563 y
O
=89 Triglyceride: n
T
= 529 y
T
= 93
Mantel –Haenszel Test – Combining Results from 2x2 Contingency Tables
Sleep apnea study. Combining results from h = 3 tables. Exposed = Modanafil Unexposed = Placebo
Outcome = Presence / Absence of Good Event
N1
7
N2
6.3
N1-N2 D
0.7 1.094211
38
21
59
36
19
55
7
7
14
37
58
95
16
43
59
2
4
6
52
62
114
75
79
154
9
11
20
36 25.08772 10.91228 7.123962
38 28.73377 9.266234 9.152318
MHChiSq 25.09499
MHCC 23.90743

 2
MH



 h q 

1 q  h

1


 n h 11
 n n h 1
 h

1 n h
 n n n n h 1
 h 2
 h

1 h

2 n
2 h

 n h









2
 2
MH
 

2
,1
P


1
2   2
MH

^
OR
MH

R
S
R
 q  h

1 n n h 11 h 22 n h

S
 h q 

1 n n h 12 h 21 n h

^ ^ v V OR

MH




S
2 h q 

1
S h
2 





n n n n h 11 h 12 h 21 h 22






^
OR
MH e

1.96
v
^
OR
MH e
1.96
v 

