Worksheet – Correlation – Estimated and True Winning Probabilities in Poker
Subjects: Group of 28 Seasoned Texas Hold-em Poker Players (Data are averages across players)
Study Variables: X = True Probability of Winning Hand, Y = Mean Assessed Probability Among Players
Factors Affecting True Probability: Time of decision(early,mid,late), #opponents(unknown to player), strength of hand
(weak, strong)
Data: n = 25 Combinations of factors assessed by players (Plot on worksheet webpage)
TrueProb
7.4
2.3
3.5
6
12.6
EstProb
10.5
11.8
11.9
12.5
14.9
TrueProb
5.6
9.7
11.8
20.5
14
T-bar
31.932
E-bar
33.996
SSTT
SSEE
SSTE
16595.81 11273.19 13076.84

25
i 1

Ti  T
2
EstProb
15.8
17.2
19.1
20.1
21.9
TrueProb
18.9
13.3
27.5
34.8
36.2

25
 16595.81
i 1
Ei  E
EstProb
22.8
24.8
26.7
31
31.5

2
TrueProb
22.1
43.4
31.3
48
73.2
 11273.19
EstProb
31.7
34.4
48.2
49.1
49.1
TrueProb
62.7
61.5
59.9
84.9
87.2
EstProb
56.9
58.8
64.1
78.3
86.8
 T  T  E  E   13076.84
25
i 1
i
i
Compute the sample correlation: rTE:
 T  T  E  E 
25
rTE 
i
i 1
i
 T  T    E  E 
25
i 1
2 25
i
i 1


2
i
Compute the test statistic and give rejection region for testing H 0: TE = 0 vs HA: TE ≠ 0:
Test Statistic:
Rejection Region:
Compute Fisher’s z-transformation and 95% CI for  
1  1  rTE
z '  ln 
2  1  rTE



1  1  TE 
ln 
:
2  1  TE 
95%CI for  : z ' z (.975)
1

n 3
Back-Transform to obtain 95% CI for TE:
TE
e 2  1
 2
e 1
e 2 LB =____________________
TE ( LB )  ___________________________
e 2 UB =__________________
TE (UB )  ___________________________