ST 372
Lecture Worksheet #11
Reiland
Chi-Square Worksheet
1. Is birth month related to success on the basketball court? In the table below the birth
months (Jan. = 1) of all men's D1 basketball players from 1996 through 2008 are
shown. Also shown (from national demographic statistics) is the percentage of all
people born in each month. Is the random fluctuation of the players' birth months a
real deviation from the national pattern?
Month
1
2
3
4
5
6
Obs. Count
1685
1538
1651
1519
1490
1506
Nat'l %
8%
7%
8%
8%
8%
8%
Expected
1525.6
1334.9
1525.6
1525.6
1525.6
1525.6
Month
7
8
9
10
11
12
Total
Obs. Count
1607
1569
1662
1667
1548
1628
19070
Nat'l %
9%
9%
9%
9%
8%
9%
100%
Expected
1716.3
1716.3
1716.3
1716.3
1525.6
1716.3
L! À The distribution of birth months for D1 players is the same as the distribution for
the general population
LE À The distribution of birth months for D1 players differs from the distribution for the
general population
This is a chi-square goodness of fit test with hypotheses as stated above.
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Test Statistic: \ #
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œ

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œ "'Þ'&&  $!Þ*!"  "!Þ$!)  !Þ!#*
 !Þ)$"  !Þ#&#  'Þ*'"  "#Þ'%#
 "Þ(")  "Þ%"'  !Þ$#*  %Þ&%$ œ )'Þ&)#
degrees of freedom œ "#  " œ ""; from ;# table, ;#Þ!& œ "*Þ'(&.
Since )'Þ&)#  "*Þ'(&, we reject L! and conclude that the distribution of birth months
for D1 basketball players differes from the distribution for the general population.
In which months do the birth month distributions differ? To answer this, for each cell
calculate
9,=/<@/./B:/->/.
(these are frequently referred to as the cell residuals); the values are
 /B:/->/.
Jan
4.081
Feb
5.559
Mar
3.211
Apr
-0.169
May
-0.911
June
-0.502
July
-2.638
Aug
-3.556
Sept
-1.311
Oct
-1.190
Nov
0.573
Dec
-2.131
ST 372
Lecture Worksheet 11
page 2
It appears that the months in which more D1 basketball players are born compared to the
general population are Jan, Feb, Mar. The months in which fewer D1 basketball
players are born compared to the general population are July, Aug, and, to a lesser
extent, Dec.
2. An urban economist want to determine whether the region of the US a resident lives
in is associated with their general level of satisfaction with life. He randomly selects
150 residents 18 years of age or older from 4 geographic regions of the US: East,
South, Midwest, and West. He asks: “are you satisfied or dissatisfied with the way
things are going in the United States at this time?”
Satisfaction
Satisfied
Dissatisfied
Total
East
77
73
150
South
84
66
150
Midwest
93
57
150
West
83
67
150
Total
337
263
600
L! À Satisfaction ratings are the same for all four regions
LE : Satisfaction ratings are not the same in all four regions
This is a chi-square test of homogeneity with hypotheses as stated above.
/B:/->/.Ñ
œ  Ð9,=/<@/.
/B:/->/.
)
Test Statistic: \ #
#
3œ"
‡ -96?78 >9>+6
The expected count in each cell is <9A >9>+6
.
1<+8. >9>+6
The table below shows the observed and expected counts in each cell:
Satisfaction
East
South
Midwest
West
Total
Satisfied
77 (84.25) 84 (84.25) 93 (84.25) 83 (84.25)
337
Dissatisfied 73 (65.75) 66 (65.75) 57 (65.75) 67 (65.75)
263
Total
150
150
150
150
600
/B:/->/.Ñ
œ  Ð9,=/<@/.
/B:/->/.
)
Test Statistic: \ #
#
3œ"
œ
Ð(()%Þ#&Ñ#
)%Þ#&
Ð($'&Þ(&Ñ#
'&Þ(&

Ð)%)%Þ#&Ñ#
)%Þ#&
Ð'''&Þ(&Ñ#
'&Þ(&
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Ð*$)%Þ#&Ñ#
)%Þ#&
Ð&('&Þ(&Ñ#
'&Þ(&
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Ð)$)%Þ#&Ñ#
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Ð'('&Þ(&Ñ#
'&Þ(&

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œ !Þ'#$*  !Þ!!!(  !Þ*!))  !Þ!")&
 !Þ(**%  !Þ!!"!  "Þ"'%%  !Þ!#$) œ $Þ&%!&
degrees of freedom œ (2  ")*(4  1) œ $; from ;# table, ;#Þ!& œ (Þ)"&.
Since $Þ&%!&  (Þ)"&, we .9 89> reject L! ; there is no evidence that the distributions of
satisfaction ratings differ significantly from region to region.