1. Business Statistics, 4e by Ken Black
Chapter 12
Analysis of Categorical Data
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

2. Learning Objectives
Understand the 2 goodness-of-fit test and how to use it.
Analyze data using the 2 test of independence.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 2

3. 2 Goodness-of-Fit Test
The 2 goodness-of-fit test compares
expected (theoretical) frequencies
of categories from a population distribution
to the observed (actual) frequencies
from a distribution to determine whether
there is a difference between what was
expected and what was observed.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 6

4. 2 Goodness-of-Fit Test
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 7

5. Milk Sales Data for Demonstration Problem 12.1
Month
Gallons
January
1,610
February
1,585
March
1,649
April
1,590
May
1,540
June
1,397
July
1,410
August
1,350
September
1,495
October
1,564
November
1,602
December
1,655
18,447
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 8

6. Hypotheses and Decision Rules for Demonstration Problem 12.1
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9

7. Calculations for Demonstration Problem 12.1
Month fo fe
(fo - fe)2/fe
January
1,610
1,537.25
3.44
February
1,585
1.48
March
1,649
8.12
April
1,590
1.81
May
1,540
0.00
June
1,397
12.80
July
1,410
10.53
August
1,350
22.81
September
1,495
1.16
October
1,564
0.47
November
1,602
2.73
December
1,655
9.02
18,447
18,447.00
74.38
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10

8. Demonstration Problem 12.1: Conclusion
0.01
df = 11
24.725
Non Rejection
region
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 11

9. Bank Customer Arrival Data for Demonstration Problem 12.2
Number of
Arrivals
Observed
Frequencies 7 1 18 2 25 3 17 4 12 5 5
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 12

10. Hypotheses and Decision Rules for Demonstration Problem 12.2
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13

11. Calculations for Demonstration Problem 12
Calculations for Demonstration Problem 12.2: Estimating the Mean Arrival Rate
Number of
Arrivals X
Observed
Frequencies f
f·X 7 1 18 2 25 50 3 17 51 4 12 48 5 5
192
Mean
Arrival
Rate
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14

12. Calculations for Demonstration Problem 12
Calculations for Demonstration Problem 12.2: Poisson Probabilities for  = 2.3
Number of
Arrivals X
Expected
Probabilities
P(X)
Frequencies
n·P(X)
0.1003
8.42 1
0.2306
19.37 2
0.2652
22.28 3
0.2033
17.08 4
0.1169
9.82 
0.0838
7.04
Poisson
Probabilities
for  = 2.3
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 15

13. 2 Calculations for Demonstration Problem 12.2
Number of
Arrivals X
Observed
Frequencies f
Expected
nP(X)
(fo - fe)2 fe 1 2 3 4 5 7
8.42 18
19.37 25
22.28 17
17.08 12
9.82 5
7.04 84
84.00
0.24
0.10
0.33
0.00
0.48
0.59
1.74
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16

14. Demonstration Problem 12.2: Conclusion
0.05
df = 4
9.488
Non Rejection
region
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17

15. Using a 2 Goodness-of-Fit Test to Test a Population Proportion
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 18

16. Using a 2 Goodness-of-Fit Test to Test a Population Proportion: Calculations  2 33 16
167
184      o e f = fo fe
Defects
Nondefects
200
n =
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 19

17. Using a 2 Goodness-of-Fit Test to Test a Population Proportion: Conclusion
0.05
df = 1
3.841
Non Rejection
region
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 20

18. Qualitative Variables
2 Test of Independence
Qualitative Variables
Nominal Data
Used to analyze the frequencies of two variables with multiple categories to determine whether the two variables are independent.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 21

19. 2 Test of Independence: Investment Example
In which region of the country do you reside?
A. Northeast B. Midwest C. South D. West
Which type of financial investment are you most likely to make today?
E. Stocks F. Bonds G. Treasury bills
Type of financial
Investment E F G A
O13 nA
Geographic B nB
Region C nC D nD nE nF nG N
Contingency Table
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 22

20. 2 Test of Independence: Investment Example
Type of Financial
Investment E F G A
e12 nA
Geographic B nB
Region C nC D nD nE nF nG N
Contingency Table
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 23

21. 2 Test of Independence: Formulas  ij i j e n N
where  : =
the row
the column
the total
of row i
of column
of all fr
equencies 2    o f df (r -
1)(c 1) r
the number
r of rows c
r of columns
Expected
Frequencies
Calculated 
(Observed )
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 24

22. 2 Test of Independence: Gasoline Preference Versus Income Category
Type of
Gasoline
Income
Regular
Premium
Extra
Less than $30,000
$30,000 to $49,999
$50,000 to $99,000
At least $100,000
r = 4
c = 3
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 25

23. Gasoline Preference Versus Income Category: Observed Frequencies
Type of
Gasoline
Income
Regular
Premium
Extra
Less than $30,000 85 16 6
107
$30,000 to $49,999
102 27 13
142
$50,000 to $99,000 36 22 15 73
At least $100,000 23 25 63
238 88 59
385
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 26

24. Gasoline Preference Versus Income Category: Expected Frequencies
Type of
Gasoline
Income
Regular
Premium
Extra
Less than $30,000
(66.15)
(24.46)
(16.40) 85 16 6
107
$30,000 to $49,999
(87.78)
(32.46)
(21.76)
102 27 13
142
$50,000 to $99,000
(45.13)
(16.69)
(11.19) 36 22 15 73
At least $100,000
(38.95)
(14.40)
(9.65) 23 25 63
238 88 59
385   ij i j e n N  11 12 66 24 46 40 .
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 27

25. Gasoline Preference Versus Income Category: 2 Calculation  2 88 66 15 16 24 46 6 40
102 87 78 27 32 13 21 76 36 45 22 69 11 19 38 95 23 14 25 9 65 70      o e f .
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 28

26. Gasoline Preference Versus Income Category: Conclusion
0.01
df = 6
16.812
Non rejection
region
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 29