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Business Statistics, 5th ed.by Ken Black
Chapter 12
Analysis ofCategorical Data
Discrete Distributions
PowerPoint presentations prepared by Lloyd Jaisingh,Morehead State University
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Learning Objectives
Understand the 2 goodness-of-fit test and how touse it.
Analyze data using the 2 test of independence.
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2 Goodness-of-Fit TestThe 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.
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2 Goodness-of-Fit Test
datasamplethefromestimatedparametersofnumber=
categoriesofnumber
valuesexpectedoffrequency
valuesobservedoffrequency:
-1-=df
2
0
02
c
k
where
ck
e
e
e
f
f
f
ff
The formula which is used to compute the test statistic fora chi-square goodness-of-fit test is given below.
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Month Gallons
January 1,610
February 1,585
March 1,649
April
1,590
May
1,540
June
1,397July
1,410
August
1,350
September
1,495
October
1,564
November
1,602December
1,655
18,447
Milk Sales Datafor Demonstration
Problem 12.1
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Hypotheses and Decision Rules
for Demonstration Problem 12.1
ddistributeuniformlynotare
salesmilkforfiguresmonthlyThe:H
ddistributeuniformlyare
salesmilkforfiguresmonthlyThe:H
a
o
.
.. ,
01
1
12 1 0
11
24 7250111
2
df k cIf reject H .
If do not reject H .
Cal
2
o
Cal
2
o
24 725
24 725
. ,
. ,
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Calculations
for Demonstration Problem 12.1Month fo fe (fo - fe)
2/feJanuary 1,610 1,537.25 3.44
February 1,585 1,537.25 1.48
March 1,649 1,537.25 8.12
April 1,590 1,537.25 1.81May 1,540 1,537.25 0.00
June 1,397 1,537.25 12.80
July 1,410 1,537.25 10.53
August 1,350 1,537.25 22.81
September 1,495 1,537.25 1.16
October 1,564 1,537.25 0.47
November 1,602 1,537.25 2.73
December 1,655 1,537.25 9.02
18,447 18,447.00 74.38
ef
18447
12
153725.
Cal
2
74 37 .
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The observed chi-square value of 74.37 isgreater than the critical value of 24.725.
The decision is to reject the null
hypothesis. The data provides enoughevidence to indicate that the distributionof milk sales is not uniform.
Calculations
for Demonstration Problem 12.1
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Calculations
for Demonstration Problem 12.1
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Bank Customer Arrival Data
for Demonstration Problem 12.2
Number of
Arrivals
Observed
Frequencies
0 7
1 18
2 25
3 17
4 12
5 5
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Calculations
for Demonstration Problem 12.2:
Estimating the Mean Arrival RateNumber of
Arrivals
X
Observed
Frequencies
f fX
0 7 0
1 18 18
2 25 50
3 17 51
4 12 485 5 25192
f X
f
192
84
2 3. customers per minute
Mean
Arrival
Rate
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Calculations for Demonstration Problem
12.2: Poisson Probabilities for = 2.3Number of
Arrivals X
Expected
Probabilities
P(X)
Expected
Frequencies
nP(X)
0 0.1003 8.421 0.2306 19.37
2 0.2652 22.28
3 0.2033 17.08
4 0.1169 9.82
0.0838 7.04
n f
84
Poisson
Probabilities
for = 2.3
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2 Calculationsfor Demonstration Problem 12.2
Cal
2
174 .Number ofArrivalsX
Observed
Frequencies
f
Expected
Frequencies
nP(X)
(fo - fe)2
fe
0
1
2
3
45
7 8.42
18 19.37
25 22.28
17 17.08
12 9.825 7.04
84 84.00
0.24
0.10
0.33
0.00
0.480.59
1.74
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The observed chi-square value of 1.74 isless than the critical value of 9.4877.
The decision is not to reject the null
hypothesis. The data does not provideenough evidence to indicate that thedistribution of bank arrivals is Poisson.
Calculations
for Demonstration Problem 12.2
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Using a 2 Goodness-of-Fit Testto Test a Population Proportion
.08:H
.08=:
ap
pHo
.
.. ,
05
1
2 1 0
1
384105 1
2
df k c
If reject H .
If do not reject H .
Cal
2
o
Cal
2
o
3841
3841
. ,
. ,
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Using a 2 Goodness-of-Fit Test to Testa Population Proportion: Calculations
fo feDefects 33 16
Nondefects 167 184
200 200n =
184
92.200
1
16
08.200
f
f
f
f
e
e
e
e
PnNondef ects
PnDef ects
6332.19
5707.10625.18
184
)184167(
16
)1633( 22
2
02
e
e
f
ff
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The observed chi-square value of 19.63 isgreater than the critical value of 3.8415.
The decision is to reject the nullhypothesis. The data does provideenough evidence to indicate that themanufacturer does not produce 8% ofdefective items.
Observing the actual sample result, in
which 0.165 of the sample was defective,indicates that the proportion of thepopulation that is defective might begreater than 8%.
Using a 2 Goodness-of-Fit Test toTest a Population Proportion
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Using a 2 Goodness-of-Fit Testto Test a Population Proportion
MINITAB Solution
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2 Test of IndependenceUsed to analyze the frequencies of two
variables with multiple categories todetermine whether the two variables
are independent.
Qualitative Variables
Nominal Data
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2 Test of Independence: Investment Example
In whichregion of the country do you reside?A. Northeast B. Midwest C. South D. West Whichtype of financial investment are you most likely to
make today?E. Stocks F. Bonds G. Treasury bills
Type of financialInvestment
E F G
A O13 nAGeographic B nBRegion C nC
D nDnE nF nG N
Contingency Table
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2 Test of Independence: Investment Example
Type of FinancialInvestment
E F G
A e12 nA
Geographic B nBRegion C nC
D nDnE nF nG N
Contingency Table
If A and F are independent,
P A F P A P F
P AN
P FN
P A FN N
A F
A F
n n
n n
AF
A F
A F
en n
n n
N P A F
NN N
N
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2 Test of Independence: Formulas
ij
i j
en n
N
where
: i = the row
j = the columnn
the total of row ithe total of column j
N = the total of all frequencies
i
j
nn
2 2 o ewhere
f ff
e
: df = (r - 1)(c - 1)r = the numberr of rowsc = the numberr of columns
ExpectedFrequencies
Calculated (Observed )
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Type ofGasoline
Income Regular PremiumExtra
Premium
Less than $30,000$30,000 to $49,999$50,000 to $99,000
At least $100,000
r = 4 c = 3
2 Test of Independence: GasolinePreference Versus Income Category
incomeoftindependen
notisgasolineofType:H
incomeoftindependen
isgasolineofType:
a
oH
.
.. ,
01
1 1
4 1 3 1
6
1681201 6
2
df r c
If reject H .
If do not reject H .
Cal
2o
Cal
2
o
16812
16812
. ,
. ,
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Gasoline Preference Versus Income
Category: Observed Frequencies
Type of
Gasoline
Income Regular PremiumExtra
Premium
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 15 23 25 63238 88 59 385
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Gasoline Preference Versus Income
Category: Expected Frequencies
Type ofGasoline
Income Regular Premium
Extra
PremiumLess 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)
15 23 25 63
238 88 59 385
ij
i j
en n
e
e
e
N
11
12
13
107 238385
66 15
107 88
385
24 46107 59
385
16 40
.
.
.
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2
2
88 66 15 16 24 46 6 16 40
102 87 78 27 32 46 13 21 76
36 45 13 22 16 69 15 11 19
15 38 95 23 14 40 25 9 65
66 15 24 46 16 40
87 78 32 46 21 76
45 13 16 69 11 19
38 95 14 40 9 6570 78
o ef f
fe
2 2 2
2 2 2
2 2 2
2 2 2
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . ..
Gasoline Preference Versus Income
Category: 2 Calculation
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Gasoline Preference Versus Income
Category: 2 Calculation
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Gasoline Preference Versus Income
Category: MINITAB Output
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