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1Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
MARIO F. TRIOLAEIGHTH
EDITION
ELEMENTARY STATISTICSChapter 10 Multinomial Experiments and
Contingency Tables
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2Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Chapter 10
Multinomial Experiments and
Contingency Tables
10-1 Overview10-2 Multinomial Experiments:
Goodness-of-fit
10-3 Contingency Tables:Independence and Homogeneity
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3Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
10-1 Overview
Focus on analysis of categorical (qualitative orattribute) data that can be separated into
different categories (often called cells)
Use the X2(chi-square) test statistic (Table A-4)
One-way frequency table (single row or column)
Two-way frequency table or contingency table
(two or more rows and columns)
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4Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
MARIO
F.
TRIOLA EIGHTH
EDITIO
ELEMENTARY STATISTICSSection 10-2 Goodness of Fit
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5Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
10-2 Multinomial Experiments:
Goodness-of-Fit
Assumptionswhen testing hypothesis that the population
proportion for each of the categories is as claimed:
1. The data have been randomly selected.
2. The sample data consist of frequency counts
for each of the different categories.
3. The expected frequency is at least 5. (There is
no requirement that the observed frequency
for each category must be at least 5.)
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6Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Multinomial ExperimentAn experiment that meets the following conditions:
1. The number of trials is fixed.
2. The trials are independent.
3. All outcomes of each trial must beclassified into exactly one of several different
categories.
4. The probabilities for the differentcategories remain constant for each trial.
Definition
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Definition
Goodness-of-fit test
used to test the hypothesis that an
observed frequency distribution fits(or conforms to) some claimed
distribution
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8/538Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
0 represents the observed frequencyof an outcome
E represents the expected frequency of an outcome
k represents the number of different categoriesor
outcomes
n represents the total number of trials
Goodness-of-Fit Test
Notation
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Expected Frequencies
If all expected frequencies are equal:
the sum of all observed frequencies divided
by the number of categories
nE =k
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10/5310Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Expected Frequencies
If all expected frequencies are not all equal:
each expected frequency is found by multiplying
the sum of all observed frequencies by the
probability for the category
E = n p
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11/5311Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Goodness-of-fit Test in Multinomial Experiments
Test Statistic
Critical Values
1. Found in Table A-4 using k-1 degrees of
freedom
where k=number of categories
2. Goodness-of-fit hypothesis tests are always
right-tailed.
X2=
(O- E)2
E
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A large disagreementbetween observed
and expected values will lead to a largevalue of X2and a small P-value.
A significantly largevalue of
2
will causea rejectionof the null hypothesis of no
difference between the observed and the
expected.
A close agreement between observed
and expected values will lead to a small
value of X2 and a large P-value.
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Figure 10-3
Relationships Among
Components in
Goodness-of-Fit
Hypothesis Test
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Categories with Equal
Frequencies
H0: p1= p2= p3= . . . = pk
H1: at least one of the probabilities is
different from the others
(Probabilities)
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15Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
H0: p1, p2, p3, . . . , pkare as claimed
H1: at least one of the above proportions
is different from the claimed value
Categories with Unequal
Frequencies(Probabilities)
E l
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16Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Example: Mars, Inc. claims its M&M candies are distributed withthe color percentages of 30% brown, 20% yellow, 20% red, 10% orange,10% green, and 10% blue. At the 0.05 significance level, test the claimthat the color distribution is as claimed by Mars, Inc.
E l
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17Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Example: Mars, Inc. claims its M&M candies are distributed withthe color percentages of 30% brown, 20% yellow, 20% red, 10% orange,10% green, and 10% blue. At the 0.05 significance level, test the claimthat the color distribution is as claimed by Mars, Inc.
Claim: p1= 0.30, p2= 0.20, p3= 0.20, p4= 0.10,p5= 0.10, p6= 0.10
H0 : p1= 0.30, p2= 0.20, p3= 0.20, p4= 0.10,p5= 0.10, p6= 0.10
H1: At least one of the proportions is
different from the claimed value.
E l
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18Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Example: Mars, Inc. claims its M&M candies are distributed withthe color percentages of 30% brown, 20% yellow, 20% red, 10% orange,10% green, and 10% blue. At the 0.05 significance level, test the claimthat the color distribution is as claimed by Mars, Inc.
Brown Yellow Red Orange Green Blue
Observed frequency 33 26 21 8 7 5
Frequencies of M&Ms
n= 100
E l
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19Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Example: Mars, Inc. claims its M&M candies are distributed withthe color percentages of 30% brown, 20% yellow, 20% red, 10% orange,10% green, and 10% blue. At the 0.05 significance level, test the claimthat the color distribution is as claimed by Mars, Inc.
Brown Yellow Red Orange Green Blue
Observed frequency 33 26 21 8 7 5
Frequencies of M&Ms
Brown E=np= (100)(0.30) = 30Yellow E=np= (100)(0.20) = 20
Red E=np= (100)(0.20) = 20Orange E=np= (100)(0.10) = 10Green E=np= (100)(0.10) = 10
Blue E=np= (100)(0.10) = 10
n= 100
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20Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Brown Yellow Red Orange Green Blue
Observed frequency 33 26 21 8 7 5
Frequencies of M&Ms
Expected frequency 30 20 20 10 10 10
(O -E)2/E 0.3 1.8 0.05 0.4 0.9 2.5
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21Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Brown Yellow Red Orange Green Blue
Observed frequency 33 26 21 8 7 5
Frequencies of M&Ms
Expected frequency 30 20 20 10 10 10
(O -E)2/E 0.3 1.8 0.05 0.4 0.9 2.5
X2= = 5.95
(O- E)2
E
Test Statistic
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22Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Brown Yellow Red Orange Green Blue
Observed frequency 33 26 21 8 7 5
Frequencies of M&Ms
Expected frequency 30 20 20 10 10 10
(O -E)2/E 0.3 1.8 0.05 0.4 0.9 2.5
X2= = 5.95
(O- E)2
E
Test StatisticCritical Value X
2=11.071
(with k-1 = 5 and = 0.05)
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23Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Test Statistic does not fall within critical region;Fail to reject H0: percentages are as claimed
There is not sufficient evidence to warrant rejection of theclaim that the colors are distributed with the givenpercentages.
0
Sample data: X2= 5.95
= 0.05
X2= 11.071
Fail to Reject Reject
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24Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Comparison of Claimed and Observed Proportions
0.30
0.20
0.10
0
Green
Yellow
Red
Orange
Brown
Blue
Claimed proportions
Observed proportions
Proportions
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25Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
MARIO F. TRIOLA EIGHTHEDITIO
ELEMENTARY STATISTICSSection 10-3 Contingency Tables: Independence
and Homogeneity
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26Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Definition
Contingency Table (or two-way frequency table)
a table in which frequencies
correspond to two variables.
(One variable is used to categorize rows,and a second variable is used to
categorize columns.)
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27Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Definition
Contingency Table (or two-way frequency table)
a table in which frequencies
correspond to two variables.
(One variable is used to categorize rows,and a second variable is used to
categorize columns.)
Contingency tables have at least tworows and at least two columns.
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28Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Test of Independence
tests the null hypothesis that
the row variable and columnvariable in a contingency table arenot related. (The null hypothesis
is the statement thatthe row and column variables areindependent.)
Definition
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29Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Assumptions
1. The sample data are randomly selected.
2. The null hypothesis H0is the statement that
the row and column variables
are independent; the alternative
hypothesis H1is the statement that the row
and column variables are dependent.
3. For every cell in the contingency table, the
expectedfrequency E is at least 5. (There is
no requirement that everyobserved
frequency must be at least 5.)
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30Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Test of Independence
Test Statistic
Critical Values
1. Found in Table A-4 using
degrees of freedom = (r - 1)(c - 1)
r is the number of rows and c is the number of columns
2. Tests of Independence are always right-tailed.
X2=
(O- E)2
E
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31Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
(row total) (column total)
(grand total)E=
Total number of all observed frequencies
in the table
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32Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Tests of Independence
H0: The row variable is independent of thecolumn variable
H1: The row variable is dependent (related to)
the column variable
This procedure cannot be used to establish adirect cause-and-effect link between variables inquestion.
Dependence means only there is a relationshipbetween the two variables.
E t d F f
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33Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Expected Frequency for
Contingency Tables
E t d F f
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34Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
E= row total column total
grand total
Expected Frequency for
Contingency Tables
grand totalgrand total
E t d F f
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35Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
n p
E= row total column total
grand total
Expected Frequency for
Contingency Tables
grand totalgrand total
(probability of a cell)
E pected Freq enc for
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36Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
n p
E= row total column total
grand total
Expected Frequency for
Contingency Tables
grand totalgrand total
(probability of a cell)
Expected Frequency for
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37Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
n p
E= row total column total
grand total
Expected Frequency for
Contingency Tables
grand totalgrand total
(probability of a cell)
E= (row total) (column total)(grand total)
I th t f i i d d t f h th th
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38Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Is the type of crime independent of whether thecriminal is a stranger?
Stranger
Acquaintance
or Relative
12
39
379
106
727
642
Homicide Robbery Assault
I th t f i i d d t f h th th
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39Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Row Total
Column Total
Stranger
Acquaintance
or Relative
1118
787
1905
12
39
51
379
106
485
727
642
1369
Homicide Robbery Assault
Is the type of crime independent of whether thecriminal is a stranger?
I th t f i i d d t f h th th
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40Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Row Total
Column Total
E= (row total) (column total)(grand total)
Stranger
Acquaintance
or Relative
Homicide Robbery Assault
Is the type of crime independent of whether thecriminal is a stranger?
1118
787
1905
12
39
51
379
106
485
727
642
1369
I th t f i i d d t f h th th
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41Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Row Total
(29.93)
Column Total
E= (row total) (column total)(grand total)
E= (1118)(5
1) 1905= 29.93
Stranger
Acquaintance
or Relative
Homicide Robbery Assault
Is the type of crime independent of whether thecriminal is a stranger?
1118
787
1905
12
39
51
379
106
485
727
642
1369
I th t f i i d d t f h th th
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42Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Row Total
(29.93)
(21.07)
(284.64)
(200.36)
(803.43)
(565.57)
Column Total
E= (row total) (column total)(grand total)
E= (1118)(5
1) 1905= 29.93 E= (1118)(485)
1905= 284.64
etc.
Stranger
Acquaintance
or Relative
Homicide Robbery Assault
Is the type of crime independent of whether thecriminal is a stranger?
1118
787
1905
12
39
51
379
106
485
727
642
1369
Is the t pe of crime independent of hether the
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43Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
12
39
379
106
727
642
Homicide Robbery Forgery
(29.93)
(21.07)
(284.64)
(200.36)
(803.43)
(565.57
[10.741]Stranger
Acquaintance
or Relative
X2=
(O - E )2E
(O -E )2
EUpper left cell: = = 10.741
(12 -29.93)2
29.93
(E)
(O - E )2
E
Is the type of crime independent of whether thecriminal is a stranger?
Is the type of crime independent of whether the
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44Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
12
39
379
106
727
642
Homicide Robbery Forgery
(29.93)
(21.07)
[15.258]
(284.64)
[31.281]
(200.36)
[44.439]
(803.43)
[7.271]
(565.57)
[10.329]
[10.741]Stranger
Acquaintance
or Relative
X2=
(O - E )2E
(O -E )2
EUpper left cell: = = 10.741
(12 -29.93)2
29.93
(E)
(O - E )2
E
Is the type of crime independent of whether thecriminal is a stranger?
Is the type of crime independent of whether the
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45Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
12
39
379
106
727
642
Homicide Robbery Forgery
(29.93)
(21.07)
[15.258]
(284.64)
[31.281]
(200.36)
[44.439]
(803.43)
[7.271]
(565.57)
[10.329]
[10.741]Stranger
Acquaintance
or Relative
X2=
(O - E )2E
(E)
(O - E )2
E
Is the type of crime independent of whether thecriminal is a stranger?
Test Statistic X2= 10.741 + 31.281 + ... + 10.329 =
119.319
Test Statistic X2
= 119 319
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46Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Test Statistic X=119.319
with = 0.05 and (r -1) (c-1) = (2 -1) (3 -1) = 2 degrees offreedomCritical Value X
2=5.991 (from Table A-4)
Test Statistic X2= 119 319
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47Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Test Statistic X=119.319
with = 0.05 and (r -1) (c-1) = (2 -1) (3 -1) = 2 degrees offreedom
0
= 0.05
X2= 5.991
RejectIndependence
Critical Value X2=5.991 (from Table A-4)
Sample data: X2=119.319
Fail to RejectIndependence
Test Statistic X2= 119 319
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48Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Test Statistic X=119.319
with = 0.05 and (r -1) (c-1) = (2 -1) (3 -1) = 2 degrees offreedom
0
= 0.05
X2= 5.991
RejectIndependence
Critical Value X2=5.991 (from Table A-4)
Reject independence
Sample data: X2=119.319
Fail to RejectIndependence
Test Statistic X2= 119 319
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49Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Test Statistic X=119.319
with = 0.05 and (r -1) (c-1) = (2 -1) (3 -1) = 2 degrees offreedom
0
= 0.05
X2= 5.991
RejectIndependence
Critical Value X2=5.991 (from Table A-4)
Reject independence
Sample data: X2=119.319
Fail to RejectIndependence
Claim: The type of crime and knowledge of criminal are independentHo: The type of crime and knowledge of criminal are independentH1: The type of crime and knowledge of criminal are dependent
Test Statistic X2= 119 319
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50Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Test Statistic X=119.319
with = 0.05 and (r -1) (c-1) = (2 -1) (3 -1) = 2 degrees offreedom
It appears that the type of crime andknowledge of the criminal are related.
0
= 0.05
X2= 5.991
RejectIndependence
Critical Value X2=5.991 (from Table A-4)
Reject independence
Sample data: X2=119.319
Fail to RejectIndependence
Relationships Among Components in X2 Test
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51Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Figure 10-8
Relationships Among Components in XTest
of Independence
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52Chapter 10. Section 10.1 and 10.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Definition
Test of Homogeneity
test the claim that di f ferent popu lat ions
have the same proportions of somecharacteristics
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How to distinguish between a
test of homogeneity and a
test for independence:
Werepredetermined
sample sizesused for different populations (test of
homogeneity), or was one big sample
drawn so both row and column totalswere determined randomly (test of
independence)?
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