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1 Chapter 10. Section Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY S TATISTICS Section 10-3 Contingency Tables: Independence and Homogeneity

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2 Chapter 10. Section Triola, Elementary Statistics, Eighth Edition. Copyright 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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3 Chapter 10. Section Triola, Elementary Statistics, Eighth Edition. Copyright 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 two rows and at least two columns.

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4 Chapter 10. Section Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Test of Independence tests the null hypothesis that the row variable and column variable in a contingency table are not related. (The null hypothesis is the statement that the row and column variables are independent.) Definition

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5 Chapter 10. Section Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Assumptions 1. The sample data are randomly selected. 2.The null hypothesis H 0 is the statement that the row and column variables are independent; the alternative hypothesis H 1 is the statement that the row and column variables are dependent. 3. For every cell in the contingency table, the expected frequency E is at least 5. (There is no requirement that every observed frequency must be at least 5.)

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6 Chapter 10. Section Triola, Elementary Statistics, Eighth Edition. Copyright 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. X 2 = ( O - E ) 2 E

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7 Chapter 10. Section Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman (row total) (column total) (grand total) E = Total number of all observed frequencies in the table

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8 Chapter 10. Section Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Tests of Independence H 0 : The row variable is independent of the column variable H 1 : The row variable is dependent (related to) the column variable This procedure cannot be used to establish a direct cause-and-effect link between variables in question. Dependence means only there is a relationship between the two variables.

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9 Chapter 10. Section Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Expected Frequency for Contingency Tables

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10 Chapter 10. Section Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman E = row total column total grand total Expected Frequency for Contingency Tables grand total

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11 Chapter 10. Section Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman n p E = row total column total grand total Expected Frequency for Contingency Tables grand total (probability of a cell)

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12 Chapter 10. Section Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman n p E = row total column total grand total Expected Frequency for Contingency Tables grand total (probability of a cell)

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13 Chapter 10. Section Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman n p E = row total column total grand total Expected Frequency for Contingency Tables grand total (probability of a cell) E = (row total) (column total) (grand total)

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14 Chapter 10. Section Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Is the type of crime independent of whether the criminal is a stranger? Stranger Acquaintance or Relative Homicide Robbery Assault

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15 Chapter 10. Section Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Row Total Column Total Stranger Acquaintance or Relative Homicide Robbery Assault Is the type of crime independent of whether the criminal is a stranger?

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16 Chapter 10. Section Triola, Elementary Statistics, Eighth Edition. Copyright 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 the criminal is a stranger?

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17 Chapter 10. Section Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Row Total (29.93) Column Total E = (row total) (column total) (grand total) E = (1118)(51) 1905 = Stranger Acquaintance or Relative Homicide Robbery Assault Is the type of crime independent of whether the criminal is a stranger?

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18 Chapter 10. Section Triola, Elementary Statistics, Eighth Edition. Copyright 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)(51) 1905 = E = (1118)(485) 1905 = etc. Stranger Acquaintance or Relative Homicide Robbery Assault Is the type of crime independent of whether the criminal is a stranger?

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19 Chapter 10. Section Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Homicide Robbery Forgery (29.93) (21.07) (284.64) (200.36) (803.43) ( [ ] Stranger Acquaintance or Relative X 2 = (O - E ) 2 E E Upper left cell: = = ( ) (E)(E) (O - E ) 2 E Is the type of crime independent of whether the criminal is a stranger?

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20 Chapter 10. Section Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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] [ ] Stranger Acquaintance or Relative X 2 = (O - E ) 2 E E Upper left cell: = = ( ) (E)(E) (O - E ) 2 E Is the type of crime independent of whether the criminal is a stranger?

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21 Chapter 10. Section Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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] [ ] Stranger Acquaintance or Relative X 2 = (O - E ) 2 E (E)(E) E Is the type of crime independent of whether the criminal is a stranger? Test Statistic X 2 = =

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22 Chapter 10. Section Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Test Statistic X 2 = with = 0.05 and ( r -1) ( c -1) = (2 -1) (3 -1) = 2 degrees of freedom Critical Value X 2 = (from Table A-4)

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23 Chapter 10. Section Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Test Statistic X 2 = with = 0.05 and ( r -1) ( c -1) = (2 -1) (3 -1) = 2 degrees of freedom 0 = 0.05 X 2 = Reject Independence Critical Value X 2 = (from Table A-4) Sample data: X 2 = Fail to Reject Independence

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24 Chapter 10. Section Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Test Statistic X 2 = with = 0.05 and ( r -1) ( c -1) = (2 -1) (3 -1) = 2 degrees of freedom 0 = 0.05 X 2 = Reject Independence Critical Value X 2 = (from Table A-4) Reject independence Sample data: X 2 = Fail to Reject Independence

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25 Chapter 10. Section Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Test Statistic X 2 = with = 0.05 and ( r -1) ( c -1) = (2 -1) (3 -1) = 2 degrees of freedom 0 = 0.05 X 2 = Reject Independence Critical Value X 2 = (from Table A-4) Reject independence Sample data: X 2 = Fail to Reject Independence Claim: The type of crime and knowledge of criminal are independent H o : The type of crime and knowledge of criminal are independent H 1 : The type of crime and knowledge of criminal are dependent

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26 Chapter 10. Section Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Test Statistic X 2 = with = 0.05 and ( r -1) ( c -1) = (2 -1) (3 -1) = 2 degrees of freedom It appears that the type of crime and knowledge of the criminal are related. 0 = 0.05 X 2 = Reject Independence Critical Value X 2 = (from Table A-4) Reject independence Sample data: X 2 = Fail to Reject Independence

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27 Chapter 10. Section Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Figure 10-8 Relationships Among Components in X 2 Test of Independence

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28 Chapter 10. Section Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Definition Test of Homogeneity test the claim that different populations have the same proportions of some characteristics

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29 Chapter 10. Section Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman How to distinguish between a test of homogeneity and a test for independence: Were predetermined sample sizes used for different populations (test of homogeneity), or was one big sample drawn so both row and column totals were determined randomly (test of independence)?

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