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Contingency Tables 10-3

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Definition Contingency Table (or two-way frequency table) Contingency Table (or two-way frequency table) a table in which frequencies correspond to two variables. 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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Definition Contingency Table (or two-way frequency table) Contingency Table (or two-way frequency table) a table in which frequencies correspond to two variables. 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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Contingency Table Stranger Acquaintance or Relative 12 39 379 106 727 642 Homicide Robbery Assault

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Test of Independence Test of Independence tests the null hypothesis that there is no association between the row variable and the column variable. tests the null hypothesis that there is no association between the row variable and the column variable. (The null hypothesis is the statement that the row and column variables are independent.) Definition

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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 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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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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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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(row total) (column total) (grand total) E = Total number of all observed frequencies in the table

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Contingency Table Stranger Acquaintance or Relative 12 39 379 106 727 642 Homicide Robbery Assault

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E = row total column total grand total Expected Frequency for Contingency Tables grand total

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n p E = row total column total grand total Expected Frequency for Contingency Tables grand total (probability of a cell)

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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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Stranger Acquaintance or Relative Homicide Robbery Assault Is the type of crime independent of whether the criminal is a stranger? 12 39 379 106 727 642

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Stranger Acquaintance or Relative Homicide Robbery Assault Is the type of crime independent of whether the criminal is a stranger? 12 39 379 106 727 642 H 0 : Type of crime is independent of knowing the criminal H 1 : Type of crime is dependent with knowing the criminal

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Row Total Column Total Stranger Acquaintance or Relative Homicide Robbery Assault Is the type of crime independent of whether the criminal is a stranger? 1118 787 12 39 51 379 106 485 727 642 1369 H 0 : Type of crime is independent of knowing the criminal H 1 : Type of crime is dependent with knowing the criminal

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Row Total Column Total Stranger Acquaintance or Relative Homicide Robbery Assault Is the type of crime independent of whether the criminal is a stranger? 1118 787 1905 12 39 51 379 106 485 727 642 1369 H 0 : Type of crime is independent of knowing the criminal H 1 : Type of crime is dependent with knowing the criminal

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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? 1118 787 1905 12 39 51 379 106 485 727 642 1369

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Row Total (29.93) Column Total E = (row total) (column total) (grand total) E = (1118)(51) 1905 = 29.93 Stranger Acquaintance or Relative Homicide Robbery Assault Is the type of crime independent of whether the criminal is a stranger? 1118 787 1905 12 39 51 379 106 485 727 642 1369

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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 = 29.93 E = (1118)(485) 1905 = 284.64 etc. Stranger Acquaintance or Relative Homicide Robbery Assault Is the type of crime independent of whether the criminal is a stranger? 1118 787 1905 12 39 51 379 106 485 727 642 1369

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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 X 2 = (O - E ) 2 E E Upper left cell: = = 10.741 (12 -29.93) 2 29.93 (E)(E) (O - E ) 2 E Is the type of crime independent of whether the criminal is a stranger?

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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 X 2 = (O - E ) 2 E E Upper left cell: = = 10.741 (12 -29.93) 2 29.93 (E)(E) (O - E ) 2 E Is the type of crime independent of whether the criminal is a stranger?

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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 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 = 10.741 + 31.281 +... + 10.329 = 119.319 X 2 = 10.741 + 31.281 +... + 10.329 = 119.319

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Test Statistic: X 2 = 119.319 X 2 = 119.319 with = 0.05 and ( r -1) ( c -1) = (2 -1) (3 -1) = 2 degrees of freedom Critical Value: X 2 = 5.991 (from Table A-4)

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0 = 0.05 X 2 = 5.991 Reject Independence Reject independence Sample data: X 2 =119.319 Fail to Reject Independence Test Statistic: X 2 = 119.319 X 2 = 119.319 with = 0.05 and ( r -1) ( c -1) = (2 -1) (3 -1) = 2 degrees of freedom Critical Value: X 2 = 5.991 (from Table A-4)

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0 = 0.05 X 2 = 5.991 Reject Independence Reject independence Sample data: X 2 =119.319 Fail to Reject Independence H o : The type of crime and knowing the criminal are independent H 1 : The type of crime and knowing the criminal are dependent Test Statistic: X 2 = 119.319 X 2 = 119.319 with = 0.05 and ( r -1) ( c -1) = (2 -1) (3 -1) = 2 degrees of freedom Critical Value: X 2 = 5.991 (from Table A-4)

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It appears that the type of crime and knowing the criminal are related. 0 = 0.05 X 2 = 5.991 Reject Independence Test Statistic: X 2 = 119.319 X 2 = 119.319 with = 0.05 and ( r -1) ( c -1) = (2 -1) (3 -1) = 2 degrees of freedom Critical Value: X 2 = 5.991 (from Table A-4) Reject independence Sample data: X 2 =119.319 Fail to Reject Independence

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Figure 10-8 Relationships Among Components in X 2 Test of Independence

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Definition Test of Homogeneity Test of Homogeneity tests the claim that different populations have the same proportions of some characteristics tests the claim that different populations have the same proportions of some characteristics

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Example - Test of Homogeneity 3 74 Taxi has usable seat belt? New York Chicago Pittsburgh Yes No 42 87 2 70 Seat Belt Use in Taxi Cabs Claim: The 3 cities have the same proportion of taxis with usable seat belts H 0 : The 3 cities have the same proportion of taxis with usable seat belts H 0 : The 3 cities have the same proportion of taxis with usable seat belts H 1 : The proportion of taxis with usable seat belts is not the same in all 3 cities H 1 : The proportion of taxis with usable seat belts is not the same in all 3 cities

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Example - Test of Homogeneity 3 74 Taxi has usable seat belt? New York Chicago Pittsburgh Yes No 42 87 2 70 Seat Belt Use in Taxi Cabs Claim: The 3 cities have the same proportion of taxis with usable seat belts H 0 : The 3 cities have the same proportion of taxis with usable seat belts H 0 : The 3 cities have the same proportion of taxis with usable seat belts H 1 : The proportion of taxis with usable seat belts is not the same in all 3 cities H 1 : The proportion of taxis with usable seat belts is not the same in all 3 cities 0 Sample data: X 2 = 42.004 = 0.05 X 2 = 5.991 Fail to Reject homogeneity Reject homogeneity

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Example - Test of Homogeneity 3 74 Taxi has usable seat belt? New York Chicago Pittsburgh Yes No 42 87 2 70 Seat Belt Use in Taxi Cabs Claim: The 3 cities have the same proportion of taxis with usable seat belts H 0 : The 3 cities have the same proportion of taxis with usable seat belts H 0 : The 3 cities have the same proportion of taxis with usable seat belts H 1 : The proportion of taxis with usable seat belts is not the same in all 3 cities H 1 : The proportion of taxis with usable seat belts is not the same in all 3 cities 0 Sample data: X 2 = 42.004 = 0.05 X 2 = 5.991 Fail to Reject homogeneity Reject homogeneity There is sufficient evidence to warrant rejection of the claim that the 3 cities have the same proportion of usable seat belts in taxis; appears from Table Chicago has a much higher proportion.

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