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##### M ARIO F . T RIOLA

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**STATISTICS**ELEMENTARY Section 10-3 Contingency Tables: Independence and Homogeneity MARIO F. TRIOLA EIGHTH EDITION**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.) 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. Definition**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**1. The sample data are randomly selected.**2. The null hypothesis H0 is the statement that the row and column variables are independent; the alternative hypothesis H1 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.) Assumptions**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 Test of IndependenceTest Statistic**E =**• (row total) (column total) • (grand total) • Total number of all observed frequencies in the table**H0: The row variable is independent of the column**variable H1: 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. Tests of Independence**Expected Frequency for Contingency Tables**• row total column total E= • • • grand total • grand total • grand total**Expected Frequency for Contingency Tables**• row total column total E= • • • grand total • grand total • grand total n • p • (probability of a cell)**Expected Frequency for Contingency Tables**• row total column total E= • • • grand total • grand total • grand total n • p • (probability of a cell)**E=**• (row total) (column total) • (grand total) Expected Frequency for Contingency Tables • row total column total E= • • • grand total • grand total • grand total n • p • (probability of a cell)**Assault**Robbery Homicide 12 39 379 106 727 642 Is the type of crime independent of whether the criminal is a stranger? Stranger Acquaintance or Relative**12**39 51 379 106 485 727 642 1369 1118 787 1905 Is the type of crime independent of whether the criminal is a stranger? Assault Row Total Robbery Homicide Stranger Acquaintance or Relative Column Total**12**39 51 379 106 485 727 642 1369 1118 787 1905 Is the type of crime independent of whether the criminal is a stranger? Assault Row Total Robbery Homicide Stranger Acquaintance or Relative Column Total • (row total) (column total) E= • (grand total)**12**39 51 379 106 485 727 642 1369 1118 787 1905 Is the type of crime independent of whether the criminal is a stranger? Assault Row Total Robbery Homicide Stranger Acquaintance or Relative (29.93) Column Total • (row total) (column total) E= • (grand total) • (1118)(51) E= = 29.93 • 1905**12**39 51 379 106 485 727 642 1369 1118 787 1905 Is the type of crime independent of whether the criminal is a stranger? Assault Row Total Robbery Homicide Stranger Acquaintance or Relative (29.93) (284.64) (803.43) (565.57) (21.07) (200.36) Column Total • (row total) (column total) E= • (grand total) • (1118)(485) E= = 284.64 • (1118)(51) E= = 29.93 • 1905 • 1905 etc.**Is the type of crime independent of whether the criminal is**a stranger? X2= (O - E )2 E Robbery Homicide Forgery (E) 12 39 379 106 727 642 (803.43) (284.64) (29.93) (O - E )2 Stranger Acquaintance or Relative [10.741] E (21.07) (565.57 (200.36) (O -E )2 (12 -29.93)2 Upper left cell: = = 10.741 E 29.93**Is the type of crime independent of whether the criminal is**a stranger? X2= (O - E )2 E Robbery Homicide Forgery (E) 12 39 379 106 727 642 (803.43) [7.271] (284.64) [31.281] (29.93) (O - E )2 Stranger Acquaintance or Relative [10.741] E (21.07) [15.258] (565.57) [10.329] (200.36) [44.439] (O -E )2 (12 -29.93)2 Upper left cell: = = 10.741 E 29.93**Is the type of crime independent of whether the criminal is**a stranger? X2= (O - E )2 E Robbery Homicide Forgery (E) 12 39 379 106 727 642 (803.43) [7.271] (284.64) [31.281] (29.93) (O - E )2 Stranger Acquaintance or Relative [10.741] E (21.07) [15.258] (565.57) [10.329] (200.36) [44.439] X2= 10.741 + 31.281 + ... + 10.329 = 119.319 Test Statistic**X2=119.319**Test Statistic with = 0.05 and (r -1) (c -1) = (2 -1) (3 -1) = 2degrees of freedom X2=5.991 (from Table A-4) Critical Value**X2=119.319**Test Statistic with = 0.05 and (r -1) (c -1) = (2 -1) (3 -1) = 2degrees of freedom X2=5.991 (from Table A-4) Critical Value Fail to Reject Independence Reject Independence = 0.05 0 X2 = 5.991 Sample data: X2=119.319**X2=119.319**Test Statistic with = 0.05 and (r -1) (c -1) = (2 -1) (3 -1) = 2degrees of freedom X2=5.991 (from Table A-4) Critical Value Fail to Reject Independence Reject Independence = 0.05 Reject independence 0 X2 = 5.991 Sample data: X2=119.319**X2=119.319**Test Statistic with = 0.05 and (r -1) (c -1) = (2 -1) (3 -1) = 2degrees of freedom X2=5.991 (from Table A-4) Critical Value Fail to Reject Independence Reject Independence = 0.05 Reject independence 0 X2 = 5.991 Sample data: X2=119.319 Claim: The type of crime and knowledge of criminal are independent Ho : The type of crime and knowledge of criminal are independent H1 : The type of crime and knowledge of criminal are dependent**X2=119.319**Test Statistic with = 0.05 and (r -1) (c -1) = (2 -1) (3 -1) = 2degrees of freedom X2=5.991 (from Table A-4) Critical Value Fail to Reject Independence Reject Independence = 0.05 Reject independence 0 X2 = 5.991 Sample data: X2=119.319 It appears that the type of crime and knowledge of the criminal are related.**Relationships Among Components in X2 Test of Independence**Figure 10-8**Test of Homogeneity**test the claim that different populations have the same proportions of some characteristics Definition**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)? How to distinguish between a test of homogeneity and a test for independence: