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M ARIO F . T RIOLA

S TATISTICS. E LEMENTARY. Section 10-3 Contingency Tables: Independence and Homogeneity. M ARIO F . T RIOLA. E IGHTH. E DITION. Contingency Table (or two-way frequency table) a table in which frequencies correspond to two variables.

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M ARIO F . T RIOLA

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  1. STATISTICS ELEMENTARY Section 10-3 Contingency Tables: Independence and Homogeneity MARIO F. TRIOLA EIGHTH EDITION

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

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

  8. 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

  9. Expected Frequency for Contingency Tables

  10. Expected Frequency for Contingency Tables • row total column total E= • • • grand total • grand total • grand total

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

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

  13. 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)

  14. 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

  15. 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

  16. 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)

  17. 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

  18. 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.

  19. 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

  20. 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

  21. 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

  22. 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

  23. 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

  24. 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

  25. 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

  26. 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.

  27. Relationships Among Components in X2 Test of Independence Figure 10-8

  28. Test of Homogeneity test the claim that different populations have the same proportions of some characteristics Definition

  29. 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:

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