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Test of Homogeneity

Test of Homogeneity. Lecture 45 Section 14.4 Wed, Apr 19, 2006. Homogeneous Populations. Two distributions are called homogeneous if they exhibit the same proportions within categories.

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Test of Homogeneity

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  1. Test of Homogeneity Lecture 45 Section 14.4 Wed, Apr 19, 2006

  2. Homogeneous Populations • Two distributions are called homogeneous if they exhibit the same proportions within categories. • For example, if two colleges’ student bodies are each 55% female and 45% male, then the distributions are homogeneous.

  3. Example • Suppose a teacher teaches two sections of Statistics and uses two different teaching methods. • At the end of the semester, he gives both sections the same final exam and he compares the grade distributions. • He wants to know if the differences that he observes are significant.

  4. Example • Does there appear to be a difference? • Or are the two sets (plausibly) homogeneous?

  5. The Test of Homogeneity • The null hypothesis is that the populations are homogeneous. • The alternative hypothesis is that the populations are not homogeneous. H0: The populations are homogeneous. H1: The populations are not homogeneous. • Notice that H0 does not specify a distribution; it just says that whatever the distribution is, it is the same in all rows.

  6. The Test Statistic • The test statistic is the chi-square statistic, computed as • The question now is, how do we compute the expected counts?

  7. Expected Counts • Under the assumption of homogeneity (H0), the rows should exhibit the same proportions. • We can get the best estimate of those proportions by pooling the rows. • That is, add the rows (i.e., find the column totals), and then compute the column proportions from them.

  8. Row and Column Proportions

  9. Row and Column Proportions

  10. Row and Column Proportions

  11. Expected Counts • Similarly, the columns should exhibit the same proportions, so we can get the best estimate by pooling the columns. • That is, add the columns (i.e., find the row totals), and then compute the row proportions from them.

  12. Row and Column Proportions

  13. Row and Column Proportions

  14. Row and Column Proportions

  15. Row and Column Proportions Grand Total

  16. Expected Counts • Now apply the appropriate row and column proportions to each cell to get the expected count. • Let’s use the upper-left cell as an example. • According to the row and column proportions, it should contain 60% of 10% of the grand total of 120. • That is, the expected count is 0.60 0.10  120 = 7.2

  17. Expected Counts • Notice that this can be obtained more quickly by the following formula. • In the upper-left cell, this formula produces (72  12)/120 = 7.2

  18. Expected Counts • Apply that formula to each cell to find the expected counts and add them to the table.

  19. The Test Statistic • Now compute 2 in the usual way.

  20. Degrees of Freedom • The number of degrees of freedom is df = (no. of rows – 1)  (no. of cols – 1). • In our example, df = (2 – 1)  (5 – 1) = 4. • To find the p-value, calculate 2cdf(7.2106, E99, 4) = 0.1252. • At the 5% level of significance, the differences are not statistically significant.

  21. TI-83 – Test of Homogeneity • The tables in these examples are not lists, so we can’t use the lists in the TI-83. • Instead, the tables are matrices. • The TI-83 can handle matrices.

  22. TI-83 – Test of Homogeneity • Enter the observed counts into a matrix. • Press MATRIX. • Select EDIT. • Use the arrow keys to select the matrix to edit, say [A]. • Press ENTER to edit that matrix. • Enter the number of rows and columns. (Press ENTER to advance.) • Enter the observed counts in the cells. • Press 2nd Quit to exit the matrix editor.

  23. TI-83 – Test of Homogeneity • Perform the test of homogeneity. • Select STATS > TESTS > 2-Test… • Press ENTER. • Enter the name of the matrix of observed counts. • Enter the name (e.g., [E]) of a matrix for the expected counts. These will be computed for you by the TI-83. • Select Calculate. • Press ENTER.

  24. TI-83 – Test of Homogeneity • The window displays • The title “2-Test”. • The value of 2. • The p-value. • The number of degrees of freedom. • See the matrix of expected counts. • Press MATRIX. • Select matrix [E]. • Press ENTER.

  25. Example • Is the color distribution in Skittles candy the same as in M & M candy? • One package of Skittles: • Red: 12 • Orange: 14 • Yellow: 10 • Green: 10 • Purple: 12

  26. Example • One package of M & Ms: • Red: 8 • Orange: 19 • Yellow: 4 • Green: 8 • Blue: 10 • Brown: 6

  27. The Table

  28. The Table • df = 4 • 2 = 4.787 • p-value = 0.3099

  29. Example • Let’s gather more evidence. Buy a second package of Skittles and add it to the first package. • Second package of Skittles: • Red: 10 • Orange: 13 • Yellow: 15 • Green: 13 • Purple: 7

  30. Example • Buy a second package of M & Ms and add it to the first package. • Second package of M & Ms: • Red: 5 • Orange: 12 • Yellow: 16 • Green: 9 • Blue: 8 • Brown: 8

  31. The Table

  32. The Table • df = 4 • 2 = 2.740 • p-value = 0.6022

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