Tests of Homogeneity and Independence

Tests of Homogeneity and Independence Lecture 52 Sections 14.5 Robb T. Koether Hampden-Sydney College Mon, Apr 26, 2010 Robb T. Koether (Hampden-Sydney College) Tests of Homogeneity and Independence Mon, Apr 26, 2010 1 / 22

Outline 1 Homework Review 2 Degrees of Freedom 3 The Traffic Fatality Example 4 Practice 5 Assignment Robb T. Koether (Hampden-Sydney College) Tests of Homogeneity and Independence Mon, Apr 26, 2010 2 / 22

Homework Review Exercise 14.20, page 949. A certain job in a car assembly plant involves a great deal of stress. A study was conducted to assess whether there is any difference in how men versus women adjust to the stress. A random sample of 25 men and 25 women employed in this job were surveyed. The results are given as follows: Well Adjusted Not Well Adjusted Men 18 7 Women 13 12 At the 1% significance level, test whether the adjustment status is the same for men versus women. Robb T. Koether (Hampden-Sydney College) Tests of Homogeneity and Independence Mon, Apr 26, 2010 4 / 22

Homework Review Solution (1) H 0 : The populations are homogeneous. H 1 : The populations are not homogeneous. Robb T. Koether (Hampden-Sydney College) Tests of Homogeneity and Independence Mon, Apr 26, 2010 5 / 22

Homework Review Solution (1) H 0 : The populations are homogeneous. H 1 : The populations are not homogeneous. (2) α = 0.01. Robb T. Koether (Hampden-Sydney College) Tests of Homogeneity and Independence Mon, Apr 26, 2010 5 / 22

Homework Review Solution (1) H 0 : The populations are homogeneous. H 1 : The populations are not homogeneous. (2) α = 0.01. (3) χ 2 = all cells (O E) 2. E Robb T. Koether (Hampden-Sydney College) Tests of Homogeneity and Independence Mon, Apr 26, 2010 5 / 22

Homework Review Solution (4) Enter the matrix [ 18 7 13 12 ] as matrix A in the TI-83. Robb T. Koether (Hampden-Sydney College) Tests of Homogeneity and Independence Mon, Apr 26, 2010 6 / 22

Homework Review Solution (4) Enter the matrix [ 18 7 13 12 Use STAT > TEST > χ 2 -Test. ] as matrix A in the TI-83. Robb T. Koether (Hampden-Sydney College) Tests of Homogeneity and Independence Mon, Apr 26, 2010 6 / 22

Homework Review Solution (4) Enter the matrix [ 18 7 13 12 Use STAT > TEST > χ 2 -Test. The calculator reports that χ 2 = 2.122. ] as matrix A in the TI-83. Robb T. Koether (Hampden-Sydney College) Tests of Homogeneity and Independence Mon, Apr 26, 2010 6 / 22

Homework Review Solution (4) Enter the matrix [ 18 7 13 12 Use STAT > TEST > χ 2 -Test. The calculator reports that χ 2 = 2.122. ] as matrix A in the TI-83. (5) The calculator also reports that p-value = 0.1452. Robb T. Koether (Hampden-Sydney College) Tests of Homogeneity and Independence Mon, Apr 26, 2010 6 / 22

Homework Review Solution (4) Enter the matrix [ 18 7 13 12 Use STAT > TEST > χ 2 -Test. The calculator reports that χ 2 = 2.122. ] as matrix A in the TI-83. (5) The calculator also reports that p-value = 0.1452. (6) Accept H 0. Robb T. Koether (Hampden-Sydney College) Tests of Homogeneity and Independence Mon, Apr 26, 2010 6 / 22

Homework Review Solution (4) Enter the matrix [ 18 7 13 12 Use STAT > TEST > χ 2 -Test. The calculator reports that χ 2 = 2.122. ] as matrix A in the TI-83. (5) The calculator also reports that p-value = 0.1452. (6) Accept H 0. (7) The adjustment statuses of men and women are the same. Robb T. Koether (Hampden-Sydney College) Tests of Homogeneity and Independence Mon, Apr 26, 2010 6 / 22

Degrees of Freedom Consider the mortality data. 55-64 65-74 75-84 Row Tot Heart disease 207 490 1383 2080 Cancer 321 727 1264 2312 Stroke 33 96 335 464 Respiratory 132 149 363 644 Col Total 693 1462 3345 5500 What if we had only the row and column totals? How many values could we fill in arbitrarily? Robb T. Koether (Hampden-Sydney College) Tests of Homogeneity and Independence Mon, Apr 26, 2010 8 / 22

Degrees of Freedom 55-64 65-74 75-84 Row Tot Heart disease 2080 Cancer 2312 Stroke 464 Respiratory 644 Col Total 693 1462 3345 5500 Robb T. Koether (Hampden-Sydney College) Tests of Homogeneity and Independence Mon, Apr 26, 2010 9 / 22

Degrees of Freedom 55-64 65-74 75-84 Row Tot Heart disease 200 400 2080 Cancer 2312 Stroke 464 Respiratory 644 Col Total 693 1462 3345 5500 Robb T. Koether (Hampden-Sydney College) Tests of Homogeneity and Independence Mon, Apr 26, 2010 10 / 22

Degrees of Freedom 55-64 65-74 75-84 Row Tot Heart disease 200 400 1480 2080 Cancer 2312 Stroke 464 Respiratory 644 Col Total 693 1462 3345 5500 Robb T. Koether (Hampden-Sydney College) Tests of Homogeneity and Independence Mon, Apr 26, 2010 11 / 22

Degrees of Freedom 55-64 65-74 75-84 Row Tot Heart disease 200 400 1480 2080 Cancer 300 800 2312 Stroke 464 Respiratory 644 Col Total 693 1462 3345 5500 Robb T. Koether (Hampden-Sydney College) Tests of Homogeneity and Independence Mon, Apr 26, 2010 12 / 22

Degrees of Freedom 55-64 65-74 75-84 Row Tot Heart disease 200 400 1480 2080 Cancer 300 800 1212 2312 Stroke 464 Respiratory 644 Col Total 693 1462 3345 5500 Robb T. Koether (Hampden-Sydney College) Tests of Homogeneity and Independence Mon, Apr 26, 2010 13 / 22

Degrees of Freedom 55-64 65-74 75-84 Row Tot Heart disease 200 400 1480 2080 Cancer 300 800 1212 2312 Stroke 50 100 464 Respiratory 644 Col Total 693 1462 3345 5500 Robb T. Koether (Hampden-Sydney College) Tests of Homogeneity and Independence Mon, Apr 26, 2010 14 / 22

Degrees of Freedom 55-64 65-74 75-84 Row Tot Heart disease 200 400 1480 2080 Cancer 300 800 1212 2312 Stroke 50 100 314 464 Respiratory 644 Col Total 693 1462 3345 5500 Robb T. Koether (Hampden-Sydney College) Tests of Homogeneity and Independence Mon, Apr 26, 2010 15 / 22

Degrees of Freedom 55-64 65-74 75-84 Row Tot Heart disease 200 400 1480 2080 Cancer 300 800 1212 2312 Stroke 50 100 314 464 Respiratory 143 162 644 Col Total 693 1462 3345 5500 Robb T. Koether (Hampden-Sydney College) Tests of Homogeneity and Independence Mon, Apr 26, 2010 16 / 22

Degrees of Freedom 55-64 65-74 75-84 Row Tot Heart disease 200 400 1480 2080 Cancer 300 800 1212 2312 Stroke 50 100 314 464 Respiratory 143 162 339 644 Col Total 693 1462 3345 5500 Robb T. Koether (Hampden-Sydney College) Tests of Homogeneity and Independence Mon, Apr 26, 2010 17 / 22

The Traffic Fatality Example Example (The Traffic Fatality Example) Are the variables time of day and alcohol-impaired status of the driver independent in fatal traffic accidents? Does the value of one variable appear to influence the proportions of the other variable? 12-3am 3-6 6-9 9-12 12-3pm 3-6 6-9 9-12 Impaired 2496 1221 343 182 332 635 1189 1616 Not Impaired 1337 1100 1507 1421 1906 2203 2337 1924 Test the hypotheses at the 5% level. Show the expected counts. Robb T. Koether (Hampden-Sydney College) Tests of Homogeneity and Independence Mon, Apr 26, 2010 19 / 22

Practice What type of problem is it? p. 967: 37. Robb T. Koether (Hampden-Sydney College) Tests of Homogeneity and Independence Mon, Apr 26, 2010 21 / 22

Practice What type of problem is it? p. 967: 37. p. 967: 38. Robb T. Koether (Hampden-Sydney College) Tests of Homogeneity and Independence Mon, Apr 26, 2010 21 / 22

Practice What type of problem is it? p. 967: 37. p. 967: 38. p. 969: 42. Robb T. Koether (Hampden-Sydney College) Tests of Homogeneity and Independence Mon, Apr 26, 2010 21 / 22

Practice What type of problem is it? p. 967: 37. p. 967: 38. p. 969: 42. p. 971: 45. Robb T. Koether (Hampden-Sydney College) Tests of Homogeneity and Independence Mon, Apr 26, 2010 21 / 22

Assignment Homework Read Section 14.5, pages 951-959. Let s Do It! 14.6, 14.7. (14.6 is an example of Simpson s paradox.) Exercises 23-27, 31, 34, page 959. (Ex. 31 and 34 are examples of Simpson s paradox.) Chapter Review 35-40, 42, 47, page 966. Robb T. Koether (Hampden-Sydney College) Tests of Homogeneity and Independence Mon, Apr 26, 2010 23 / 22