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Inference for Categorical Data 297
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The correct answer is (b) The expected values for the cells are exactly equal to the 25 observed values (eg, for the 1st row, 1st column, Exp = (8) = 5 , so X 2 must 40 equal 0 the variables are independent, and are not related
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The correct answer is (e) The expected values for this two-way table are given by the 125 275 matrix: 125 275 Then, X2= (15 12 5)2 ( 25 27 5)2 (10 12 5)2 + + 12 5 27 5 12 5 2 (30 27 5) + = 1 45 27 5
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The correct answer is (c) In a 2 test for independence, we are interested in whether or not two categorical variables, measured on a single population, are related In a 2 test for homogeneity of proportions, we are interested in whether two or more populations have the same proportions for each of the values of a single categorical variable The correct answer is (e) Using Ethnicity as the row variable, there are five rows (r = 5) and four columns (c = 4) The number of degrees of freedom for an r c table is (r 1)(c 1) In this question, (5 1)(4 1) = 4 3 = 12 The correct answer is (c) In III, the expected count for each category in a goodness-offit test is found by multiplying the proportion of the distribution of each category by the sample size The expected count for a test of independence is found by multiplying the row total by the column total and then dividing by n
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1 I Let p1 = the proportion of 0 s, p2 = the proportion of 1 s, p3 = the proportion of 2 s, p4 = the proportion of 3 s, and p5 = the proportion of 4 s H0: p1 = 024, p2 = 041, p3 = 027, p4 = 007, p5 = 001 HA: Not all of the proportions stated in H0 are correct II We will use a chi-square goodness-of-fit test The observed and expected values for the 500 trials are shown in the following table:
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(024)(500) = 120 (041)(500) = 205 (027)(500) = 135 (007)(500) = 35 (001)(500) = 5 Continued
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298 U Step 4 Review the Knowledge You Need to Score High
We note that all expected values are at least 5, so the conditions necessary for the chi-square test are present III X 2 = (O E )2 (110 120)2 = + E 120 + ( 4 5)2 = 679 , df = 4 010 < P-value 5
< 015 (from Table C) Using the TI-83/84, 2 cdf(679,1000,40)= 0147 IV The P-value is greater than any commonly accepted significance level Hence, we do not reject H0 and conclude that we do not have good evidence that the calculator is not correctly generating values from B(4, 03) 2 For a 3 4 two-way table, df = (3 1)(4 1) = 6 0005 < P-value < 001 (from Table C) The finding is significant at the 001 level of significance Using the TI-83/84, P-value = 2 cdf(172,1000,6) = 0009 (a) Is not correct For a given set of observations, they both do produce the same value of chi-square However, they differ in that they are different ways to design a study (b) is correct A test of independence hypothesizes that two categorical variables are independent within a given population A test for homogeneity of proportions hypothesizes that the proportions of the values of a single categorical variable are the same for more than one population (c) Is incorrect It is a reversal of the actual difference between the two designs (d) Is incorrect You always use count data when computing chi-square The expected value of the cell with the frog is 128 (96) = 3977 309
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