2 Worksheet Name: ________________________________ A 2 goodness-of-fit compares the distribution of observed outcomes for a single categorical variable to the expected outcomes predicted by a probability model to see if the model is viable. One variable, one sample. The null hypothesis is that the probability model is correct. (All faces of the die are equally likely to occur.) A 2 test of homogeneity compares observed distributions for several groups to each other to see if there is evidence of difference among the respective populations. Several groups, but one variable. The null hypothesis is that the two groups are drawn from populations in which the variable has the same model. (The distribution of eye color among boys is the same as it is among girls.) A 2 test of independence cross-categorizes one group on two variables to see if there is an association between them. One sample, but two variables. The null hypothesis is that there is no association between the two variables. (Smoking is independent of level of education.) For # 1-3, state the type of test, the hypotheses, the test statistic, p-value, df, and decision. 1) Polly and Pete go through one page from a telephone book and write down the last digit of 50 numbers. Could the last digit from the phone numbers on this page be used as random digits? Digit: 0 1 2 3 4 5 6 7 8 9 Freq: 7 6 3 2 5 7 2 10 8 0 X2 Goodness of fit test x2 = 18 p-value = .0352 df = 9 telephone Reject Ho Ho: The proportion for each digit 0-9 is the same. Ha: The proportion for at least one category is different. 2) The counseling unit of Woodrock College is interested in the relationship between anxiety level and the need to succeed. A random sample of 200 college freshmen was taken. The freshmen were given tests to measure their anxiety level and their need to succeed. Is there evidence that anxiety level and need to succeed are dependent? Anxiety Level Need High Medium Low High 30 40 5 Medium 17 50 33 Low 3 10 12 X2 Test for Independence x2 = 29.6 p-value = .0000059 df = 4 Reject Ho Ho: Anxiety level and need to succeed are independent for college freshmen. Ha: Anxiety level and need to succeed are not independent. 3) In July 1991 and again in April 2001, the Gallup Poll asked random samples of 1015 adults about their opinions on working parents. The table summarizes responses to the question “Considering the needs of both parents and children, which of the following do you see as the ideal family in today’s society?” Based on these results, do you think there was a 1991 2001 change in people’s attitudes during the 10 years Both work full time 142 131 between these polls? One works full time, other part time 274 244 One works, other works at home One works, other stays home for kids No opinion 152 396 51 173 416 51 X2 test for homogeneity x2 = 4.03 p-value = .4019 to reject Ho Ho: Adult opinions on working parents are the same for the two samples. Ha: Adult opinions on working parents are different for the two samples. df = 4 Fail 4) “Buddies” is a volunteer social service that works with disadvantaged children. Adults, 18 years & older, volunteer from 1 to 6 hours per week to take disadvantaged children to the zoo, to a museum, to a movie, or to some other activity. The program recruits adults from three main groups: college students, non-students living in the inner city, and nonstudents living in the suburbs. A random sample of adult volunteers gave the following information. Is there an association between the number of hours volunteered and the type of volunteer? Hours Volunteered 1-2 3-4 5-6 College student 115 (90.7) 93 (114.6) 47 (49.7) Inner-city resident 88 (104.7) 150 (132.6) 56 (57.3) Suburban resident 95 (102.5) 133 (129.4) 60 (56.1) Assumptions: Ho: The number of hours volunteered and the type of volunteer are independent. Random sample stated Ha: The number of hours volunteered and the type of volunteer are not independent. All expected counts > 5 as shown in table 2 (obs exp)2 16.7 with 4 df exp Chi-square test for independence = .05 p-value = x2cdf(16.7, ∞, 4) = .0022 Since p-value < α, we reject the null hypothesis. There is sufficient evidence to suggest that there is an association between hours volunteered and type of volunteer. 5) In November 1997, the Gallup Organization released the results of a poll on family values. One of the questions asked was: “For you personally, do you think it is necessary or not necessary to have a child at some point in your life in order to feel fulfilled?” Results from adults in different countries appear below. Is there evidence that the proportion of adults who would give each answer is different for each country? U.S. India Mexico Canada Germany Total Yes 460 (616) 930 (616) 610 (616) 590 (616) 490 (616) 3080 No 510 (354) 60 (354) 380 (354) 370 (354) 450 (354) 1770 Undecided 30 (30) 10 (30) 10 (30) 40 (30) 60 (30) 1000 1000 Total 1000 1000 1000 150 5000 Assumptions: Ho: The distribution of answers is the same for each country. Random sample stated Ha: The distribution of answers is different for at least one country. All expected counts > 5 as shown in table 2 (obs exp)2 628.08 with 8 df exp Chi-square test for homogeneity = .05 p-value = x2cdf(628.08, ∞, 8) = 0 Since p-value < α, we reject the null hypothesis. There is sufficient evidence to suggest that the distribution of answers is not the same for at least one country.