質問: problem 1 the director of gainesville sun newspaper is studying the relationship between the type of... problem 1 the director of gainesville sun newspaper is studying the relationship between the type of com- munity in which a subscriber resides and the section of the newspaper he or she reads first. for a sample of readers, she collected the sample information in the following table: city suburb rural national news sports 170 124 120 130 130 9 0 comics 90 100 88 a. at the 95% confidence level, can we conclude that there is a relationship between the type of community where the person resides and the section of the paper read first, i.e. are they dependent? b. what is the largest chi-square value? how would you interpret this? c. is the number of people that read national news in rural areas greater or less than expected? d. is the number of people that read comics in cities greater or less than expected?
回答: the contingency table is given as follows: observed frequency national news sports comics total city 32 30 28 90 suburb 74 64 42 180 rural 110 25 15 150 total 216 119 85 420 null hypothesis: ho – the type of community and section of newspaper are independent alternative hypothesis: h1 – the type of community and section of newspaper are dependent determining test statistics χ2test = σ[(f0 – fe)2/fe] where, f0 = observed frequency of the category fe = expected frequency of the category the expected frequency of the category is calculated as follows: fe = (∑rf0)(∑cf0)/(n) ∑rf0 = total observed frequency of respective row ∑cf0 = total observed frequency of respective column n = total number of observation = 420 expected frequency for city resident and national news subscriber category = (90)(216)/(420) = 46.29 similarly expected frequencies for the remaining category are as follows: expected frequency national news sports comics total city 46.29 25.50 18.21 90 suburb 92.57 51.00 36.43 180 rural 77.14 42.50 30.36 150 total 216 119 85 420 contribution of category to chi-square test is calculated as follows: contribution = (f0 – fe)2/fe for city and national news category the contribution is calculated as follows: contribution = (32 – 46.29)2/(46.29) = 4.41 similarly, contributions of other categories are obtained as follows: contribution national news sports comics total city 4.41 0.79 5.26 10.46 suburb 3.73 3.31 0.85 7.89 rural 13.99 7.21 7.77 28.97 total 22.13 11.31 13.88 47.32 test statistics χ2test = σ[(f0 – fe)2/fe] value of test statistics = χ2test = 47.32 critical value degree of freedom = (r – 1)(c – 1) = (3 – 1)(3 – 1) = 4 significance level = 1 – 0.95 = 0.05 from table of chi-square, χ2critical = 9.488 using the critical value approach, the decision rule is to reject the null hypothesis if χ2test >= χ2critical since, χ2test (47.32) >= χ2critical (9.488), we reject null hypothesis that the type of community and the section of the newspaper subscriber reads are independent at a = 0.05. from table, χ2test = 60.44 and degree of freedom = 4, p-value < 0.005 since p-value < a (0.05), reject null hypothesis. thus, there is sufficient data to conclude that there is relationship between the type of community and the section of the newspaper subscriber reads at 95% confidence level.