Question

Problem 3 Independent random samples of American adults in urban, suburban, and rural regions, are asked “Do you support or oppose requiring background checks for all gun buyers?" Here are the results. Region Suburban Urban Rural Total Support 335 348 318 1001 Response Oppose 35 23 50 | 108 Total 370 371 368 | 1109 At the 5% significance level, do the data provide sufficient evidence to conclude that a difference exists in the proportions of supporters among the three regions?

Answer 1

The null and alternative hypotheses are; Null hypothesis: There does not exists a difference in the proportions of supporters among the three regions. Alternative hypothesis: There exists a difference in the proportions of supporters among the three regions. The row and columns total are; Row Totals Support Oppose Urban 835 25 Suburban 348 23 Column and Row Totals Rural 1318 50 1001 108 Column Totals 370 371 388 1109 (Grand Total) To find the chi-square test statistic, first we need to find the expected frequency for each of the observed frequency. Total of Support x Total of Urban E(Support Urban) = Grand Total 1001x 370 1109 = 333.97

Similarly, all the expected frequencies are calculated which is shown in the table below; L l Support Oppose Results Urban Suburban Rural 335 (333.07) 10.001 348 (334.87) [0.511 318 (332.16) (0.80 35 (30.03) (0.03) 23 (30.13) [4.77] 50 (35.84) (5.00) Row Totals 1001 108 Column Totals 3 70 1109 (Grand Total)

The chi-square test statistic can be obtained as: (0-E)? X =< E (335–333.97) (348–334.87)? (50-35.84)? 333.97 334.87 * ** 35.84 = 11.5195. The degrees of freedom for the critical value is, df =(r-1)(c-1) = (2-1)(3-1) = 2 The level of significance is a = 0.05. In the chi-square distribution table, locate 2 degrees of freedom column and xe. in level of significance row. The intersecting value of row and column is 5.991. The obtained critical value is less than the chi-square test statistic, that is, 5.991<11.5195. So, the null hypothesis is rejected. Therefore, it can be concluded that there exists a difference in the proportions of supporters among the three regions, that is, urban, suburban and rural.