The data summarized in the accompanying table is from a paper. Suppose that the data resulted from classifying each person in a random sample of 46 male students and each person in a random sample of 94 female students at a particular college according to their response to a question about whether they usually eat three meals a day or rarely eat three meals a day. Male Female Usually Eat 3 Meals a Day Male 25 Female 39 (a) Is there evidence that the proportions falling into each of the two response categories are not the same for males and females? Use the x² statistic to test the relevant hypotheses with a significance level of 0.05. Calculate the test statistic. (Round your answer to two decimal places.) x² = [ Rarely Eat 3 Meals a Day Use technology to calculate the P-value. (Round your answer to four decimal places.) P-value= O Yes O No What can you conclude? O Reject Ho. There is not convincing evidence that the proportions falling into the two response categories are not the same for males and females. O Reject Ho. There is convincing evidence that the proportions falling into the two response categories are not the same for males and females. O Fail to reject Ho. There is not convincing evidence that the proportions falling into the two response categories are not the same for males and females. O Fail to reject Ho. There is convincing evidence that the proportions falling into the two response categories are not the same for males and females. 21 (b) Are the calculations and conclusions from part (a) consistent with the accompanying Minitab output? Expected counts are printed below observed counts Chi-Square contributions are printed below expected counts Usually Rarely Total 55 O Yes O No 25 21.03 0.750 39 42.97 0.367 64 Total Chi-Sq = 2.06, DF1, P-Value = 0.1514 21 24.97 0.632 55 51.03 0.309 76 (c) Because the response variable in this exercise has only two categories (usually and rarely), we could have also answered the question posed in part (a) by carrying out a two-sample z test of Ho: P₁ P₂ = 0 versus H: P₁ P₂ = 0, where p, is the proportion who usually eat three meals a day for males and p₂ is the proportion who usually eat three meals a day for females. Minitab output from the two-sample z test is shown below. Using a significance level of 0.05, does the two-sample z test lead to the same conclusion as in part (a)? Test for Two Proportions. Sample Male Female X 25 39 Difference p(1) - p (2) Test for difference 0 (vs not = 0): 2-1.43 F-Value 0.1514 46 94 Sample p 0.543478 0.414894 140 (d) How do the P-values from the tests in parts (a) and (c) compare? Does this surprise you? Explain? O The two P-values are equal when rounded to three decimal places. It is surprising that the P-values are at least similar, since the P-value from the chi-squared test is measuring the probability of getting sample proportions at least as far from the expected proportions as what was observed, given that the proportions who usually eat three meals per day are the same for the two populations and the z-test is measuring the probability of getting sample proportions closer to the expected proportions than what was observed, given that the proportions who usually eat three meals per day are the same for the two populations. O The two P-values are not equal when rounded to three decimal places. It is not surprising that the P-values are different, since the P-value from the chi-squared test is measuring the probability of getting sample proportions at least as far from the expected proportions as what was observed, given that the proportions who usually eat three meals per day are the same for the two populations and the z-test is measuring the probability of getting sample proportions closer to the expected proportions than what was observed, given that the proportions who usually eat three meals per day are the same for the two populations. O The two P-values are very different. It is quite surprising that the P-values are this different, since both measure the probability of getting sample proportions at least as far from the expected proportions as what was observed, given that the proportions who usually eat three meals per day are the same for the two populations. O The two P-values are equal when rounded to three decimal places. It is not surprising that the P-values are at least similar, since both measure the probability of getting sample proportions at least as far from the expected proportions as what was observed, given that the proportions who usually eat three meals per day are the same for the two populations.