CSSSI'SOCJ'STAT 221 Autumn 2021 Name: Collaborators: Student number: Problem Set 4: Using a chi-square test to identify differences in survival rates between different populations Early on the morning of 15 April 1912, the ocean liner RMS Titanic sank in the North Atlantic, on the ship's rst voyage. Tragically, approximately 68% of the ship's passengers and crew perished. The table below describes the frequencies of those who survived the catastrophe and those who perished, separated according to passenger class (1\CSSSISOCJSTAT 221 Autumn 2021 325 285 706 01 885 22 (1) Given this equation, calculate the missing values for the crew to complete the table below, to the second decimal place. Based on the values that you calculate for the crew, did more crew survive than expected, or less? By how much? The chi-squared test is valid only if the expected frequency for all cells is greater than 5, so we can use the test in this case. If you compare the tables of observed frequencies (aid) and expected frequencies (EM) above, you should notice that the observed frequencies of survivors between classes are not exactly what we would expect given the null hypothesis that survival is independent of passenger class. For example, we would have expected approximately 105 l"-class passengers to survive if the null hypothesis were true, yet in reality 203 survived. To measure how different our observed data are from what we would expect under our null hypothesis, we rst need to calculate eight squared Z-scores (or \"squared standardized differences\"), one each for each cell in the table below, given the following equation: 2 22 _ (0L! 5:4) {J _ E! j These scores measure the groups that contribute the most and least to the pattern of disproportionate survivorship. For example, the 1"t class passengers are far more different from expectation than the 2nd class passengers. Squared standardized differences for the crew are missing. 91. 50 7. 31 10. CSSSISOCI'STAT 221 Autumn 2021 (2) Given this equation, calculate the missing values for the crew to complete the table above. Calculate to the second decimal place. Based on the values that you calculate for the crew, are the crew more different from expectation than the l\"t class passengers? Why or why not? To calculate the observed chi-square test statistic, we sum these eight squared standardized differences: x3; = 190.40 To identify a critical value for this test statistic, we will use the chi-square distribution. To do so, we need to calculate this distribution's degrees-of-freedom (d. f.) parameter, which is: (3) d.f.= (number of rows 1) x (number of columns 1) = (Note that the number of rows is the number of groups on the ship, while the number of columns is the number of different survival outcomes 'perish' or 'survive.') To evaluate whether the difference between observed and expected data is significant, you need to identify a critical value 162,.\" based on the d. f. you calculated above, as well as a pre-specied level of signicance. Let's use an exceptionally low level of signicance, a = 0.001 . Choosing such a low it makes it very hard to reject the null hypothesis, implying a high standard for the alternative hypothesis that survivorship varied between classes. Based on the chi-squared table presented on the next page, what is 13m? (4) xczrtr. = (5) Based on x3\" and 1%,.\" above, should you accept or reject the null hypothesis? (6) Depending on your decision about the null hypothesis in Question 5, what does this imply about the relationship between the passengerfcrew status of the individual and their survival rates? In other words, are these two variables associated or unassociated