Question 2 10 marks The file GW Data.mwx can be used to produce a contingency table giving the GWTC run and the M1 source mass range. The data values for runs 1 and 2 have been combined. This contingency table is shown in Table 1. A copy of this contingency table can be found in the file M1 contingency.mwx. Table 1 M1 source masses for GW events in runs 1 and 2 combined and run 3 M1 source range (Mo) 0-20 GWTC runs 1 and 2 12 GWTC run 3 10 5 20-30 30-40 40+ - 16 11 20 9 Note that for the first three ranges, the upper limit of each M1 source range is not included in the range itself. For example, the range 20-30 includes values that are greater than or equal to 20, and less than 30. (a) What is the probability that a randomly selected GW event has a mass in the range 30 or over? (ie. that they are in the 30-40 or 40+ mass ranges) [1] (b) Given that a randomly selected GW event is in GWTC run 3, what is the probability that the M1 mass is 30 or over? [1] (c) Is the event that a randomly selected GW event has a M1 source range of 30 or over statistically independent from the event that a randomly selected GW event is in GWTC run 3? Justify your answer. [1] Suppose a researcher is interested in using this table to answer the following question: Is there a relationship between GWTC run and M1 source mass? (d) Write down suitable null and alternative hypotheses. Do not quote the researcher's question verbatim; instead use appropriate statistical terminology to express your hypothesis. [1] (e) Calculate by hand the expected value for GWTC runs 1 and 2 for the M1 source mass range 20-30'; show your working. [1] (f) Using Minitab, or otherwise, carry out the X2 test for contingency tables and interpret the results. You should explicitly state: the value of the test statistic the degrees of freedom the p-value, or the values of CV5 and CV1 your conclusion from the test. [4] (g) Should there be any reservations about the use of this test here? Justify your answer. [1]