2. (4 marks) A group of researchers were interested in comparing lengths of patient hos- pital stays after surgery, across two hospitals. They constructed the following two-way table, using data from several months: 2 Length of Stay Less than 3 days 3 days or more Total 750 250 Hospital 1,000 800 200 1,000 Total 1.550 450 2.000 (a) By hand, calculate the conditional probability that a patient stays less than 3 days, for each of the two hospitals. Based on these values, which hospital appears to have shorter hospital stays? (b) However, after collecting further data on patients, the researchers discovered that a possible confounding factor may be severity of the surgery. The following three- way table displays the relationship between hospital and length of stay, for two categorizations of severity: High-Severity Surgeries Length of Stay Less than 3 days 3 days or more Total 500 225 Hospital A 725 B 150 100 250 Total 650 325 975 Low-Severity Surgeries Length of Stay Less than 3 days 3 days or more Total Hospital A 250 25 275 B 650 100 750 Total 900 125 1,025 Create a three-way table of this data in R. Print the table. (Hint: Recall that, if you wish to force R to keep the ordering of the rows and columns the tray you want them, you can define the categorical variables as factors, and specify the level ordering using levels=.. .) (c) Use R to compute the conditional probability of a patient staying for less than 3 days, for each of the two hospitals, within each severity level separately. Explain what you observe. (d) Use R to conduct the Mantel-Haenszel test for association between hospital and length of stay, when controlling for severity of surgery. Report the final conclusion. 3