Question 2 (50 marks) Collusion occurs in online exams when multiple students share inputs and or outputs regarding exam questions before and or during the examination. Students may collude on the belief that doing so would give them a higher mark or a higher likelihood of passing a module. You are asked to examine whether such possible advantages existed in the imaginary case of one exam in one university. The sample size is 112 students. The variables are as listed in Table 2 below. The variables Mark and Pass are calculated for all students as if collusion carried no penalties. Table 2: Variable definitions and descriptive statistics Variable Definition Mean SD Mark Exam mark out of 100 61.34 17.22 Pass 1 = module passed; 0 = module failed 0.80 0.40 Colluder 1 = student colluded; 0.31 0 = student did not collude 0.47 Male 1 = male student; 0 = female student 0.55 0.50 a. How many students failed the module? How many students colluded? How many female students are in the sample? (5 marks) The first claim that you would like to explore is this: Colluding students on average have a higher exam mark. To this end, you have estimated a regression equation of Mark as a function of Colluder. The estimates are shown in Table 3 below, which you should use to help answer questions b through d. Table 3: Regression estimates of Mark Dependent variable: Mark R Square 0.0 Coefficients Standard Error Intercept 61.23 2.15 Colluder 0.34 3.10 b. Calculate the 95% confidence interval for the Intercept. What information does this confidence interval convey? (5 marks)c. What is the average mark among the sampled students that colluded? (5 marks) d. Do the estimates support the claim that colluding students on average have a higher exam mark? Use a 95% confidence interval. (5 marks) [3] e. If you wanted to use a technique other than linear regression, then which technique discussed during the module would allow you to examine whether statistically meaningful differences existed between the marks of colluding and other students? (5 marks) The second claim that you would like to explore is this: Colluding students are more likely to pass the module. To this end, you have estimated a regression equation of Pass as a function of Colluder. The estimates are shown in Table 4 below, which you should use to help answer questions f through h. Table 4: Regression estimates of Pass Dependent variable: Pass R Square 0.02 Coefficients |Standard Error Intercept 0.7 0.05 Colluder 0.12 0.07 f. Among the sampled students who did not collude, what proportion failed the module? (5 marks) g. Do the estimates support the claim that colluding students were more likely to pass the module? Use the critical value approach; assume a one-tailed test and an alpha of 0.05. (5 marks) h. If collusion carried no penalties, then how many colluding students in the sample would have passed the module? (5 marks)Might there have been a gender difference in the likelihood of colluding? To explore this question, you have estimated a regression equation of Colluder as a function of Male. The estimates are shown in Table 5 below, which you should use to help answer questions i and j. Table 5: Regression estimates of Colluder Dependent variable: Colluder R Square 0.000 Coefficients Standard Error Intercept 0.34 0.07 Male -0.05 0.09 i. How many female students in the sample colluded? (5 marks) j. Do the estimates suggest a gender difference in the likelihood of colluding? Use the critical value approach and assume an alpha of 0.05