Notes Use automated RStudio routines to produce the output need to answer the questions. Report (copy/paste) your RStudio output from the console to support your answers. There is no need to check your answers by manual (formula-based) calculation, but you can if you wish (and this might give you partial credit if your RStudio output/command is incorrect). Include your R code at the end of your assignment; failure to include this (or substantial parts of it) may incur a penalty of up to five marks. Conduct all hypothesis tests at the 5% level of significance unless otherwise stated. When writing up conclusions from hypothesis-testing results, please answer in the context of what's being investigated, not merely in terms of whether the null is rejected or not. Where asked for critical values, use RStudio and report the commands used.

Data description and background (data file: marks.csv) A researcher obtained data on 400 BCom students' recent exam marks on level-1 and level-2 statistics subjects. (The level-1 subject is the main prerequisite for the level-2 subject; think of introductory and intermediate statistics.) On the level-2 course, students were allowed to complete their assignments in groups of up to two (i.e., pairs) or individually (alone). Of the 400 students, half (200) completed their assignments in groups of two (pairs) and the other half (200) completed their assignments alone. The researcher is primarily interested in whether assignment-pairing students performed better in the level-2 exam than students who did their assignments individually (i.e., assignment-nonpairing students). The data file contains information on the variables listed below. XNGi: level-2 exam mark for assignment-nonpairing student i; XGi: level-2 exam mark for assignment-pairing student i; PXNG:: level-1 (prerequisite) exam mark for assignment-nonpairing student i; PXGi: level-1 (prerequisite) exam mark for assignment-pairing student i; The researcher includes the level-1 (prerequisite) exam marks data to investigate whether students who did well in the prerequisite-subject's exam also did well on the level-2 exam (as one would expect). All exam marks are out of 100. Note: no tests of normality are needed because we have 200 observations on both assignment- pairing and nonpairing students. (See lecture 2.)Question 1 (15 marks; 5 per part) Here, we investigate whether the variances of assignment-pairing students' level-2 exam marks and assignment-nonpairing students' level-2 exam marks differ. (a) Specify suitable null and alternative hypotheses for the test and define your notation. (b) Compute an appropriate test statistic and report appropriate critical value(s) for the test (latter just for practice!). (c) Based on the p-value from the RStudio output, what do you conclude? Question 2 (15 marks; 5 per part) Here, we investigate whether assignment-pairing level-2 students average higher level-2 exam marks than assignment-nonpairing students. (a) Specify suitable null and alternative hypotheses for the test and define your notation. (b) Taking account of your result from question 1, compute the p-value for testing the hypotheses specified in part (a) of this question. (c) What do you conclude? Question 3 (15 marks) In this question, we use data on assignment-pairing students only to investigate whether there is evidence that above-average performers on the level-1 exam do better in the level-2 exam (as one would expect). To do this, follow the instructions below. . Use only the data on the students who paired for assignments. Create two new data frames: one for assignment-pairing students whose level-1 exam mark is above the median (among assignment-pairing students) and one for assignment-pairing students whose level-1 exam mark is below the median. (Note that there are 98 students in the former category and 99 students in the latter category; three have the median mark.) Call these two groups 'quant-skilled' (QS) and 'quant-unskilled' (QU), respectively. Having done the above in RStudio (tutorial 1 should help), perform the test described below, and explain what conclusion you draw from the p-value for the test. Test whether (assignment-pairing) QS students do better (on average) on the level-2 exam than do (assignment-pairing) QU students. (Assume unequal population variances for this test.)