PXNG	PXG	XNG	XG
76	70	65	73
83	81	94	83
93	40	82	62
68	49	74	70
79	84	83	89
72	44	36	59
86	54	83	79
77	76	76	88
75	78	47	85
68	79	61	92
82	89	83	65
88	66	55	89
78	79	81	88
73	63	60	79
88	82	96	91
94	81	81	100
98	85	77	73
68	93	13	91
88	56	73	81
76	56	73	98
94	93	47	93
83	89	92	92
82	56	35	65
67	76	71	84
97	52	62	87
97	57	55	78
94	88	73	80
76	85	82	71
75	64	86	84
94	70	74	77
74	61	53	77
85	52	82	56
67	61	73	67
87	54	92	57
73	34	87	71
59	43	79	73
64	44	82	84
70	80	88	64
50	49	53	70
70	49	87	65
50	54	43	78
56	71	81	82
49	78	82	93
51	74	75	53
59	59	73	55
51	66	66	78
51	58	87	59
52	85	65	92
53	75	93	73
53	62	88	83
54	60	97	52
56	87	56	83
56	34	66	66
56	50	70	59
56	64	55	64
57	50	80	76
57	44	92	81
58	61	86	70
58	65	85	87
58	86	76	90
59	88	87	83
59	62	64	81
59	84	84	72
61	89	72	94
61	73	94	83
61	92	51	84
61	94	63	98
61	75	85	87
61	91	83	80
61	87	61	85
62	90	79	69
64	60	79	62
64	64	87	84
65	73	76	59
65	91	71	76
65	62	74	79
65	55	89	71
66	91	77	97
67	85	72	73
67	64	88	80
67	96	86	90
67	61	79	94
68	77	80	89
68	66	68	63
68	68	85	85
69	82	87	69
70	93	96	98
70	89	85	84
71	66	83	83
72	84	92	87
73	85	84	86
73	79	67	82
74	76	90	78
74	78	56	52
74	87	100	85
75	70	82	91
75	93	96	92
75	87	57	84
76	89	81	90
76	82	88	91
76	74	82	72
76	85	70	76
77	85	74	89
77	84	92	80
78	74	82	74
78	97	82	94
78	65	80	58
78	51	94	82
79	84	64	88
80	98	74	91
60	82	88	92
61	72	70	85
62	71	88	76
63	66	70	67
64	86	89	93
65	94	93	91
66	84	56	86
67	72	58	86
68	50	48	71
69	79	36	92
70	54	41	57
71	84	75	69
72	70	58	75
73	86	59	95
74	80	61	87
75	89	54	76
76	68	84	76
77	89	61	82
78	59	44	79
79	84	90	90
80	87	78	90
81	79	31	78
82	91	41	92
83	59	58	75
84	76	38	80
85	63	34	77
76	73	81	82
58	79	38	73
60	68	74	69
67	50	57	61
66	49	65	77
75	95	61	98
57	92	51	80
66	53	52	70
48	82	44	86
93	82	44	76
56	51	64	76
58	71	45	92
48	75	80	73
36	65	61	76
41	84	47	82
75	83	56	94
58	84	78	89
59	77	61	83
61	81	71	54
54	86	57	84
84	49	44	66
61	88	53	93
44	82	86	94
90	93	76	86
78	96	65	88
31	77	77	80
41	70	84	77
58	67	65	82
38	58	84	72
34	92	62	87
81	94	78	83
38	80	94	84
74	74	85	78
57	97	65	78
65	88	67	82
61	86	80	86
51	82	94	90
52	69	90	87
44	79	88	63
44	85	61	56
64	61	69	60
45	85	67	84
80	50	53	69
61	95	61	86
47	60	49	70
56	60	56	81
78	68	82	84
61	64	74	82
71	82	66	64
57	80	82	84
62	54	74	38
78	61	83	79
94	90	36	88
61	86	83	64
69	89	76	78
67	72	47	70
53	81	61	77
61	84	83	80
49	78	55	84
56	76	81	90
82	76	60	70
74	89	96	72
66	66	35	76
61	86	71	81

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.)