Provide R code, output and written interpretation for parts a) to d) of this question. Provide only output that is directly relevant to address each section.

Test for multivariate normality (MVN) by:

a). Provide output from the structure function and describe the structure of the ‘film_2023.txt’ data.
b). Produce and interpret univariate QQ plots, histograms and univariate ShapiroWilks tests of normality for each of the four film thickness variables. What is the default univariate test produced by the mvn function?
c). Produce and interpret perspective and contour plots for the TopRight and TopLeft film t hickness v ariables. W hat is a n i nherent p roblem w ith using these plots to assess MVN?
d). Do the analysis necessary to provide the results of the Mardia, Henze-Zirkler and Royston tests of MVN based on all four film thickness v ariables. Include in your i nterpretation:

• The Chi-Square QQ plot and interpretation.
• Describe how the QQ plot is constructed and its relationship to the univariate normal QQ plots.
• Output and interpretation for the 3 tests.
• What is a key limitation of these MVN statistical tests?

e). If your data does not meet MVN, why might you need to consider the ratio of cases to variables? (This question does not necessarily relate specifically to this particular data set)

NNumber TopRight TopLeft BottomRight BottomLeft
1 807 436 555 267
2 681 176 413 84
3 589 469 462 593
4 1012 1056 764 1118
5 587 612 477 686
6 654 685 672 743
7 448 10 493 25
8 678 182 727 336
9 501 579 455 337
10 234 774 165 620
11 379 724 153 748
12 650 788 337 972
13 865 250 509 639
14 922 86 680 156
15 860 13 714 166
16 620 301 422 480
17 328 277 179 518
18 505 934 261 1147
19 835 56 639 146
20 1131 322 995 182
21 755 256 850 202
22 613 594 606 486
23 227 177 208 65
24 382 622 264 573
25 274 357 60 458
26 655 474 249 218
27 936 517 539 765
28 656 805 518 1068
29 540 841 339 974
30 620 610 338 777
31 454 148 272 366
32 912 387 651 201
33 486 232 313 9
34 795 1110 750 1086
35 555 1440 579 1245
36 555 1123 738 896
37 146 881 272 397
38 80 332 320 11
39 352 400 374 89
40 26 692 29 597
41 250 1408 248 1262
42 118 1283 190 1404
43 350 1075 390 1255
44 842 320 572 461
45 612 206 448 120
46 768 380 617 394
47 649 659 665 655
48 453 917 544 808
49 515 981 683 728
50 529 418 897 182
51 853 51 1093 454
52 626 319 857 141
53 463 856 601 408
54 244 516 335 300
55 410 643 379 729
56 747 0 577 229
57 1141 269 992 158
58 1264 42 1088 94
59 1256 596 1229 571
60 536 439 502 398
61 446 523 418 543
62 508 225 546 184
63 567 488 798 600
64 616 354 870 244
65 686 1038 953 589
66 617 829 830 396
67 771 486 805 191
68 225 107 181 92
69 829 769 455 635
70 1080 146 834 15
71 1168 567 957 800
72 855 784 669 814
73 796 597 617 732
74 842 753 545 857
75 674 390 481 165
76 806 345 598 187
77 675 482 524 851
78 667 1242 425 1410
79 427 964 258 1156
80 291 750 23 921
81 548 517 189 764
82 670 309 229 346
83 352 882 49 958
84 153 1096 156 1226
85 237 880 23 1267
86 301 891 9 1223
87 762 246 514 460
88 505 130 448 175
89 572 265 553 317
90 389 938 355 906
91 44 972 5 915
92 73 1297 20 1255
93 348 454 467 399
94 732 607 631 467
95 685 400 362 453
96 459 1070 141 1075
97 342 784 23 722
98 394 772 103 912
99 393 754 37 1080
100 496 22 315 148
101 560 439 440 468
102 600 1006 641 795
103 146 1086 222 893
104 527 1107 602 950
105 405 636 544 654
106 579 214 584 85
107 371 180 303 100
108 679 515 596 500
109 813 1000 720 779
110 666 566 686 510
111 507 596 331 529
112 561 103 309 54
113 1212 158 859 240
114 861 396 672 643
115 656 727 574 809
116 708 858 565 947
117 548 792 444 867
118 509 303 490 391
119 498 194 410 127
120 765 544 800 430
121 493 837 660 677
122 543 915 764 733
123 35 953 151 686
124 527 264 706 40
125 624 61 738 392
126 391 414 521 180
127 466 884 361 955
128 499 381 274 666
129 824 505 612 759
130 909 219 626 417
131 903 552 601 354
132 741 27 617 33
133 755 958 750 698
134 352 1159 420 888
135 176 946 388 719
136 371 859 710 613
137 322 35 626 222
138 84 197 360 106
139 155 855 384 608
140 360 953 317 704