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