1 Divisibility

(a). How many numbers from 1 to 1000 are divisible by 3? How many numbers from 1 to 1000 are not divisible by 7?

for (x in 1:1000) {
  if (x%%3 == 0) {
    print(x)
  }
}
## [1] 3
## [1] 6
## [1] 9
## [1] 12
## [1] 15
## [1] 18
## [1] 21
## [1] 24
## [1] 27
## [1] 30
## [1] 33
## [1] 36
## [1] 39
## [1] 42
## [1] 45
## [1] 48
## [1] 51
## [1] 54
## [1] 57
## [1] 60
## [1] 63
## [1] 66
## [1] 69
## [1] 72
## [1] 75
## [1] 78
## [1] 81
## [1] 84
## [1] 87
## [1] 90
## [1] 93
## [1] 96
## [1] 99
## [1] 102
## [1] 105
## [1] 108
## [1] 111
## [1] 114
## [1] 117
## [1] 120
## [1] 123
## [1] 126
## [1] 129
## [1] 132
## [1] 135
## [1] 138
## [1] 141
## [1] 144
## [1] 147
## [1] 150
## [1] 153
## [1] 156
## [1] 159
## [1] 162
## [1] 165
## [1] 168
## [1] 171
## [1] 174
## [1] 177
## [1] 180
## [1] 183
## [1] 186
## [1] 189
## [1] 192
## [1] 195
## [1] 198
## [1] 201
## [1] 204
## [1] 207
## [1] 210
## [1] 213
## [1] 216
## [1] 219
## [1] 222
## [1] 225
## [1] 228
## [1] 231
## [1] 234
## [1] 237
## [1] 240
## [1] 243
## [1] 246
## [1] 249
## [1] 252
## [1] 255
## [1] 258
## [1] 261
## [1] 264
## [1] 267
## [1] 270
## [1] 273
## [1] 276
## [1] 279
## [1] 282
## [1] 285
## [1] 288
## [1] 291
## [1] 294
## [1] 297
## [1] 300
## [1] 303
## [1] 306
## [1] 309
## [1] 312
## [1] 315
## [1] 318
## [1] 321
## [1] 324
## [1] 327
## [1] 330
## [1] 333
## [1] 336
## [1] 339
## [1] 342
## [1] 345
## [1] 348
## [1] 351
## [1] 354
## [1] 357
## [1] 360
## [1] 363
## [1] 366
## [1] 369
## [1] 372
## [1] 375
## [1] 378
## [1] 381
## [1] 384
## [1] 387
## [1] 390
## [1] 393
## [1] 396
## [1] 399
## [1] 402
## [1] 405
## [1] 408
## [1] 411
## [1] 414
## [1] 417
## [1] 420
## [1] 423
## [1] 426
## [1] 429
## [1] 432
## [1] 435
## [1] 438
## [1] 441
## [1] 444
## [1] 447
## [1] 450
## [1] 453
## [1] 456
## [1] 459
## [1] 462
## [1] 465
## [1] 468
## [1] 471
## [1] 474
## [1] 477
## [1] 480
## [1] 483
## [1] 486
## [1] 489
## [1] 492
## [1] 495
## [1] 498
## [1] 501
## [1] 504
## [1] 507
## [1] 510
## [1] 513
## [1] 516
## [1] 519
## [1] 522
## [1] 525
## [1] 528
## [1] 531
## [1] 534
## [1] 537
## [1] 540
## [1] 543
## [1] 546
## [1] 549
## [1] 552
## [1] 555
## [1] 558
## [1] 561
## [1] 564
## [1] 567
## [1] 570
## [1] 573
## [1] 576
## [1] 579
## [1] 582
## [1] 585
## [1] 588
## [1] 591
## [1] 594
## [1] 597
## [1] 600
## [1] 603
## [1] 606
## [1] 609
## [1] 612
## [1] 615
## [1] 618
## [1] 621
## [1] 624
## [1] 627
## [1] 630
## [1] 633
## [1] 636
## [1] 639
## [1] 642
## [1] 645
## [1] 648
## [1] 651
## [1] 654
## [1] 657
## [1] 660
## [1] 663
## [1] 666
## [1] 669
## [1] 672
## [1] 675
## [1] 678
## [1] 681
## [1] 684
## [1] 687
## [1] 690
## [1] 693
## [1] 696
## [1] 699
## [1] 702
## [1] 705
## [1] 708
## [1] 711
## [1] 714
## [1] 717
## [1] 720
## [1] 723
## [1] 726
## [1] 729
## [1] 732
## [1] 735
## [1] 738
## [1] 741
## [1] 744
## [1] 747
## [1] 750
## [1] 753
## [1] 756
## [1] 759
## [1] 762
## [1] 765
## [1] 768
## [1] 771
## [1] 774
## [1] 777
## [1] 780
## [1] 783
## [1] 786
## [1] 789
## [1] 792
## [1] 795
## [1] 798
## [1] 801
## [1] 804
## [1] 807
## [1] 810
## [1] 813
## [1] 816
## [1] 819
## [1] 822
## [1] 825
## [1] 828
## [1] 831
## [1] 834
## [1] 837
## [1] 840
## [1] 843
## [1] 846
## [1] 849
## [1] 852
## [1] 855
## [1] 858
## [1] 861
## [1] 864
## [1] 867
## [1] 870
## [1] 873
## [1] 876
## [1] 879
## [1] 882
## [1] 885
## [1] 888
## [1] 891
## [1] 894
## [1] 897
## [1] 900
## [1] 903
## [1] 906
## [1] 909
## [1] 912
## [1] 915
## [1] 918
## [1] 921
## [1] 924
## [1] 927
## [1] 930
## [1] 933
## [1] 936
## [1] 939
## [1] 942
## [1] 945
## [1] 948
## [1] 951
## [1] 954
## [1] 957
## [1] 960
## [1] 963
## [1] 966
## [1] 969
## [1] 972
## [1] 975
## [1] 978
## [1] 981
## [1] 984
## [1] 987
## [1] 990
## [1] 993
## [1] 996
## [1] 999
for (x in 1:1000){
  if (x %% 7 != 0){
    print(x)
  }
}
## [1] 1
## [1] 2
## [1] 3
## [1] 4
## [1] 5
## [1] 6
## [1] 8
## [1] 9
## [1] 10
## [1] 11
## [1] 12
## [1] 13
## [1] 15
## [1] 16
## [1] 17
## [1] 18
## [1] 19
## [1] 20
## [1] 22
## [1] 23
## [1] 24
## [1] 25
## [1] 26
## [1] 27
## [1] 29
## [1] 30
## [1] 31
## [1] 32
## [1] 33
## [1] 34
## [1] 36
## [1] 37
## [1] 38
## [1] 39
## [1] 40
## [1] 41
## [1] 43
## [1] 44
## [1] 45
## [1] 46
## [1] 47
## [1] 48
## [1] 50
## [1] 51
## [1] 52
## [1] 53
## [1] 54
## [1] 55
## [1] 57
## [1] 58
## [1] 59
## [1] 60
## [1] 61
## [1] 62
## [1] 64
## [1] 65
## [1] 66
## [1] 67
## [1] 68
## [1] 69
## [1] 71
## [1] 72
## [1] 73
## [1] 74
## [1] 75
## [1] 76
## [1] 78
## [1] 79
## [1] 80
## [1] 81
## [1] 82
## [1] 83
## [1] 85
## [1] 86
## [1] 87
## [1] 88
## [1] 89
## [1] 90
## [1] 92
## [1] 93
## [1] 94
## [1] 95
## [1] 96
## [1] 97
## [1] 99
## [1] 100
## [1] 101
## [1] 102
## [1] 103
## [1] 104
## [1] 106
## [1] 107
## [1] 108
## [1] 109
## [1] 110
## [1] 111
## [1] 113
## [1] 114
## [1] 115
## [1] 116
## [1] 117
## [1] 118
## [1] 120
## [1] 121
## [1] 122
## [1] 123
## [1] 124
## [1] 125
## [1] 127
## [1] 128
## [1] 129
## [1] 130
## [1] 131
## [1] 132
## [1] 134
## [1] 135
## [1] 136
## [1] 137
## [1] 138
## [1] 139
## [1] 141
## [1] 142
## [1] 143
## [1] 144
## [1] 145
## [1] 146
## [1] 148
## [1] 149
## [1] 150
## [1] 151
## [1] 152
## [1] 153
## [1] 155
## [1] 156
## [1] 157
## [1] 158
## [1] 159
## [1] 160
## [1] 162
## [1] 163
## [1] 164
## [1] 165
## [1] 166
## [1] 167
## [1] 169
## [1] 170
## [1] 171
## [1] 172
## [1] 173
## [1] 174
## [1] 176
## [1] 177
## [1] 178
## [1] 179
## [1] 180
## [1] 181
## [1] 183
## [1] 184
## [1] 185
## [1] 186
## [1] 187
## [1] 188
## [1] 190
## [1] 191
## [1] 192
## [1] 193
## [1] 194
## [1] 195
## [1] 197
## [1] 198
## [1] 199
## [1] 200
## [1] 201
## [1] 202
## [1] 204
## [1] 205
## [1] 206
## [1] 207
## [1] 208
## [1] 209
## [1] 211
## [1] 212
## [1] 213
## [1] 214
## [1] 215
## [1] 216
## [1] 218
## [1] 219
## [1] 220
## [1] 221
## [1] 222
## [1] 223
## [1] 225
## [1] 226
## [1] 227
## [1] 228
## [1] 229
## [1] 230
## [1] 232
## [1] 233
## [1] 234
## [1] 235
## [1] 236
## [1] 237
## [1] 239
## [1] 240
## [1] 241
## [1] 242
## [1] 243
## [1] 244
## [1] 246
## [1] 247
## [1] 248
## [1] 249
## [1] 250
## [1] 251
## [1] 253
## [1] 254
## [1] 255
## [1] 256
## [1] 257
## [1] 258
## [1] 260
## [1] 261
## [1] 262
## [1] 263
## [1] 264
## [1] 265
## [1] 267
## [1] 268
## [1] 269
## [1] 270
## [1] 271
## [1] 272
## [1] 274
## [1] 275
## [1] 276
## [1] 277
## [1] 278
## [1] 279
## [1] 281
## [1] 282
## [1] 283
## [1] 284
## [1] 285
## [1] 286
## [1] 288
## [1] 289
## [1] 290
## [1] 291
## [1] 292
## [1] 293
## [1] 295
## [1] 296
## [1] 297
## [1] 298
## [1] 299
## [1] 300
## [1] 302
## [1] 303
## [1] 304
## [1] 305
## [1] 306
## [1] 307
## [1] 309
## [1] 310
## [1] 311
## [1] 312
## [1] 313
## [1] 314
## [1] 316
## [1] 317
## [1] 318
## [1] 319
## [1] 320
## [1] 321
## [1] 323
## [1] 324
## [1] 325
## [1] 326
## [1] 327
## [1] 328
## [1] 330
## [1] 331
## [1] 332
## [1] 333
## [1] 334
## [1] 335
## [1] 337
## [1] 338
## [1] 339
## [1] 340
## [1] 341
## [1] 342
## [1] 344
## [1] 345
## [1] 346
## [1] 347
## [1] 348
## [1] 349
## [1] 351
## [1] 352
## [1] 353
## [1] 354
## [1] 355
## [1] 356
## [1] 358
## [1] 359
## [1] 360
## [1] 361
## [1] 362
## [1] 363
## [1] 365
## [1] 366
## [1] 367
## [1] 368
## [1] 369
## [1] 370
## [1] 372
## [1] 373
## [1] 374
## [1] 375
## [1] 376
## [1] 377
## [1] 379
## [1] 380
## [1] 381
## [1] 382
## [1] 383
## [1] 384
## [1] 386
## [1] 387
## [1] 388
## [1] 389
## [1] 390
## [1] 391
## [1] 393
## [1] 394
## [1] 395
## [1] 396
## [1] 397
## [1] 398
## [1] 400
## [1] 401
## [1] 402
## [1] 403
## [1] 404
## [1] 405
## [1] 407
## [1] 408
## [1] 409
## [1] 410
## [1] 411
## [1] 412
## [1] 414
## [1] 415
## [1] 416
## [1] 417
## [1] 418
## [1] 419
## [1] 421
## [1] 422
## [1] 423
## [1] 424
## [1] 425
## [1] 426
## [1] 428
## [1] 429
## [1] 430
## [1] 431
## [1] 432
## [1] 433
## [1] 435
## [1] 436
## [1] 437
## [1] 438
## [1] 439
## [1] 440
## [1] 442
## [1] 443
## [1] 444
## [1] 445
## [1] 446
## [1] 447
## [1] 449
## [1] 450
## [1] 451
## [1] 452
## [1] 453
## [1] 454
## [1] 456
## [1] 457
## [1] 458
## [1] 459
## [1] 460
## [1] 461
## [1] 463
## [1] 464
## [1] 465
## [1] 466
## [1] 467
## [1] 468
## [1] 470
## [1] 471
## [1] 472
## [1] 473
## [1] 474
## [1] 475
## [1] 477
## [1] 478
## [1] 479
## [1] 480
## [1] 481
## [1] 482
## [1] 484
## [1] 485
## [1] 486
## [1] 487
## [1] 488
## [1] 489
## [1] 491
## [1] 492
## [1] 493
## [1] 494
## [1] 495
## [1] 496
## [1] 498
## [1] 499
## [1] 500
## [1] 501
## [1] 502
## [1] 503
## [1] 505
## [1] 506
## [1] 507
## [1] 508
## [1] 509
## [1] 510
## [1] 512
## [1] 513
## [1] 514
## [1] 515
## [1] 516
## [1] 517
## [1] 519
## [1] 520
## [1] 521
## [1] 522
## [1] 523
## [1] 524
## [1] 526
## [1] 527
## [1] 528
## [1] 529
## [1] 530
## [1] 531
## [1] 533
## [1] 534
## [1] 535
## [1] 536
## [1] 537
## [1] 538
## [1] 540
## [1] 541
## [1] 542
## [1] 543
## [1] 544
## [1] 545
## [1] 547
## [1] 548
## [1] 549
## [1] 550
## [1] 551
## [1] 552
## [1] 554
## [1] 555
## [1] 556
## [1] 557
## [1] 558
## [1] 559
## [1] 561
## [1] 562
## [1] 563
## [1] 564
## [1] 565
## [1] 566
## [1] 568
## [1] 569
## [1] 570
## [1] 571
## [1] 572
## [1] 573
## [1] 575
## [1] 576
## [1] 577
## [1] 578
## [1] 579
## [1] 580
## [1] 582
## [1] 583
## [1] 584
## [1] 585
## [1] 586
## [1] 587
## [1] 589
## [1] 590
## [1] 591
## [1] 592
## [1] 593
## [1] 594
## [1] 596
## [1] 597
## [1] 598
## [1] 599
## [1] 600
## [1] 601
## [1] 603
## [1] 604
## [1] 605
## [1] 606
## [1] 607
## [1] 608
## [1] 610
## [1] 611
## [1] 612
## [1] 613
## [1] 614
## [1] 615
## [1] 617
## [1] 618
## [1] 619
## [1] 620
## [1] 621
## [1] 622
## [1] 624
## [1] 625
## [1] 626
## [1] 627
## [1] 628
## [1] 629
## [1] 631
## [1] 632
## [1] 633
## [1] 634
## [1] 635
## [1] 636
## [1] 638
## [1] 639
## [1] 640
## [1] 641
## [1] 642
## [1] 643
## [1] 645
## [1] 646
## [1] 647
## [1] 648
## [1] 649
## [1] 650
## [1] 652
## [1] 653
## [1] 654
## [1] 655
## [1] 656
## [1] 657
## [1] 659
## [1] 660
## [1] 661
## [1] 662
## [1] 663
## [1] 664
## [1] 666
## [1] 667
## [1] 668
## [1] 669
## [1] 670
## [1] 671
## [1] 673
## [1] 674
## [1] 675
## [1] 676
## [1] 677
## [1] 678
## [1] 680
## [1] 681
## [1] 682
## [1] 683
## [1] 684
## [1] 685
## [1] 687
## [1] 688
## [1] 689
## [1] 690
## [1] 691
## [1] 692
## [1] 694
## [1] 695
## [1] 696
## [1] 697
## [1] 698
## [1] 699
## [1] 701
## [1] 702
## [1] 703
## [1] 704
## [1] 705
## [1] 706
## [1] 708
## [1] 709
## [1] 710
## [1] 711
## [1] 712
## [1] 713
## [1] 715
## [1] 716
## [1] 717
## [1] 718
## [1] 719
## [1] 720
## [1] 722
## [1] 723
## [1] 724
## [1] 725
## [1] 726
## [1] 727
## [1] 729
## [1] 730
## [1] 731
## [1] 732
## [1] 733
## [1] 734
## [1] 736
## [1] 737
## [1] 738
## [1] 739
## [1] 740
## [1] 741
## [1] 743
## [1] 744
## [1] 745
## [1] 746
## [1] 747
## [1] 748
## [1] 750
## [1] 751
## [1] 752
## [1] 753
## [1] 754
## [1] 755
## [1] 757
## [1] 758
## [1] 759
## [1] 760
## [1] 761
## [1] 762
## [1] 764
## [1] 765
## [1] 766
## [1] 767
## [1] 768
## [1] 769
## [1] 771
## [1] 772
## [1] 773
## [1] 774
## [1] 775
## [1] 776
## [1] 778
## [1] 779
## [1] 780
## [1] 781
## [1] 782
## [1] 783
## [1] 785
## [1] 786
## [1] 787
## [1] 788
## [1] 789
## [1] 790
## [1] 792
## [1] 793
## [1] 794
## [1] 795
## [1] 796
## [1] 797
## [1] 799
## [1] 800
## [1] 801
## [1] 802
## [1] 803
## [1] 804
## [1] 806
## [1] 807
## [1] 808
## [1] 809
## [1] 810
## [1] 811
## [1] 813
## [1] 814
## [1] 815
## [1] 816
## [1] 817
## [1] 818
## [1] 820
## [1] 821
## [1] 822
## [1] 823
## [1] 824
## [1] 825
## [1] 827
## [1] 828
## [1] 829
## [1] 830
## [1] 831
## [1] 832
## [1] 834
## [1] 835
## [1] 836
## [1] 837
## [1] 838
## [1] 839
## [1] 841
## [1] 842
## [1] 843
## [1] 844
## [1] 845
## [1] 846
## [1] 848
## [1] 849
## [1] 850
## [1] 851
## [1] 852
## [1] 853
## [1] 855
## [1] 856
## [1] 857
## [1] 858
## [1] 859
## [1] 860
## [1] 862
## [1] 863
## [1] 864
## [1] 865
## [1] 866
## [1] 867
## [1] 869
## [1] 870
## [1] 871
## [1] 872
## [1] 873
## [1] 874
## [1] 876
## [1] 877
## [1] 878
## [1] 879
## [1] 880
## [1] 881
## [1] 883
## [1] 884
## [1] 885
## [1] 886
## [1] 887
## [1] 888
## [1] 890
## [1] 891
## [1] 892
## [1] 893
## [1] 894
## [1] 895
## [1] 897
## [1] 898
## [1] 899
## [1] 900
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## [1] 997
## [1] 998
## [1] 999
## [1] 1000

(b). How many numbers from 1 to 1000 are divisible by both 3 and 7? How many numbers from 1 to 1000 are divisible by 3 but not 7?

for (x in 1:1000){
  if (x %%3 ==0 && x %% 7 ==0) {
    print(x)
  }
}
## [1] 21
## [1] 42
## [1] 63
## [1] 84
## [1] 105
## [1] 126
## [1] 147
## [1] 168
## [1] 189
## [1] 210
## [1] 231
## [1] 252
## [1] 273
## [1] 294
## [1] 315
## [1] 336
## [1] 357
## [1] 378
## [1] 399
## [1] 420
## [1] 441
## [1] 462
## [1] 483
## [1] 504
## [1] 525
## [1] 546
## [1] 567
## [1] 588
## [1] 609
## [1] 630
## [1] 651
## [1] 672
## [1] 693
## [1] 714
## [1] 735
## [1] 756
## [1] 777
## [1] 798
## [1] 819
## [1] 840
## [1] 861
## [1] 882
## [1] 903
## [1] 924
## [1] 945
## [1] 966
## [1] 987
for (x in 1:1000){
  if (x %%3 ==0 && x %% 7 != 0 )
    print(x)
}
## [1] 3
## [1] 6
## [1] 9
## [1] 12
## [1] 15
## [1] 18
## [1] 24
## [1] 27
## [1] 30
## [1] 33
## [1] 36
## [1] 39
## [1] 45
## [1] 48
## [1] 51
## [1] 54
## [1] 57
## [1] 60
## [1] 66
## [1] 69
## [1] 72
## [1] 75
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## [1] 81
## [1] 87
## [1] 90
## [1] 93
## [1] 96
## [1] 99
## [1] 102
## [1] 108
## [1] 111
## [1] 114
## [1] 117
## [1] 120
## [1] 123
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## [1] 135
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## [1] 579
## [1] 582
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## [1] 591
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## [1] 603
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## [1] 621
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## [1] 975
## [1] 978
## [1] 981
## [1] 984
## [1] 990
## [1] 993
## [1] 996
## [1] 999

2 Wallis product

(a). Calculate \[ 2\cdot \left(\frac{2}{1}\cdot \frac{2}{3}\right) \left(\frac{4}{3}\cdot \frac{4}{5}\right)\left(\frac{6}{5}\cdot \frac{6}{7}\right) \]

k <- (2*2*2*4*4*6*6)/(3*3*5*5*7)
k
## [1] 2.925714
#(b). Use the help page to find out what `prod()` function does. Define `a <- c(2,4,6)`, `b <- c(1,3,5)`,  `c <- c(3,5,7)`, can you use `a`,`b`,`c` and `prod()` to calculate the result in (a)?'''

#<!-- Please write down your code into the R chunk below -->

#```{r}
a <- c(2,4,6)
b <- c(1,3,5)
c <- c(3,5,7)
prod(a,b,c)
## [1] 75600
w <- (prod(c(2,2,2,4,4,6,6)))/(prod(c(1,3,3,5,5,7)))
w
## [1] 2.925714

(c). (optional) Calculate \[ 2\cdot \left(\frac{2}{1}\cdot \frac{2}{3}\right) \left(\frac{4}{3}\cdot \frac{4}{5}\right)\cdots \left(\frac{100}{99}\cdot \frac{100}{101}\right) \] Replace 100, 99, 101 with larger numbers. For example, calculate \[ 2\cdot \left(\frac{2}{1}\cdot \frac{2}{3}\right) \left(\frac{4}{3}\cdot \frac{4}{5}\right)\cdots \left(\frac{1000}{999}\cdot \frac{1000}{1001}\right) \]

What do you find from the result? 

result <- 2.0
for (t in range(2:1001)){
  left <- (2.*t)/(2.*t-1)
  right <- (2.*t)/(2.*t+1)
  result = result * left * right
}
print(result)
## [1] 2.133334
#Result gets closer to pi/2

2*(prod(seq(2,1000,2),c(2,1000,2))/prod(seq(1,999,2)*seq(3,1001,2)))
## [1] NaN
#This is my attempt to use the prod() function to find the same result. I don't get an answer, but wanted to show some of my thought process out before I switched to iterations.

3 Creating a data frame

(a). Creating a data frame (shown in the orginal homework file) 

City <- c("New York", "New York", "London", "London", "Beijing", "Beijing")
Size <- c("large", "small", "large", "small", "large", "small")
Amount <- c(23,14,22,16,121,56)
df <- data.frame(City, Size, Amount)
df
##       City  Size Amount
## 1 New York large     23
## 2 New York small     14
## 3   London large     22
## 4   London small     16
## 5  Beijing large    121
## 6  Beijing small     56

(b). Extract the rows about the large partical size. Extract the rows with amount larger than 50. 

z <- df[df$Size=="large",]
z
##       City  Size Amount
## 1 New York large     23
## 3   London large     22
## 5  Beijing large    121
a<-df[df$Amount>50,]
a
##      City  Size Amount
## 5 Beijing large    121
## 6 Beijing small     56
library(ggplot2)
ggplot(data=cars)+geom_point(aes(x=speed, y=dist))

?cars

4 Exploring cars data

(a). How many observations and variables are there in the dataset? 50 observations and 2 variables

(b). What do the variables represent, and what is the meaninng of their values? (use the help page)
Speed is the numeric speed in mph and dist is the numeric stopping distance in feet given a correlating speed.

(c). Make a scatterplot of speed vs dist What do you find for the relationship between these two variables? Interpret your findings.
The faster the car is driving, the more feet it takes to come to a complete stop.

(d). (optional) The unit of speed is mph (miles per hour) and the unit of dist is ft (feet). Many countries use km (kilometers) instead of miles and m (meters) insteal of feet. Create a different version of cars data in which the unit of speed is kmph (kilometers per hour) and the unit of dist is m. (1 mile \(\approx\) 1.609 kilometers, 1 foot \(\approx\) 0.305 meter )

5 Scatterplot of mpg data

(a). Make a scatterplot of cyl vs displ. What do you find for the relationship between these two variables? Interpret your findings.

library(ggplot2)
ggplot(data=mpg, aes(x=cyl,y=displ)) + geom_point()

?mpg

The correlation is the more cylinders a vehicle has, the higher engine displacement there is, in liters.

(b). Make a scatterplot of cty vs hwy. What do you find for the relationship between these two variables? Interpret your findings.

ggplot(data=mpg, aes(x=cty,y=hwy)) + geom_point()

It appears that the higher city miles per gallon a vehicle has, a vehicle also has a positive relationship in terms of highway miles per gallon.

(c). Which model is most efficient in terms of the city miles per gallon? Which model is most efficient in terms of the highway miles per gallon?

The 1999 Volkswagen New Beetle is the most efficient in city miles per gallon with 35 city mpg. The 1999 models of the Volkswagen New Beetle and the Jetta is the most efficient in highway miles per gallon with 44 mpg.

6 Calculating the distance between Paris and New York

Suppose you have geographical data and want to calculate the distance between two places on earth, given by their latitude and longitude coordinates. Consider the coordinates for:

You can look up “Great-circle distance” on Wikipedia, and use the spherical law of cosines to find the distance. Let’s do this in four steps. We will use the following common abbreviations:

(a). Create 4 objects phi.paris, phi.ny, lambda.paris, lambda.ny, representing these coordinates. Because New York is located in the West, you have to enter this as a negative value (-74.0060).

phi.paris <- 48.8566
lamda.paris <- 2.3522
phi.ny <- 40.7128
lamda.ny <- (-74.0060)

(b). Convert the 4 coordinates from degrees to radians: \[ radians = degrees \times \frac{\pi}{180}. \]

rad.phi.paris <- (phi.paris)*(pi/180)
rad.lambda.paris <- (lamda.paris)*(pi/180)
rad.phi.ny <- (phi.ny)*(pi/180)
rad.lambda.ny <- (lamda.ny)*(pi/180)

(c). Calculate the central angle (\(\Delta \sigma\)) between both cities, using the spherical law of cosines: \[ \Delta \sigma = \arccos(\sin \phi_1 \sin \phi_2+\cos \phi_1 \cos \phi_2 cos(\Delta \lambda)), \] where: \(\phi_1\) is the latitude of Paris, \(\phi_2\) the latitude of New York, and \(\Delta \lambda\) is the absolute difference between both longitudes. (Hint: For this calculation you need the following mathematical functions in R: sin, cos, acos, and abs.)

h<- sin(rad.phi.paris)
j <-sin(rad.phi.ny)
k <-cos(rad.phi.paris)
l <-cos(rad.phi.ny)
m <-cos(abs(rad.lambda.paris - rad.lambda.ny))
o <-acos(prod(h,j)+prod(k,l,m))

(d). To find the distance, multiply \(\Delta \sigma\) by the radius of the earth (6371 km.)

d <- o*(6371)
d
## [1] 5837.241