-
Notifications
You must be signed in to change notification settings - Fork 0
/
resampling_method.html
2286 lines (2246 loc) · 119 KB
/
resampling_method.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"><head>
<meta charset="utf-8">
<meta name="generator" content="quarto-1.6.1">
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
<title>2 The resampling method – Resampling statistics</title>
<style>
code{white-space: pre-wrap;}
span.smallcaps{font-variant: small-caps;}
div.columns{display: flex; gap: min(4vw, 1.5em);}
div.column{flex: auto; overflow-x: auto;}
div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
ul.task-list{list-style: none;}
ul.task-list li input[type="checkbox"] {
width: 0.8em;
margin: 0 0.8em 0.2em -1em; /* quarto-specific, see https://github.com/quarto-dev/quarto-cli/issues/4556 */
vertical-align: middle;
}
/* CSS for syntax highlighting */
pre > code.sourceCode { white-space: pre; position: relative; }
pre > code.sourceCode > span { line-height: 1.25; }
pre > code.sourceCode > span:empty { height: 1.2em; }
.sourceCode { overflow: visible; }
code.sourceCode > span { color: inherit; text-decoration: inherit; }
div.sourceCode { margin: 1em 0; }
pre.sourceCode { margin: 0; }
@media screen {
div.sourceCode { overflow: auto; }
}
@media print {
pre > code.sourceCode { white-space: pre-wrap; }
pre > code.sourceCode > span { display: inline-block; text-indent: -5em; padding-left: 5em; }
}
pre.numberSource code
{ counter-reset: source-line 0; }
pre.numberSource code > span
{ position: relative; left: -4em; counter-increment: source-line; }
pre.numberSource code > span > a:first-child::before
{ content: counter(source-line);
position: relative; left: -1em; text-align: right; vertical-align: baseline;
border: none; display: inline-block;
-webkit-touch-callout: none; -webkit-user-select: none;
-khtml-user-select: none; -moz-user-select: none;
-ms-user-select: none; user-select: none;
padding: 0 4px; width: 4em;
}
pre.numberSource { margin-left: 3em; padding-left: 4px; }
div.sourceCode
{ }
@media screen {
pre > code.sourceCode > span > a:first-child::before { text-decoration: underline; }
}
</style>
<script src="site_libs/quarto-nav/quarto-nav.js"></script>
<script src="site_libs/quarto-nav/headroom.min.js"></script>
<script src="site_libs/clipboard/clipboard.min.js"></script>
<script src="site_libs/quarto-search/autocomplete.umd.js"></script>
<script src="site_libs/quarto-search/fuse.min.js"></script>
<script src="site_libs/quarto-search/quarto-search.js"></script>
<meta name="quarto:offset" content="./">
<link href="./what_is_probability.html" rel="next">
<link href="./intro.html" rel="prev">
<script src="site_libs/quarto-html/quarto.js"></script>
<script src="site_libs/quarto-html/popper.min.js"></script>
<script src="site_libs/quarto-html/tippy.umd.min.js"></script>
<script src="site_libs/quarto-html/anchor.min.js"></script>
<link href="site_libs/quarto-html/tippy.css" rel="stylesheet">
<link href="site_libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
<script src="site_libs/bootstrap/bootstrap.min.js"></script>
<link href="site_libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
<link href="site_libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
<script id="quarto-search-options" type="application/json">{
"location": "sidebar",
"copy-button": false,
"collapse-after": 3,
"panel-placement": "start",
"type": "textbox",
"limit": 50,
"keyboard-shortcut": [
"f",
"/",
"s"
],
"show-item-context": false,
"language": {
"search-no-results-text": "No results",
"search-matching-documents-text": "matching documents",
"search-copy-link-title": "Copy link to search",
"search-hide-matches-text": "Hide additional matches",
"search-more-match-text": "more match in this document",
"search-more-matches-text": "more matches in this document",
"search-clear-button-title": "Clear",
"search-text-placeholder": "",
"search-detached-cancel-button-title": "Cancel",
"search-submit-button-title": "Submit",
"search-label": "Search"
}
}</script>
<script type="text/javascript">
$(document).ready(function() {
$("table").addClass('lightable-paper lightable-striped lightable-hover')
});
</script>
<script src="site_libs/kePrint-0.0.1/kePrint.js"></script>
<link href="site_libs/lightable-0.0.1/lightable.css" rel="stylesheet">
<link rel="stylesheet" href="style.css">
<link rel="stylesheet" href="font-awesome.min.css">
</head>
<body class="nav-sidebar floating">
<div id="quarto-search-results"></div>
<header id="quarto-header" class="headroom fixed-top">
<nav class="quarto-secondary-nav">
<div class="container-fluid d-flex">
<button type="button" class="quarto-btn-toggle btn" data-bs-toggle="collapse" role="button" data-bs-target=".quarto-sidebar-collapse-item" aria-controls="quarto-sidebar" aria-expanded="false" aria-label="Toggle sidebar navigation" onclick="if (window.quartoToggleHeadroom) { window.quartoToggleHeadroom(); }">
<i class="bi bi-layout-text-sidebar-reverse"></i>
</button>
<nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item"><a href="./resampling_method.html"><span class="chapter-number">2</span> <span class="chapter-title">The resampling method</span></a></li></ol></nav>
<a class="flex-grow-1" role="navigation" data-bs-toggle="collapse" data-bs-target=".quarto-sidebar-collapse-item" aria-controls="quarto-sidebar" aria-expanded="false" aria-label="Toggle sidebar navigation" onclick="if (window.quartoToggleHeadroom) { window.quartoToggleHeadroom(); }">
</a>
<button type="button" class="btn quarto-search-button" aria-label="Search" onclick="window.quartoOpenSearch();">
<i class="bi bi-search"></i>
</button>
</div>
</nav>
</header>
<!-- content -->
<div id="quarto-content" class="quarto-container page-columns page-rows-contents page-layout-article">
<!-- sidebar -->
<nav id="quarto-sidebar" class="sidebar collapse collapse-horizontal quarto-sidebar-collapse-item sidebar-navigation floating overflow-auto">
<div class="pt-lg-2 mt-2 text-left sidebar-header">
<div class="sidebar-title mb-0 py-0">
<a href="./">Resampling statistics</a>
</div>
</div>
<div class="mt-2 flex-shrink-0 align-items-center">
<div class="sidebar-search">
<div id="quarto-search" class="" title="Search"></div>
</div>
</div>
<div class="sidebar-menu-container">
<ul class="list-unstyled mt-1">
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./index.html" class="sidebar-item-text sidebar-link">
<span class="menu-text">R version</span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./preface_third.html" class="sidebar-item-text sidebar-link">
<span class="menu-text">Preface to the third edition</span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./preface_second.html" class="sidebar-item-text sidebar-link">
<span class="menu-text">Preface to the second edition</span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./intro.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">1</span> <span class="chapter-title">Introduction</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./resampling_method.html" class="sidebar-item-text sidebar-link active">
<span class="menu-text"><span class="chapter-number">2</span> <span class="chapter-title">The resampling method</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./what_is_probability.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">3</span> <span class="chapter-title">What is probability?</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./about_technology.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">4</span> <span class="chapter-title">Introducing R and the Jupyter notebook</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./resampling_with_code.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">5</span> <span class="chapter-title">Resampling with code</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./resampling_with_code2.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">6</span> <span class="chapter-title">More resampling with code</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./sampling_tools.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">7</span> <span class="chapter-title">Tools for samples and sampling</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./probability_theory_1a.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">8</span> <span class="chapter-title">Probability Theory, Part 1</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./probability_theory_1b.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">9</span> <span class="chapter-title">Probability Theory Part I (continued)</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./more_sampling_tools.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">10</span> <span class="chapter-title">Two puzzles and more tools</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./probability_theory_2_compound.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">11</span> <span class="chapter-title">Probability Theory, Part 2: Compound Probability</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./probability_theory_3.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">12</span> <span class="chapter-title">Probability Theory, Part 3</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./probability_theory_4_finite.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">13</span> <span class="chapter-title">Probability Theory, Part 4: Estimating Probabilities from Finite Universes</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./sampling_variability.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">14</span> <span class="chapter-title">On Variability in Sampling</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./monte_carlo.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">15</span> <span class="chapter-title">The Procedures of Monte Carlo Simulation (and Resampling)</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./standard_scores.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">16</span> <span class="chapter-title">Ranks, Quantiles and Standard Scores</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./inference_ideas.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">17</span> <span class="chapter-title">The Basic Ideas in Statistical Inference</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./inference_intro.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">18</span> <span class="chapter-title">Introduction to Statistical Inference</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./point_estimation.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">19</span> <span class="chapter-title">Point Estimation</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./framing_questions.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">20</span> <span class="chapter-title">Framing Statistical Questions</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./testing_counts_1.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">21</span> <span class="chapter-title">Hypothesis-Testing with Counted Data, Part 1</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./significance.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">22</span> <span class="chapter-title">The Concept of Statistical Significance in Testing Hypotheses</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./testing_counts_2.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">23</span> <span class="chapter-title">The Statistics of Hypothesis-Testing with Counted Data, Part 2</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./testing_measured.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">24</span> <span class="chapter-title">The Statistics of Hypothesis-Testing With Measured Data</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./testing_procedures.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">25</span> <span class="chapter-title">General Procedures for Testing Hypotheses</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./confidence_1.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">26</span> <span class="chapter-title">Confidence Intervals, Part 1: Assessing the Accuracy of Samples</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./confidence_2.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">27</span> <span class="chapter-title">Confidence Intervals, Part 2: The Two Approaches to Estimating Confidence Intervals</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./reliability_average.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">28</span> <span class="chapter-title">Some Last Words About the Reliability of Sample Averages</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./correlation_causation.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">29</span> <span class="chapter-title">Correlation and Causation</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./how_big_sample.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">30</span> <span class="chapter-title">How Large a Sample?</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./bayes_simulation.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">31</span> <span class="chapter-title">Bayesian Analysis by Simulation</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./references.html" class="sidebar-item-text sidebar-link">
<span class="menu-text">References</span></a>
</div>
</li>
<li class="sidebar-item sidebar-item-section">
<div class="sidebar-item-container">
<a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" role="navigation" aria-expanded="true">
<span class="menu-text">Appendices</span></a>
<a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" role="navigation" aria-expanded="true" aria-label="Toggle section">
<i class="bi bi-chevron-right ms-2"></i>
</a>
</div>
<ul id="quarto-sidebar-section-1" class="collapse list-unstyled sidebar-section depth1 show">
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./exercise_solutions.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">A</span> <span class="chapter-title">Exercise Solutions</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./technical_note.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">B</span> <span class="chapter-title">Technical Note to the Professional Reader</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./acknowlegements.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">C</span> <span class="chapter-title">Acknowledgements</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./code_topics.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">D</span> <span class="chapter-title">Code topics</span></span></a>
</div>
</li>
<li class="sidebar-item">
<div class="sidebar-item-container">
<a href="./errors_suggestions.html" class="sidebar-item-text sidebar-link">
<span class="menu-text"><span class="chapter-number">E</span> <span class="chapter-title">Errors and suggestions</span></span></a>
</div>
</li>
</ul>
</li>
</ul>
</div>
</nav>
<div id="quarto-sidebar-glass" class="quarto-sidebar-collapse-item" data-bs-toggle="collapse" data-bs-target=".quarto-sidebar-collapse-item"></div>
<!-- margin-sidebar -->
<div id="quarto-margin-sidebar" class="sidebar margin-sidebar">
<nav id="TOC" role="doc-toc" class="toc-active">
<h2 id="toc-title">Table of contents</h2>
<ul>
<li><a href="#the-resampling-approach-in-action" id="toc-the-resampling-approach-in-action" class="nav-link active" data-scroll-target="#the-resampling-approach-in-action"><span class="header-section-number">2.1</span> The resampling approach in action</a>
<ul class="collapse">
<li><a href="#randomness-from-physical-methods" id="toc-randomness-from-physical-methods" class="nav-link" data-scroll-target="#randomness-from-physical-methods"><span class="header-section-number">2.1.1</span> Randomness from physical methods</a></li>
</ul></li>
<li><a href="#sec-randomness-computer" id="toc-sec-randomness-computer" class="nav-link" data-scroll-target="#sec-randomness-computer"><span class="header-section-number">2.2</span> Randomness from your computer</a></li>
<li><a href="#solving-the-problem-using" id="toc-solving-the-problem-using" class="nav-link" data-scroll-target="#solving-the-problem-using"><span class="header-section-number">2.3</span> Solving the problem using R</a>
<ul class="collapse">
<li><a href="#repeating-trials" id="toc-repeating-trials" class="nav-link" data-scroll-target="#repeating-trials">2.3.1 Repeating trials</a></li>
<li><a href="#the-solution" id="toc-the-solution" class="nav-link" data-scroll-target="#the-solution">2.3.2 The solution</a></li>
</ul></li>
<li><a href="#sec-resamp-differs" id="toc-sec-resamp-differs" class="nav-link" data-scroll-target="#sec-resamp-differs"><span class="header-section-number">2.4</span> How resampling differs from the conventional approach</a></li>
</ul>
</nav>
</div>
<!-- main -->
<main class="content" id="quarto-document-content">
<header id="title-block-header" class="quarto-title-block default">
<div class="quarto-title">
<h1 class="title"><span id="sec-resampling-method" class="quarto-section-identifier"><span class="chapter-number">2</span> <span class="chapter-title">The resampling method</span></span></h1>
</div>
<div class="quarto-title-meta">
</div>
</header>
<p>This chapter is a brief introduction to the resampling method of solving problems in probability and statistics. We’re going to dive right in and solve a problem hands-on.</p>
<p>You will see that the resampling method is easy to understand and apply: all it requires is to understand the physical problem. You then simulate a statistical model of the physical problem with techniques that are intuitively obvious, and estimate the probability sought with repeated random sampling.</p>
<p>After finding a solution, we will look at the more conventional formulaic approach, and how that compares. Here’s the spoiler: it requires you to understand complex formulas, and to choose the correct one from many.</p>
<p>After reading this chapter, you will understand why we are excited about the resampling method, and why it will allow you to approach even even hard problems without knowing sophisticated statistic techniques.</p>
<section id="the-resampling-approach-in-action" class="level2" data-number="2.1">
<h2 data-number="2.1" class="anchored" data-anchor-id="the-resampling-approach-in-action"><span class="header-section-number">2.1</span> The resampling approach in action</h2>
<p>Recall the problem from section <a href="intro.html#sec-what-problems" class="quarto-xref"><span>Section 1.2</span></a> in which the contractor owns 16 ambulances:</p>
<blockquote class="blockquote">
<p><em>You are the manager and part owner</em> of one of several contractors providing ambulances to a hospital. You own 16 ambulances. Based on past experience, the chance that any one ambulance will be unfit for service on any given day is about one in ten. You want to know the chance on a particular day — tomorrow — that three or more of them will be out of action.</p>
</blockquote>
<p>The resampling approach produces the estimate as follows.</p>
<section id="randomness-from-physical-methods" class="level3" data-number="2.1.1">
<h3 data-number="2.1.1" class="anchored" data-anchor-id="randomness-from-physical-methods"><span class="header-section-number">2.1.1</span> Randomness from physical methods</h3>
<p>We collect 10 coins, and mark one of them with a pen or pencil as being the coin that represents “out-of-order;” the other nine coins stand for “in operation”. For any one ambulance, this set of 10 coins provides a “model” for the one-in-ten chance — a probability of .10 (10 percent) — of it being out of order on a given day. We put the coins into a little jar or bucket.</p>
<p>For ambulance #1, we draw a single coin from the bucket. This coin represents whether that ambulance is going to be broken tomorrow. If we draw the marked coin, we label this ambulance as out-of-order, otherwise we label the ambulance as in-operation. After replacing the coin and shaking the bucket, we repeat the same procedure for ambulance #2, ambulance #3 and so forth. Having repeated the procedure <em>16</em> times, we now have a representation of all ambulances for a single day.</p>
<p>We can now repeat this whole process as many times as we like: each time, we come up with a representation for a different day, telling us how many ambulances will be out-of-service on that day.</p>
<p>After collecting evidence for, say, 50 experimental days we determine the proportion of the experimental days on which three or more ambulances are out of order. That proportion is an estimate of the probability that three or more ambulances will be out of order on a given day — the answer we seek. This procedure is an example of Monte Carlo simulation, which is the heart of the resampling method of statistical estimation.</p>
<p>A more direct way to answer this question would be to examine the firm’s actual records for the past 100 days or, better, 500 days (if that’s available) to determine how many days had three or more ambulances out of order. But the resampling procedure described above gives us an estimate even if we do not have such long-term information. This is realistic; it is frequently the case in the real world that we must make estimates on the basis of insufficient history about an event.</p>
<p>A quicker resampling method than the coins could be obtained with 16 ten-sided dice or spinners (like those found in Dungeons & Dragons games). For each die, we identify one of its ten sides as “out-of-order”.</p>
<p>Standard 10-sided dice have the numbers 0 through 9 on their faces, rather than 1 through 10. <a href="#fig-one-d10s" class="quarto-xref">Figure <span>2.1</span></a> shows a standard 10-sided die:</p>
<div id="fig-one-d10s" class="quarto-float quarto-figure quarto-figure-center anchored">
<figure class="quarto-float quarto-float-fig figure">
<div aria-describedby="fig-one-d10s-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
<img src="images/one_d10s.jpg" class="img-fluid figure-img">
</div>
<figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-one-d10s-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Figure 2.1: 10-sided die
</figcaption>
</figure>
</div>
<p>We decide, arbitrarily, that the 9 side means “out-of-order”. We could even put a little bit of paint on the 9 side to remind us. The die represents an ambulance. If we roll the die, and get this face, this indicates that the ambulance was out of order. If we get any of the other faces — 0 through 8 — this ambulance was in working order. A single throw of all 16 dice will be our experimental trial that represents a single day; we just have to count whether three or more ambulances turn up “out of order”. <a href="#fig-16-d10s" class="quarto-xref">Figure <span>2.2</span></a> show the result of one trial — throwing 16 dice:</p>
<div id="fig-16-d10s" class="quarto-float quarto-figure quarto-figure-center anchored">
<figure class="quarto-float quarto-float-fig figure">
<div aria-describedby="fig-16-d10s-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
<img src="images/16_d10s.jpg" class="img-fluid figure-img">
</div>
<figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-16-d10s-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Figure 2.2: 16 10-sided dice
</figcaption>
</figure>
</div>
<p>As you can see, the trial in <a href="#fig-16-d10s" class="quarto-xref">Figure <span>2.2</span></a> gave us a single 9, so there was only one ambulance out of order.</p>
<p>In a hundred throws of the 16 dice — which probably takes less than 5 minutes — we can get a fast and reasonably accurate answer to our question.</p>
</section>
</section>
<section id="sec-randomness-computer" class="level2" data-number="2.2">
<h2 data-number="2.2" class="anchored" data-anchor-id="sec-randomness-computer"><span class="header-section-number">2.2</span> Randomness from your computer</h2>
<p>Computers make it easy to generate random numbers for resampling.</p>
<div class="callout callout-style-default callout-note callout-titled">
<div class="callout-header d-flex align-content-center">
<div class="callout-icon-container">
<i class="callout-icon"></i>
</div>
<div class="callout-title-container flex-fill">
What do we mean by random?
</div>
</div>
<div class="callout-body-container callout-body">
<p>Random numbers are numbers where it is impossible to predict which number is coming next. If we ask the computer for a number between 0 and 9, we will get one of the numbers 0 though 9, but we cannot do any better than that in predicting <em>which</em> number it will give us. There is an equal (10%) chance we will get any of the numbers 0 through 9 — just as there is when we roll a fair 10-sided die. We will go into more detail about what exactly we mean by <em>random</em> and <em>chance</em> later in the book (<a href="what_is_probability.html#sec-what-is-chance" class="quarto-xref"><span>Section 3.8</span></a>).</p>
</div>
</div>
<!---
Press Your Luck
The Michael Larsen Incident
https://web.archive.org/web/20170712041017/http://www.rotten.com/library/conspiracy/press-your-luck/
-->
<p>We can use random numbers from computers to simulate our problem. For example, we can ask the computer to choose a random number between 0 and 9 to represent one ambulance. Let’s say the number 9 represents “out-of-order” and 0 through 8 “in operation”, then any one random number gives us a trial observation for a single ambulance. To get an experimental trial for a single day we look at 16 numbers and count how many of them are 9. We then look at, say, one hundred sets of 16 numbers and count the proportion of sets whose 16 numbers show three or more ambulances being “out-of-order”. Once again, that proportion estimates the probability that three or more ambulances will be out-of-order on any given day.</p>
<p>Soon we will do all these steps with some R code, but for now, consider <a href="#tbl-veh-numbers" class="quarto-xref">Table <span>2.1</span></a>. In each row, we placed 16 numbers, each one representing an ambulance. We added 25 such rows, each representing a simulation of one day.</p>
<div class="cell" data-layout-align="center">
<div id="tbl-veh-numbers" class="cell quarto-float quarto-figure quarto-figure-center anchored" data-layout-align="center">
<figure class="quarto-float quarto-float-tbl figure">
<figcaption class="quarto-float-caption-top quarto-float-caption quarto-float-tbl" id="tbl-veh-numbers-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Table 2.1: 25 simulations of 16 ambulances
</figcaption>
<div aria-describedby="tbl-veh-numbers-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
<div class="cell-output-display">
<table class="lightable-paper lightable-striped lightable-hover do-not-create-environment cell caption-top table table-sm table-striped small" data-quarto-postprocess="true">
<thead>
<tr class="header">
<th data-quarto-table-cell-role="th" style="text-align: right; font-weight: bold;">Day</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A1</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A2</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A3</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A4</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A5</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A6</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A7</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A8</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A9</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A10</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A11</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A12</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A13</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A14</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A15</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A16</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: right; font-weight: bold;">1</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">6</td>
</tr>
<tr class="even">
<td style="text-align: right; font-weight: bold;">2</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">1</td>
</tr>
<tr class="odd">
<td style="text-align: right; font-weight: bold;">3</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">3</td>
</tr>
<tr class="even">
<td style="text-align: right; font-weight: bold;">4</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">6</td>
</tr>
<tr class="odd">
<td style="text-align: right; font-weight: bold;">5</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">4</td>
</tr>
<tr class="even">
<td style="text-align: right; font-weight: bold;">6</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">5</td>
</tr>
<tr class="odd">
<td style="text-align: right; font-weight: bold;">7</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">7</td>
</tr>
<tr class="even">
<td style="text-align: right; font-weight: bold;">8</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">0</td>
</tr>
<tr class="odd">
<td style="text-align: right; font-weight: bold;">9</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">7</td>
</tr>
<tr class="even">
<td style="text-align: right; font-weight: bold;">10</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">1</td>
</tr>
<tr class="odd">
<td style="text-align: right; font-weight: bold;">11</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">3</td>
</tr>
<tr class="even">
<td style="text-align: right; font-weight: bold;">12</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">9</td>
</tr>
<tr class="odd">
<td style="text-align: right; font-weight: bold;">13</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">0</td>
</tr>
<tr class="even">
<td style="text-align: right; font-weight: bold;">14</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">4</td>
</tr>
<tr class="odd">
<td style="text-align: right; font-weight: bold;">15</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">3</td>
</tr>
<tr class="even">
<td style="text-align: right; font-weight: bold;">16</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">8</td>
</tr>
<tr class="odd">
<td style="text-align: right; font-weight: bold;">17</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">4</td>
</tr>
<tr class="even">
<td style="text-align: right; font-weight: bold;">18</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">1</td>
</tr>
<tr class="odd">
<td style="text-align: right; font-weight: bold;">19</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">0</td>
</tr>
<tr class="even">
<td style="text-align: right; font-weight: bold;">20</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">6</td>
</tr>
<tr class="odd">
<td style="text-align: right; font-weight: bold;">21</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">0</td>
</tr>
<tr class="even">
<td style="text-align: right; font-weight: bold;">22</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">2</td>
</tr>
<tr class="odd">
<td style="text-align: right; font-weight: bold;">23</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">0</td>
</tr>
<tr class="even">
<td style="text-align: right; font-weight: bold;">24</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">0</td>