-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathresampling_with_code2.html
1554 lines (1521 loc) · 107 KB
/
resampling_with_code2.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>6 More resampling with code – 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; }
}
/* CSS for citations */
div.csl-bib-body { }
div.csl-entry {
clear: both;
margin-bottom: 0em;
}
.hanging-indent div.csl-entry {
margin-left:2em;
text-indent:-2em;
}
div.csl-left-margin {
min-width:2em;
float:left;
}
div.csl-right-inline {
margin-left:2em;
padding-left:1em;
}
div.csl-indent {
margin-left: 2em;
}</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="./sampling_tools.html" rel="next">
<link href="./resampling_with_code.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>
<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_with_code2.html"><span class="chapter-number">6</span> <span class="chapter-title">More resampling with code</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">Python 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">
<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 Python 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 active">
<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 1 (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="#a-question-of-life-and-death" id="toc-a-question-of-life-and-death" class="nav-link active" data-scroll-target="#a-question-of-life-and-death"><span class="header-section-number">6.1</span> A question of life and death</a></li>
<li><a href="#a-small-disparity-and-a-hypothetical-world" id="toc-a-small-disparity-and-a-hypothetical-world" class="nav-link" data-scroll-target="#a-small-disparity-and-a-hypothetical-world"><span class="header-section-number">6.2</span> A small disparity and a hypothetical world</a></li>
<li><a href="#designing-the-experiment" id="toc-designing-the-experiment" class="nav-link" data-scroll-target="#designing-the-experiment"><span class="header-section-number">6.3</span> Designing the experiment</a>
<ul class="collapse">
<li><a href="#one-trial" id="toc-one-trial" class="nav-link" data-scroll-target="#one-trial"><span class="header-section-number">6.3.1</span> One trial</a></li>
<li><a href="#using-code-to-simulate-a-trial" id="toc-using-code-to-simulate-a-trial" class="nav-link" data-scroll-target="#using-code-to-simulate-a-trial">6.3.2 Using code to simulate a trial</a></li>
<li><a href="#sec-random-zero-through-ninety-nine" id="toc-sec-random-zero-through-ninety-nine" class="nav-link" data-scroll-target="#sec-random-zero-through-ninety-nine">6.3.3 Random numbers from 0 through 99</a></li>
</ul></li>
<li><a href="#three-simulation-steps" id="toc-three-simulation-steps" class="nav-link" data-scroll-target="#three-simulation-steps">6.4 Three simulation steps</a>
<ul class="collapse">
<li><a href="#more-on-s" id="toc-more-on-s" class="nav-link" data-scroll-target="#more-on-s">6.4.1 More on arrays</a></li>
</ul></li>
<li><a href="#sec-array-length" id="toc-sec-array-length" class="nav-link" data-scroll-target="#sec-array-length">6.5 Array length</a></li>
<li><a href="#sec-array-indexing" id="toc-sec-array-indexing" class="nav-link" data-scroll-target="#sec-array-indexing">6.6 Indexing into arrays with integers</a>
<ul class="collapse">
<li><a href="#repeating-trials" id="toc-repeating-trials" class="nav-link" data-scroll-target="#repeating-trials">6.6.1 Repeating trials</a></li>
<li><a href="#sec-for-loops" id="toc-sec-for-loops" class="nav-link" data-scroll-target="#sec-for-loops">6.6.2 For-loops in Python</a></li>
<li><a href="#sec-for-range" id="toc-sec-for-range" class="nav-link" data-scroll-target="#sec-for-range">6.6.3 <code>range</code> in Python <code>for</code> loops</a></li>
<li><a href="#putting-it-all-together" id="toc-putting-it-all-together" class="nav-link" data-scroll-target="#putting-it-all-together">6.6.4 Putting it all together</a></li>
</ul></li>
<li><a href="#many-many-trials" id="toc-many-many-trials" class="nav-link" data-scroll-target="#many-many-trials">6.7 Many many trials</a></li>
<li><a href="#conclusion" id="toc-conclusion" class="nav-link" data-scroll-target="#conclusion"><span class="header-section-number">6.8</span> Conclusion</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-two" class="quarto-section-identifier"><span class="chapter-number">6</span> <span class="chapter-title">More resampling with code</span></span></h1>
</div>
<div class="quarto-title-meta">
</div>
</header>
<p><a href="resampling_with_code.html" class="quarto-xref"><span>Chapter 5</span></a> introduced a problem in probability, that was also a problem in statistics. We asked how surprised we should be at the results of a trial of a new cancer treatment regime.</p>
<p>Here we study another urgent problem in the real world - racial bias and the death penalty.</p>
<section id="a-question-of-life-and-death" class="level2" data-number="6.1">
<h2 data-number="6.1" class="anchored" data-anchor-id="a-question-of-life-and-death"><span class="header-section-number">6.1</span> A question of life and death</h2>
<p>This example comes from the excellent Berkeley introduction to data science <span class="citation" data-cites="adhikari2021data8">(<a href="references.html#ref-adhikari2021data8" role="doc-biblioref">Ani Adhikari and Wagner 2021</a>)</span>.</p>
<p>Robert Swain was a young black man who was sentenced to death in the early 60s. Swain’s trial was held in Talladega County, Alabama. At the time, 26% of the eligible jurors in that county were black, but every member of Swain’s jury was white. Swain and his legal team appealed to the Alabama Supreme Court, and then to the <a href="https://en.wikipedia.org/wiki/Swain_v._Alabama">US Supreme Court</a>, arguing that there was racial bias in the jury selection. They noted that there had been no black jurors in Talladega county since 1950, even though they made up about a quarter of the eligible pool of jurors. The US Supreme Court rejected this argument, in a 6 to 3 opinion, writing that “The overall percentage disparity has been small and reflects no studied attempt to include or exclude a specified number of Negros.”.</p>
<p>Swain’s team presented a variety of evidence on bias in jury selection, but here we will look at the obvious and apparently surprising fact that Swain’s jury was entirely white. The Supreme Court decided that the “disparity” between selection of white and black jurors “has been small”. In this particular case the disparity is between 0% of selected black jurors and 26% of eligible black jurors. How would the Supreme Court judges, and how would we, make a rational decision about whether this particular disparity really was “small”?</p>
<p>You might be wondering about the result of this decision for Robert Swain. In fact his death sentence was invalidated by a <a href="https://en.wikipedia.org/wiki/Furman_v._Georgia">later, unrelated decision</a> and he served a long prison sentence instead. In 1986, the Supreme Court overturned the precedent set by Swain’s case, in <a href="https://supreme.justia.com/cases/federal/us/476/79">Batson v. Kentucky, 476 U.S. 79</a>.</p>
</section>
<section id="a-small-disparity-and-a-hypothetical-world" class="level2" data-number="6.2">
<h2 data-number="6.2" class="anchored" data-anchor-id="a-small-disparity-and-a-hypothetical-world"><span class="header-section-number">6.2</span> A small disparity and a hypothetical world</h2>
<p>To answer the question that the Supreme Court asked, we return to the method we used in the last chapter.</p>
<p>Let us imagine a hypothetical world, in which each individual black or white person had an equal chance of being selected for the jury. Call this world Hypothetical County, Alabama.</p>
<p>Just as in 1960’s Talladega County, 26% of eligible jurors in Hypothetical County are black. Hypothetical County jury selection has no bias against black people, so we expect around 26% of the jury to be black. 26% of 12 is 3.12 (0.26 * 12 = 3.12), so we expect that, on average, just over 3 out of 12 jurors in a jury from Hypothetical County will be black. But, if we select each juror at random from the population, that means that, sometimes, by chance, we will have fewer than 3 black jurors, and sometimes will have more than 3 black jurors. And, by chance, sometimes we will have no black jurors. But, if the jurors really are selected at random, how often would we expect this to happen — that there are no black jurors? We would like to estimate the <em>probability</em> that we will get no black jurors. If that probability is small, then we have some evidence that the disparity in selection between black and white jurors, was not “small”.</p>
<div class="question">
<p>What is the probability of an <em>all white</em> jury being randomly selected out of a population having 26% black people?</p>
</div>
</section>
<section id="designing-the-experiment" class="level2" data-number="6.3">
<h2 data-number="6.3" class="anchored" data-anchor-id="designing-the-experiment"><span class="header-section-number">6.3</span> Designing the experiment</h2>
<p>Before we start, we need to figure out three things:</p>
<ol type="1">
<li>What do we mean by one trial?</li>
<li>What is the outcome of interest from the trial?</li>
<li>How do we simulate one trial?</li>
</ol>
<p>We then take <strong>three steps</strong> to calculate the desired probability:</p>
<ol type="1">
<li><em>Repeat</em> the simulated trial procedure N times.</li>
<li><em>Count</em> M, the number of trials with an outcome that matches the outcome we are interested in.</li>
<li>Calculate the <em>proportion</em>, M/N. This is an estimate of the probability in question.</li>
</ol>
<p>For this problem, our task is made a little easier by the fact that our <em>trial</em> (in the resampling sense) is a simulated <em>trial</em> (in the legal sense). One trial requires 12 simulated jurors, each labeled by race (white or black).</p>
<p>The outcome we are interested in is the number of black jurors.</p>
<p>Now comes the harder part. How do we simulate one trial?</p>
<section id="one-trial" class="level3" data-number="6.3.1">
<h3 data-number="6.3.1" class="anchored" data-anchor-id="one-trial"><span class="header-section-number">6.3.1</span> One trial</h3>
<p>One trial requires 12 jurors, and we are interested only in the race of each juror. In Hypothetical County, where selection by race is entirely random, each juror has a 26% chance of being black.</p>
<p>We need a way of simulating a 26% chance.</p>
<p>One way of doing this is by getting a random number from 0 through 99 (inclusive). There are 100 numbers in the range 0 through 99 (inclusive).</p>
<p>We will arbitrarily say that the juror is white if the random number is in the range from 0 through 73. 74 of the 100 numbers are in this range, so the juror has a 74/100 = 74% chance of getting the label “white”. We will say the juror is black if the random number is in the range 74 though 99. There are 26 such numbers, so the juror has a 26% chance of getting the label “black”.</p>
<p>Next we need a way of getting a random number in the range 0 through 99. This is an easy job for the computer, but if we had to do this with a physical device, we could get a single number by throwing <em>two</em> 10-sided dice, say a blue die and a green die. The face of the blue die will be the 10s digit, and the green face will be the ones digit. So, if the blue die comes up with 8 and the green die has 4, then the random number is 84.</p>
<p>We could then simulate 12 jurors by repeating this process 12 times, each time writing down “white” if the number is from 0 through 74, and “black” otherwise. The trial outcome is the number of times we wrote “black” for these 12 simulated jurors.</p>
<div id="nte-life_and_death" 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">
Note 6.1: Notebook: Simulating juries
</div>
</div>
<div class="callout-body-container callout-body">
<div class="nb-links">
<p><a class="notebook-link" href="notebooks/life_and_death.ipynb">Download notebook</a> <a class="interact-button" href="./interact/lab/index.html?path=life_and_death.ipynb">Interact</a></p>
</div>
</div>
</div>
<div class="nb-start" name="life_and_death" title="Simulating juries">
</div>
</section>
<section id="using-code-to-simulate-a-trial" class="level3" data-number="6.3.2">
<h3 data-number="6.3.2" class="anchored" data-anchor-id="using-code-to-simulate-a-trial">6.3.2 Using code to simulate a trial</h3>
<p>We use the same logic to simulate a trial with the computer. A little code makes the job easier, because we can ask Python to give us 12 random numbers from 0 through 99, and to count how many of these numbers are in the range from 75 through 99. Numbers in the range from 75 through 99 correspond to black jurors.</p>
</section>
<section id="sec-random-zero-through-ninety-nine" class="level3" data-number="6.3.3">
<h3 data-number="6.3.3" class="anchored" data-anchor-id="sec-random-zero-through-ninety-nine">6.3.3 Random numbers from 0 through 99</h3>
<p>We can now use NumPy and the random number functions from the last chapter to get 12 random numbers from 0 through 99.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Import the Numpy library, rename as "np"</span></span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> numpy <span class="im">as</span> np</span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Ask Numpy for a random number generator.</span></span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a>rnd <span class="op">=</span> np.random.default_rng()</span>
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-7"><a href="#cb1-7" aria-hidden="true" tabindex="-1"></a><span class="co"># All the integers from 0 up to, but not including 100.</span></span>
<span id="cb1-8"><a href="#cb1-8" aria-hidden="true" tabindex="-1"></a>zero_thru_99 <span class="op">=</span> np.arange(<span class="dv">100</span>)</span>
<span id="cb1-9"><a href="#cb1-9" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-10"><a href="#cb1-10" aria-hidden="true" tabindex="-1"></a><span class="co"># Get 12 random numbers from 0 through 99</span></span>
<span id="cb1-11"><a href="#cb1-11" aria-hidden="true" tabindex="-1"></a>a <span class="op">=</span> rnd.choice(zero_thru_99, size<span class="op">=</span><span class="dv">12</span>)</span>
<span id="cb1-12"><a href="#cb1-12" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-13"><a href="#cb1-13" aria-hidden="true" tabindex="-1"></a><span class="co"># Show the result</span></span>
<span id="cb1-14"><a href="#cb1-14" aria-hidden="true" tabindex="-1"></a>a</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>array([59, 43, 45, 58, 95, 89, 23, 99, 17, 51, 85, 23])</code></pre>
</div>
</div>
<section id="counting-the-jurors" class="level4" data-number="6.3.3.1">
<h4 data-number="6.3.3.1" class="anchored" data-anchor-id="counting-the-jurors">6.3.3.1 Counting the jurors</h4>
<p>We use <em>comparison</em> and <code>np.sum</code> to count how many numbers are greater than 74, and therefore, in the range from 75 through 99:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb3"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="co"># How many numbers are greater than 74?</span></span>
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a>b <span class="op">=</span> np.<span class="bu">sum</span>(a <span class="op">></span> <span class="dv">74</span>)</span>
<span id="cb3-3"><a href="#cb3-3" aria-hidden="true" tabindex="-1"></a><span class="co"># Show the result</span></span>
<span id="cb3-4"><a href="#cb3-4" aria-hidden="true" tabindex="-1"></a>b</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>np.int64(4)</code></pre>
</div>
</div>
</section>
<section id="a-single-simulated-trial" class="level4" data-number="6.3.3.2">
<h4 data-number="6.3.3.2" class="anchored" data-anchor-id="a-single-simulated-trial">6.3.3.2 A single simulated trial</h4>
<p>We assemble the pieces from the last few sections to make a cell that simulates a single trial:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb5"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a>rnd <span class="op">=</span> np.random.default_rng()</span>
<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a>zero_thru_99 <span class="op">=</span> np.arange(<span class="dv">100</span>)</span>
<span id="cb5-3"><a href="#cb5-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb5-4"><a href="#cb5-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Get 12 random numbers from 0 through 99</span></span>
<span id="cb5-5"><a href="#cb5-5" aria-hidden="true" tabindex="-1"></a>a <span class="op">=</span> rnd.choice(zero_thru_99, size<span class="op">=</span><span class="dv">12</span>)</span>
<span id="cb5-6"><a href="#cb5-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb5-7"><a href="#cb5-7" aria-hidden="true" tabindex="-1"></a><span class="co"># How many numbers are greater than 74?</span></span>
<span id="cb5-8"><a href="#cb5-8" aria-hidden="true" tabindex="-1"></a>b <span class="op">=</span> np.<span class="bu">sum</span>(a <span class="op">></span> <span class="dv">74</span>)</span>
<span id="cb5-9"><a href="#cb5-9" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb5-10"><a href="#cb5-10" aria-hidden="true" tabindex="-1"></a><span class="co"># Show the result</span></span>
<span id="cb5-11"><a href="#cb5-11" aria-hidden="true" tabindex="-1"></a>b</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>np.int64(4)</code></pre>
</div>
</div>
</section>
</section>
</section>
<section id="three-simulation-steps" class="level2" data-number="6.4">
<h2 data-number="6.4" class="anchored" data-anchor-id="three-simulation-steps">6.4 Three simulation steps</h2>
<p>Now we come back to the details of how we:</p>
<ol type="1">
<li>Repeat the simulated trial many times;</li>
<li>record the results for each trial;</li>
<li>calculate the required proportion as an estimate of the probability we seek.</li>
</ol>
<p>Repeating the trial many times is the job of the <code>for</code> loop, and we will come to that soon.</p>
<p>In order to record the results, we will store each trial result in an array.</p>
<section id="more-on-s" class="level3" data-number="6.4.1">
<h3 data-number="6.4.1" class="anchored" data-anchor-id="more-on-s">6.4.1 More on arrays</h3>
<p>Since we will be working with arrays a lot, it is worth knowing more about them.</p>
<p>A Numpy array is a <em>container</em> that stores many elements of the same type. You have already seen, in <a href="resampling_method.html" class="quarto-xref"><span>Chapter 2</span></a>, how we can create an array from a sequence of numbers using the <code>np.array</code> function. </p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb7"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Make an array of numbers, store with the name "some_numbers".</span></span>
<span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a>some_numbers <span class="op">=</span> np.array([<span class="dv">0</span>, <span class="dv">1</span>, <span class="dv">2</span>, <span class="dv">3</span>, <span class="dv">4</span>, <span class="dv">5</span>, <span class="dv">6</span>, <span class="dv">7</span>, <span class="dv">8</span>, <span class="dv">9</span>])</span>
<span id="cb7-3"><a href="#cb7-3" aria-hidden="true" tabindex="-1"></a><span class="co"># Show the value of "some_numbers"</span></span>
<span id="cb7-4"><a href="#cb7-4" aria-hidden="true" tabindex="-1"></a>some_numbers</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])</code></pre>
</div>
</div>
<p>Another way that we can create arrays is to use the <code>np.zeros</code> function to make a new array where all the elements are 0.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb9"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Make a new array containing 5 zeros.</span></span>
<span id="cb9-2"><a href="#cb9-2" aria-hidden="true" tabindex="-1"></a><span class="co"># store with the name "z".</span></span>
<span id="cb9-3"><a href="#cb9-3" aria-hidden="true" tabindex="-1"></a>z <span class="op">=</span> np.zeros(<span class="dv">5</span>)</span>
<span id="cb9-4"><a href="#cb9-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Show the value of "z"</span></span>
<span id="cb9-5"><a href="#cb9-5" aria-hidden="true" tabindex="-1"></a>z</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>array([0., 0., 0., 0., 0.])</code></pre>
</div>
</div>
<p>Notice the argument <code>5</code> to the <code>np.zeros</code> function. This tells the function how many zeros we want in the array that the function will return.</p>
</section>
</section>
<section id="sec-array-length" class="level2" data-number="6.5">
<h2 data-number="6.5" class="anchored" data-anchor-id="sec-array-length">6.5 Array length</h2>
<p>The are various useful things we can do with this array container. One is to ask how many elements there are in the array container. We can use the <code>len</code> function to calculate the number of elements in an array:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb11"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Show the number of elements in "z"</span></span>
<span id="cb11-2"><a href="#cb11-2" aria-hidden="true" tabindex="-1"></a><span class="bu">len</span>(z)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>5</code></pre>
</div>
</div>
</section>
<section id="sec-array-indexing" class="level2" data-number="6.6">
<h2 data-number="6.6" class="anchored" data-anchor-id="sec-array-indexing">6.6 Indexing into arrays with integers</h2>
<p>Another thing we can do with arrays is <em>set</em> the value for a particular element. To do this, we use square brackets following the array value, on the left hand side of the equals sign, like this:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb13"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Set the value of the *first* element in the array.</span></span>
<span id="cb13-2"><a href="#cb13-2" aria-hidden="true" tabindex="-1"></a>z[<span class="dv">0</span>] <span class="op">=</span> <span class="dv">99</span></span>
<span id="cb13-3"><a href="#cb13-3" aria-hidden="true" tabindex="-1"></a><span class="co"># Show the new contents of the array.</span></span>
<span id="cb13-4"><a href="#cb13-4" aria-hidden="true" tabindex="-1"></a>z</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>array([99., 0., 0., 0., 0.])</code></pre>
</div>
</div>
<p>Read the first line of code as “the element at position 0 gets a value of 99”.</p>
<div class="python">
<p>Notice that the position number of the first element in the array is 0, and the position number of the second element is 1. Think of the position as an <em>offset</em> from the beginning of the array. The first element is at the beginning of the array, and so it is at offset (position) 0. This can be a little difficult to get used to at first, but you will find that thinking of the positions of offsets in this way soon starts to come naturally, and later you will also find that it helps you to avoid some common mistakes when using positions for getting and setting values.</p>
</div>
<p>For practice, let us also set the value of the third element in the array:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb15"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb15-1"><a href="#cb15-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Set the value of the *third* element in the array.</span></span>
<span id="cb15-2"><a href="#cb15-2" aria-hidden="true" tabindex="-1"></a>z[<span class="dv">2</span>] <span class="op">=</span> <span class="dv">99</span></span>
<span id="cb15-3"><a href="#cb15-3" aria-hidden="true" tabindex="-1"></a><span class="co"># Show the new contents of the array.</span></span>
<span id="cb15-4"><a href="#cb15-4" aria-hidden="true" tabindex="-1"></a>z</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>array([99., 0., 99., 0., 0.])</code></pre>
</div>
</div>
<p>Read the first code line above as as “set the value at position 2 in the array to have the value 99”.</p>
<p>We can also <em>get</em> the value of the element at a given position, using the same square-bracket notation:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb17"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb17-1"><a href="#cb17-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Get the value of the *first* element in the array.</span></span>
<span id="cb17-2"><a href="#cb17-2" aria-hidden="true" tabindex="-1"></a><span class="co"># Store the value with name "v"</span></span>
<span id="cb17-3"><a href="#cb17-3" aria-hidden="true" tabindex="-1"></a>v <span class="op">=</span> z[<span class="dv">0</span>]</span>
<span id="cb17-4"><a href="#cb17-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Show the value we got</span></span>
<span id="cb17-5"><a href="#cb17-5" aria-hidden="true" tabindex="-1"></a>v</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>np.float64(99.0)</code></pre>
</div>
</div>
<p>Read the first code line here as “v gets the value at position 0 in the array”.</p>
<p>Using square brackets to get and set element values is called <em>indexing</em> into the array.</p>
<section id="repeating-trials" class="level3" data-number="6.6.1">
<h3 data-number="6.6.1" class="anchored" data-anchor-id="repeating-trials">6.6.1 Repeating trials</h3>
<p>As a preview, let us now imagine that we want to do 50 simulated trials of Robert Swain’s jury in Hypothetical County. We will want to store the count for each trial, to give 50 counts.</p>
<p>In order to do this, we make an array to hold the 50 counts. Call this array <code>z</code>.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb19"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb19-1"><a href="#cb19-1" aria-hidden="true" tabindex="-1"></a><span class="co"># An array to hold the 50 count values.</span></span>
<span id="cb19-2"><a href="#cb19-2" aria-hidden="true" tabindex="-1"></a>z <span class="op">=</span> np.zeros(<span class="dv">50</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>We could run a single trial to get a single simulated count. Here we just repeat the code cell you saw above. Notice that we can get a different result each time we run this code, because the numbers in <code>a</code> are <em>random</em> choices from the range 0 through 99, and different random numbers will give different counts.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb20"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb20-1"><a href="#cb20-1" aria-hidden="true" tabindex="-1"></a>rnd <span class="op">=</span> np.random.default_rng()</span>
<span id="cb20-2"><a href="#cb20-2" aria-hidden="true" tabindex="-1"></a>zero_thru_99 <span class="op">=</span> np.arange(<span class="dv">100</span>)</span>
<span id="cb20-3"><a href="#cb20-3" aria-hidden="true" tabindex="-1"></a><span class="co"># Get 12 random numbers from 0 through 99</span></span>
<span id="cb20-4"><a href="#cb20-4" aria-hidden="true" tabindex="-1"></a>a <span class="op">=</span> rnd.choice(zero_thru_99, size<span class="op">=</span><span class="dv">12</span>)</span>
<span id="cb20-5"><a href="#cb20-5" aria-hidden="true" tabindex="-1"></a><span class="co"># How many numbers are greater than 74?</span></span>
<span id="cb20-6"><a href="#cb20-6" aria-hidden="true" tabindex="-1"></a>b <span class="op">=</span> np.<span class="bu">sum</span>(a <span class="op">></span> <span class="dv">74</span>)</span>
<span id="cb20-7"><a href="#cb20-7" aria-hidden="true" tabindex="-1"></a><span class="co"># Show the result</span></span>
<span id="cb20-8"><a href="#cb20-8" aria-hidden="true" tabindex="-1"></a>b</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>np.int64(4)</code></pre>
</div>
</div>
<p>Now we have the result of a single trial, we can store it as the first number in the <code>z</code> array:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb22"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb22-1"><a href="#cb22-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Store the single trial count as the first value in the "z" array.</span></span>
<span id="cb22-2"><a href="#cb22-2" aria-hidden="true" tabindex="-1"></a>z[<span class="dv">0</span>] <span class="op">=</span> b</span>
<span id="cb22-3"><a href="#cb22-3" aria-hidden="true" tabindex="-1"></a><span class="co"># Show all the values in the "z" array.</span></span>
<span id="cb22-4"><a href="#cb22-4" aria-hidden="true" tabindex="-1"></a>z</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>array([4., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.])</code></pre>
</div>
</div>
<p>Of course we could just keep doing this: run the cell corresponding to a trial, above, to get a new count, and then store it at the next position in the <code>z</code> array. For example, we could store the counts for the first three trials with:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb24"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb24-1"><a href="#cb24-1" aria-hidden="true" tabindex="-1"></a><span class="co"># First trial</span></span>
<span id="cb24-2"><a href="#cb24-2" aria-hidden="true" tabindex="-1"></a>a <span class="op">=</span> rnd.choice(zero_thru_99, size<span class="op">=</span><span class="dv">12</span>)</span>
<span id="cb24-3"><a href="#cb24-3" aria-hidden="true" tabindex="-1"></a>b <span class="op">=</span> np.<span class="bu">sum</span>(a <span class="op">></span> <span class="dv">74</span>)</span>
<span id="cb24-4"><a href="#cb24-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Store the result at the first position in z</span></span>
<span id="cb24-5"><a href="#cb24-5" aria-hidden="true" tabindex="-1"></a><span class="co"># Remember, the first position is offset 0.</span></span>
<span id="cb24-6"><a href="#cb24-6" aria-hidden="true" tabindex="-1"></a>z[<span class="dv">0</span>] <span class="op">=</span> b</span>
<span id="cb24-7"><a href="#cb24-7" aria-hidden="true" tabindex="-1"></a><span class="co"># Second trial</span></span>
<span id="cb24-8"><a href="#cb24-8" aria-hidden="true" tabindex="-1"></a>a <span class="op">=</span> rnd.choice(zero_thru_99, size<span class="op">=</span><span class="dv">12</span>)</span>
<span id="cb24-9"><a href="#cb24-9" aria-hidden="true" tabindex="-1"></a>b <span class="op">=</span> np.<span class="bu">sum</span>(a <span class="op">></span> <span class="dv">74</span>)</span>
<span id="cb24-10"><a href="#cb24-10" aria-hidden="true" tabindex="-1"></a><span class="co"># Store the result at the second position in z</span></span>
<span id="cb24-11"><a href="#cb24-11" aria-hidden="true" tabindex="-1"></a>z[<span class="dv">1</span>] <span class="op">=</span> b</span>
<span id="cb24-12"><a href="#cb24-12" aria-hidden="true" tabindex="-1"></a><span class="co"># Third trial</span></span>
<span id="cb24-13"><a href="#cb24-13" aria-hidden="true" tabindex="-1"></a>a <span class="op">=</span> rnd.choice(zero_thru_99, size<span class="op">=</span><span class="dv">12</span>)</span>
<span id="cb24-14"><a href="#cb24-14" aria-hidden="true" tabindex="-1"></a>b <span class="op">=</span> np.<span class="bu">sum</span>(a <span class="op">></span> <span class="dv">74</span>)</span>
<span id="cb24-15"><a href="#cb24-15" aria-hidden="true" tabindex="-1"></a><span class="co"># Store the result at the third position in z</span></span>
<span id="cb24-16"><a href="#cb24-16" aria-hidden="true" tabindex="-1"></a>z[<span class="dv">2</span>] <span class="op">=</span> b</span>
<span id="cb24-17"><a href="#cb24-17" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb24-18"><a href="#cb24-18" aria-hidden="true" tabindex="-1"></a><span class="co"># And so on ...</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>This would get terribly long and boring to type for 50 trials. Luckily computer code is very good at repeating the same procedure many times. For example, Python can do this using a <code>for</code> loop. You have already seen a preview of the <code>for</code> loop in <a href="resampling_method.html" class="quarto-xref"><span>Chapter 2</span></a> and <a href="resampling_with_code.html" class="quarto-xref"><span>Chapter 5</span></a>. Here we dive into <code>for</code> loops in more depth.</p>
</section>
<section id="sec-for-loops" class="level3" data-number="6.6.2">
<h3 data-number="6.6.2" class="anchored" data-anchor-id="sec-for-loops">6.6.2 For-loops in Python</h3>
<p>A for-loop is a way of asking Python to:</p>
<ul>
<li>Take a sequence of things, one by one, and</li>
<li>Do the same task on each one.</li>
</ul>
<p>We often use this idea when we are trying to explain a repeating procedure. For example, imagine we wanted to explain what the supermarket checkout person does for the items in your shopping basket. You might say that they do this:</p>
<blockquote class="blockquote">
<p>For each item of shopping in your basket, they take the item off the conveyor belt, scan it, and put it on the other side of the till.</p>
</blockquote>
<p>You could also break this description up into bullet points with indentation, to say the same thing:</p>
<ul>
<li>For each item from your shopping basket, they:
<ul>
<li>Take the item off the conveyor belt.</li>
<li>Scan the item.</li>
<li>Put it on the other side of the till.</li>
</ul></li>
</ul>
<p>Notice the logic; the checkout person is repeating the same procedure for each of a series of items.</p>
<p>This is the logic of the <code>for</code> loop in Python. The procedure that Python repeats is called the <em>body of the for loop</em>. In the example of the checkout person above, the repeating procedure is:</p>
<ul>
<li>Take the item off the conveyor belt.</li>
<li>Scan the item.</li>
<li>Put it on the other side of the till.</li>
</ul>
<p>Now imagine we wanted to use Python to print out the year of birth for each of the authors for the third edition of this book:</p>
<table class="caption-top table">
<thead>
<tr class="header">
<th>Author</th>
<th>Year of birth</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>Julian Lincoln Simon</td>
<td>1932</td>
</tr>
<tr class="even">
<td>Matthew Brett</td>
<td>1964</td>
</tr>
<tr class="odd">
<td>Stéfan van der Walt</td>
<td>1980</td>
</tr>
</tbody>
</table>
<p>We want to see this output:</p>
<pre><code>Author birth year is 1932
Author birth year is 1964
Author birth year is 1980</code></pre>
<p>Of course, we could just ask Python to print out these exact lines, like this:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb26"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb26-1"><a href="#cb26-1" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'Author birth year is 1932'</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>Author birth year is 1932</code></pre>
</div>
<div class="sourceCode cell-code" id="cb28"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb28-1"><a href="#cb28-1" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'Author birth year is 1964'</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>Author birth year is 1964</code></pre>
</div>
<div class="sourceCode cell-code" id="cb30"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb30-1"><a href="#cb30-1" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'Author birth year is 1980'</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>Author birth year is 1980</code></pre>
</div>
</div>
<p>We might instead notice that we are repeating the same procedure for each of the three birth years, and decide to do the same thing using a <code>for</code> loop:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb32"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb32-1"><a href="#cb32-1" aria-hidden="true" tabindex="-1"></a>author_birth_years <span class="op">=</span> np.array([<span class="dv">1932</span>, <span class="dv">1964</span>, <span class="dv">1980</span>])</span>
<span id="cb32-2"><a href="#cb32-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb32-3"><a href="#cb32-3" aria-hidden="true" tabindex="-1"></a><span class="co"># For each birth year</span></span>
<span id="cb32-4"><a href="#cb32-4" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> birth_year <span class="kw">in</span> author_birth_years:</span>
<span id="cb32-5"><a href="#cb32-5" aria-hidden="true" tabindex="-1"></a> <span class="co"># Repeat this procedure ...</span></span>
<span id="cb32-6"><a href="#cb32-6" aria-hidden="true" tabindex="-1"></a> <span class="bu">print</span>(<span class="st">'Author birth year is'</span>, birth_year)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>Author birth year is 1932
Author birth year is 1964
Author birth year is 1980</code></pre>
</div>
</div>
<p>The <code>for</code> loop starts with a line where we tell it what items we want to repeat the procedure for:</p>
<div class="python">
<pre><code>for birth_year in author_birth_years:</code></pre>
<p>This <em>initial line</em> of the <code>for</code> loop ends with a colon.</p>
<p>The next thing in the <code>for</code> loop is the procedure Python should follow for each item. Python knows that the following lines are the procedure it should repeat, because the lines are <em>indented</em>. The <em>indented</em> lines are the <em>body of the for loop</em>.</p>
</div>
<p>The initial line of the <code>for</code> loop above tells Python that it should take <em>each item</em> in <code>author_birth_years</code>, one by one — first 1932, then 1964, then 1980. For each of these numbers it will:</p>
<ul>
<li>Put the number into the variable <code>birth_year</code>, then</li>
<li>Run the indented code.</li>
</ul>
<p>Just as the person at the supermarket checkout takes each item in turn, for each iteration (repeat) of the <code>for</code> loop, <code>birth_year</code> gets a new value from the sequence in <code>author_birth_years</code>. <code>birth_year</code> is called the <em>loop variable</em>, because it is the variable that gets a new value each time we begin a new iteration of the <code>for</code> loop procedure. As for any variable in Python, we can call our loop variable anything we like. We used <code>birth_year</code> here, but we could have used <code>y</code> or <code>year</code> or some other name.</p>
<p>Now you know what the <code>for</code> loop is doing, you can see that the <code>for</code> loop above is equivalent to the following code:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb35"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb35-1"><a href="#cb35-1" aria-hidden="true" tabindex="-1"></a>birth_year <span class="op">=</span> <span class="dv">1932</span> <span class="co"># Set the loop variable to contain the first value.</span></span>
<span id="cb35-2"><a href="#cb35-2" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'Author birth year is'</span>, birth_year) <span class="co"># Use it.</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>Author birth year is 1932</code></pre>
</div>
<div class="sourceCode cell-code" id="cb37"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb37-1"><a href="#cb37-1" aria-hidden="true" tabindex="-1"></a>birth_year <span class="op">=</span> <span class="dv">1964</span> <span class="co"># Set the loop variable to contain the next value.</span></span>
<span id="cb37-2"><a href="#cb37-2" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'Author birth year is'</span>, birth_year) <span class="co"># Use the second value.</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>Author birth year is 1964</code></pre>
</div>
<div class="sourceCode cell-code" id="cb39"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb39-1"><a href="#cb39-1" aria-hidden="true" tabindex="-1"></a>birth_year <span class="op">=</span> <span class="dv">1980</span></span>
<span id="cb39-2"><a href="#cb39-2" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'Author birth year is'</span>, birth_year)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>Author birth year is 1980</code></pre>
</div>
</div>
<p>Writing the steps in the <code>for</code> loop out like this is called <em>unrolling</em> the loop. It can be a useful exercise to do this when you come across a <code>for</code> loop, in order to work through the logic of the loop. For example, you may want to write out the unrolled equivalent of the first couple of iterations, to see what the loop variable will be, and what will happen in the body of the loop.</p>
<p>We often use <code>for</code> loops with ranges (see <a href="resampling_with_code.html#sec-ranges" class="quarto-xref"><span>Section 5.9</span></a>). Here we use a loop to print out the numbers 0 through 3:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb41"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb41-1"><a href="#cb41-1" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> n <span class="kw">in</span> np.arange(<span class="dv">4</span>):</span>
<span id="cb41-2"><a href="#cb41-2" aria-hidden="true" tabindex="-1"></a> <span class="bu">print</span>(<span class="st">'The loop variable n is'</span>, n)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>The loop variable n is 0
The loop variable n is 1
The loop variable n is 2
The loop variable n is 3</code></pre>
</div>
</div>
<p>Notice that the range ended at (the number before) 4, and that means we repeat the loop body 4 times. We can also use the loop variable value from the range as an <em>index</em>, to get or set the first, second, etc values from an array.</p>
<p>For example, maybe we would like to show the author position <em>and</em> the author year of birth.</p>
<p>Remember our author birth years:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb43"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb43-1"><a href="#cb43-1" aria-hidden="true" tabindex="-1"></a>author_birth_years</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>array([1932, 1964, 1980])</code></pre>
</div>
</div>
<p>We can get (for example) the second author birth year with:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb45"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb45-1"><a href="#cb45-1" aria-hidden="true" tabindex="-1"></a>author_birth_years[<span class="dv">1</span>]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>np.int64(1964)</code></pre>
</div>
</div>
<div class="python">
<p>Remember, for Python, the first element is position 0, so the second element is position 1.</p>
</div>
<p>Using the combination of looping over a range, and array indexing, we can print out the author position <em>and</em> the author birth year:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb47"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb47-1"><a href="#cb47-1" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> n <span class="kw">in</span> np.arange(<span class="dv">3</span>):</span>
<span id="cb47-2"><a href="#cb47-2" aria-hidden="true" tabindex="-1"></a> year <span class="op">=</span> author_birth_years[n]</span>
<span id="cb47-3"><a href="#cb47-3" aria-hidden="true" tabindex="-1"></a> <span class="bu">print</span>(<span class="st">'Birth year of author position'</span>, n, <span class="st">'is'</span>, year)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>Birth year of author position 0 is 1932
Birth year of author position 1 is 1964
Birth year of author position 2 is 1980</code></pre>
</div>
</div>
<div class="python">
<p>Again, remember Python considers 0 as the first position.</p>
</div>
<p>Just for practice, let us unroll the three iterations through this <code>for</code> loop, to remind ourselves what the code is doing:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb49"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb49-1"><a href="#cb49-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Unrolling the for loop.</span></span>
<span id="cb49-2"><a href="#cb49-2" aria-hidden="true" tabindex="-1"></a>n <span class="op">=</span> <span class="dv">0</span></span>
<span id="cb49-3"><a href="#cb49-3" aria-hidden="true" tabindex="-1"></a>year <span class="op">=</span> author_birth_years[n] <span class="co"># Will be 1932</span></span>
<span id="cb49-4"><a href="#cb49-4" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'Birth year of author position'</span>, n, <span class="st">'is'</span>, year)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>Birth year of author position 0 is 1932</code></pre>
</div>
<div class="sourceCode cell-code" id="cb51"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb51-1"><a href="#cb51-1" aria-hidden="true" tabindex="-1"></a>n <span class="op">=</span> <span class="dv">1</span></span>
<span id="cb51-2"><a href="#cb51-2" aria-hidden="true" tabindex="-1"></a>year <span class="op">=</span> author_birth_years[n] <span class="co"># Will be 1964</span></span>
<span id="cb51-3"><a href="#cb51-3" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'Birth year of author position'</span>, n, <span class="st">'is'</span>, year)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>Birth year of author position 1 is 1964</code></pre>
</div>
<div class="sourceCode cell-code" id="cb53"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb53-1"><a href="#cb53-1" aria-hidden="true" tabindex="-1"></a>n <span class="op">=</span> <span class="dv">2</span></span>
<span id="cb53-2"><a href="#cb53-2" aria-hidden="true" tabindex="-1"></a>year <span class="op">=</span> author_birth_years[n] <span class="co"># Will be 1980</span></span>
<span id="cb53-3"><a href="#cb53-3" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'Birth year of author position'</span>, n, <span class="st">'is'</span>, year)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>Birth year of author position 2 is 1980</code></pre>
</div>
</div>
</section>
<section id="sec-for-range" class="level3 python" data-number="6.6.3">
<h3 data-number="6.6.3" class="anchored" data-anchor-id="sec-for-range">6.6.3 <code>range</code> in Python <code>for</code> loops</h3>
<p>So far we have used <code>np.arange</code> to give us the sequence of integers that we feed into the <code>for</code> loop. But — as you saw in <a href="resampling_with_code.html#sec-python-range" class="quarto-xref"><span>Section 5.10</span></a> — we can also get a range of numbers from Python’s <code>range</code> function. <code>range</code> is a common and useful alternative way to provide a range of numbers to a <code>for</code> loop.</p>
<p>You have just seen how we would use <code>np.arange</code> to send the numbers 0, 1, 2, and 3 to a <code>for</code> loop, in the example above, repeated here:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb55"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb55-1"><a href="#cb55-1" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> n <span class="kw">in</span> np.arange(<span class="dv">3</span>):</span>
<span id="cb55-2"><a href="#cb55-2" aria-hidden="true" tabindex="-1"></a> year <span class="op">=</span> author_birth_years[n]</span>
<span id="cb55-3"><a href="#cb55-3" aria-hidden="true" tabindex="-1"></a> <span class="bu">print</span>(<span class="st">'Birth year of author position'</span>, n, <span class="st">'is'</span>, year)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>Birth year of author position 0 is 1932
Birth year of author position 1 is 1964
Birth year of author position 2 is 1980</code></pre>
</div>
</div>
<p>We could also use <code>range</code> instead of <code>np.arange</code> to do the same task:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb57"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb57-1"><a href="#cb57-1" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> n <span class="kw">in</span> <span class="bu">range</span>(<span class="dv">3</span>):</span>
<span id="cb57-2"><a href="#cb57-2" aria-hidden="true" tabindex="-1"></a> year <span class="op">=</span> author_birth_years[n]</span>
<span id="cb57-3"><a href="#cb57-3" aria-hidden="true" tabindex="-1"></a> <span class="bu">print</span>(<span class="st">'Birth year of author position'</span>, n, <span class="st">'is'</span>, year)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>Birth year of author position 0 is 1932
Birth year of author position 1 is 1964
Birth year of author position 2 is 1980</code></pre>
</div>
</div>
<p>In fact, you will see this pattern throughout the book, where we use <code>for</code> statements like <code>for value in range(10000):</code> to ask Python to put each number in the range 0 up to (not including) 10000 into the variable <code>value</code>, and then do something in the body of the loop. Just to be clear, we could always, and almost as easily, write <code>for value in np.arange(10000):</code> to do the same task. However, we generally prefer <code>range</code> in our Python <code>for</code> loops, because it is just a little less typing (without the <code>np.a</code> of <code>np.arange</code>), and because it is a more common pattern in standard Python code.<a href="#fn1" class="footnote-ref" id="fnref1" role="doc-noteref"><sup>1</sup></a></p>
</section>
<section id="putting-it-all-together" class="level3" data-number="6.6.4">
<h3 data-number="6.6.4" class="anchored" data-anchor-id="putting-it-all-together">6.6.4 Putting it all together</h3>
<p>Here is the code we worked out above, to implement a single trial:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb59"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb59-1"><a href="#cb59-1" aria-hidden="true" tabindex="-1"></a>rnd <span class="op">=</span> np.random.default_rng()</span>
<span id="cb59-2"><a href="#cb59-2" aria-hidden="true" tabindex="-1"></a>zero_thru_99 <span class="op">=</span> np.arange(<span class="dv">100</span>)</span>
<span id="cb59-3"><a href="#cb59-3" aria-hidden="true" tabindex="-1"></a><span class="co"># Get 12 random numbers from 0 through 99</span></span>
<span id="cb59-4"><a href="#cb59-4" aria-hidden="true" tabindex="-1"></a>a <span class="op">=</span> rnd.choice(zero_thru_99, size<span class="op">=</span><span class="dv">12</span>)</span>
<span id="cb59-5"><a href="#cb59-5" aria-hidden="true" tabindex="-1"></a><span class="co"># How many numbers are greater than 74?</span></span>
<span id="cb59-6"><a href="#cb59-6" aria-hidden="true" tabindex="-1"></a>b <span class="op">=</span> np.<span class="bu">sum</span>(a <span class="op">></span> <span class="dv">74</span>)</span>
<span id="cb59-7"><a href="#cb59-7" aria-hidden="true" tabindex="-1"></a><span class="co"># Show the result</span></span>
<span id="cb59-8"><a href="#cb59-8" aria-hidden="true" tabindex="-1"></a>b</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>np.int64(4)</code></pre>
</div>
</div>
<p>We found that we could use arrays to store the results of these trials, and that we could use <code>for</code> loops to repeat the same procedure many times.</p>
<p>Now we can put these parts together to do 50 simulated trials:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb61"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb61-1"><a href="#cb61-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Procedure for 50 simulated trials.</span></span>
<span id="cb61-2"><a href="#cb61-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb61-3"><a href="#cb61-3" aria-hidden="true" tabindex="-1"></a><span class="co"># The Numpy random number generator.</span></span>
<span id="cb61-4"><a href="#cb61-4" aria-hidden="true" tabindex="-1"></a>rnd <span class="op">=</span> np.random.default_rng()</span>
<span id="cb61-5"><a href="#cb61-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb61-6"><a href="#cb61-6" aria-hidden="true" tabindex="-1"></a><span class="co"># All the numbers from 0 through 99.</span></span>
<span id="cb61-7"><a href="#cb61-7" aria-hidden="true" tabindex="-1"></a>zero_through_99 <span class="op">=</span> np.arange(<span class="dv">100</span>)</span>
<span id="cb61-8"><a href="#cb61-8" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb61-9"><a href="#cb61-9" aria-hidden="true" tabindex="-1"></a><span class="co"># An array to store the counts for each trial.</span></span>
<span id="cb61-10"><a href="#cb61-10" aria-hidden="true" tabindex="-1"></a>z <span class="op">=</span> np.zeros(<span class="dv">50</span>)</span>
<span id="cb61-11"><a href="#cb61-11" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb61-12"><a href="#cb61-12" aria-hidden="true" tabindex="-1"></a><span class="co"># Repeat the trial procedure 50 times.</span></span>
<span id="cb61-13"><a href="#cb61-13" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> i <span class="kw">in</span> np.arange(<span class="dv">50</span>):</span>
<span id="cb61-14"><a href="#cb61-14" aria-hidden="true" tabindex="-1"></a> <span class="co"># Get 12 random numbers from 0 through 99</span></span>
<span id="cb61-15"><a href="#cb61-15" aria-hidden="true" tabindex="-1"></a> a <span class="op">=</span> rnd.choice(zero_through_99, size<span class="op">=</span><span class="dv">12</span>)</span>
<span id="cb61-16"><a href="#cb61-16" aria-hidden="true" tabindex="-1"></a> <span class="co"># How many numbers are greater than 74?</span></span>
<span id="cb61-17"><a href="#cb61-17" aria-hidden="true" tabindex="-1"></a> b <span class="op">=</span> np.<span class="bu">sum</span>(a <span class="op">></span> <span class="dv">74</span>)</span>
<span id="cb61-18"><a href="#cb61-18" aria-hidden="true" tabindex="-1"></a> <span class="co"># Store the result at the next position in the "z" array.</span></span>
<span id="cb61-19"><a href="#cb61-19" aria-hidden="true" tabindex="-1"></a> z[i] <span class="op">=</span> b</span>
<span id="cb61-20"><a href="#cb61-20" aria-hidden="true" tabindex="-1"></a> <span class="co"># Now go back and do the next trial until finished.</span></span>
<span id="cb61-21"><a href="#cb61-21" aria-hidden="true" tabindex="-1"></a><span class="co"># Show the result of all 50 trials.</span></span>
<span id="cb61-22"><a href="#cb61-22" aria-hidden="true" tabindex="-1"></a>z</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>array([4., 2., 3., 3., 4., 1., 4., 2., 7., 2., 3., 1., 6., 2., 5., 5., 3.,
1., 3., 4., 2., 2., 2., 4., 3., 4., 4., 2., 3., 3., 3., 1., 3., 1.,
2., 3., 2., 2., 3., 3., 6., 1., 3., 3., 4., 2., 4., 3., 4., 3.])</code></pre>
</div>
</div>