-
Notifications
You must be signed in to change notification settings - Fork 0
Expand file tree
/
Copy pathtesting_procedures.html
More file actions
1097 lines (1066 loc) · 65.2 KB
/
testing_procedures.html
File metadata and controls
1097 lines (1066 loc) · 65.2 KB
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>25 General Procedures for Testing Hypotheses – 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="./confidence_1.html" rel="next">
<link href="./testing_measured.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="./testing_procedures.html"><span class="chapter-number">25</span> <span class="chapter-title">General Procedures for Testing Hypotheses</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">
<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 active">
<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="#introduction" id="toc-introduction" class="nav-link active" data-scroll-target="#introduction"><span class="header-section-number">25.1</span> Introduction</a></li>
<li><a href="#canonical-question-and-answer-procedure-for-testing-hypotheses" id="toc-canonical-question-and-answer-procedure-for-testing-hypotheses" class="nav-link" data-scroll-target="#canonical-question-and-answer-procedure-for-testing-hypotheses"><span class="header-section-number">25.2</span> Canonical question-and-answer procedure for testing hypotheses</a></li>
<li><a href="#skeleton-procedure-for-testing-hypotheses" id="toc-skeleton-procedure-for-testing-hypotheses" class="nav-link" data-scroll-target="#skeleton-procedure-for-testing-hypotheses"><span class="header-section-number">25.3</span> Skeleton procedure for testing hypotheses</a></li>
<li><a href="#an-example-can-the-bio-engineer-increase-the-female-calf-rate" id="toc-an-example-can-the-bio-engineer-increase-the-female-calf-rate" class="nav-link" data-scroll-target="#an-example-can-the-bio-engineer-increase-the-female-calf-rate"><span class="header-section-number">25.4</span> An example: can the bio-engineer increase the female calf rate?</a></li>
<li><a href="#computation-of-probabilities-with-resampling" id="toc-computation-of-probabilities-with-resampling" class="nav-link" data-scroll-target="#computation-of-probabilities-with-resampling"><span class="header-section-number">25.5</span> Computation of Probabilities with Resampling</a></li>
<li><a href="#conventional-methods" id="toc-conventional-methods" class="nav-link" data-scroll-target="#conventional-methods"><span class="header-section-number">25.6</span> Conventional methods</a>
<ul class="collapse">
<li><a href="#the-sample-space-and-first-principles" id="toc-the-sample-space-and-first-principles" class="nav-link" data-scroll-target="#the-sample-space-and-first-principles"><span class="header-section-number">25.6.1</span> The Sample Space and First Principles</a></li>
<li><a href="#sample-space-calculations" id="toc-sample-space-calculations" class="nav-link" data-scroll-target="#sample-space-calculations"><span class="header-section-number">25.6.2</span> Sample Space Calculations</a></li>
<li><a href="#pascals-triangle" id="toc-pascals-triangle" class="nav-link" data-scroll-target="#pascals-triangle"><span class="header-section-number">25.6.3</span> Pascal’s Triangle</a></li>
<li><a href="#the-quincunx" id="toc-the-quincunx" class="nav-link" data-scroll-target="#the-quincunx"><span class="header-section-number">25.6.4</span> The Quincunx</a></li>
<li><a href="#table-of-binomial-coefficients" id="toc-table-of-binomial-coefficients" class="nav-link" data-scroll-target="#table-of-binomial-coefficients"><span class="header-section-number">25.6.5</span> Table of Binomial Coefficients</a></li>
<li><a href="#binomial-formula" id="toc-binomial-formula" class="nav-link" data-scroll-target="#binomial-formula"><span class="header-section-number">25.6.6</span> Binomial Formula</a></li>
<li><a href="#the-normal-approximation" id="toc-the-normal-approximation" class="nav-link" data-scroll-target="#the-normal-approximation"><span class="header-section-number">25.6.7</span> The Normal Approximation</a></li>
</ul></li>
<li><a href="#choice-of-the-benchmark-universebruce" id="toc-choice-of-the-benchmark-universebruce" class="nav-link" data-scroll-target="#choice-of-the-benchmark-universebruce"><span class="header-section-number">25.7</span> Choice of the benchmark universe</a></li>
<li><a href="#why-is-statistics-and-hypothesis-testing-so-difficult" id="toc-why-is-statistics-and-hypothesis-testing-so-difficult" class="nav-link" data-scroll-target="#why-is-statistics-and-hypothesis-testing-so-difficult"><span class="header-section-number">25.8</span> Why is statistics — and hypothesis testing — so difficult?</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-testing-procedures" class="quarto-section-identifier"><span class="chapter-number">25</span> <span class="chapter-title">General Procedures for Testing Hypotheses</span></span></h1>
</div>
<div class="quarto-title-meta">
</div>
</header>
<section id="introduction" class="level2" data-number="25.1">
<h2 data-number="25.1" class="anchored" data-anchor-id="introduction"><span class="header-section-number">25.1</span> Introduction</h2>
<p>The previous chapters have presented procedures for making statistical inferences that apply to both testing hypotheses and constructing confidence intervals: This chapter focuses on specific procedures for testing hypotheses.</p>
<p>`The general idea in testing hypotheses is to ask: Is there some <em>other</em> universe which might well have produced the observed sample? So we consider alternative hypotheses. This is a straightforward exercise in probability, asking about behavior of one or more universes. The choice of another universe(s) to examine depends upon purposes and other considerations.</p>
</section>
<section id="canonical-question-and-answer-procedure-for-testing-hypotheses" class="level2" data-number="25.2">
<h2 data-number="25.2" class="anchored" data-anchor-id="canonical-question-and-answer-procedure-for-testing-hypotheses"><span class="header-section-number">25.2</span> Canonical question-and-answer procedure for testing hypotheses</h2>
</section>
<section id="skeleton-procedure-for-testing-hypotheses" class="level2" data-number="25.3">
<h2 data-number="25.3" class="anchored" data-anchor-id="skeleton-procedure-for-testing-hypotheses"><span class="header-section-number">25.3</span> Skeleton procedure for testing hypotheses</h2>
<p><strong>Akin to skeleton procedure for questions in probability and confidence intervals shown elsewhere</strong></p>
<p>The following series of questions will be repeated below in the context of a specific inference.</p>
<p><em>What is the question? What is the purpose to be served by answering the question?</em></p>
<p><em>Is this a “probability” or a “statistics” question?</em></p>
<p><strong>Assuming the Question is a Statistical Inference Question</strong></p>
<p><em>What is the form of the statistics question?</em></p>
<p>Hypothesis test, or confidence interval, or other inference? One must first decide whether the conceptual-scientific question is of the form a) a test about the probability that some sample is likely to happen by chance rather than being very surprising (a test of a hypothesis), or b) a question about the accuracy of the estimate of a parameter of the population based upon sample evidence (a confidence interval):</p>
<p><strong>Assuming the Question Concerns Testing Hypotheses</strong></p>
<p><em>Will you state the costs and benefits of various outcomes, perhaps in the form of a “loss function”? If “yes,” what are they?</em></p>
<p><em>How many samples of data have been observed?</em></p>
<p>One, two, more than two?</p>
<p><em>What is the description of the observed sample(s)?</em></p>
<p>Raw data?</p>
<p><em>Which characteristic(s) (parameters) of the population are of interest to you?</em></p>
<p><em>What are the statistics of the sample(s) that refer to this (these) characteristics(s) in which you are interested?</em></p>
<p><em>What comparison(s) to make?</em></p>
<p>Samples to each other?</p>
<p>Sample to particular universe(s)? If so, which?</p>
<p><em>What is the benchmark (null) universe?</em></p>
<p>This may include presenting the raw data and/or such summary statistics as the computed mean, median, standard deviation, range, interquartile range, other:</p>
<p><em>If there is to be a Neyman-Pearson-type alternative universe, what is it?</em> (In most cases the answer to this technical question is “no.”)</p>
<p><em>Which symbols for the observed entities?</em></p>
<p>Discrete or continuous?</p>
<p>What values or ranges of values?</p>
<p><em>Which sample(s) do you wish to compare to which, or to the null universe (and perhaps to the alternative universe)?</em> (Answer: samples the same size as has been observed)</p>
<p>[Here one may continue with the conventional method, using perhaps a <em>t</em> or <em>f</em> or chi-square test or whatever: Everything up to now is the same whether continuing with resampling or with standard parametric test.]</p>
<p><em>What procedure will be used to produce the resampled entities?</em></p>
<p>Randomly drawn?</p>
<p>Simple (single step) or complex (multiple “if” drawings)?</p>
<p><em>What procedure to produce resample?</em></p>
<p>Which universe will you draw them from? With or without replacement?</p>
<p>What size resamples? Number of resample trials?</p>
<p><em>What to record as outcome of each resample trial?</em></p>
<p>Mean, median, or whatever of resample?</p>
<p><strong>Classifying the outcomes</strong></p>
<p><em>What is the criterion of significance to be used in evaluating the results of the test?</em></p>
<p><strong>Stating the distribution of results</strong></p>
<p>Graph of each statistic recorded — occurrences for each value.</p>
<p>Count the outcomes that exceed criterion and divide by number of trials.</p>
</section>
<section id="an-example-can-the-bio-engineer-increase-the-female-calf-rate" class="level2" data-number="25.4">
<h2 data-number="25.4" class="anchored" data-anchor-id="an-example-can-the-bio-engineer-increase-the-female-calf-rate"><span class="header-section-number">25.4</span> An example: can the bio-engineer increase the female calf rate?</h2>
<p><em>The question.</em> (from <span class="citation" data-cites="hodges1970basic">(<a href="references.html#ref-hodges1970basic" role="doc-biblioref">Hodges Jr and Lehmann 1970, 310</a>)</span>: Female calves are more valuable than male calves. A bio-engineer claims to have a method that can produce more females. He tests the procedure on ten of your pregnant cows, and the result is nine females. Should you believe that his method has some effect? That is, what is the probability of a result this surprising occurring by chance?</p>
<p><em>The purpose:</em> Female calves are more valuable than male.</p>
<p><em>Inference?</em> Yes.</p>
<p><em>Test of hypothesis?</em> Yes.</p>
<p><em>Will you state the costs and benefits</em> of various outcomes (or a loss function)? We need only say that the benefits of a method that works are very large, and if the results are promising, it is worth gathering more data to confirm results.</p>
<p><em>How many samples of data</em> are part of the significance test? One</p>
<p><em>What is the size of the first sample</em> about which you wish to make significance statements? Ten.</p>
<p><em>What comparison(s) to make?</em> Compare sample to benchmark universe.</p>
<p><em>What is the benchmark universe</em> that embodies the null hypothesis? 50-50 female, or 100/206 female.</p>
<p><em>If there is to be a Neyman-Pearson alternative universe</em>, what is it? None.</p>
<p><em>Which symbols for the observed entities?</em> Balls in bucket, or numbers.</p>
<p><em>What values or ranges of values?</em> 0-1, (1-100), or 101-206.</p>
<p><em>Finite or infinite?</em> Infinite.</p>
<p><em>Which sample(s) do you wish to compare</em> to which, or to the null universe (and perhaps to the alternative universe)? Ten calves compared to universe.</p>
<p><em>What procedure to produce entities?</em> Sampling with replacement,</p>
<p><em>Simple (single step) or complex (multiple “if” drawings)?</em> One can think of it either way.</p>
<p><em>What to record as outcome of each resample trial?</em> The proportion (or number) of females.</p>
<p><em>What is the criterion</em> to be used in the test? The probability that in a sample of ten calves, nine (or more) females would be drawn by chance from the benchmark universe of half females. (Or frame in terms of a significance level.)</p>
<p><em>“One-tail” or “two-tail” test?</em> One tail, because the farmer is only interested in females: Finding a large proportion of males would not be of interest, and would not cause one to reject the null hypothesis.</p>
<p><em>Computation of the probability sought.</em> The actual computation of probability may be done with several formulaic or sample-space methods, and with several resampling methods: I will first show a resampling method and then several conventional methods. The following material, which allows one to compare resampling and conventional methods, is more germane to the earlier explication of resampling taken altogether in earlier chapters than it is to the theory of hypothesis tests discussed in this chapter, but it is more expedient to present it here.</p>
</section>
<section id="computation-of-probabilities-with-resampling" class="level2" data-number="25.5">
<h2 data-number="25.5" class="anchored" data-anchor-id="computation-of-probabilities-with-resampling"><span class="header-section-number">25.5</span> Computation of Probabilities with Resampling</h2>
<p>We can do the problem by hand as follows:</p>
<ol type="1">
<li>Constitute a bucket with either one blue and one pink ball, or 106 blue and 100 pink balls.</li>
<li>Draw ten balls with replacement, count pinks, and record.</li>
<li>Repeat step (2) say 400 times.</li>
<li>Calculate proportion of results with 9 or 10 pinks.</li>
</ol>
<p>Or, we can take advantage of the speed and efficiency of the computer as follows:</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="im">import</span> numpy <span class="im">as</span> np</span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</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>rnd <span class="op">=</span> np.random.default_rng()</span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a>n <span class="op">=</span> <span class="dv">10000</span></span>
<span id="cb1-7"><a href="#cb1-7" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-8"><a href="#cb1-8" aria-hidden="true" tabindex="-1"></a>females <span class="op">=</span> np.zeros(n)</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="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(n):</span>
<span id="cb1-11"><a href="#cb1-11" aria-hidden="true" tabindex="-1"></a> samp <span class="op">=</span> rnd.choice([<span class="st">'female'</span>, <span class="st">'male'</span>], size<span class="op">=</span><span class="dv">10</span>, replace<span class="op">=</span><span class="va">True</span>)</span>
<span id="cb1-12"><a href="#cb1-12" aria-hidden="true" tabindex="-1"></a> females[i] <span class="op">=</span> np.<span class="bu">sum</span>(samp <span class="op">==</span> <span class="st">'female'</span>)</span>
<span id="cb1-13"><a href="#cb1-13" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-14"><a href="#cb1-14" aria-hidden="true" tabindex="-1"></a>plt.hist(females, bins<span class="op">=</span><span class="st">'auto'</span>)</span>
<span id="cb1-15"><a href="#cb1-15" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-16"><a href="#cb1-16" aria-hidden="true" tabindex="-1"></a>k <span class="op">=</span> np.<span class="bu">sum</span>(females <span class="op">>=</span> <span class="dv">9</span>)</span>
<span id="cb1-17"><a href="#cb1-17" aria-hidden="true" tabindex="-1"></a>kk <span class="op">=</span> k <span class="op">/</span> n</span>
<span id="cb1-18"><a href="#cb1-18" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'Proportion with >= 9 females:'</span>, kk)</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>Proportion with >= 9 females: 0.0127</code></pre>
</div>
<div class="cell-output-display">
<div class="quarto-figure quarto-figure-center">
<figure class="figure">
<p><img src="testing_procedures_files/figure-html/unnamed-chunk-1-1.png" class="img-fluid quarto-figure quarto-figure-center figure-img" style="width:70.0%"></p>
</figure>
</div>
</div>
</div>
<p>This outcome implies that there is roughly a one percent chance that one would observe 9 or 10 female births in a single sample of 10 calves if the probability of a female on each birth is .5. This outcome should help the decision-maker decide about the plausibility of the bio-engineer’s claim to be able to increase the probability of female calves being born.</p>
</section>
<section id="conventional-methods" class="level2" data-number="25.6">
<h2 data-number="25.6" class="anchored" data-anchor-id="conventional-methods"><span class="header-section-number">25.6</span> Conventional methods</h2>
<section id="the-sample-space-and-first-principles" class="level3" data-number="25.6.1">
<h3 data-number="25.6.1" class="anchored" data-anchor-id="the-sample-space-and-first-principles"><span class="header-section-number">25.6.1</span> The Sample Space and First Principles</h3>
<p>Assume for a moment that our problem is a smaller one and therefore much easier — the probability of getting two females in two calves if the probability of a female is .5. One could then map out what mathematicians call the “sample space,” a technique that (in its simplest form) assigns to each outcome a single point, and find the proportion of points that correspond to a “success.” We list all four possible combinations — FF, FM, MF, MM. Now we look at the ratio of the number of combinations that have 2 females to the total, which is 1/4. We may then interpret this probability.</p>
<p>We might also use this method for (say) <em>five</em> female calves <em>in a row</em>. We can make a list of possibilities such as FFFFF, MFFFF, MMFFF, MMMFFF … MFMFM … MMMMM. There will be 2*2*2*2*2 = 32 possibilities, and 64 and 128 possibilities for six and seven calves respectively. But when we get as high as ten calves, this method would become very troublesome.</p>
</section>
<section id="sample-space-calculations" class="level3" data-number="25.6.2">
<h3 data-number="25.6.2" class="anchored" data-anchor-id="sample-space-calculations"><span class="header-section-number">25.6.2</span> Sample Space Calculations</h3>
<p>For two females in a row, we could use the well known, and very simple, multiplication rule; we could do so even for ten females in a row. But calculating the probability of nine females in ten is a bit more complex.</p>
</section>
<section id="pascals-triangle" class="level3" data-number="25.6.3">
<h3 data-number="25.6.3" class="anchored" data-anchor-id="pascals-triangle"><span class="header-section-number">25.6.3</span> Pascal’s Triangle</h3>
<p>One can use Pascal’s Triangle to obtain binomial coefficients for p = .5 and a sample size of 10, focusing on those for 9 or 10 successes. Then calculate the <em>proportion of the total cases</em> with 9 or 10 “successes” in one direction, to find the proportion of cases that pass beyond the criterion of 9 females. The method of Pascal’s Triangle requires more complete understanding of the probabilistic system than does the resampling simulation described above because Pascal’s Triangle requires that one understand the entire structure; simulation requires only that you follow the rules of the model.</p>
</section>
<section id="the-quincunx" class="level3" data-number="25.6.4">
<h3 data-number="25.6.4" class="anchored" data-anchor-id="the-quincunx"><span class="header-section-number">25.6.4</span> The Quincunx</h3>
<p>The quincunx — a device that filters tiny balls through a set of bumper points not unlike a pinball machine, mentioned here simply for completeness — is more a simulation method than theoretical, but it may be considered “conventional.” Hence, it is included here.</p>
</section>
<section id="table-of-binomial-coefficients" class="level3" data-number="25.6.5">
<h3 data-number="25.6.5" class="anchored" data-anchor-id="table-of-binomial-coefficients"><span class="header-section-number">25.6.5</span> Table of Binomial Coefficients</h3>
<p>Pascal’s Triangle becomes cumbersome or impractical with large numbers — say, 17 females of 20 births — or with probabilities other than .5. One might produce the binomial coefficients by algebraic multiplication, but that, too, becomes tedious even with small sample sizes. One can also use the pre-computed table of binomial coefficients found in any standard text. But the probabilities for n = 10 and 9 or 10 females are too small to be shown.</p>
</section>
<section id="binomial-formula" class="level3" data-number="25.6.6">
<h3 data-number="25.6.6" class="anchored" data-anchor-id="binomial-formula"><span class="header-section-number">25.6.6</span> Binomial Formula</h3>
<p>For larger sample sizes, one can use the binomial formula. The binomial formula gives no deeper understanding of the statistical structure than does the Triangle (but it does yield a deeper understanding of the pure mathematics). With very large numbers, even the binomial formula is cumbersome.</p>
</section>
<section id="the-normal-approximation" class="level3" data-number="25.6.7">
<h3 data-number="25.6.7" class="anchored" data-anchor-id="the-normal-approximation"><span class="header-section-number">25.6.7</span> The Normal Approximation</h3>
<p>When the sample size becomes too large for any of the above methods, one can then use the Normal approximation, which yields results close to the binomial (as seen very nicely in the output of the quincunx). But use of the Normal distribution requires an estimate of the standard deviation, which can be derived either by formula or by resampling. (See a more extended parallel discussion in <a href="confidence_2.html" class="quarto-xref"><span>Chapter 27</span></a> on confidence intervals for the Bush-Dukakis comparison.)</p>
<p>The desired probability can be obtained from the Z formula and a standard table of the Normal distribution found in every elementary text.</p>
<p>The Z table can be made less mysterious if we generate it with simulation, or with graph paper or Archimedes’ method, using as raw material (say) five “continuous” (that is, non-binomial) distributions, many of which are skewed: 1) Draw samples of (say) 50 or 100. 2) Plot the means to see that the Normal shape is the outcome. Then 3) standardize with the standard deviation by marking the standard deviations onto the histograms.</p>
<p>The aim of the above exercise and the heart of the conventional parametric method is to compare the sample result — the mean — to a standardized plot of the means of samples drawn from the universe of interest to see how likely it is that that universe produces means deviating as much from the universe mean as does our observed sample mean. The steps are:</p>
<ol type="1">
<li>Establish the Normal shape — from the exercise above, or from the quincunx or Pascal’s Triangle or the binomial formula or the formula for the Normal approximation or some other device.</li>
<li>Standardize that shape in standard deviations.</li>
<li>Compute the Z score for the sample mean — that is, its deviation from the universe mean in standard deviations.</li>
<li>Examine the Normal (or really, tables computed from graph paper, etc.) to find the probability of a mean deviating that far by chance.</li>
</ol>
<p>This is the canon of the procedure for most parametric work in statistics. (For some small samples, accuracy is improved with an adjustment.)</p>
</section>
</section>
<section id="choice-of-the-benchmark-universebruce" class="level2" data-number="25.7">
<h2 data-number="25.7" class="anchored" data-anchor-id="choice-of-the-benchmark-universebruce"><span class="header-section-number">25.7</span> Choice of the benchmark universe<a href="#fn1" class="footnote-ref" id="fnref1" role="doc-noteref"><sup>1</sup></a></h2>
<p>In the example of the ten calves, the choice of a benchmark universe — a universe that (on average) produces equal proportions of males and females — seems rather straightforward and even automatic, requiring no difficult judgments. But in other cases the process requires more judgments.</p>
<p>Let’s consider another case where the choice of a benchmark universe requires no difficult judgments. Assume the U.S. Department of Labor’s Bureau of Labor Statistics (BLS) takes a very large sample — say, 20,000 persons — and finds a 10 percent unemployment rate. At some later time another but smaller sample is drawn — 2,000 persons — showing an 11 percent unemployment rate. Should BLS conclude that unemployment has risen, or is there a large chance that the difference between 10 percent and 11 percent is due to sample variability? In this case, it makes rather obvious sense to ask how often a sample of 2,000 drawn from a universe of 10 percent unemployment (ignoring the variability in the larger sample) will be as different as 11 percent due solely to sample variability? This problem differs from that of the calves only in the proportions and the sizes of the samples.</p>
<p>Let’s change the facts and assume that a very large sample had not been drawn and only a sample of 2,000 had been taken, indicating 11 percent unemployment. A policy-maker asks the probability that unemployment is above ten percent. It would still seem rather straightforward to ask how often a universe of 10 percent unemployment would produce a sample of 2000 with a proportion of 11 percent unemployed.</p>
<p>Still another problem where the choice of benchmark hypothesis is relatively straightforward: Say that BLS takes two samples of 2000 persons a month apart, and asks whether there is a difference in the results. Pooling the two samples and examining how often two samples drawn from the pooled universe would be as different as observed seems obvious.</p>
<p>One of the reasons that the above cases — especially the two-sample case — seem so clear-cut is that the variance of the benchmark hypothesis is not an issue, being implied by the fact that the samples deal with proportions. If the data were continuous, however, this issue would quickly arise. Consider, for example, that the BLS might take the same sorts of samples and ask unemployed persons the <em>lengths of time</em> they had been unemployed. Comparing a small sample to a very large one would be easy to decide about. And even comparing two small samples might be straightforward — simply pooling them as is.</p>
<p>But what about if you have a sample of 2,000 with data on lengths of unemployment spells with a mean of 30 days, and you are asked the probability that it comes from a universe with a mean of 25 days? Now there arises the question about the amount of variability to assume for that benchmark universe. Should it be the variability observed in the sample? That is probably an overestimate, because a universe with a smaller mean would probably have a smaller variance, too. So some judgment is required; there cannot be an automatic “objective” process here, whether one proceeds with the conventional or the resampling method.</p>
<p>The example of the comparison of liquor retailing systems in <a href="testing_measured.html#sec-liquor-public-measured" class="quarto-xref"><span>Section 24.0.2</span></a> provides more material on this subject.</p>
</section>
<section id="why-is-statistics-and-hypothesis-testing-so-difficult" class="level2" data-number="25.8">
<h2 data-number="25.8" class="anchored" data-anchor-id="why-is-statistics-and-hypothesis-testing-so-difficult"><span class="header-section-number">25.8</span> Why is statistics — and hypothesis testing — so difficult?</h2>
<p>Why is statistics such a difficult subject? The aforegoing procedural outline provides a window to the explanation. Hypothesis testing — as is also true of the construction of confidence intervals (but unlike simple probability problems) — involves a very long chain of reasoning, perhaps longer than in any other realm of systematic thinking. Furthermore, many decisions in the process require judgment that goes beyond technical analysis. All this emerges as one proceeds through the skeleton procedure above with any specific example.</p>
<p>(Bayes’ rule also is very difficult intuitively, but that probably is a result of the twists and turns required in all complex problems in conditional probability. Decision-tree analysis is counter-intuitive, too, probably because it starts at the end instead of the beginning of the story, as we are usually accustomed to doing.)</p>
<div id="refs" class="references csl-bib-body hanging-indent" data-entry-spacing="0" role="list" style="display: none">
<div id="ref-hodges1970basic" class="csl-entry" role="listitem">
Hodges Jr, Joseph Lawson, and Erich Leo Lehmann. 1970. <em>Basic Concepts of Probability and Statistics</em>. 2nd ed. San Francisco, California: <span>H</span>olden-<span>D</span>ay, <span>I</span>nc. <a href="https://archive.org/details/basicconceptsofp0000unse_m8m9">https://archive.org/details/basicconceptsofp0000unse_m8m9</a>.
</div>
</div>
</section>
<section id="footnotes" class="footnotes footnotes-end-of-document" role="doc-endnotes">
<hr>
<ol>
<li id="fn1"><p>This is one of many issues that Peter Bruce first raised, and whose treatment here reflects back-and-forth discussion between us.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
</ol>
</section>
</main> <!-- /main -->
<script id="quarto-html-after-body" type="application/javascript">
window.document.addEventListener("DOMContentLoaded", function (event) {
const toggleBodyColorMode = (bsSheetEl) => {
const mode = bsSheetEl.getAttribute("data-mode");
const bodyEl = window.document.querySelector("body");
if (mode === "dark") {
bodyEl.classList.add("quarto-dark");
bodyEl.classList.remove("quarto-light");
} else {
bodyEl.classList.add("quarto-light");
bodyEl.classList.remove("quarto-dark");
}
}
const toggleBodyColorPrimary = () => {
const bsSheetEl = window.document.querySelector("link#quarto-bootstrap");
if (bsSheetEl) {
toggleBodyColorMode(bsSheetEl);
}
}
toggleBodyColorPrimary();
const icon = "";
const anchorJS = new window.AnchorJS();
anchorJS.options = {
placement: 'right',
icon: icon
};
anchorJS.add('.anchored');
const isCodeAnnotation = (el) => {
for (const clz of el.classList) {
if (clz.startsWith('code-annotation-')) {
return true;
}
}
return false;
}
const onCopySuccess = function(e) {
// button target
const button = e.trigger;
// don't keep focus
button.blur();
// flash "checked"
button.classList.add('code-copy-button-checked');
var currentTitle = button.getAttribute("title");
button.setAttribute("title", "Copied!");
let tooltip;
if (window.bootstrap) {
button.setAttribute("data-bs-toggle", "tooltip");
button.setAttribute("data-bs-placement", "left");
button.setAttribute("data-bs-title", "Copied!");
tooltip = new bootstrap.Tooltip(button,
{ trigger: "manual",
customClass: "code-copy-button-tooltip",
offset: [0, -8]});
tooltip.show();
}
setTimeout(function() {
if (tooltip) {
tooltip.hide();
button.removeAttribute("data-bs-title");
button.removeAttribute("data-bs-toggle");
button.removeAttribute("data-bs-placement");
}
button.setAttribute("title", currentTitle);
button.classList.remove('code-copy-button-checked');
}, 1000);
// clear code selection
e.clearSelection();
}
const getTextToCopy = function(trigger) {
const codeEl = trigger.previousElementSibling.cloneNode(true);
for (const childEl of codeEl.children) {
if (isCodeAnnotation(childEl)) {
childEl.remove();
}
}
return codeEl.innerText;
}
const clipboard = new window.ClipboardJS('.code-copy-button:not([data-in-quarto-modal])', {
text: getTextToCopy
});
clipboard.on('success', onCopySuccess);
if (window.document.getElementById('quarto-embedded-source-code-modal')) {
// For code content inside modals, clipBoardJS needs to be initialized with a container option
// TODO: Check when it could be a function (https://github.com/zenorocha/clipboard.js/issues/860)
const clipboardModal = new window.ClipboardJS('.code-copy-button[data-in-quarto-modal]', {
text: getTextToCopy,
container: window.document.getElementById('quarto-embedded-source-code-modal')
});
clipboardModal.on('success', onCopySuccess);
}
var localhostRegex = new RegExp(/^(?:http|https):\/\/localhost\:?[0-9]*\//);
var mailtoRegex = new RegExp(/^mailto:/);
var filterRegex = new RegExp('/' + window.location.host + '/');
var isInternal = (href) => {
return filterRegex.test(href) || localhostRegex.test(href) || mailtoRegex.test(href);
}
// Inspect non-navigation links and adorn them if external
var links = window.document.querySelectorAll('a[href]:not(.nav-link):not(.navbar-brand):not(.toc-action):not(.sidebar-link):not(.sidebar-item-toggle):not(.pagination-link):not(.no-external):not([aria-hidden]):not(.dropdown-item):not(.quarto-navigation-tool):not(.about-link)');
for (var i=0; i<links.length; i++) {
const link = links[i];
if (!isInternal(link.href)) {
// undo the damage that might have been done by quarto-nav.js in the case of
// links that we want to consider external
if (link.dataset.originalHref !== undefined) {
link.href = link.dataset.originalHref;
}
}
}
function tippyHover(el, contentFn, onTriggerFn, onUntriggerFn) {
const config = {
allowHTML: true,
maxWidth: 500,
delay: 100,
arrow: false,
appendTo: function(el) {
return el.parentElement;
},
interactive: true,
interactiveBorder: 10,
theme: 'quarto',
placement: 'bottom-start',
};
if (contentFn) {
config.content = contentFn;
}
if (onTriggerFn) {
config.onTrigger = onTriggerFn;
}
if (onUntriggerFn) {
config.onUntrigger = onUntriggerFn;
}
window.tippy(el, config);
}
const noterefs = window.document.querySelectorAll('a[role="doc-noteref"]');
for (var i=0; i<noterefs.length; i++) {
const ref = noterefs[i];
tippyHover(ref, function() {
// use id or data attribute instead here
let href = ref.getAttribute('data-footnote-href') || ref.getAttribute('href');
try { href = new URL(href).hash; } catch {}
const id = href.replace(/^#\/?/, "");
const note = window.document.getElementById(id);
if (note) {
return note.innerHTML;
} else {
return "";
}
});
}
const xrefs = window.document.querySelectorAll('a.quarto-xref');
const processXRef = (id, note) => {
// Strip column container classes
const stripColumnClz = (el) => {
el.classList.remove("page-full", "page-columns");
if (el.children) {
for (const child of el.children) {
stripColumnClz(child);
}
}
}
stripColumnClz(note)
if (id === null || id.startsWith('sec-')) {
// Special case sections, only their first couple elements
const container = document.createElement("div");
if (note.children && note.children.length > 2) {
container.appendChild(note.children[0].cloneNode(true));
for (let i = 1; i < note.children.length; i++) {
const child = note.children[i];
if (child.tagName === "P" && child.innerText === "") {
continue;
} else {
container.appendChild(child.cloneNode(true));
break;
}
}
if (window.Quarto?.typesetMath) {
window.Quarto.typesetMath(container);
}
return container.innerHTML
} else {
if (window.Quarto?.typesetMath) {
window.Quarto.typesetMath(note);
}
return note.innerHTML;
}
} else {
// Remove any anchor links if they are present
const anchorLink = note.querySelector('a.anchorjs-link');
if (anchorLink) {
anchorLink.remove();
}
if (window.Quarto?.typesetMath) {
window.Quarto.typesetMath(note);
}
// TODO in 1.5, we should make sure this works without a callout special case
if (note.classList.contains("callout")) {
return note.outerHTML;
} else {
return note.innerHTML;
}
}
}
for (var i=0; i<xrefs.length; i++) {
const xref = xrefs[i];
tippyHover(xref, undefined, function(instance) {
instance.disable();
let url = xref.getAttribute('href');
let hash = undefined;
if (url.startsWith('#')) {
hash = url;
} else {
try { hash = new URL(url).hash; } catch {}
}
if (hash) {
const id = hash.replace(/^#\/?/, "");
const note = window.document.getElementById(id);
if (note !== null) {
try {
const html = processXRef(id, note.cloneNode(true));
instance.setContent(html);
} finally {
instance.enable();
instance.show();
}
} else {
// See if we can fetch this
fetch(url.split('#')[0])
.then(res => res.text())
.then(html => {
const parser = new DOMParser();
const htmlDoc = parser.parseFromString(html, "text/html");
const note = htmlDoc.getElementById(id);
if (note !== null) {
const html = processXRef(id, note);
instance.setContent(html);
}
}).finally(() => {
instance.enable();
instance.show();
});
}
} else {
// See if we can fetch a full url (with no hash to target)
// This is a special case and we should probably do some content thinning / targeting
fetch(url)
.then(res => res.text())
.then(html => {
const parser = new DOMParser();
const htmlDoc = parser.parseFromString(html, "text/html");
const note = htmlDoc.querySelector('main.content');
if (note !== null) {
// This should only happen for chapter cross references
// (since there is no id in the URL)
// remove the first header
if (note.children.length > 0 && note.children[0].tagName === "HEADER") {
note.children[0].remove();
}
const html = processXRef(null, note);
instance.setContent(html);
}
}).finally(() => {
instance.enable();
instance.show();
});
}
}, function(instance) {
});
}
let selectedAnnoteEl;
const selectorForAnnotation = ( cell, annotation) => {
let cellAttr = 'data-code-cell="' + cell + '"';
let lineAttr = 'data-code-annotation="' + annotation + '"';
const selector = 'span[' + cellAttr + '][' + lineAttr + ']';
return selector;
}
const selectCodeLines = (annoteEl) => {
const doc = window.document;
const targetCell = annoteEl.getAttribute("data-target-cell");
const targetAnnotation = annoteEl.getAttribute("data-target-annotation");
const annoteSpan = window.document.querySelector(selectorForAnnotation(targetCell, targetAnnotation));
const lines = annoteSpan.getAttribute("data-code-lines").split(",");
const lineIds = lines.map((line) => {
return targetCell + "-" + line;
})
let top = null;
let height = null;
let parent = null;
if (lineIds.length > 0) {
//compute the position of the single el (top and bottom and make a div)
const el = window.document.getElementById(lineIds[0]);
top = el.offsetTop;
height = el.offsetHeight;
parent = el.parentElement.parentElement;
if (lineIds.length > 1) {
const lastEl = window.document.getElementById(lineIds[lineIds.length - 1]);
const bottom = lastEl.offsetTop + lastEl.offsetHeight;
height = bottom - top;
}
if (top !== null && height !== null && parent !== null) {
// cook up a div (if necessary) and position it
let div = window.document.getElementById("code-annotation-line-highlight");
if (div === null) {
div = window.document.createElement("div");
div.setAttribute("id", "code-annotation-line-highlight");
div.style.position = 'absolute';
parent.appendChild(div);
}
div.style.top = top - 2 + "px";
div.style.height = height + 4 + "px";
div.style.left = 0;
let gutterDiv = window.document.getElementById("code-annotation-line-highlight-gutter");
if (gutterDiv === null) {
gutterDiv = window.document.createElement("div");
gutterDiv.setAttribute("id", "code-annotation-line-highlight-gutter");
gutterDiv.style.position = 'absolute';
const codeCell = window.document.getElementById(targetCell);
const gutter = codeCell.querySelector('.code-annotation-gutter');
gutter.appendChild(gutterDiv);
}
gutterDiv.style.top = top - 2 + "px";
gutterDiv.style.height = height + 4 + "px";
}
selectedAnnoteEl = annoteEl;
}
};
const unselectCodeLines = () => {
const elementsIds = ["code-annotation-line-highlight", "code-annotation-line-highlight-gutter"];
elementsIds.forEach((elId) => {
const div = window.document.getElementById(elId);
if (div) {
div.remove();
}
});
selectedAnnoteEl = undefined;
};
// Handle positioning of the toggle
window.addEventListener(
"resize",
throttle(() => {
elRect = undefined;
if (selectedAnnoteEl) {