-
Notifications
You must be signed in to change notification settings - Fork 0
/
appendexA.html
1502 lines (1464 loc) · 129 KB
/
appendexA.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 lang="" xml:lang="">
<head>
<meta charset="utf-8" />
<meta http-equiv="X-UA-Compatible" content="IE=edge" />
<title>A Answers to chapter exercises | Fundamental statistical concepts and techniques in the biological and environmental sciences: With jamovi</title>
<meta name="description" content="This is an introductory statistics textbook for students in the biological and environmental sciences with examples using jamovi statistical software." />
<meta name="generator" content="bookdown 0.39 and GitBook 2.6.7" />
<meta property="og:title" content="A Answers to chapter exercises | Fundamental statistical concepts and techniques in the biological and environmental sciences: With jamovi" />
<meta property="og:type" content="book" />
<meta property="og:image" content="/img/cover.png" />
<meta property="og:description" content="This is an introductory statistics textbook for students in the biological and environmental sciences with examples using jamovi statistical software." />
<meta name="github-repo" content="bradduthie/stats" />
<meta name="twitter:card" content="summary" />
<meta name="twitter:title" content="A Answers to chapter exercises | Fundamental statistical concepts and techniques in the biological and environmental sciences: With jamovi" />
<meta name="twitter:description" content="This is an introductory statistics textbook for students in the biological and environmental sciences with examples using jamovi statistical software." />
<meta name="twitter:image" content="/img/cover.png" />
<meta name="author" content="A. Bradley Duthie" />
<meta name="date" content="2024-08-06" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<meta name="apple-mobile-web-app-capable" content="yes" />
<meta name="apple-mobile-web-app-status-bar-style" content="black" />
<link rel="prev" href="Chapter_35.html"/>
<link rel="next" href="uncertainty_derivation.html"/>
<script src="libs/jquery-3.6.0/jquery-3.6.0.min.js"></script>
<script src="https://cdn.jsdelivr.net/npm/fuse.js@6.4.6/dist/fuse.min.js"></script>
<link href="libs/gitbook-2.6.7/css/style.css" rel="stylesheet" />
<link href="libs/gitbook-2.6.7/css/plugin-table.css" rel="stylesheet" />
<link href="libs/gitbook-2.6.7/css/plugin-bookdown.css" rel="stylesheet" />
<link href="libs/gitbook-2.6.7/css/plugin-highlight.css" rel="stylesheet" />
<link href="libs/gitbook-2.6.7/css/plugin-search.css" rel="stylesheet" />
<link href="libs/gitbook-2.6.7/css/plugin-fontsettings.css" rel="stylesheet" />
<link href="libs/gitbook-2.6.7/css/plugin-clipboard.css" rel="stylesheet" />
<link href="libs/anchor-sections-1.1.0/anchor-sections.css" rel="stylesheet" />
<link href="libs/anchor-sections-1.1.0/anchor-sections-hash.css" rel="stylesheet" />
<script src="libs/anchor-sections-1.1.0/anchor-sections.js"></script>
<style type="text/css">
pre > code.sourceCode { white-space: pre; position: relative; }
pre > code.sourceCode > span { display: inline-block; line-height: 1.25; }
pre > code.sourceCode > span:empty { height: 1.2em; }
.sourceCode { overflow: visible; }
code.sourceCode > span { color: inherit; text-decoration: inherit; }
pre.sourceCode { margin: 0; }
@media screen {
div.sourceCode { overflow: auto; }
}
@media print {
pre > code.sourceCode { white-space: pre-wrap; }
pre > code.sourceCode > span { 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; }
}
code span.al { font-weight: bold; } /* Alert */
code span.an { font-style: italic; } /* Annotation */
code span.cf { font-weight: bold; } /* ControlFlow */
code span.co { font-style: italic; } /* Comment */
code span.cv { font-style: italic; } /* CommentVar */
code span.do { font-style: italic; } /* Documentation */
code span.dt { text-decoration: underline; } /* DataType */
code span.er { font-weight: bold; } /* Error */
code span.in { font-style: italic; } /* Information */
code span.kw { font-weight: bold; } /* Keyword */
code span.pp { font-weight: bold; } /* Preprocessor */
code span.wa { font-style: italic; } /* Warning */
</style>
<style type="text/css">
div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
</style>
<style type="text/css">
/* Used with Pandoc 2.11+ new --citeproc when CSL is used */
div.csl-bib-body { }
div.csl-entry {
clear: both;
}
.hanging 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>
<link rel="stylesheet" href="style.css" type="text/css" />
</head>
<body>
<div class="book without-animation with-summary font-size-2 font-family-1" data-basepath=".">
<div class="book-summary">
<nav role="navigation">
<ul class="summary">
<li><a href="./">Statistics with jamovi</a></li>
<li class="divider"></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html"><i class="fa fa-check"></i>Preface</a>
<ul>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#structure"><i class="fa fa-check"></i>How this book is structured</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#datasets"><i class="fa fa-check"></i>Datasets used in this book</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#acknowledgements"><i class="fa fa-check"></i>Acknowledgements</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#author"><i class="fa fa-check"></i>About the author</a></li>
</ul></li>
<li class="chapter" data-level="1" data-path="Chapter_1.html"><a href="Chapter_1.html"><i class="fa fa-check"></i><b>1</b> Background mathematics</a>
<ul>
<li class="chapter" data-level="1.1" data-path="Chapter_1.html"><a href="Chapter_1.html#numbers-and-operations"><i class="fa fa-check"></i><b>1.1</b> Numbers and operations</a></li>
<li class="chapter" data-level="1.2" data-path="Chapter_1.html"><a href="Chapter_1.html#logarithms"><i class="fa fa-check"></i><b>1.2</b> Logarithms</a></li>
<li class="chapter" data-level="1.3" data-path="Chapter_1.html"><a href="Chapter_1.html#order-of-operations"><i class="fa fa-check"></i><b>1.3</b> Order of operations</a></li>
</ul></li>
<li class="chapter" data-level="2" data-path="Chapter_2.html"><a href="Chapter_2.html"><i class="fa fa-check"></i><b>2</b> Data organisation</a>
<ul>
<li class="chapter" data-level="2.1" data-path="Chapter_2.html"><a href="Chapter_2.html#tidy-data"><i class="fa fa-check"></i><b>2.1</b> Tidy data</a></li>
<li class="chapter" data-level="2.2" data-path="Chapter_2.html"><a href="Chapter_2.html#data-files"><i class="fa fa-check"></i><b>2.2</b> Data files</a></li>
<li class="chapter" data-level="2.3" data-path="Chapter_2.html"><a href="Chapter_2.html#managing-data-files"><i class="fa fa-check"></i><b>2.3</b> Managing data files</a></li>
</ul></li>
<li class="chapter" data-level="3" data-path="Chapter_3.html"><a href="Chapter_3.html"><i class="fa fa-check"></i><b>3</b> <em>Practical</em>. Preparing data</a>
<ul>
<li class="chapter" data-level="3.1" data-path="Chapter_3.html"><a href="Chapter_3.html#transferring-data-to-a-spreadsheet"><i class="fa fa-check"></i><b>3.1</b> Transferring data to a spreadsheet</a></li>
<li class="chapter" data-level="3.2" data-path="Chapter_3.html"><a href="Chapter_3.html#making-spreadsheet-data-tidy"><i class="fa fa-check"></i><b>3.2</b> Making spreadsheet data tidy</a></li>
<li class="chapter" data-level="3.3" data-path="Chapter_3.html"><a href="Chapter_3.html#making-data-tidy-again"><i class="fa fa-check"></i><b>3.3</b> Making data tidy again</a></li>
<li class="chapter" data-level="3.4" data-path="Chapter_3.html"><a href="Chapter_3.html#tidy-data-and-spreadsheet-calculations"><i class="fa fa-check"></i><b>3.4</b> Tidy data and spreadsheet calculations</a></li>
<li class="chapter" data-level="3.5" data-path="Chapter_3.html"><a href="Chapter_3.html#summary"><i class="fa fa-check"></i><b>3.5</b> Summary</a></li>
</ul></li>
<li class="chapter" data-level="4" data-path="Chapter_4.html"><a href="Chapter_4.html"><i class="fa fa-check"></i><b>4</b> Populations and samples</a></li>
<li class="chapter" data-level="5" data-path="Chapter_5.html"><a href="Chapter_5.html"><i class="fa fa-check"></i><b>5</b> Types of variables</a></li>
<li class="chapter" data-level="6" data-path="Chapter_6.html"><a href="Chapter_6.html"><i class="fa fa-check"></i><b>6</b> Accuracy, precision, and units</a>
<ul>
<li class="chapter" data-level="6.1" data-path="Chapter_6.html"><a href="Chapter_6.html#accuracy"><i class="fa fa-check"></i><b>6.1</b> Accuracy</a></li>
<li class="chapter" data-level="6.2" data-path="Chapter_6.html"><a href="Chapter_6.html#precision"><i class="fa fa-check"></i><b>6.2</b> Precision</a></li>
<li class="chapter" data-level="6.3" data-path="Chapter_6.html"><a href="Chapter_6.html#systems-of-units"><i class="fa fa-check"></i><b>6.3</b> Systems of units</a></li>
</ul></li>
<li class="chapter" data-level="7" data-path="Chapter_7.html"><a href="Chapter_7.html"><i class="fa fa-check"></i><b>7</b> Uncertainty propagation</a>
<ul>
<li class="chapter" data-level="7.1" data-path="Chapter_7.html"><a href="Chapter_7.html#adding-or-subtracting-errors"><i class="fa fa-check"></i><b>7.1</b> Adding or subtracting errors</a></li>
<li class="chapter" data-level="7.2" data-path="Chapter_7.html"><a href="Chapter_7.html#multiplying-or-dividing-errors"><i class="fa fa-check"></i><b>7.2</b> Multiplying or dividing errors</a></li>
</ul></li>
<li class="chapter" data-level="8" data-path="Chapter_8.html"><a href="Chapter_8.html"><i class="fa fa-check"></i><b>8</b> <em>Practical</em>. Introduction to jamovi</a>
<ul>
<li class="chapter" data-level="8.1" data-path="Chapter_8.html"><a href="Chapter_8.html#summary_statistics_02"><i class="fa fa-check"></i><b>8.1</b> Summary statistics</a></li>
<li class="chapter" data-level="8.2" data-path="Chapter_8.html"><a href="Chapter_8.html#transforming_variables_02"><i class="fa fa-check"></i><b>8.2</b> Transforming variables</a></li>
<li class="chapter" data-level="8.3" data-path="Chapter_8.html"><a href="Chapter_8.html#computing_variables_02"><i class="fa fa-check"></i><b>8.3</b> Computing variables</a></li>
<li class="chapter" data-level="8.4" data-path="Chapter_8.html"><a href="Chapter_8.html#summary-1"><i class="fa fa-check"></i><b>8.4</b> Summary</a></li>
</ul></li>
<li class="chapter" data-level="9" data-path="Chapter_9.html"><a href="Chapter_9.html"><i class="fa fa-check"></i><b>9</b> Decimal places, significant figures, and rounding</a>
<ul>
<li class="chapter" data-level="9.1" data-path="Chapter_9.html"><a href="Chapter_9.html#decimal-places-and-significant-figures"><i class="fa fa-check"></i><b>9.1</b> Decimal places and significant figures</a></li>
<li class="chapter" data-level="9.2" data-path="Chapter_9.html"><a href="Chapter_9.html#rounding"><i class="fa fa-check"></i><b>9.2</b> Rounding</a></li>
</ul></li>
<li class="chapter" data-level="10" data-path="Chapter_10.html"><a href="Chapter_10.html"><i class="fa fa-check"></i><b>10</b> Graphs</a>
<ul>
<li class="chapter" data-level="10.1" data-path="Chapter_10.html"><a href="Chapter_10.html#histograms"><i class="fa fa-check"></i><b>10.1</b> Histograms</a></li>
<li class="chapter" data-level="10.2" data-path="Chapter_10.html"><a href="Chapter_10.html#barplots-and-pie-charts"><i class="fa fa-check"></i><b>10.2</b> Barplots and pie charts</a></li>
<li class="chapter" data-level="10.3" data-path="Chapter_10.html"><a href="Chapter_10.html#box-whisker-plots"><i class="fa fa-check"></i><b>10.3</b> Box-whisker plots</a></li>
</ul></li>
<li class="chapter" data-level="11" data-path="Chapter_11.html"><a href="Chapter_11.html"><i class="fa fa-check"></i><b>11</b> Measures of central tendency</a>
<ul>
<li class="chapter" data-level="11.1" data-path="Chapter_11.html"><a href="Chapter_11.html#the-mean"><i class="fa fa-check"></i><b>11.1</b> The mean</a></li>
<li class="chapter" data-level="11.2" data-path="Chapter_11.html"><a href="Chapter_11.html#the-mode"><i class="fa fa-check"></i><b>11.2</b> The mode</a></li>
<li class="chapter" data-level="11.3" data-path="Chapter_11.html"><a href="Chapter_11.html#the-median-and-quantiles"><i class="fa fa-check"></i><b>11.3</b> The median and quantiles</a></li>
</ul></li>
<li class="chapter" data-level="12" data-path="Chapter_12.html"><a href="Chapter_12.html"><i class="fa fa-check"></i><b>12</b> Measures of spread</a>
<ul>
<li class="chapter" data-level="12.1" data-path="Chapter_12.html"><a href="Chapter_12.html#the-range"><i class="fa fa-check"></i><b>12.1</b> The range</a></li>
<li class="chapter" data-level="12.2" data-path="Chapter_12.html"><a href="Chapter_12.html#the-inter-quartile-range"><i class="fa fa-check"></i><b>12.2</b> The inter-quartile range</a></li>
<li class="chapter" data-level="12.3" data-path="Chapter_12.html"><a href="Chapter_12.html#the-variance"><i class="fa fa-check"></i><b>12.3</b> The variance</a></li>
<li class="chapter" data-level="12.4" data-path="Chapter_12.html"><a href="Chapter_12.html#the-standard-deviation"><i class="fa fa-check"></i><b>12.4</b> The standard deviation</a></li>
<li class="chapter" data-level="12.5" data-path="Chapter_12.html"><a href="Chapter_12.html#the-coefficient-of-variation"><i class="fa fa-check"></i><b>12.5</b> The coefficient of variation</a></li>
<li class="chapter" data-level="12.6" data-path="Chapter_12.html"><a href="Chapter_12.html#the-standard-error"><i class="fa fa-check"></i><b>12.6</b> The standard error</a></li>
</ul></li>
<li class="chapter" data-level="13" data-path="Chapter_13.html"><a href="Chapter_13.html"><i class="fa fa-check"></i><b>13</b> Skew and kurtosis</a>
<ul>
<li class="chapter" data-level="13.1" data-path="Chapter_13.html"><a href="Chapter_13.html#skew"><i class="fa fa-check"></i><b>13.1</b> Skew</a></li>
<li class="chapter" data-level="13.2" data-path="Chapter_13.html"><a href="Chapter_13.html#kurtosis"><i class="fa fa-check"></i><b>13.2</b> Kurtosis</a></li>
<li class="chapter" data-level="13.3" data-path="Chapter_13.html"><a href="Chapter_13.html#moments"><i class="fa fa-check"></i><b>13.3</b> Moments</a></li>
</ul></li>
<li class="chapter" data-level="14" data-path="Chapter_14.html"><a href="Chapter_14.html"><i class="fa fa-check"></i><b>14</b> <em>Practical</em>. Plotting and statistical summaries in jamovi</a>
<ul>
<li class="chapter" data-level="14.1" data-path="Chapter_14.html"><a href="Chapter_14.html#reorganise-the-dataset-into-a-tidy-format"><i class="fa fa-check"></i><b>14.1</b> Reorganise the dataset into a tidy format</a></li>
<li class="chapter" data-level="14.2" data-path="Chapter_14.html"><a href="Chapter_14.html#histograms-and-box-whisker-plots"><i class="fa fa-check"></i><b>14.2</b> Histograms and box-whisker plots</a></li>
<li class="chapter" data-level="14.3" data-path="Chapter_14.html"><a href="Chapter_14.html#calculate-summary-statistics"><i class="fa fa-check"></i><b>14.3</b> Calculate summary statistics</a></li>
<li class="chapter" data-level="14.4" data-path="Chapter_14.html"><a href="Chapter_14.html#reporting-decimals-and-significant-figures"><i class="fa fa-check"></i><b>14.4</b> Reporting decimals and significant figures</a></li>
<li class="chapter" data-level="14.5" data-path="Chapter_14.html"><a href="Chapter_14.html#comparing-across-sites"><i class="fa fa-check"></i><b>14.5</b> Comparing across sites</a></li>
</ul></li>
<li class="chapter" data-level="15" data-path="Chapter_15.html"><a href="Chapter_15.html"><i class="fa fa-check"></i><b>15</b> Introduction to probability models</a>
<ul>
<li class="chapter" data-level="15.1" data-path="Chapter_15.html"><a href="Chapter_15.html#instructive-example"><i class="fa fa-check"></i><b>15.1</b> Instructive example</a></li>
<li class="chapter" data-level="15.2" data-path="Chapter_15.html"><a href="Chapter_15.html#biological-applications"><i class="fa fa-check"></i><b>15.2</b> Biological applications</a></li>
<li class="chapter" data-level="15.3" data-path="Chapter_15.html"><a href="Chapter_15.html#sampling-with-and-without-replacement"><i class="fa fa-check"></i><b>15.3</b> Sampling with and without replacement</a></li>
<li class="chapter" data-level="15.4" data-path="Chapter_15.html"><a href="Chapter_15.html#probability-distributions"><i class="fa fa-check"></i><b>15.4</b> Probability distributions</a>
<ul>
<li class="chapter" data-level="15.4.1" data-path="Chapter_15.html"><a href="Chapter_15.html#binomial-distribution"><i class="fa fa-check"></i><b>15.4.1</b> Binomial distribution</a></li>
<li class="chapter" data-level="15.4.2" data-path="Chapter_15.html"><a href="Chapter_15.html#poisson-distribution"><i class="fa fa-check"></i><b>15.4.2</b> Poisson distribution</a></li>
<li class="chapter" data-level="15.4.3" data-path="Chapter_15.html"><a href="Chapter_15.html#uniform-distribution"><i class="fa fa-check"></i><b>15.4.3</b> Uniform distribution</a></li>
<li class="chapter" data-level="15.4.4" data-path="Chapter_15.html"><a href="Chapter_15.html#normal-distribution"><i class="fa fa-check"></i><b>15.4.4</b> Normal distribution</a></li>
</ul></li>
<li class="chapter" data-level="15.5" data-path="Chapter_15.html"><a href="Chapter_15.html#summary-2"><i class="fa fa-check"></i><b>15.5</b> Summary</a></li>
</ul></li>
<li class="chapter" data-level="16" data-path="Chapter_16.html"><a href="Chapter_16.html"><i class="fa fa-check"></i><b>16</b> Central Limit Theorem</a>
<ul>
<li class="chapter" data-level="16.1" data-path="Chapter_16.html"><a href="Chapter_16.html#the-distribution-of-means-is-normal"><i class="fa fa-check"></i><b>16.1</b> The distribution of means is normal</a></li>
<li class="chapter" data-level="16.2" data-path="Chapter_16.html"><a href="Chapter_16.html#probability-and-z-scores"><i class="fa fa-check"></i><b>16.2</b> Probability and z-scores</a></li>
</ul></li>
<li class="chapter" data-level="17" data-path="Chapter_17.html"><a href="Chapter_17.html"><i class="fa fa-check"></i><b>17</b> <em>Practical</em>. Probability and simulation</a>
<ul>
<li class="chapter" data-level="17.1" data-path="Chapter_17.html"><a href="Chapter_17.html#probabilities-from-a-dataset"><i class="fa fa-check"></i><b>17.1</b> Probabilities from a dataset</a></li>
<li class="chapter" data-level="17.2" data-path="Chapter_17.html"><a href="Chapter_17.html#probabilities-from-a-normal-distribution"><i class="fa fa-check"></i><b>17.2</b> Probabilities from a normal distribution</a></li>
<li class="chapter" data-level="17.3" data-path="Chapter_17.html"><a href="Chapter_17.html#central-limit-theorem"><i class="fa fa-check"></i><b>17.3</b> Central limit theorem</a></li>
</ul></li>
<li class="chapter" data-level="18" data-path="Chapter_18.html"><a href="Chapter_18.html"><i class="fa fa-check"></i><b>18</b> Confidence intervals</a>
<ul>
<li class="chapter" data-level="18.1" data-path="Chapter_18.html"><a href="Chapter_18.html#normal-distribution-cis"><i class="fa fa-check"></i><b>18.1</b> Normal distribution CIs</a></li>
<li class="chapter" data-level="18.2" data-path="Chapter_18.html"><a href="Chapter_18.html#binomial-distribution-cis"><i class="fa fa-check"></i><b>18.2</b> Binomial distribution CIs</a></li>
</ul></li>
<li class="chapter" data-level="19" data-path="Chapter_19.html"><a href="Chapter_19.html"><i class="fa fa-check"></i><b>19</b> The t-interval</a></li>
<li class="chapter" data-level="20" data-path="Chapter_20.html"><a href="Chapter_20.html"><i class="fa fa-check"></i><b>20</b> <em>Practical</em>. z- and t-intervals</a>
<ul>
<li class="chapter" data-level="20.1" data-path="Chapter_20.html"><a href="Chapter_20.html#confidence-intervals-with-distraction"><i class="fa fa-check"></i><b>20.1</b> Confidence intervals with distrACTION</a></li>
<li class="chapter" data-level="20.2" data-path="Chapter_20.html"><a href="Chapter_20.html#confidence-intervals-from-z--and-t-scores"><i class="fa fa-check"></i><b>20.2</b> Confidence intervals from z- and t-scores</a></li>
<li class="chapter" data-level="20.3" data-path="Chapter_20.html"><a href="Chapter_20.html#confidence-intervals-for-different-sample-sizes"><i class="fa fa-check"></i><b>20.3</b> Confidence intervals for different sample sizes</a></li>
<li class="chapter" data-level="20.4" data-path="Chapter_20.html"><a href="Chapter_20.html#proportion-confidence-intervals"><i class="fa fa-check"></i><b>20.4</b> Proportion confidence intervals</a></li>
<li class="chapter" data-level="20.5" data-path="Chapter_20.html"><a href="Chapter_20.html#another-proportion-confidence-interval"><i class="fa fa-check"></i><b>20.5</b> Another proportion confidence interval</a></li>
</ul></li>
<li class="chapter" data-level="21" data-path="Chapter_21.html"><a href="Chapter_21.html"><i class="fa fa-check"></i><b>21</b> What is hypothesis testing?</a>
<ul>
<li class="chapter" data-level="21.1" data-path="Chapter_21.html"><a href="Chapter_21.html#how-ridiculous-is-our-hypothesis"><i class="fa fa-check"></i><b>21.1</b> How ridiculous is our hypothesis?</a></li>
<li class="chapter" data-level="21.2" data-path="Chapter_21.html"><a href="Chapter_21.html#statistical-hypothesis-testing"><i class="fa fa-check"></i><b>21.2</b> Statistical hypothesis testing</a></li>
<li class="chapter" data-level="21.3" data-path="Chapter_21.html"><a href="Chapter_21.html#p-values-false-positives-and-power"><i class="fa fa-check"></i><b>21.3</b> P-values, false positives, and power</a></li>
</ul></li>
<li class="chapter" data-level="22" data-path="Chapter_22.html"><a href="Chapter_22.html"><i class="fa fa-check"></i><b>22</b> The t-test</a>
<ul>
<li class="chapter" data-level="22.1" data-path="Chapter_22.html"><a href="Chapter_22.html#one-sample-t-test"><i class="fa fa-check"></i><b>22.1</b> One sample t-test</a></li>
<li class="chapter" data-level="22.2" data-path="Chapter_22.html"><a href="Chapter_22.html#independent-samples-t-test"><i class="fa fa-check"></i><b>22.2</b> Independent samples t-test</a></li>
<li class="chapter" data-level="22.3" data-path="Chapter_22.html"><a href="Chapter_22.html#paired-samples-t-test"><i class="fa fa-check"></i><b>22.3</b> Paired samples t-test</a></li>
<li class="chapter" data-level="22.4" data-path="Chapter_22.html"><a href="Chapter_22.html#assumptions-of-t-tests"><i class="fa fa-check"></i><b>22.4</b> Assumptions of t-tests</a></li>
<li class="chapter" data-level="22.5" data-path="Chapter_22.html"><a href="Chapter_22.html#non-parametric-alternatives"><i class="fa fa-check"></i><b>22.5</b> Non-parametric alternatives</a>
<ul>
<li class="chapter" data-level="22.5.1" data-path="Chapter_22.html"><a href="Chapter_22.html#wilcoxon-test"><i class="fa fa-check"></i><b>22.5.1</b> Wilcoxon test</a></li>
<li class="chapter" data-level="22.5.2" data-path="Chapter_22.html"><a href="Chapter_22.html#mann-whitney-u-test"><i class="fa fa-check"></i><b>22.5.2</b> Mann-Whitney U test</a></li>
</ul></li>
<li class="chapter" data-level="22.6" data-path="Chapter_22.html"><a href="Chapter_22.html#summary-3"><i class="fa fa-check"></i><b>22.6</b> Summary</a></li>
</ul></li>
<li class="chapter" data-level="23" data-path="Chapter_23.html"><a href="Chapter_23.html"><i class="fa fa-check"></i><b>23</b> <em>Practical</em>. Hypothesis testing and t-tests</a>
<ul>
<li class="chapter" data-level="23.1" data-path="Chapter_23.html"><a href="Chapter_23.html#one-sample-t-test-1"><i class="fa fa-check"></i><b>23.1</b> One sample t-test</a></li>
<li class="chapter" data-level="23.2" data-path="Chapter_23.html"><a href="Chapter_23.html#paired-t-test"><i class="fa fa-check"></i><b>23.2</b> Paired t-test</a></li>
<li class="chapter" data-level="23.3" data-path="Chapter_23.html"><a href="Chapter_23.html#wilcoxon-test-1"><i class="fa fa-check"></i><b>23.3</b> Wilcoxon test</a></li>
<li class="chapter" data-level="23.4" data-path="Chapter_23.html"><a href="Chapter_23.html#independent-samples-t-test-1"><i class="fa fa-check"></i><b>23.4</b> Independent samples t-test</a></li>
<li class="chapter" data-level="23.5" data-path="Chapter_23.html"><a href="Chapter_23.html#mann-whitney-u-test-1"><i class="fa fa-check"></i><b>23.5</b> Mann-Whitney U Test</a></li>
</ul></li>
<li class="chapter" data-level="24" data-path="Chapter_24.html"><a href="Chapter_24.html"><i class="fa fa-check"></i><b>24</b> Analysis of variance</a>
<ul>
<li class="chapter" data-level="24.1" data-path="Chapter_24.html"><a href="Chapter_24.html#f-distribution"><i class="fa fa-check"></i><b>24.1</b> F-distribution</a></li>
<li class="chapter" data-level="24.2" data-path="Chapter_24.html"><a href="Chapter_24.html#one-way-anova"><i class="fa fa-check"></i><b>24.2</b> One-way ANOVA</a>
<ul>
<li class="chapter" data-level="24.2.1" data-path="Chapter_24.html"><a href="Chapter_24.html#anova-mean-variance-among-groups"><i class="fa fa-check"></i><b>24.2.1</b> ANOVA mean variance among groups</a></li>
<li class="chapter" data-level="24.2.2" data-path="Chapter_24.html"><a href="Chapter_24.html#anova-mean-variance-within-groups"><i class="fa fa-check"></i><b>24.2.2</b> ANOVA mean variance within groups</a></li>
<li class="chapter" data-level="24.2.3" data-path="Chapter_24.html"><a href="Chapter_24.html#anova-f-statistic-calculation"><i class="fa fa-check"></i><b>24.2.3</b> ANOVA F-statistic calculation</a></li>
</ul></li>
<li class="chapter" data-level="24.3" data-path="Chapter_24.html"><a href="Chapter_24.html#assumptions-of-anova"><i class="fa fa-check"></i><b>24.3</b> Assumptions of ANOVA</a></li>
</ul></li>
<li class="chapter" data-level="25" data-path="Chapter_25.html"><a href="Chapter_25.html"><i class="fa fa-check"></i><b>25</b> Multiple comparisons</a></li>
<li class="chapter" data-level="26" data-path="Chapter_26.html"><a href="Chapter_26.html"><i class="fa fa-check"></i><b>26</b> Kruskal-Wallis H test</a></li>
<li class="chapter" data-level="27" data-path="Chapter_27.html"><a href="Chapter_27.html"><i class="fa fa-check"></i><b>27</b> Two-way ANOVA</a></li>
<li class="chapter" data-level="28" data-path="Chapter_28.html"><a href="Chapter_28.html"><i class="fa fa-check"></i><b>28</b> <em>Practical</em>. ANOVA and associated tests</a>
<ul>
<li class="chapter" data-level="28.1" data-path="Chapter_28.html"><a href="Chapter_28.html#one-way-anova-site"><i class="fa fa-check"></i><b>28.1</b> One-way ANOVA (site)</a></li>
<li class="chapter" data-level="28.2" data-path="Chapter_28.html"><a href="Chapter_28.html#one-way-anova-profile"><i class="fa fa-check"></i><b>28.2</b> One-way ANOVA (profile)</a></li>
<li class="chapter" data-level="28.3" data-path="Chapter_28.html"><a href="Chapter_28.html#multiple-comparisons"><i class="fa fa-check"></i><b>28.3</b> Multiple comparisons</a></li>
<li class="chapter" data-level="28.4" data-path="Chapter_28.html"><a href="Chapter_28.html#kruskal-wallis-h-test"><i class="fa fa-check"></i><b>28.4</b> Kruskal-Wallis H test</a></li>
<li class="chapter" data-level="28.5" data-path="Chapter_28.html"><a href="Chapter_28.html#two-way-anova"><i class="fa fa-check"></i><b>28.5</b> Two-way ANOVA</a></li>
</ul></li>
<li class="chapter" data-level="29" data-path="Chapter_29.html"><a href="Chapter_29.html"><i class="fa fa-check"></i><b>29</b> Frequency and count data</a>
<ul>
<li class="chapter" data-level="29.1" data-path="Chapter_29.html"><a href="Chapter_29.html#chi-square-distribution"><i class="fa fa-check"></i><b>29.1</b> Chi-square distribution</a></li>
<li class="chapter" data-level="29.2" data-path="Chapter_29.html"><a href="Chapter_29.html#chi-square-goodness-of-fit"><i class="fa fa-check"></i><b>29.2</b> Chi-square goodness of fit</a></li>
<li class="chapter" data-level="29.3" data-path="Chapter_29.html"><a href="Chapter_29.html#chi-square-test-of-association"><i class="fa fa-check"></i><b>29.3</b> Chi-square test of association</a></li>
</ul></li>
<li class="chapter" data-level="30" data-path="Chapter_30.html"><a href="Chapter_30.html"><i class="fa fa-check"></i><b>30</b> Correlation</a>
<ul>
<li class="chapter" data-level="30.1" data-path="Chapter_30.html"><a href="Chapter_30.html#scatterplots"><i class="fa fa-check"></i><b>30.1</b> Scatterplots</a></li>
<li class="chapter" data-level="30.2" data-path="Chapter_30.html"><a href="Chapter_30.html#correlation-coefficient"><i class="fa fa-check"></i><b>30.2</b> Correlation coefficient</a>
<ul>
<li class="chapter" data-level="30.2.1" data-path="Chapter_30.html"><a href="Chapter_30.html#pearson-product-moment-correlation-coefficient"><i class="fa fa-check"></i><b>30.2.1</b> Pearson product moment correlation coefficient</a></li>
<li class="chapter" data-level="30.2.2" data-path="Chapter_30.html"><a href="Chapter_30.html#spearmans-rank-correlation-coefficient"><i class="fa fa-check"></i><b>30.2.2</b> Spearman’s rank correlation coefficient</a></li>
</ul></li>
<li class="chapter" data-level="30.3" data-path="Chapter_30.html"><a href="Chapter_30.html#correlation-hypothesis-testing"><i class="fa fa-check"></i><b>30.3</b> Correlation hypothesis testing</a></li>
</ul></li>
<li class="chapter" data-level="31" data-path="Chapter_31.html"><a href="Chapter_31.html"><i class="fa fa-check"></i><b>31</b> <em>Practical</em>. Analysis of counts and correlations</a>
<ul>
<li class="chapter" data-level="31.1" data-path="Chapter_31.html"><a href="Chapter_31.html#survival-goodness-of-fit"><i class="fa fa-check"></i><b>31.1</b> Survival goodness of fit</a></li>
<li class="chapter" data-level="31.2" data-path="Chapter_31.html"><a href="Chapter_31.html#colony-goodness-of-fit"><i class="fa fa-check"></i><b>31.2</b> Colony goodness of fit</a></li>
<li class="chapter" data-level="31.3" data-path="Chapter_31.html"><a href="Chapter_31.html#chi-square-test-of-association-1"><i class="fa fa-check"></i><b>31.3</b> Chi-Square test of association</a></li>
<li class="chapter" data-level="31.4" data-path="Chapter_31.html"><a href="Chapter_31.html#pearson-product-moment-correlation-test"><i class="fa fa-check"></i><b>31.4</b> Pearson product moment correlation test</a></li>
<li class="chapter" data-level="31.5" data-path="Chapter_31.html"><a href="Chapter_31.html#spearmans-rank-correlation-test"><i class="fa fa-check"></i><b>31.5</b> Spearman’s rank correlation test</a></li>
<li class="chapter" data-level="31.6" data-path="Chapter_31.html"><a href="Chapter_31.html#untidy-goodness-of-fit"><i class="fa fa-check"></i><b>31.6</b> Untidy goodness of fit</a></li>
</ul></li>
<li class="chapter" data-level="32" data-path="Chapter_32.html"><a href="Chapter_32.html"><i class="fa fa-check"></i><b>32</b> Simple linear regression</a>
<ul>
<li class="chapter" data-level="32.1" data-path="Chapter_32.html"><a href="Chapter_32.html#visual-interpretation-of-regression"><i class="fa fa-check"></i><b>32.1</b> Visual interpretation of regression</a></li>
<li class="chapter" data-level="32.2" data-path="Chapter_32.html"><a href="Chapter_32.html#intercepts-slopes-and-residuals"><i class="fa fa-check"></i><b>32.2</b> Intercepts, slopes, and residuals</a></li>
<li class="chapter" data-level="32.3" data-path="Chapter_32.html"><a href="Chapter_32.html#regression-coefficients"><i class="fa fa-check"></i><b>32.3</b> Regression coefficients</a></li>
<li class="chapter" data-level="32.4" data-path="Chapter_32.html"><a href="Chapter_32.html#regression-line-calculation"><i class="fa fa-check"></i><b>32.4</b> Regression line calculation</a></li>
<li class="chapter" data-level="32.5" data-path="Chapter_32.html"><a href="Chapter_32.html#coefficient-of-determination"><i class="fa fa-check"></i><b>32.5</b> Coefficient of determination</a></li>
<li class="chapter" data-level="32.6" data-path="Chapter_32.html"><a href="Chapter_32.html#regression-assumptions"><i class="fa fa-check"></i><b>32.6</b> Regression assumptions</a></li>
<li class="chapter" data-level="32.7" data-path="Chapter_32.html"><a href="Chapter_32.html#regression-hypothesis-testing"><i class="fa fa-check"></i><b>32.7</b> Regression hypothesis testing</a>
<ul>
<li class="chapter" data-level="32.7.1" data-path="Chapter_32.html"><a href="Chapter_32.html#overall-model-significance"><i class="fa fa-check"></i><b>32.7.1</b> Overall model significance</a></li>
<li class="chapter" data-level="32.7.2" data-path="Chapter_32.html"><a href="Chapter_32.html#significance-of-the-intercept"><i class="fa fa-check"></i><b>32.7.2</b> Significance of the intercept</a></li>
<li class="chapter" data-level="32.7.3" data-path="Chapter_32.html"><a href="Chapter_32.html#significance-of-the-slope"><i class="fa fa-check"></i><b>32.7.3</b> Significance of the slope</a></li>
<li class="chapter" data-level="32.7.4" data-path="Chapter_32.html"><a href="Chapter_32.html#simple-regression-output"><i class="fa fa-check"></i><b>32.7.4</b> Simple regression output</a></li>
</ul></li>
<li class="chapter" data-level="32.8" data-path="Chapter_32.html"><a href="Chapter_32.html#prediction-with-linear-models"><i class="fa fa-check"></i><b>32.8</b> Prediction with linear models</a></li>
<li class="chapter" data-level="32.9" data-path="Chapter_32.html"><a href="Chapter_32.html#conclusion"><i class="fa fa-check"></i><b>32.9</b> Conclusion</a></li>
</ul></li>
<li class="chapter" data-level="33" data-path="Chapter_33.html"><a href="Chapter_33.html"><i class="fa fa-check"></i><b>33</b> Multiple regression</a>
<ul>
<li class="chapter" data-level="33.1" data-path="Chapter_33.html"><a href="Chapter_33.html#adjusted-coefficient-of-determination"><i class="fa fa-check"></i><b>33.1</b> Adjusted coefficient of determination</a></li>
</ul></li>
<li class="chapter" data-level="34" data-path="Chapter_34.html"><a href="Chapter_34.html"><i class="fa fa-check"></i><b>34</b> <em>Practical</em>. Using regression</a>
<ul>
<li class="chapter" data-level="34.1" data-path="Chapter_34.html"><a href="Chapter_34.html#predicting-pyrogenic-carbon-from-soil-depth"><i class="fa fa-check"></i><b>34.1</b> Predicting pyrogenic carbon from soil depth</a></li>
<li class="chapter" data-level="34.2" data-path="Chapter_34.html"><a href="Chapter_34.html#predicting-pyrogenic-carbon-from-fire-frequency"><i class="fa fa-check"></i><b>34.2</b> Predicting pyrogenic carbon from fire frequency</a></li>
<li class="chapter" data-level="34.3" data-path="Chapter_34.html"><a href="Chapter_34.html#multiple-regression-depth-and-fire-frequency"><i class="fa fa-check"></i><b>34.3</b> Multiple regression depth and fire frequency</a></li>
<li class="chapter" data-level="34.4" data-path="Chapter_34.html"><a href="Chapter_34.html#large-multiple-regression"><i class="fa fa-check"></i><b>34.4</b> Large multiple regression</a></li>
<li class="chapter" data-level="34.5" data-path="Chapter_34.html"><a href="Chapter_34.html#predicting-temperature-from-fire-frequency"><i class="fa fa-check"></i><b>34.5</b> Predicting temperature from fire frequency</a></li>
</ul></li>
<li class="chapter" data-level="35" data-path="Chapter_35.html"><a href="Chapter_35.html"><i class="fa fa-check"></i><b>35</b> Randomisation</a>
<ul>
<li class="chapter" data-level="35.1" data-path="Chapter_35.html"><a href="Chapter_35.html#summary-of-parametric-hypothesis-testing"><i class="fa fa-check"></i><b>35.1</b> Summary of parametric hypothesis testing</a></li>
<li class="chapter" data-level="35.2" data-path="Chapter_35.html"><a href="Chapter_35.html#randomisation-approach"><i class="fa fa-check"></i><b>35.2</b> Randomisation approach</a></li>
<li class="chapter" data-level="35.3" data-path="Chapter_35.html"><a href="Chapter_35.html#randomisation-for-hypothesis-testing"><i class="fa fa-check"></i><b>35.3</b> Randomisation for hypothesis testing</a></li>
<li class="chapter" data-level="35.4" data-path="Chapter_35.html"><a href="Chapter_35.html#randomisation-assumptions"><i class="fa fa-check"></i><b>35.4</b> Randomisation assumptions</a></li>
<li class="chapter" data-level="35.5" data-path="Chapter_35.html"><a href="Chapter_35.html#bootstrapping"><i class="fa fa-check"></i><b>35.5</b> Bootstrapping</a></li>
<li class="chapter" data-level="35.6" data-path="Chapter_35.html"><a href="Chapter_35.html#randomisation-conclusions"><i class="fa fa-check"></i><b>35.6</b> Randomisation conclusions</a></li>
</ul></li>
<li class="appendix"><span><b>Appendix</b></span></li>
<li class="chapter" data-level="A" data-path="appendexA.html"><a href="appendexA.html"><i class="fa fa-check"></i><b>A</b> Answers to chapter exercises</a>
<ul>
<li class="chapter" data-level="A.1" data-path="appendexA.html"><a href="appendexA.html#chapter-3"><i class="fa fa-check"></i><b>A.1</b> Chapter 3</a>
<ul>
<li class="chapter" data-level="A.1.1" data-path="appendexA.html"><a href="appendexA.html#exercise-3.1"><i class="fa fa-check"></i><b>A.1.1</b> Exercise 3.1:</a></li>
<li class="chapter" data-level="A.1.2" data-path="appendexA.html"><a href="appendexA.html#exercise-3.2"><i class="fa fa-check"></i><b>A.1.2</b> Exercise 3.2</a></li>
<li class="chapter" data-level="A.1.3" data-path="appendexA.html"><a href="appendexA.html#exercise-3.3"><i class="fa fa-check"></i><b>A.1.3</b> Exercise 3.3</a></li>
<li class="chapter" data-level="A.1.4" data-path="appendexA.html"><a href="appendexA.html#exercise-3.4"><i class="fa fa-check"></i><b>A.1.4</b> Exercise 3.4</a></li>
</ul></li>
<li class="chapter" data-level="A.2" data-path="appendexA.html"><a href="appendexA.html#chapter-8"><i class="fa fa-check"></i><b>A.2</b> Chapter 8</a>
<ul>
<li class="chapter" data-level="A.2.1" data-path="appendexA.html"><a href="appendexA.html#exercise-8.1"><i class="fa fa-check"></i><b>A.2.1</b> Exercise 8.1</a></li>
<li class="chapter" data-level="A.2.2" data-path="appendexA.html"><a href="appendexA.html#exercise-8.2"><i class="fa fa-check"></i><b>A.2.2</b> Exercise 8.2</a></li>
<li class="chapter" data-level="A.2.3" data-path="appendexA.html"><a href="appendexA.html#exercise-8.3"><i class="fa fa-check"></i><b>A.2.3</b> Exercise 8.3</a></li>
</ul></li>
<li class="chapter" data-level="A.3" data-path="appendexA.html"><a href="appendexA.html#chapter-14"><i class="fa fa-check"></i><b>A.3</b> Chapter 14</a>
<ul>
<li class="chapter" data-level="A.3.1" data-path="appendexA.html"><a href="appendexA.html#exercise-14.1"><i class="fa fa-check"></i><b>A.3.1</b> Exercise 14.1</a></li>
<li class="chapter" data-level="A.3.2" data-path="appendexA.html"><a href="appendexA.html#exercise-14.2"><i class="fa fa-check"></i><b>A.3.2</b> Exercise 14.2</a></li>
<li class="chapter" data-level="A.3.3" data-path="appendexA.html"><a href="appendexA.html#exercise-14.3"><i class="fa fa-check"></i><b>A.3.3</b> Exercise 14.3</a></li>
<li class="chapter" data-level="A.3.4" data-path="appendexA.html"><a href="appendexA.html#exercise-14.4"><i class="fa fa-check"></i><b>A.3.4</b> Exercise 14.4</a></li>
<li class="chapter" data-level="A.3.5" data-path="appendexA.html"><a href="appendexA.html#exercise-14.5"><i class="fa fa-check"></i><b>A.3.5</b> Exercise 14.5</a></li>
</ul></li>
<li class="chapter" data-level="A.4" data-path="appendexA.html"><a href="appendexA.html#chapter-17"><i class="fa fa-check"></i><b>A.4</b> Chapter 17</a>
<ul>
<li class="chapter" data-level="A.4.1" data-path="appendexA.html"><a href="appendexA.html#exercise-17.1"><i class="fa fa-check"></i><b>A.4.1</b> Exercise 17.1</a></li>
<li class="chapter" data-level="A.4.2" data-path="appendexA.html"><a href="appendexA.html#exercise-17.2"><i class="fa fa-check"></i><b>A.4.2</b> Exercise 17.2</a></li>
<li class="chapter" data-level="A.4.3" data-path="appendexA.html"><a href="appendexA.html#exercise-17.3"><i class="fa fa-check"></i><b>A.4.3</b> Exercise 17.3</a></li>
</ul></li>
<li class="chapter" data-level="A.5" data-path="appendexA.html"><a href="appendexA.html#chapter-20"><i class="fa fa-check"></i><b>A.5</b> Chapter 20</a>
<ul>
<li class="chapter" data-level="A.5.1" data-path="appendexA.html"><a href="appendexA.html#exercise-20.1"><i class="fa fa-check"></i><b>A.5.1</b> Exercise 20.1</a></li>
<li class="chapter" data-level="A.5.2" data-path="appendexA.html"><a href="appendexA.html#exercise-20.2"><i class="fa fa-check"></i><b>A.5.2</b> Exercise 20.2</a></li>
<li class="chapter" data-level="A.5.3" data-path="appendexA.html"><a href="appendexA.html#exercise-20.3"><i class="fa fa-check"></i><b>A.5.3</b> Exercise 20.3</a></li>
<li class="chapter" data-level="A.5.4" data-path="appendexA.html"><a href="appendexA.html#exercise-20.4"><i class="fa fa-check"></i><b>A.5.4</b> Exercise 20.4</a></li>
<li class="chapter" data-level="A.5.5" data-path="appendexA.html"><a href="appendexA.html#exercise-20.5"><i class="fa fa-check"></i><b>A.5.5</b> Exercise 20.5</a></li>
</ul></li>
<li class="chapter" data-level="A.6" data-path="appendexA.html"><a href="appendexA.html#chapter-23"><i class="fa fa-check"></i><b>A.6</b> Chapter 23</a>
<ul>
<li class="chapter" data-level="A.6.1" data-path="appendexA.html"><a href="appendexA.html#exercise-23.1"><i class="fa fa-check"></i><b>A.6.1</b> Exercise 23.1</a></li>
<li class="chapter" data-level="A.6.2" data-path="appendexA.html"><a href="appendexA.html#exercise-23.2"><i class="fa fa-check"></i><b>A.6.2</b> Exercise 23.2</a></li>
<li class="chapter" data-level="A.6.3" data-path="appendexA.html"><a href="appendexA.html#exercise-23.3"><i class="fa fa-check"></i><b>A.6.3</b> Exercise 23.3</a></li>
<li class="chapter" data-level="A.6.4" data-path="appendexA.html"><a href="appendexA.html#exercise-23.4"><i class="fa fa-check"></i><b>A.6.4</b> Exercise 23.4</a></li>
<li class="chapter" data-level="A.6.5" data-path="appendexA.html"><a href="appendexA.html#exercise-23.5"><i class="fa fa-check"></i><b>A.6.5</b> Exercise 23.5</a></li>
</ul></li>
<li class="chapter" data-level="A.7" data-path="appendexA.html"><a href="appendexA.html#chapter-28"><i class="fa fa-check"></i><b>A.7</b> Chapter 28</a>
<ul>
<li class="chapter" data-level="A.7.1" data-path="appendexA.html"><a href="appendexA.html#exercise-28.1"><i class="fa fa-check"></i><b>A.7.1</b> Exercise 28.1</a></li>
<li class="chapter" data-level="A.7.2" data-path="appendexA.html"><a href="appendexA.html#exercise-28.2"><i class="fa fa-check"></i><b>A.7.2</b> Exercise 28.2</a></li>
<li class="chapter" data-level="A.7.3" data-path="appendexA.html"><a href="appendexA.html#exercise-28.3"><i class="fa fa-check"></i><b>A.7.3</b> Exercise 28.3</a></li>
<li class="chapter" data-level="A.7.4" data-path="appendexA.html"><a href="appendexA.html#exercise-28.4"><i class="fa fa-check"></i><b>A.7.4</b> Exercise 28.4</a></li>
</ul></li>
<li class="chapter" data-level="A.8" data-path="appendexA.html"><a href="appendexA.html#chapter-31"><i class="fa fa-check"></i><b>A.8</b> Chapter 31</a>
<ul>
<li class="chapter" data-level="A.8.1" data-path="appendexA.html"><a href="appendexA.html#exercise-31.1"><i class="fa fa-check"></i><b>A.8.1</b> Exercise 31.1</a></li>
<li class="chapter" data-level="A.8.2" data-path="appendexA.html"><a href="appendexA.html#exercise-31.2"><i class="fa fa-check"></i><b>A.8.2</b> Exercise 31.2</a></li>
<li class="chapter" data-level="A.8.3" data-path="appendexA.html"><a href="appendexA.html#exercise-31.3"><i class="fa fa-check"></i><b>A.8.3</b> Exercise 31.3</a></li>
<li class="chapter" data-level="A.8.4" data-path="appendexA.html"><a href="appendexA.html#exercise-31.4"><i class="fa fa-check"></i><b>A.8.4</b> Exercise 31.4</a></li>
<li class="chapter" data-level="A.8.5" data-path="appendexA.html"><a href="appendexA.html#exercise-31.5"><i class="fa fa-check"></i><b>A.8.5</b> Exercise 31.5</a></li>
</ul></li>
<li class="chapter" data-level="A.9" data-path="appendexA.html"><a href="appendexA.html#chapter-34"><i class="fa fa-check"></i><b>A.9</b> Chapter 34</a>
<ul>
<li class="chapter" data-level="A.9.1" data-path="appendexA.html"><a href="appendexA.html#exercise-34.1"><i class="fa fa-check"></i><b>A.9.1</b> Exercise 34.1</a></li>
<li class="chapter" data-level="A.9.2" data-path="appendexA.html"><a href="appendexA.html#exercise-34.2"><i class="fa fa-check"></i><b>A.9.2</b> Exercise 34.2</a></li>
<li class="chapter" data-level="A.9.3" data-path="appendexA.html"><a href="appendexA.html#exercise-34.3"><i class="fa fa-check"></i><b>A.9.3</b> Exercise 34.3</a></li>
<li class="chapter" data-level="A.9.4" data-path="appendexA.html"><a href="appendexA.html#exercise-34.4"><i class="fa fa-check"></i><b>A.9.4</b> Exercise 34.4</a></li>
<li class="chapter" data-level="A.9.5" data-path="appendexA.html"><a href="appendexA.html#exercise-33.5"><i class="fa fa-check"></i><b>A.9.5</b> Exercise 33.5</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="B" data-path="uncertainty_derivation.html"><a href="uncertainty_derivation.html"><i class="fa fa-check"></i><b>B</b> Uncertainty derivation</a>
<ul>
<li class="chapter" data-level="B.1" data-path="uncertainty_derivation.html"><a href="uncertainty_derivation.html#propagation-of-error-for-addition-and-subtraction"><i class="fa fa-check"></i><b>B.1</b> Propagation of error for addition and subtraction</a></li>
<li class="chapter" data-level="B.2" data-path="uncertainty_derivation.html"><a href="uncertainty_derivation.html#propagation-of-error-for-multiplication-and-division"><i class="fa fa-check"></i><b>B.2</b> Propagation of error for multiplication and division</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="references.html"><a href="references.html"><i class="fa fa-check"></i>References</a></li>
<li class="divider"></li>
<li><a href="https://github.com/rstudio/bookdown" target="blank">Published with bookdown</a></li>
</ul>
</nav>
</div>
<div class="book-body">
<div class="body-inner">
<div class="book-header" role="navigation">
<h1>
<i class="fa fa-circle-o-notch fa-spin"></i><a href="./">Fundamental statistical concepts and techniques in the biological and environmental sciences: With jamovi</a>
</h1>
</div>
<div class="page-wrapper" tabindex="-1" role="main">
<div class="page-inner">
<section class="normal" id="section-">
<div id="appendexA" class="section level1 hasAnchor" number="36">
<h1><span class="header-section-number">A</span> Answers to chapter exercises<a href="appendexA.html#appendexA" class="anchor-section" aria-label="Anchor link to header"></a></h1>
<p>Answers to exercises in chapters are provided below.
Many questions are open-ended and intended to be thought provoking, and the answers to these questions that are provided below are mostly intended to explain the motivation underlying the question.
Numerical answers are also provided.
Answers are written in bold.
All datasets that are made tidy in the process of doing exercises can be found at <a href="https://osf.io/dxwyv" class="uri">https://osf.io/dxwyv</a> in the ‘exercise_answer_datasets’, and in the links below.</p>
<div id="chapter-3" class="section level2 hasAnchor" number="36.1">
<h2><span class="header-section-number">A.1</span> Chapter 3<a href="appendexA.html#chapter-3" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<div id="exercise-3.1" class="section level3 hasAnchor" number="36.1.1">
<h3><span class="header-section-number">A.1.1</span> Exercise 3.1:<a href="appendexA.html#exercise-3.1" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<ol style="list-style-type: decimal">
<li><strong>Seeds are tallied incorrectly.</strong></li>
<li>Tallies are not counted correctly in the lab notebook.</li>
<li><strong>Counts are not correctly input into spreadsheet.</strong></li>
</ol>
<p><strong>You can download the tidy dataset from this exercise here:</strong></p>
<p><a href="https://bradduthie.github.io/stats/data/Ch3_Exercise_1.csv" class="uri">https://bradduthie.github.io/stats/data/Ch3_Exercise_1.csv</a></p>
</div>
<div id="exercise-3.2" class="section level3 hasAnchor" number="36.1.2">
<h3><span class="header-section-number">A.1.2</span> Exercise 3.2<a href="appendexA.html#exercise-3.2" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>How many columns did you need to create the new dataset? <strong>2 columns.</strong></p>
<p>Are there any missing data in this dataset? <strong>There are no missing data.</strong></p>
<p><strong>The tidy dataset should include two columns, one for Species and the other for egg loads. It should have 54 rows (plus the header). You can download the dataset here:</strong></p>
<p><a href="https://bradduthie.github.io/stats/data/Ch3_Exercise_2.csv" class="uri">https://bradduthie.github.io/stats/data/Ch3_Exercise_2.csv</a></p>
</div>
<div id="exercise-3.3" class="section level3 hasAnchor" number="36.1.3">
<h3><span class="header-section-number">A.1.3</span> Exercise 3.3<a href="appendexA.html#exercise-3.3" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p><strong>The tidy dataset should include three columns, one for Species, one for Fruit, and one for Count. It should be have 25 rows (plus the header). You can download the dataset here:</strong></p>
<p><a href="https://bradduthie.github.io/stats/data/Ch3_Exercise_3.csv" class="uri">https://bradduthie.github.io/stats/data/Ch3_Exercise_3.csv</a></p>
</div>
<div id="exercise-3.4" class="section level3 hasAnchor" number="36.1.4">
<h3><span class="header-section-number">A.1.4</span> Exercise 3.4<a href="appendexA.html#exercise-3.4" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>What columns should this new dataset include? <strong>Species, Wasp number, Head length (mm), Head width (mm), Thorax length (mm), Thorax width (mm), Abdomen length (mm), Abdomen width (mm).</strong></p>
<p>How many rows are needed? <strong>26 rows of data (plus the column header).</strong></p>
<p><strong>You can download the dataset here:</strong></p>
<p><a href="https://bradduthie.github.io/stats/data/Ch3_Exercise_4.csv" class="uri">https://bradduthie.github.io/stats/data/Ch3_Exercise_4.csv</a></p>
<p>What formula will you type into your empty spreadsheet cell to calculate <span class="math inline">\(V_{thorax}\)</span>? <strong>=(4/3) * 3.14 * (D2/2) * ( ( E2/2)^2 )</strong></p>
<p>What are some reasons that we might want to be cautious about our calculated wasp volumes? <strong>There is error associated with the measurement of fig wasp dimensions (e.g., length, width). There is also error because we are assuming that the head is a sphere and the thorax and abdomen are ellipses.</strong></p>
</div>
</div>
<div id="chapter-8" class="section level2 hasAnchor" number="36.2">
<h2><span class="header-section-number">A.2</span> Chapter 8<a href="appendexA.html#chapter-8" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<div id="exercise-8.1" class="section level3 hasAnchor" number="36.2.1">
<h3><span class="header-section-number">A.2.1</span> Exercise 8.1<a href="appendexA.html#exercise-8.1" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>Mean: <strong>6.52 g C / kg soil</strong></p>
<p>Minimum: <strong>0.600 g C / kg soil</strong></p>
<p>Maximum: <strong>16.2 g C / kg soil</strong></p>
<p>Topsoil Mean: <strong>9.75 g C / kg soil</strong></p>
<p>Topsoil Minimum: <strong>4.00 g C / kg soil</strong></p>
<p>Topsoil Maximum: <strong>16.2 g C / kg soil</strong></p>
<p>Subsoil Mean: <strong>2.43 g C / kg soil</strong></p>
<p>Subsoil Minimum: <strong>0.600 g C / kg soil</strong></p>
<p>Subsoil Maximum: <strong>4.60 g C / kg soil</strong></p>
<p>Based on these samples in the dataset, can we really say for certain that the population mean of topsoil is higher than the population mean of subsoil?</p>
<p><strong>The sample means themselves do not tell us whether the population mean of topsoil will be bigger or smaller than that of subsoil, just that the sample means are different. We can use the data to calculate a standard error associated with the mean, which will give an indication of the level of confidence that is appropriate.</strong></p>
<p>What would make you more (or less) confident that topsoil and subsoil population means are different?</p>
<p><strong>The larger the sample size, the more confidence we have that the sample mean is close to the population mean. Also, the narrower the spread of the data, the more confidence we should have that the sample mean is close to the population mean.</strong></p>
</div>
<div id="exercise-8.2" class="section level3 hasAnchor" number="36.2.2">
<h3><span class="header-section-number">A.2.2</span> Exercise 8.2<a href="appendexA.html#exercise-8.2" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>Grand Mean length: <strong>1.59 cm</strong></p>
<p>Grand Mean height: <strong>1.46 cm</strong></p>
<p>Grand Mean width: <strong>1.57 cm</strong></p>
<p>Missing width row: <strong>Row 62</strong></p>
<p>Missing height row: <strong>Row 22</strong></p>
<p>Grand Mean length (mm): <strong>15.9</strong></p>
<p>Grand Mean height (mm): <strong>14.6</strong></p>
<p>Grand Mean width (mm): <strong>15.7</strong></p>
<p>Do the differences between means in cm and the means in mm make sense? <strong>Yes. Note that means in millimetres are ten times the value of the means in centimetres, as expected.</strong></p>
</div>
<div id="exercise-8.3" class="section level3 hasAnchor" number="36.2.3">
<h3><span class="header-section-number">A.2.3</span> Exercise 8.3<a href="appendexA.html#exercise-8.3" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>In this case, how might assuming that figs are perfectly spherical affect the accuracy of our estimated fig volume?</p>
<p><strong>As figs are not perfectly spherical, the estimation will be inaccurate. It could be systematically inaccurate, i.e., it would likely consistently over- or under-estimate the true volume of the figs, but it could also be randomly inaccurate as each fig will differ in shape a little so the approximation to a sphere will be differently wrong for each fig. Note that the measuring equipment used (a ruler in this case) will also limit the precision of the estimates.</strong></p>
<p>Mean: <strong>2150 mm<span class="math inline">\(\mathbf{^3}\)</span></strong></p>
<p>Minimum: <strong>697 mm<span class="math inline">\(\mathbf{^3}\)</span></strong></p>
<p>Maximum: <strong>4847 mm<span class="math inline">\(\mathbf{^3}\)</span></strong></p>
<p>Check the option for ‘Histogram’ and see the new histogram plotted in the window to the
right. Draw a rough sketch of the histogram in the area below. <strong>Your drawing should include a histogram with fig volume ranging from under 1000 to about 5000, with most values between 1000 and 2500.</strong></p>
</div>
</div>
<div id="chapter-14" class="section level2 hasAnchor" number="36.3">
<h2><span class="header-section-number">A.3</span> Chapter 14<a href="appendexA.html#chapter-14" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<div id="exercise-14.1" class="section level3 hasAnchor" number="36.3.1">
<h3><span class="header-section-number">A.3.1</span> Exercise 14.1<a href="appendexA.html#exercise-14.1" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p><strong>You can download the tidy dataset from this exercise here:</strong></p>
<p><a href="https://bradduthie.github.io/stats/data/Nymphaea_alba_tidy.csv" class="uri">https://bradduthie.github.io/stats/data/Nymphaea_alba_tidy.csv</a></p>
</div>
<div id="exercise-14.2" class="section level3 hasAnchor" number="36.3.2">
<h3><span class="header-section-number">A.3.2</span> Exercise 14.2<a href="appendexA.html#exercise-14.2" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>Just looking at the histogram, write down what you think the following summary statistics will be.</p>
<p><strong>While it will never be necessary (or recommended) to try to work out the mean, median, and standard deviation of a distribution directly from a histogram, the point of this is to help you connect the numerical summary statistics with the visualisation of the data in the histogram. You should be able to recognise that the mean and median are likely somewhere between 5 to 6 by recognising this as the centre of the distribution. Working out the standard deviation is a bit more challenging, but the average deviation from the centre looks to be about 2 (i.e., most points are probably about 2 mm from the mean). If your own answer was a bit different, this is not a cause for concern, but if you are able to interpret the histogram successfully, it probably should not be off by more than 3–4 mm.</strong></p>
<p>Based on the histogram, do you think that the mean and median are the same? Why or why
not?</p>
<p><strong>The mean and median appear to be quite similar. The distribution is mostly symmetrical, so the mean and median will likely be very close, although it might have a bit of a positive skew to it.</strong></p>
<p>Write a caption for the histogram below.</p>
<p><strong>Figure: Distribution of petiole diameter (mm) from white water lillies (<em>Nymphaea alba</em>) collected from 7 Scottish lochs.</strong></p>
<p>Highest: <strong>Lily loch</strong></p>
<p>Lowest: <strong>Linne</strong></p>
</div>
<div id="exercise-14.3" class="section level3 hasAnchor" number="36.3.3">
<h3><span class="header-section-number">A.3.3</span> Exercise 14.3<a href="appendexA.html#exercise-14.3" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<ul>
<li>N: <strong>140</strong></li>
<li>Std. deviation: <strong>1.83</strong></li>
<li>Variance: <strong>3.36</strong></li>
<li>Minimum: <strong>1.57</strong></li>
<li>Maximum: <strong>9.93</strong></li>
<li>Range: <strong>8.36</strong></li>
<li>IQR: <strong>2.90</strong></li>
<li>Mean: <strong>5.52</strong></li>
<li>Median: <strong>5.56</strong></li>
<li>Mode: <strong>3.2</strong></li>
<li>Std. error of mean: <strong>0.155</strong></li>
</ul>
<p>Which of the 7 sites in the dataset has the highest mean petiole diameter, and what is its
mean?</p>
<p>Site: <strong>Beag</strong></p>
<p>Mean: <strong>5.94 mm</strong></p>
<p>Which of the 7 sites has the lowest variance in petiole diameter, and what is its variance?</p>
<p>Site: <strong>Fidhle</strong></p>
<p>Variance: <strong>2.51 mm<span class="math inline">\(\mathbf{^2}\)</span></strong></p>
<p>Can you find the first and third quartiles for each site?</p>
<p>Beag: <strong>7.13 mm</strong></p>
<p>Buic: <strong>6.33 mm</strong></p>
<p>Choille-Bharr: <strong>6.83 mm</strong></p>
<p>Creig-Moire: <strong>7.24 mm</strong></p>
<p>Fidhle: <strong>6.53 mm</strong></p>
<p>Lily_Loch: <strong>7.24 mm</strong></p>
<p>Linne: <strong>6.52 mm</strong></p>
</div>
<div id="exercise-14.4" class="section level3 hasAnchor" number="36.3.4">
<h3><span class="header-section-number">A.3.4</span> Exercise 14.4<a href="appendexA.html#exercise-14.4" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<ul>
<li>N: <strong>140</strong></li>
<li>Std. deviation: <strong>1.8</strong></li>
<li>Variance: <strong>3.4</strong></li>
<li>Minimum: <strong>1.6</strong></li>
<li>Maximum: <strong>9.9</strong></li>
<li>Range: <strong>8.4</strong></li>
<li>IQR: <strong>2.9</strong></li>
<li>Mean: <strong>5.5</strong></li>
<li>Median: <strong>5.6</strong></li>
<li>Mode: <strong>3.2</strong></li>
<li>Std. error of mean: <strong>0.15</strong></li>
</ul>
<p>Were you able to get a similar value from the histogram as calculated in jamovi from the data? What can you learn from the histogram that you cannot from the summary statistics, and what can you learn from the summary statistics that you cannot from the histogram?</p>
<p><strong>Values might or might not be similar (it is not important that you can guess a mean or median to any degree of accuracy just by looking at a histogram). Note, however, that with the histogram, we can see the full shape of the distribution, which is not really possible (or at least, not easy), with the summary statistics alone. The summary statistics, in contrast, can give us specific numbers of central tendency or spread (e.g., mean, median, variance).</strong></p>
</div>
<div id="exercise-14.5" class="section level3 hasAnchor" number="36.3.5">
<h3><span class="header-section-number">A.3.5</span> Exercise 14.5<a href="appendexA.html#exercise-14.5" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>Recall back from Chapter 12; what information do these error bars convey about the estimated mean petiole diameter?</p>
<p><strong>Standard errors tell us how far our sample mean is expected to deviate from the true mean. Specifically, the standard error of the mean is the standard deviation of the sample means around the true mean. It is a measure for evaluating the uncertainty of the mean.</strong></p>
<p>What can you say about the mean petiole diameters across the different sites? Do these sites appear to have very different mean petiole diameters?</p>
<p><strong>Different sites appear to have similar means. Given the uncertainty indicated by the standard error, it is unclear if different sites have different mean petiole diameters (note, an ANOVA, which is introduced in Chapter 24, does not reject the null hypothesis that the means are the same).</strong></p>
<p>There were 20 total petiole diameters sampled from each site. If we were to go back out to these 7 sites and sample another 20 petiole diameters, could we really expect to get the exact same site means? Assuming the site means would be at least a bit different for our new sample, is it possible that the sites with the highest or lowest petiole diameters might also be different in our new sample? If so, then what does this say about our ability to make conclusions about the differences in petiole diameter among sites?</p>
<p><strong>If we were to go out and sample another 20 petiole diameters from each site, then we would not expect to get the exact same site means. The site means would be a bit different. The distribution of these sample means (with each repeated resampling of 20 and mean calculation) would be normally distributed around the true mean with a standard deviation approximately equal to the standard error of any given sample (such as the one in the barplots). It is possible that the rank order of mean petiole diameters might change entirely. Given this uncertainty, we cannot really say for sure which site population mean is really the highest or lowest.</strong></p>
</div>
</div>
<div id="chapter-17" class="section level2 hasAnchor" number="36.4">
<h2><span class="header-section-number">A.4</span> Chapter 17<a href="appendexA.html#chapter-17" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<div id="exercise-17.1" class="section level3 hasAnchor" number="36.4.1">
<h3><span class="header-section-number">A.4.1</span> Exercise 17.1<a href="appendexA.html#exercise-17.1" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>Now, fill in Table 17.1 with counts, percentage, and the estimated probability of a player selecting a small, medium, or large dam.</p>
<table>
<caption><strong>TABLE 17.1</strong> Statistics of Power Up! decisions for dam size with answers.</caption>
<thead>
<tr class="header">
<th>Dam Size</th>
<th>Counts</th>
<th>Percentage</th>
<th>Estimated Probability</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>Small</td>
<td><strong>21</strong></td>
<td><strong>28.4</strong></td>
<td><strong>0.284</strong></td>
</tr>
<tr class="even">
<td>Medium</td>
<td><strong>13</strong></td>
<td><strong>17.6</strong></td>
<td><strong>0.176</strong></td>
</tr>
<tr class="odd">
<td>Large</td>
<td><strong>40</strong></td>
<td><strong>54.1</strong></td>
<td><strong>0.541</strong></td>
</tr>
</tbody>
</table>
<p>What is the probability that this player chooses a small or a large dam?</p>
<p><strong>To get this probability, calculate the probability that the player chooses a small dam plus the probability that the player chooses a large dam: <em>P</em>(small or large) = 0.284 + 0.541 = 0.825.</strong></p>
<p>Now suppose that 3 new players arrive and decide to play the game. What is the probability that all 3 of these new players choose a large dam?</p>
<p><strong>To get this probability, calculate the probability of choosing a large dam raised to the power of 3: <em>P</em>(3 large) = 0.541 <span class="math inline">\(\times\)</span> 0.541 <span class="math inline">\(\times\)</span> 0.541 = 0.541<span class="math inline">\(\mathbf{^{3}}\)</span> = 0.158.</strong></p>
<p>What is the probability that the first player chooses a small dam, the second player chooses a medium dam, and the third player chooses a large dam?</p>
<p><strong>To get this probability, calculate the probability of choosing small dam, times a medium dam, times a large dam: <em>P</em>(Player 1 = small, Player 2 = medium, Player 3 = large) = 0.541 <span class="math inline">\(\times\)</span> 0.176 <span class="math inline">\(\times\)</span> 0.284 = 0.027.</strong></p>
<p>Imagine that you randomly choose one of the 74 players with equal probability (i.e., every player is equally likely to be chosen). What is the probability that you choose player 20?</p>
<p><strong>To get this probability, we just need to calculate one divided by 74: <em>P</em>(Player 20) = 1/74 = 0.01351.</strong></p>
<p>What is the probability that you choose player 20, <em>then</em> choose a different player with a large dam? As a hint, remember that you are now sampling <em>without replacement</em>. The second choice cannot be player 20 again, so the probability of choosing a player with a large dam has changed from the estimated probability in Table 17.1.</p>
<p><strong>To get this probability, we need to first recognise that there is a 1/74 probability (0.01351) of choosing Player 20. Player 20 chose a large dam, so the number of remaining players in the dataset is now 73, and now only 39 of them chose large dams. Hence, the probability of choosing a large dam from the remaining players is 39/73 (0.53425). To calculate the probability of both events happening, we need to multiply: <em>P</em>(Player 20, Large) = 0.01351 <span class="math inline">\(\times\)</span> 0.53425 = 0.00722.</strong></p>
<p>Now, recreate the table in Figure 17.3 and estimate the probability that an Android user will choose to build a large dam.</p>
<p><strong>To get this probability, divide the number of Android players that choose a large dam (31) by the total number of Android users (56): <em>P</em>(Large | Android) = 31/56 = 0.554.</strong></p>
<p>Is P(Large|Android) much different from the probability that any player chooses a large dam, as calculated in Table 17.1? Do you think that the difference is significant?</p>
<p><strong>This is a small difference, but not that much. It is probably not significant.</strong></p>
</div>
<div id="exercise-17.2" class="section level3 hasAnchor" number="36.4.2">
<h3><span class="header-section-number">A.4.2</span> Exercise 17.2<a href="appendexA.html#exercise-17.2" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>Use jamovi to find the mean and the standard deviation of player score (note, we can just say that score is unitless, so no need to include units).</p>
<p><strong>Mean score: 95.9</strong></p>
<p><strong>Standard deviation score: 22.3</strong></p>
<p>What is the probability of a player getting a score between 80 and 120?</p>
<p><strong><span class="math inline">\(P(80 \leq X \leq 120)\)</span> = 0.6222</strong></p>
<p>What is the probability of a player getting a score greater than 130?</p>
<p><strong><span class="math inline">\(P(X \geq 130)\)</span> = 0.0631</strong></p>
<p>Now try the following probabilities for different scores.</p>
<p><strong><span class="math inline">\(P(X \geq 120)\)</span> = 0.1399</strong></p>
<p><strong><span class="math inline">\(P(X \leq 100)\)</span> = 0.5729</strong></p>
<p><strong><span class="math inline">\(P(100 \leq X \leq 120)\)</span> = 0.2872</strong></p>
<p>What is the probability of a player getting a score lower than 70 or higher than 130?</p>
<p><strong><span class="math inline">\(P(X \leq 70 \: \cup \: X \geq 130)\)</span> = <span class="math inline">\(1 - 0.8142\)</span> = 0.1858</strong></p>
<p>There is more than one way to figure this last one out. How did you do it, and what was your reasoning?</p>
<p><strong>We can find the total area under the curve between 70 and 130 using jamovi. Since the entire area under the curve must sum to 1, if we subtract this area (0.8142) from 1, then we are left with the area in the tails of the distribution (i.e., lower than 70 or higher than 130).</strong></p>
</div>
<div id="exercise-17.3" class="section level3 hasAnchor" number="36.4.3">
<h3><span class="header-section-number">A.4.3</span> Exercise 17.3<a href="appendexA.html#exercise-17.3" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>How would you describe the shape of the distribution of v1?</p>
<p><strong>The distribution of v1 is approximately uniform.</strong></p>
<p>Sketch what you predict the shape of its distribution will be below.</p>
<p><strong>The ‘all_means’ values should have the shape of a normal distribution (roughly) as it is the distribution of the sample means. The CLT states that the original distribution (in this case uniform) does not matter; when a set of sample means are individually calculated, the dataset will form a normal distribution.</strong></p>
<p>As best you can, explain why the shapes of the two distributions differ.</p>
<p><strong>A consequence of the central limit theorem is that the distribution of sample means should be normal regardless of the distribution of the sample data. In this case, the sample data were uniformly distributed, but the means of the 40 datasets is normally distributed.</strong></p>
<p>Now try increasing the number of trials to 200. What happens to the histogram? What about when you increase the number of trials to 2000?</p>
<p><strong>The distribution of sample means appears to get closer to a normal distribution.</strong></p>
<p>Try playing around with different source distributions, sample sizes, and numbers of trials. What general conclusion can you make about the distribution of sample means from the different distributions?</p>
<p><strong>Using clt-Demonstrations: Increasing number of trials shows dataset looking more and more normally distributed. General conclusion should be that the distribution of sample means is approximately normal, regardless of the original distribution.</strong></p>
</div>
</div>
<div id="chapter-20" class="section level2 hasAnchor" number="36.5">
<h2><span class="header-section-number">A.5</span> Chapter 20<a href="appendexA.html#chapter-20" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<div id="exercise-20.1" class="section level3 hasAnchor" number="36.5.1">
<h3><span class="header-section-number">A.5.1</span> Exercise 20.1<a href="appendexA.html#exercise-20.1" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>Do these data appear to be roughly normal? Why or why not?</p>
<p><strong>The data appear to be normal. The distribution is mostly symmetric with no clear outliers.</strong></p>
<p>Next, calculate the grand mean and standard deviation of tree DBH (i.e., the mean and standard deviation of trees across all sites).</p>
<p>Grand Mean: <strong>36.93</strong></p>
<p>Grand Standard Deviation: <strong>10.96</strong></p>
<p>Using the same principles, what is the cumulative 0.4 quantile for the DBH data? <strong>34.11 cm.</strong></p>
<p>From the Results table on the right, what interval of DBH values will contain 95% of the probability density around the mean? <strong><span class="math inline">\(\mathbf{15.34-58.46}\)</span> cm</strong></p>
<p>From the Descriptives panel in jamovi (recall that this is under the ‘Exploration’
button), find the standard error of DBH. Std. error of Mean: <strong>1.000.</strong></p>
<p>Based on the Results table, what can you infer are the lower and upper 95%
confidence intervals (CIs) around the mean?</p>
<p>Lower 95% CI: <strong>34.94</strong></p>
<p>Upper 95% CI: <strong>38.86</strong></p>
<p>From this Descriptives table now, write the lower and upper 95% CIs below.</p>
<p>Lower 95% CI: <strong>34.95</strong></p>
<p>Upper 95% CI: <strong>38.91</strong></p>
</div>
<div id="exercise-20.2" class="section level3 hasAnchor" number="36.5.2">
<h3><span class="header-section-number">A.5.2</span> Exercise 20.2<a href="appendexA.html#exercise-20.2" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>From these quantiles, what is the proper z-score to use in the equations for LCI
and UCI above?</p>
<p>z-score: <strong>1.96</strong></p>
<p>Now, use the values of <span class="math inline">\(\bar{x}\)</span>, z, and SE for DBH in the equations above to calculate lower and upper 95% confidence intervals again.</p>
<p>Lower 95% CI: <strong>34.94</strong></p>
<p>Upper 95% CI: <strong>38.86</strong></p>
<p>Are these confidence intervals the same as what you calculated in Exercise 19.1?</p>
<p><strong>These confidence intervals are the same as those calculated from the normal distribution, bu they are slightly different from those from the ‘Descriptives’ menu as those ones are calculated with t-scores (which are more accurate), rather than z-scores.</strong></p>
<p>What are the appropriate df for DBH? <strong>df: </strong></p>
<p>From the Results table, what is the proper t-score to use in the equations for LCI and UCI? <strong>t-score: 1.980.</strong></p>
<p>Again, use the values of <span class="math inline">\(\bar{x}\)</span>, <span class="math inline">\(t\)</span>, and <span class="math inline">\(SE\)</span> for DBH in the equations above to calculate lower and upper 95% confidence intervals.</p>
<p>Lower 95% CI: <strong>34.92</strong></p>
<p>Upper 95% CI: <strong>38.99</strong></p>
<p>Reflect on any similarities or differences that you see in all of these different ways of calculating confidence intervals.</p>
<p><strong>The confidence intervals are very similar, but not exactly the same as those calculated with z-scores. This is because the sample size is 120 and degrees of freedom is therefore <span class="math inline">\(\mathbf{120 - 1 = 119}\)</span>, which is quite large, and as the sample size becomes larger the t-scores and z-scores become more similar. While there is no fixed threshold, usually when the sample size is more than about 30, the difference between the two methods is sufficiently small as to not make an important difference.</strong></p>
</div>
<div id="exercise-20.3" class="section level3 hasAnchor" number="36.5.3">
<h3><span class="header-section-number">A.5.3</span> Exercise 20.3<a href="appendexA.html#exercise-20.3" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>From the Descriptives tool in jamovi, write the sample sizes for DBH split by site below.</p>
<p>Site 1182: N = <strong>4</strong></p>
<p>Site 1223: N = <strong>22</strong></p>
<p>Site 3008: N = <strong>10</strong></p>
<p>Site 10922: N = <strong>84</strong></p>
<p>For which of these sites would you predict CIs calculated from z-scores versus t- scores to differ the most? <strong>Site: 1182 (note that this site has the lowest sample size).</strong></p>
<p>Now, fill in the table below reporting 95% CIs calculated using each distribution from the 4 sites using any method you prefer.</p>
<table>
<caption><strong>TABLE 20.1</strong> 95% Confidence intervals calculated for tree diameter at breast height (DBH) in centimetres.</caption>
<colgroup>
<col width="13%" />
<col width="12%" />
<col width="30%" />
<col width="43%" />
</colgroup>
<thead>
<tr class="header">
<th>Site</th>
<th>N</th>
<th>95% CIs (Normal)</th>
<th>95% CIs (t-distribution)</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>1182</td>
<td><strong>4</strong></td>
<td><strong><span class="math inline">\(\mathbf{42.73-57.57}\)</span></strong></td>
<td><strong><span class="math inline">\(\mathbf{38.09-62.21}\)</span></strong></td>
</tr>
<tr class="even">
<td>1223</td>
<td><strong>22</strong></td>
<td><strong><span class="math inline">\(\mathbf{21.56-24.16}\)</span></strong></td>
<td><strong><span class="math inline">\(\mathbf{21.48-24.24}\)</span></strong></td>
</tr>
<tr class="odd">
<td>3008</td>
<td><strong>10</strong></td>
<td><strong><span class="math inline">\(\mathbf{51.49-61.13}\)</span></strong></td>
<td><strong><span class="math inline">\(\mathbf{50.75-61.87}\)</span></strong></td>
</tr>
<tr class="even">
<td>10922</td>
<td><strong>84</strong></td>
<td><strong><span class="math inline">\(\mathbf{36.09-39.25}\)</span></strong></td>
<td><strong><span class="math inline">\(\mathbf{36.07-39.27}\)</span></strong></td>
</tr>
</tbody>
</table>
<p>Next, do the same, but now calculate 99% CIs instead of 95% CIs.</p>
<table>
<caption><strong>TABLE 20.2</strong> 99% Confidence intervals calculated for tree diameter at breast height (DBH) in centimetres.</caption>
<colgroup>
<col width="13%" />
<col width="12%" />
<col width="30%" />
<col width="43%" />
</colgroup>
<thead>
<tr class="header">
<th>Site</th>
<th>N</th>
<th>99% CIs (Normal)</th>
<th>99% CIs (t-distribution)</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>1182</td>
<td><strong>4</strong></td>
<td><strong><span class="math inline">\(\mathbf{40.38-59.92}\)</span></strong></td>
<td><strong><span class="math inline">\(\mathbf{28.02-72.28}\)</span></strong></td>
</tr>
<tr class="even">
<td>1223</td>
<td><strong>22</strong></td>
<td><strong><span class="math inline">\(\mathbf{21.14-24.58}\)</span></strong></td>
<td><strong><span class="math inline">\(\mathbf{20.98-24.74}\)</span></strong></td>
</tr>
<tr class="odd">
<td>3008</td>
<td><strong>10</strong></td>
<td><strong><span class="math inline">\(\mathbf{49.96-62.66}\)</span></strong></td>
<td><strong><span class="math inline">\(\mathbf{48.32-64.30}\)</span></strong></td>
</tr>
<tr class="even">
<td>10922</td>
<td><strong>84</strong></td>
<td><strong><span class="math inline">\(\mathbf{35.60-39.74}\)</span></strong></td>
<td><strong><span class="math inline">\(\mathbf{35.55-39.79}\)</span></strong></td>
</tr>
</tbody>
</table>
<p>What do you notice about the difference between CIs calculated from the normal distribution versus the t-distribution across the different sites?</p>
<p><strong>The t-distribution gives a slightly wider spread than the normal distribution, and the difference increases as sample size gets smaller.</strong></p>
<p>In your own words, what do these CIs <em>actually mean?</em></p>
<p><strong>Confidence intervals: if you take a sample and calculate the 95% (or 99%) confidence interval, and then go to the same population and resample that population numerous times, then there is a 95% chance the new sample mean will be within the CIs calculated at the 95% level (and 99% chance that it would be within the CIs calculated at the 99% level).</strong></p>
</div>
<div id="exercise-20.4" class="section level3 hasAnchor" number="36.5.4">
<h3><span class="header-section-number">A.5.4</span> Exercise 20.4<a href="appendexA.html#exercise-20.4" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>From the Descriptives options, find the number of sites grazed versus not grazed.</p>
<p>Grazed: <strong>4</strong></p>
<p>Not Grazed: <strong>20</strong></p>
<p>From these counts above, what is the estimate (<span class="math inline">\(p\)</span>, or more technically <span class="math inline">\(\hat{p}\)</span>, with the hat indicating that it is an estimate) of the proportion of sites that are grazed?</p>
<p>p: <strong>4 / (20 + 4) = 0.166667</strong></p>
<p>We can estimate <span class="math inline">\(p\)</span> using <span class="math inline">\(p\)</span>, and <span class="math inline">\(N\)</span> is the total sample size. Using the above equation, what is the standard error of <span class="math inline">\(p\)</span>?</p>
<p><span class="math display">\[\mathbf{SE(p) = \sqrt{\frac{0.166667(1 - 0.166667)}{24}} = 0.0761}\]</span></p>
<p>Using this standard error, what are the Wald lower and upper 95% confidence intervals around <span class="math inline">\(p\)</span>?</p>
<p>Wald <span class="math inline">\(LCI_{95\%} = 0.166667 - (1.96 \times 0.0761) =\)</span> <strong>0.0175</strong></p>
<p>Wald <span class="math inline">\(UCI_{95\%} = 0.166667 + (1.96 \times 0.0761) =\)</span> <strong>0.3158</strong></p>
<p>Next, find the lower and upper 99% CIs around <span class="math inline">\(p\)</span> and report them below.</p>
<p>Wald <span class="math inline">\(LCI_{99\%} = 0.166667 - (2.58 \times 0.0761) =\)</span> <strong><span class="math inline">\(\mathbf{-0.0297}\)</span></strong></p>
<p>Wald <span class="math inline">\(UCI_{99\%} = 0.166667 + (2.58 \times 0.0761) =\)</span> <strong>0.3630</strong></p>
<p>Do you notice anything unusual about the lower 99% CI? <strong>The lower 99% CI is a negative number, which is not possible for a proportion.</strong></p>
<p><span class="math inline">\(p\)</span>: <strong>0.16667</strong></p>
<p>Clopper-Pearson <span class="math inline">\(LCI_{95\%} =\)</span> <strong>0.04735</strong></p>
<p>Clopper-Pearson <span class="math inline">\(UCI_{95\%} =\)</span> <strong>0.37384</strong></p>
<p>To calculate 99% CIs, change the number in the Interval box from 95 to 99.
Report the 99% CIs below.</p>
<p>Clopper-Pearson <span class="math inline">\(LCI_{99\%} =\)</span> <strong>0.02947</strong></p>
<p>Clopper-Pearson <span class="math inline">\(UCI_{99\%} =\)</span> <strong>0.43795</strong></p>
<p>What do you notice about the difference between the Wald CIs and the Clopper-Pearson CIs?</p>
<p><strong>The Clopper-Pearson CIs are a bit wider than the Wald CIs, thereby suggesting that a wider range of values is needed to encompass the means for a given level of confidence. The Clopper-Pearson CIs have higher upper confidence intervals, but also higher lower confidence intervals (and the 99% Clopper-Pearson CI does not overlap zero).</strong></p>
</div>
<div id="exercise-20.5" class="section level3 hasAnchor" number="36.5.5">
<h3><span class="header-section-number">A.5.5</span> Exercise 20.5<a href="appendexA.html#exercise-20.5" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>First consider an 80% CI.</p>
<p><span class="math inline">\(LCI_{80\%} =\)</span> <strong>0.47359</strong></p>
<p><span class="math inline">\(UCI_{80\%} =\)</span> <strong>0.75942</strong></p>
<p>Next, calculate 95% CIs for the proportion of sites classified as Ancient woodland.</p>
<p><span class="math inline">\(LCI_{95\%} =\)</span> <strong>0.40594</strong></p>
<p><span class="math inline">\(UCI_{95\%} =\)</span> <strong>0.81201</strong></p>
<p>Finally, calculate 99% CIs for the proportion of sites classified as Ancient woodland.</p>
<p><span class="math inline">\(LCI_{99\%} =\)</span> <strong>0.34698</strong></p>
<p><span class="math inline">\(UCI_{99\%} =\)</span> <strong>0.85353</strong></p>
</div>
</div>
<div id="chapter-23" class="section level2 hasAnchor" number="36.6">
<h2><span class="header-section-number">A.6</span> Chapter 23<a href="appendexA.html#chapter-23" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<div id="exercise-23.1" class="section level3 hasAnchor" number="36.6.1">
<h3><span class="header-section-number">A.6.1</span> Exercise 23.1<a href="appendexA.html#exercise-23.1" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>Report these below.</p>
<p>N: <strong>21</strong></p>
<p><span class="math inline">\(\bar{x}\)</span>: <strong>58.76</strong></p>
<p><span class="math inline">\(s\)</span>: <strong>8.687</strong></p>
<p>What kind(s) of statistical test would be most appropriate to use in this case, and what is the null hypothesis (<span class="math inline">\(H_{0}\)</span>) of the test?</p>
<p>Test to use: <strong>One sample t-test</strong></p>
<p><span class="math inline">\(H_{0}\)</span>: <strong>The overall student scores were sampled from a population with a mean of 60.1.</strong></p>
<p>What is the alternative hypothesis (<span class="math inline">\(H_{A}\)</span>), and should you use a one- or two-tailed test?</p>
<p><span class="math inline">\(H_{A}\)</span>: <strong>The overall student scores were sampled from a population with a mean lower than 60.1.</strong></p>
<p>One- or two-tailed? <strong>One-tailed.</strong></p>
<p>From the Normality Test table, what is the p-value of the Shapiro-Wilk test? <strong><em>P</em> = 0.112</strong></p>
<p>Based on this p-value, should we reject the null hypothesis?</p>
<p><strong>No, we do not reject the null hypothesis that the data are normally distributed.</strong></p>
<p>On the right panel of jamovi, you will see a table with the t-statistic, degrees of freedom, and p-value of the one sample t-test. Write these values down below.</p>
<p>t-statistic: <strong><span class="math inline">\(\mathbf{-0.7067}\)</span></strong></p>
<p>degrees of freedom: <strong>20</strong></p>
<p>p-value: <strong>0.244</strong></p>
<p>Based on the p-value, should you reject the null hypothesis that your students’ mean overall grade is the same as the national average? Why or why not?</p>
<p><strong>We should not reject the null hypothesis because our p-value is greater than our threshold Type I error rate of 0.05 (i.e., <em>P</em> > 0.05). Assuming that the null hypothesis is true, the probability getting a t-statistic as extreme as the one we observed is only about 1 in 4, which is not especially unlikely.</strong></p>
<p>Based on this test, how would you respond to your colleague who is concerned that your students are performing below the national average?</p>
<p><strong>There is no evidence that students in this class are performing below the national average.</strong></p>
<p>Is there an assumption that might be particularly suspect when comparing the scores of students in a single classroom with a national average? Why or why not?</p>
<p><strong>The students in this classroom are unlikely to be a random sample from the overall population. There may be other factors affecting student test scores that have nothing to do with the quality of the instruction.</strong></p>
</div>
<div id="exercise-23.2" class="section level3 hasAnchor" number="36.6.2">
<h3><span class="header-section-number">A.6.2</span> Exercise 23.2<a href="appendexA.html#exercise-23.2" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>Is there any reason to believe that the data are not normally distributed?</p>
<p><strong>No, the Shapiro-Wilk test gives us no reason to reject the null hypothesis that the data are normally distributed (<span class="math inline">\(\mathbf{P > 0.05}\)</span>).</strong></p>
<p>We want to know if student grades have improved. What is the null hypothesis (<span class="math inline">\(H_{0}\)</span>) and alternative hypothesis (<span class="math inline">\(H_{A}\)</span>) in this case?</p>
<p><span class="math inline">\(H_{0}\)</span>: <strong>The mean change in student grade is 0.</strong></p>
<p><span class="math inline">\(H_{A}\)</span>: <strong>The mean change in student grade is greater than 0.</strong></p>
<p>Write these values down below.</p>
<p>t-statistic: <strong><span class="math inline">\(\mathbf{-8.18}\)</span></strong></p>
<p>degrees of freedom: <strong>20</strong></p>
<p>p-value: <strong><em>P</em> < 0.001</strong></p>
<p>Based on this p-value, should you reject or fail to reject your null hypothesis? What can you then conclude about student test scores?</p>
<p><strong>Because <em>P</em> < 0.05, we reject the null hypothesis. We can conclude that the mean score for Test 1 is less than the mean score for Test 2, so the grades appear to have improved.</strong></p>
</div>
<div id="exercise-23.3" class="section level3 hasAnchor" number="36.6.3">
<h3><span class="header-section-number">A.6.3</span> Exercise 23.3<a href="appendexA.html#exercise-23.3" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>We are not interested in whether the scores are higher or lower than 62, just that
they are different. Consequently, what should our alternative hypothesis (<span class="math inline">\(H_{A}\)</span>) be?</p>
<p><span class="math inline">\(H_{A}\)</span>: <strong>Test 3 scores were sampled from a population with a mean not equal to 62.</strong></p>
<p>What is the p-value of the Shaprio-Wilk test this time? <strong><em>P</em> = 0.022.</strong></p>
<p>What inference can you make from the Q-Q plot? Do the points fall along the diagonal line?</p>
<p><strong>The points appear to be a bit curved. They do not cleanly fall along the diagonal line.</strong></p>
<p>Based on the Shapiro-Wilk test and Q-Q plot, is it safe to assume that the Test 3 scores are normally distributed?</p>
<p><strong>Based on the Shapiro-Wilk test and the Q-Q plot, we should not assume that the data are normally distributed.</strong></p>
<p>What are the null and alternative hypotheses of this test?</p>
<p><span class="math inline">\(H_{0}\)</span>: <strong>Test 3 scores were sampled from a population with a median of 62.</strong></p>
<p><span class="math inline">\(H_{A}\)</span>: <strong>Test 3 scores were sampled from a population with a median not equal to 62.</strong></p>
<p>What is the test statistic (not the p-value) for the Wilcoxon test? Test statistic: <strong>53.</strong></p>
<p>Based on what you learned in Section 22.5.1, what does this test statistic actually mean?</p>
<p><strong>It means that if we subtract 62 from the Test 3 values, then rank each by its absolute value, the sum of ranks that came from positive values should equal 53.</strong></p>
<p>Now look at the p-value for the Wilcoxon test. What is the p-value, and what
should you conclude from it? <strong><em>P</em> = 0.055.</strong></p>
<p>Conclusion: <strong>We do not reject the null hypothesis that the median Test 3 score is 62.</strong></p>
</div>
<div id="exercise-23.4" class="section level3 hasAnchor" number="36.6.4">
<h3><span class="header-section-number">A.6.4</span> Exercise 23.4<a href="appendexA.html#exercise-23.4" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>One- or two-tailed? <strong>Two-tailed.</strong></p>
<p>Based on the Assumption Checks in jamovi (and Figure 23.5), what can you conclude about the t-test assumptions?</p>
<p><strong>The data appear to be normally distributed, but the groups do not have equal variances.</strong></p>
<p>What is the p-value for Levene’s test? <strong><em>P</em> = 0.021.</strong></p>
<p>Based on what you learnt in Chapter 22.2, what is the appropriate test to run in this case? Test: <strong>Welch’s independent samples t-test.</strong></p>
<p>Check the box for the correct test, then report the test statistic and p-value from the table that appears in the right panel.</p>
<p>Test statistic: <strong><span class="math inline">\(\mathbf{-0.3279}\)</span></strong></p>
<p><strong>P* = 0.745</strong>*</p>
<p>What can you conclude from this t-test?</p>
<p><strong>There is no evidence to reject the null hypothesis that both years were sampled from a population from the same mean score. The overall scores appear to be the same between years.</strong></p>
</div>
<div id="exercise-23.5" class="section level3 hasAnchor" number="36.6.5">
<h3><span class="header-section-number">A.6.5</span> Exercise 23.5<a href="appendexA.html#exercise-23.5" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>Below, summarise the hypotheses for this new test.</p>
<p><span class="math inline">\(H_{0}\)</span>: <strong>Test 3 scores in 2022 and 2023 were sampled from a population with the same mean.</strong></p>
<p><span class="math inline">\(H_{A}\)</span>: <strong>Test 3 scores in 2022 and 2023 were sampled from a population with different means.</strong></p>
<p>Is this a one or two tailed test?: <strong>Two-tailed test.</strong></p>
<p>Do the variances appear to be the same for Test 3 scores in 2022 versus 2023? How can you make this conclusion?</p>
<p><strong>The homogeneity of variances test (i.e., Levene’s test) show a test statistic of <em>F</em> = 0.1379 and a p-value of <em>P</em> = 0.712, so there is no evidence to reject the null hypothesis that the two years have different variances.</strong></p>
<p>What is the p-value of the Shapiro-Wilk test? <strong><em>P</em> < 0.001</strong></p>
<p>Now, have a look at the Q-Q plot. What can you infer from this plot about the normality of the data, and why?</p>
<p><strong>The data appear to deviate from the diagonal line, suggesting non-normality.</strong></p>
<p>Based on what you found from testing the model assumptions above, and the material in Chapter 22, what test is the most appropriate one to use?</p>
<p>Test: <strong>Mann-Whitney U test.</strong></p>
<p>Run the above test in jamovi, then report the test statistic and p-value below.</p>
<p>Test statistic: <strong>221.0</strong></p>
<p>p-value: <strong>0.487</strong></p>
<p>Based on what you learned in Section 22.5.2, what does this test statistic actually mean?</p>
<p><strong>If we rank the full dataset (all Test 3 scores regardless of year), then sum up the ranks of 2022, the rank sum would be 221.</strong></p>
<p>Finally, what conclusions can you make about Test 3 scores in 2022 versus 2023?</p>
<p><strong>There appears to be no difference in Test 3 scores between 2022 and 2023. We do not reject the null hypothesis that the medians (technically, the distributions) differ between years.</strong></p>
<p>What could you do to test the null hypothesis that the change in scores from Test 1 to Test 2 is the same between years?</p>
<p><strong>Because the paired samples t-test is really just a one-sample t-test, we could first calculate the change from Test 1 to Test 2. That is, create a new column of data that is Change = Test 1 – Test 2. We could then use an independent samples t-test to check if the change differs between 2022 and 2023.</strong></p>