This problem illustrates an interesting variation of simple random sampling. a.

Question

This problem illustrates an interesting variation of simple random sampling.
a. Open a blank spreadsheet and use the RAND() function to create a column of 1000 random numbers. Don’t freeze them. This is actually a simple random sample from the uniform distribution between 0 and 1. Use the COUNTIF function to count the number of values between 0 and 0.1, between 0.1 and 0.2, and so on. Each such
interval should contain about 1/10 of all values. Do they? (Keep pressing the F9 key to see how the results change.)

b. Repeat part a, generating a second column of random numbers, but now generate the first 100 as uniform between 0 and 0.1, the next 100 as uniform between 0.1 and 0.2, and so on, up to 0.9 to 1. (Hint: For example, to create a random number uniformly distributed between 0.5 and 0.6, use the formula =0.5+0.1*RAND(). Do you see why?) Again, use COUNTIF to find the number of the 1000 values in each of the intervals, although there shouldn’t be any surprises this time. Why might this type of random sampling be preferable to the random sampling in part a?

Explanation / Answer

x
1 0.3325443575
2 0.7247514937
3 0.1634753060
4 0.9401272442
5 0.5450557594
6 0.5019670038
7 0.5785779057
8 0.7851169878
9 0.9999295331
10 0.2772184524
11 0.1125063745
12 0.9926788346
13 0.8240645318
14 0.1343678948
15 0.8317973779
16 0.0831834667
17 0.3854754628
18 0.2027516826
19 0.0003481396
20 0.6250518290
21 0.7367168344
22 0.5978668188
23 0.8088948370
24 0.5336591948
25 0.0981268333
26 0.3700629456
27 0.2310021087
28 0.6296199451
29 0.2863955768
30 0.3158983111
31 0.8144590696
32 0.4923294839
33 0.0671753678
34 0.5068043268
35 0.4520526670
36 0.9373750915
37 0.8753380454
38 0.8070233169
39 0.8206404836
40 0.3859070076
41 0.2871543334
42 0.2393018410
43 0.5671322644
44 0.4542971340
45 0.6723089097
46 0.7809727401
47 0.8253992426
48 0.5983048866
49 0.0977255816
50 0.1383026394
51 0.5379747723
52 0.9635981813
53 0.3445273505
54 0.5009856957
55 0.3351906340
56 0.2179405598
57 0.8868271888
58 0.3893996878
59 0.5834470149
60 0.0970292820
61 0.9884484401
62 0.6791741662
63 0.8102494313
64 0.3616988023
65 0.9795720305
66 0.5537558533
67 0.6625573121
68 0.7659501869
69 0.9564866938
70 0.4014586571
71 0.1422354521
72 0.8929169709
73 0.2663332482
74 0.9508952252
75 0.4989547990
76 0.6088071181
77 0.4898674756
78 0.0297706018
79 0.3823039474
80 0.8006835745
81 0.9864334608
82 0.8907168456
83 0.5248052571
84 0.8586626283
85 0.1754182205
86 0.1106646997
87 0.1518517474
88 0.3369041409
89 0.7199400067
90 0.8858155115
91 0.9023355574
92 0.3101038644
93 0.2918599886
94 0.1778534278
95 0.0376769307
96 0.4022159379
97 0.7577081430
98 0.8570476898
99 0.1524570452
100 0.9769787854
101 0.9491313025
102 0.8109758233
103 0.1035710138
104 0.1402996380
105 0.9018654730
106 0.1726521447
107 0.1423088466
108 0.1506086444
109 0.4821114526
110 0.6901253189
111 0.7140489849
112 0.5975184783
113 0.2194195199
114 0.0117036179
115 0.5383841535
116 0.1999526639
117 0.0763070146
118 0.6011056572
119 0.5064041016
120 0.3584296922
121 0.2177200394
122 0.6164777547
123 0.0488996971
124 0.2915893202
125 0.3969092127
126 0.1630412464
127 0.6540064495
128 0.6811323331
129 0.0918531611
130 0.4552664803
131 0.9644416627
132 0.6227579829
133 0.0116311677
134 0.8346922344
135 0.4678441670
136 0.0450644807
137 0.0442454540
138 0.0225734648
139 0.0572018432
140 0.3775931476
141 0.1756467386
142 0.5910173722
143 0.2487590953
144 0.5399395479
145 0.4375573618
146 0.5434396656
147 0.2777570353
148 0.0954415926
149 0.5895790516
150 0.7532835158
151 0.7409923445
152 0.3874323759
153 0.7068056769
154 0.4388231959
155 0.6017357784
156 0.6836004066
157 0.3379717262
158 0.0576234304
159 0.5698176986
160 0.8185640755
161 0.0161257319
162 0.7400877357
163 0.1629811919
164 0.8442518052
165 0.8790356477
166 0.1455241353
167 0.0212460549
168 0.5709289056
169 0.2756997356
170 0.3216913680
171 0.4109107256
172 0.4239074723
173 0.8947256880
174 0.6279374373
175 0.0492445158
176 0.5244147643
177 0.8903637852
178 0.5329815976
179 0.2614644975
180 0.0464437408
181 0.1475680987
182 0.6330235172
183 0.4313738863
184 0.0131312872
185 0.0012763769
186 0.1346489694
187 0.7913922584
188 0.6329005882
189 0.1128749014
190 0.6406299311
191 0.7411456027
192 0.5748610061
193 0.4956940757
194 0.9406025838
195 0.1873415110
196 0.5467771620
197 0.7697985205
198 0.3141526799
199 0.1131112031
200 0.9948836700
201 0.4065185343
202 0.0700389612
203 0.2897287905
204 0.1343826444
205 0.9527507473
206 0.1311520997
207 0.5369968419
208 0.6240354816
209 0.4421895547
210 0.5144631653
211 0.3333937302
212 0.9686309730
213 0.7883659552
214 0.2349665081
215 0.0932864067
216 0.9117737210
217 0.0074819576
218 0.0562582531
219 0.9361436698
220 0.7902676030
221 0.0197430677
222 0.7253545783
223 0.9398914508
224 0.7780485167
225 0.3131493942
226 0.2607827617
227 0.7834957165
228 0.9889911262
229 0.0708101010
230 0.8945257687
231 0.4399032334
232 0.3472099451
233 0.7124968232
234 0.2745891968
235 0.2977427610
236 0.8677616564
237 0.4313158928
238 0.1980452454
239 0.3309276963
240 0.8353899301
241 0.9988238192
242 0.7229023059
243 0.8564665755
244 0.2378845487
245 0.0779942553
246 0.3223031866
247 0.1438560076
248 0.4738736090
249 0.2823747715
250 0.7184123364
251 0.4655465728
252 0.8069709465
253 0.8536875113
254 0.3115624425
255 0.0922639032
256 0.8474401017
257 0.6743899765
258 0.3327869072
259 0.0475908769
260 0.6341012786
261 0.0084765318
262 0.5411775429
263 0.1315515998
264 0.2705805041
265 0.6369405244
266 0.8222520719
267 0.8516331434
268 0.1704583552
269 0.2826547651
270 0.5402343015
271 0.5925920941
272 0.9442992129
273 0.1855964502
274 0.9848259985
275 0.2767289625
276 0.8969082567
277 0.3921832209
278 0.3187520048
279 0.6980387243
280 0.5098909298
281 0.1561653528
282 0.0312623428
283 0.5920154063
284 0.8074301272
285 0.2724702973
286 0.4413064595
287 0.0490828240
288 0.5578683515
289 0.5219146211
290 0.1915161191
291 0.9396332572
292 0.1722530490
293 0.9470815191
294 0.3026815455
295 0.3622461043
296 0.5048289602
297 0.4955212490
298 0.3070028962
299 0.9302375889
300 0.8344497439
301 0.7179822228
302 0.2627234326
303 0.1172985658
304 0.8676624289
305 0.4216493783
306 0.7504679216
307 0.0845758480
308 0.6113933632
309 0.0431076509
310 0.6137779979
311 0.3096307833
312 0.9892471670
313 0.6190225061
314 0.9749798847
315 0.5047357685
316 0.4935692793
317 0.6290260588
318 0.2956384218
319 0.2720489271
320 0.7041394364
321 0.9723392853
322 0.8557358207
323 0.8296378923
324 0.3578976896
325 0.8631689989
326 0.2902295210
327 0.7869407511
328 0.7891384405
329 0.0390795236
330 0.3079195414
331 0.4902183877
332 0.8261392799
333 0.5663500535
334 0.0672813635
335 0.3989724203
336 0.2678426616
337 0.2493886943
338 0.1520716560
339 0.7958269487
340 0.2957895182
341 0.9545353455
342 0.8047289045
343 0.6218772738
344 0.3052414251
345 0.5528933874
346 0.7683817036
347 0.3369517706
348 0.3010441661
349 0.0568233179
350 0.5656903049
351 0.4929248928
352 0.6449574635
353 0.0555398541
354 0.3061512732
355 0.3466952965
356 0.3707083683
357 0.5715350555
358 0.5531960363
359 0.0910871825
360 0.2944495606
361 0.5138349843
362 0.6496224147
363 0.7580147830
364 0.8389201595
365 0.4388607019
366 0.7451172939
367 0.3858785764
368 0.6155504296
369 0.4496805223
370 0.1976552242
371 0.0982087287
372 0.9238576062
373 0.9784874786
374 0.3643327230
375 0.9333076724
376 0.6408922314
377 0.0826982087
378 0.1016781512
379 0.0092935753
380 0.4743782501
381 0.2110007387
382 0.3173407237
383 0.2081351804
384 0.0942711893
385 0.0516475344
386 0.6592223949
387 0.2621821084
388 0.7776496732
389 0.1913026001
390 0.8276979949
391 0.4119376172
392 0.1850870117
393 0.3686671034
394 0.1380947405
395 0.8357657867
396 0.9718208590
397 0.2558770229
398 0.3317683702
399 0.5399253722
400 0.5253702430
401 0.1446428841
402 0.0213594036
403 0.9185380219
404 0.8715843316
405 0.9476454936
406 0.5883761628
407 0.3276195300
408 0.5431865822
409 0.6278604807
410 0.8861640575
411 0.4886485341
412 0.3654147529
413 0.0075349498
414 0.9539385187
415 0.4236643298
416 0.4873639785
417 0.8001789700
418 0.9670820355
419 0.5666810237
420 0.9838232100
421 0.5206092198
422 0.9059801616
423 0.2177207628
424 0.0364688612
425 0.1797034307
426 0.7005534596
427 0.4584409562
428 0.3475567591
429 0.1415097942
430 0.1073510551
431 0.0459690886
432 0.0741392183
433 0.7848176265
434 0.3764135372
435 0.7490323186
436 0.0420731672
437 0.7117010860
438 0.3744592511
439 0.1741489870
440 0.5629358198
441 0.8562263995
442 0.6970591294
443 0.3640823516
444 0.8641834052
445 0.0886756892
446 0.2751704259
447 0.9866691621
448 0.3651785492
449 0.0969412792
450 0.8419340274
451 0.0429870551
452 0.4817660975
453 0.0059752248
454 0.6394466348
455 0.9320866037
456 0.0129272097
457 0.4505645966
458 0.8153190229
459 0.9591208987
460 0.2185362489
461 0.1472316140
462 0.8280016368
463 0.7039041950
464 0.4061378189
465 0.7730082409
466 0.8292353714
467 0.9420485285
468 0.1870081902
469 0.7862193878
470 0.2787873815
471 0.2374999714
472 0.4839701839
473 0.7386460253
474 0.7970135952
475 0.2245047202
476 0.0558278440
477 0.4774754057
478 0.9231287076
479 0.7980522478
480 0.7675812833
481 0.4060059667
482 0.2130700161
483 0.0277674182
484 0.5544047724
485 0.3728032096
486 0.4531767240
487 0.1140805727
488 0.4088595526
489 0.7431255896
490 0.1938041695
491 0.9381317410
492 0.0637960935
493 0.4364793492
494 0.0332827519
495 0.8279570467
496 0.9090758625
497 0.4279185343
498 0.1806394449
499 0.9779490554
500 0.2664125995
501 0.2058979864
502 0.8439661989
503 0.8753384682
504 0.8940005603
505 0.4796928323
506 0.7203943476
507 0.9335281688
508 0.4849739561
509 0.1219003717
510 0.2847154278
511 0.8272163789
512 0.2950252609
513 0.1826584432
514 0.3525267125
515 0.7903231075
516 0.6725351505
517 0.3405998701
518 0.6869299482
519 0.3206840167
520 0.8618656285
521 0.4420733037
522 0.3217215117
523 0.5578417152
524 0.5031923319
525 0.4615082522
526 0.9693154676
527 0.7692111717
528 0.5162816278
529 0.6832933656
530 0.0815997333
531 0.9087666473
532 0.9591014495
533 0.0360004473
534 0.8256485094
535 0.2900643605
536 0.6230129865
537 0.4778477352
538 0.9704182190
539 0.3705067316
540 0.9567312994
541 0.0251108343
542 0.5721361365
543 0.3741185023
544 0.2945357095
545 0.8130957668
546 0.4481631289
547 0.8366254452
548 0.9609894108
549 0.8245791355
550 0.0649121895
551 0.9364674592
552 0.6569167417
553 0.4896659686
554 0.3579268751
555 0.5358097169
556 0.6223085076
557 0.5374843634
558 0.4778186842
559 0.4499458843
560 0.0566363763
561 0.0213201006
562 0.8717487596
563 0.0895245818
564 0.8179605089
565 0.8344457920
566 0.8749564600
567 0.1307501288
568 0.5608091669
569 0.3909150346
570 0.3681160768
571 0.1627886437
572 0.5536387670
573 0.5368309421
574 0.6754840720
575 0.1363347352
576 0.0186945440
577 0.7673370531
578 0.2743331508
579 0.7936085600
580 0.7381169682
581 0.9409979100
582 0.9759853613
583 0.5950884258
584 0.8494581943
585 0.0232735733
586 0.2156254530
587 0.4564626103
588 0.9035528749
589 0.5791971760
590 0.1111021687
591 0.8492041528
592 0.3360738445
593 0.7198114197
594 0.1538461228
595 0.0924450785
596 0.7157741708
597 0.8830606444
598 0.1289982928
599 0.9475354853
600 0.2520179865
601 0.2504612061
602 0.6354350424
603 0.7023301544
604 0.6100567835
605 0.1596349976
606 0.2964432482
607 0.8349757313
608 0.9991602465
609 0.4178710743
610 0.4896978226
611 0.0667237951
612 0.3288988972
613 0.0039971359
614 0.4459252856
615 0.9048769069
616 0.7440516434
617 0.0334242911
618 0.0942290004
619 0.2194448563
620 0.8554191522
621 0.8724308799
622 0.0954507659
623 0.3105320085
624 0.8851241169
625 0.8131785356
626 0.3343868724
627 0.7628926917
628 0.8774112547
629 0.7176255325
630 0.1898530028
631 0.1788042528
632 0.4666378086
633 0.4887561002
634 0.8185008941
635 0.6460022843
636 0.0534296890
637 0.4498128847
638 0.5763213972
639 0.0487151814
640 0.2291730929
641 0.9891927857
642 0.3981206836
643 0.5483566120
644 0.0489506924
645 0.0451917336
646 0.1242406142
647 0.3995156863
648 0.7997206473
649 0.3512779451
650 0.6285241444
651 0.2032332977
652 0.0845275368
653 0.2422923243
654 0.7604801974
655 0.2267834793
656 0.6897929278
657 0.4449285313
658 0.2590974031
659 0.8705794010
660 0.8212391797
661 0.0696802288
662 0.7266834176
663 0.2066611259
664 0.1974739684
665 0.7104891865
666 0.6904740697
667 0.6573168237
668 0.0285589085
669 0.3204822734
670 0.2047666921
671 0.4014522349
672 0.0332142888
673 0.8956031606
674 0.1836874369
675 0.3310428062
676 0.3167737497
677 0.0675838972
678 0.3311658967
679 0.3156364921
680 0.4458417159
681 0.3304156945
682 0.5914308471
683 0.9008673069
684 0.1683682054
685 0.4265354171
686 0.7491676856
687 0.0432181989
688 0.5703400483
689 0.2874544652
690 0.3198967236
691 0.0414607776
692 0.9136631968
693 0.4678574982
694 0.9629573657
695 0.6236788763
696 0.1536582711
697 0.4873248709
698 0.1944335324
699 0.7099323864
700 0.4233159230
701 0.9516338727
702 0.8829679030
703 0.7524368940
704 0.0844653924
705 0.3169644829
706 0.5320565216
707 0.7872669988
708 0.6398045677
709 0.0136619266
710 0.8303085899
711 0.2928196832
712 0.3757708892
713 0.3903168114
714 0.5491588768
715 0.7117996672
716 0.6233499935
717 0.7983560031
718 0.6912225990
719 0.5561182851
720 0.1887180223
721 0.7666367525
722 0.9908146041
723 0.8781658714
724 0.8208937596
725 0.5719885279
726 0.7718865741
727 0.5100156246
728 0.3708490934
729 0.5133962503
730 0.3462732616
731 0.0645413804
732 0.4719649749
733 0.1389508815
734 0.7582277276
735 0.3711068346
736 0.8072847675
737 0.4684792901
738 0.1544157513
739 0.0887527100
740 0.5168090370
741 0.3145076889
742 0.8683827482
743 0.9831337337
744 0.9677211537
745 0.7223926096
746 0.1262552817
747 0.7038037588
748 0.5357868487
749 0.7162344414
750 0.5591250120
751 0.0156246487
752 0.7623422723
753 0.2956805262
754 0.3578438587
755 0.4654905121
756 0.6692100086
757 0.7549335794
758 0.4038357169
759 0.9190711933
760 0.9939260310
761 0.9825271571
762 0.3863099099
763 0.0563115040
764 0.2224658129
765 0.7114184275
766 0.6759798357
767 0.4789257511
768 0.9005629537
769 0.9993464113
770 0.2666670692
771 0.0034603958
772 0.6191473731
773 0.7101388092
774 0.2898918968
775 0.3494584428
776 0.0587183153
777 0.9764067268
778 0.7024593700
779 0.3788258771
780 0.8877179166
781 0.5404919123
782 0.2405228934
783 0.6271070184
784 0.8859973932
785 0.5500599088
786 0.0393053237
787 0.4197818516
788 0.6904334174
789 0.4431113675
790 0.9831671484
791 0.1617054155
792 0.0777042669
793 0.9357410416
794 0.2476292138
795 0.6974725157
796 0.7342098469
797 0.4295117292
798 0.8693518711
799 0.6010170346
800 0.8288101712
801 0.9688200434
802 0.3777873293
803 0.0648177476
804 0.3338714039
805 0.9099370746
806 0.2429538271
807 0.9767195366
808 0.9196899291
809 0.3521866906
810 0.2964174799
811 0.9778834367
812 0.4843403304
813 0.6946219879
814 0.2552282503
815 0.3251233608
816 0.5434228259
817 0.0171121724
818 0.0133071716
819 0.2794197728
820 0.2251762128
821 0.1163474701
822 0.3833917347
823 0.2312618250
824 0.2283003980
825 0.5723108072
826 0.1482174147
827 0.3447301751
828 0.2785170011
829 0.9107813314
830 0.5812587780
831 0.6438988969
832 0.9841040918
833 0.5247802383
834 0.2210200028
835 0.8090038472
836 0.2067870377
837 0.3253095001
838 0.0396977302
839 0.6303946511
840 0.0292507424
841 0.4070713357
842 0.0515598536
843 0.0303108932
844 0.5511123429
845 0.1345180608
846 0.5075264634
847 0.2648129482
848 0.8280266961
849 0.6138304621
850 0.5598058954
851 0.7991803598
852 0.5006577137
853 0.1632848566
854 0.9333637366
855 0.0549082316
856 0.4157633423
857 0.1124140553
858 0.3285515693
859 0.7461266092
860 0.3614100185
861 0.3663540885
862 0.4375504784
863 0.4844602104
864 0.9727155811
865 0.2299373890
866 0.5872628044
867 0.8646770197
868 0.4237545342
869 0.4837187950
870 0.9384470729
871 0.7285071593
872 0.1109414853
873 0.3422435089
874 0.6904054771
875 0.6030009147
876 0.0085704743
877 0.6349028507
878 0.9310646793
879 0.5258507873
880 0.6103508072
881 0.2401604026
882 0.3006961134
883 0.6984622406
884 0.3078753275
885 0.2007548532
886 0.6003928636
887 0.0824150282
888 0.9432075846
889 0.4656638457
890 0.5788328934
891 0.1107612825
892 0.7562355024
893 0.3735184246
894 0.8925910057
895 0.0225693586
896 0.0167195159
897 0.9653968117
898 0.8676494793
899 0.4719143014
900 0.7822820635
901 0.3484406562
902 0.4097531794
903 0.4403530539
904 0.2173598420
905 0.5993583428
906 0.3026280929
907 0.4181934749
908 0.9310367082
909 0.1099210070
910 0.9334625301
911 0.3924892787
912 0.3074208000
913 0.0500739862
914 0.1832373401
915 0.3760367003
916 0.9840205545
917 0.0071173497
918 0.5911827276
919 0.6048450724
920 0.4082555058
921 0.6863152450
922 0.6892368777
923 0.7381885713
924 0.3206626158
925 0.7111169638
926 0.9572377731
927 0.0020731329
928 0.3366380786
929 0.0545559640
930 0.0975022465
931 0.7066168743
932 0.9847547151
933 0.5980233885
934 0.6992221610
935 0.9894166484
936 0.0268847113
937 0.5305307165
938 0.2623521709
939 0.0480555317
940 0.1311219514
941 0.1199137382
942 0.8160068414
943 0.7908763867
944 0.4045032638
945 0.8567988295
946 0.8230779129
947 0.5794570369
948 0.2066906705
949 0.0681866247
950 0.4091017998
951 0.5622208922
952 0.7890299684
953 0.7337831033
954 0.3563053086
955 0.4785253683
956 0.3754812398
957 0.4584744822
958 0.6140172768
959 0.4084368220
960 0.2053469385
961 0.1233099625
962 0.4228741187
963 0.1927089361
964 0.9028868584
965 0.5971696174
966 0.9140805635
967 0.3152810936
968 0.4224151135
969 0.8498049655
970 0.5307344827
971 0.6822636717
972 0.2125899573
973 0.4742278163
974 0.5208497653
975 0.9472192714
976 0.5695140725
977 0.8422010376
978 0.0142064029
979 0.2912742880
980 0.0906376629
981 0.2189252288
982 0.3495960690
983 0.5249084244
984 0.6909320001
985 0.8629586636
986 0.9622162110
987 0.5430584787
988 0.2680571673
989 0.5900416081
990 0.3577267060
991 0.0438476049
992 0.6853607129
993 0.0802923737
994 0.4209953386
995 0.9682450874
996 0.7760396542
997 0.5925334070
998 0.7563799748
999 0.4588214871
1000 0.3139134280

so after geting this data we are count the number of values between 0 and 0.1, between 0.1 and 0.2, and so on

so we get

1 98
2 107
3 103
4 91
5 96
6 108
7 104
8 110
9 86
10 97

as the intervals are not same

in part b ) the all interval are same i.e 100 samples contains

this samplig process is worth while as we want to get 100 sample from each interval and we get it 100 whereas in part a we dont have exact sample 100 from each interval.

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