Question

The file medinc.mtw contains data on the median incomes (medinc) of census dissemination areas in Toronto.
(a) Treating this set of data as the population, use Minitab to calculate the population mean and the population standard deviation for the medinc variable. Set aside all population information until parts (d) and (e).

(b) Use Minitab (Calc Menu – Random Data – Sample from Columns) to draw twenty samples of size n = 40 from the Toronto medinc population. This procedure must be replicated twenty times (note that if you open up the same sampling dialog box each time from the menu, then you only have to replace the last destination column with the next one. In Minitab Express, you must specify the sample size each time but you do not need to specify the destination column). For each sample, use Minitab to calculate the sample mean. Note that when you can use the Descriptive Statistics function, you can specify the calculation of only the sample means of all your samples at once.

(c) Copy the sample means into a new column and calculate the mean and standard deviation of your sample means.

(d) The sampling distribution of the sample mean has an average equal to the population mean. Explain why the mean in part (c) may be different from the population mean in part (a).

(e) How would you use the sample standard deviation in part (c) to estimate the population standard deviation in part (a)? How close is your estimate to the true value?

medinc
18891
24869
22773
25482
17282
11836
25110
25995
51411
15146
27686
21540
22899
22582
23676
32112
26216
18741
8332
22090
14644
17478
22083
26601
14854
33627
21575
26818
31424
18678
44126
20810
25622
23244
24554
16750
26526
22691
16603
19595
21458
35239
39952
17450
25387
17014
23444
34851
14317
24367
21778
17225
28573
26054
19798
22645
31970
19513
32413
36183
18047
25271
18537
22591
32495
15060
18184
27693
16556
29242
38764
20496
27339
20639
32242
18293
20252
33210
17190
24152
28859
19333
27176
25770
34968
22854
51951
21080
38063
45627
14837
17525
30153
18934
19343
14081
38193
15669
28529
15435
31025
17429
14696
47892
16408
22608
51036
46652
16307
30266
29869
19238
34197
22468
28160
41114
21094
27640
24748
21044
52260
18245
23819
24265
24544
30152
13563
20013
16475
33190
19185
24186
18183
24709
28220
35050
20970
39776
18262
23419
34076
27468
22592
33561
22640
29932
11792
25104
20321
18924
13057
19951
20032
22716
26119
25841
23155
20931
14694
31407
21169
26829
22796
23938
33519
25216
19367
22117
33532
57993
34730
35951
26754
19562
22320
25566
23399
16901
15833
25749
73435
30527
36008
23429
19307
17447
10931
45915
28973
19128
22765
21991
20314
20815
23632
22430
32459
30546
48076
14046
22777
42572
19423
18177
16763
24386
22519
20621
20063
21961
18274
20256
27484
23375
26560
25772
21981
14441
22300
25899
17384
17072
26842
20715
40010
15025
28960
28752
22004
36297
29474
15545
36459
21610
32447
18654
37467
38553
22811
55977
18430
20504
24949
17476
15400
30009
19143
25784
26948
13865
18742
22383
24690
22387
30052
31049
29330
30601
31256
24586
23623
16297
31413
23080
29322
38585
31228
26110
45892
30523
10085
30191
18849
25940
28875
32676
27521
14599
37905
26271
32356
20961
33854
23438
21067
13873
20675
17296
34883
36889
49841
17159
17292
14880
24073
28967
42015
15470
28941
28735
20575
16946
15658
17400
22477
16580
20579
24797
19304
18353
19969
24130
40251
16577
26938
49666
15524
23229
15389
23235
15592
26758
20762
18525
17456
14090
20039
31535
24762
20023
18520
16044
19923
21366
18174
24340
35679
23167
23819
55973
16081
13834
22718
20177
19008
36130
28704
23652
31163
31606
21263
21429
21391
33804
20811
14898
39252
30761
25231
17412
21234
18096
31029
28505
20165
28074
24508
18393
33989
28366
12796
23572
22529
23444
21575
28639
24717
50396
19383
13372
11275
22570
22544
32749
26719
24958
23681
24427
28741
14851
13886
18015
18694
17155
30356
38927
26894
15738
53770
23230
24339
38620
15707
20972
21521
24572
31684
21323
21427
16638
15603
39975
16181
19740
28677
34623
20560
16068
17404
23339
40763
26389
40801
11839
15913
34734
19549
28092
21434
16152
24282
29816
23315
19357
31553
23981
30424
37708
16218
19969
26756
18960
41441
14838
22370
38118
19031
23275
38960
19218
15921
23253
23188
17394
18771
47166
27701
24168
14036
19582
21101
35141
33114
29246
24025
15040
23950
47688
33255
18005
37046
31198
19889
28269
31676
18923
35795
20715
35517
14058
19173
39173
19159
16986
33336
56892
23621
24682
25566
27833
47018
12579
18218
44499
20780
34628
20066
18775
19337
24953
28824
27931
20699
45343
35866
37194
20190
14335
35867
25997
42012
25501
17807
40209
21276
16023
30347
14397
25976
32513
15761
23188
36366
36062
15432
30927
20999
17617
34028
43828
26326
21143
27981
15006
18227
26280
31140
18333
41419
23413
32199
31088
13490
17294
25560
35705
55712
25559
35430
22027
45669
21615
27648
21562
25722
30102
23411
45871
18195
15019
25588
24527
28794
23606
15454
15616
16846
22430
37577
24320
26900
21714
23221
18106
17858
11040
27351
33524
14367
13419
25029
43744
39506
52737
31363
20981
11569
17418
48469
33967
18887
27079
27583
23012
28011
18992
24934
18659
14808
33838
19244
12519
58310
9647
14327
26336
20019
34335
18399
20022
24724
24650
34598
33696
34078
38755
17130
22226
54420
11508
18735
18612
36748
19220
22286
20120
60798
29776
14334
31358
26233
16602
26324
23435
20589
21418
19383
28330
16264
51397
22597
38878
45115
16695
39940
21173
33390
26134
25016
22398
22379
18085
18343
38934
15341
36266
13780
18829
28953
27605
52768
21351
48469
28017
27489
21768
24040
28257
23606
18999
69929
19541
37006
21260
22017
11794
36682
30062
10612
22463
23092
20017
24237
15759
13910
29917
50012
29872
24161
53612
30856
18491
17202
35360
37598
25414
16793
29637
18857
19976
25227
21407
36592
25779
13831
22703
15409
52188
17629
17358
26821
21098
46517
25579
19025
13910
17466
28729
16961
23584
20032
23415
38084
19623
22738
32654
19458
30048
16678
15141
15831
18588
26971
27635
25867
17345
25003
41735
37109
26102
17261
18515
19604
16348
26957
16826
22829
14275
18137
58328
17094
32708
17324
22819
34862
32215
22282
18360
15515
20494
30337
23146
28553
19689
21533
29220
25565
52354
14305
23446
25947
22932
19483
24231
21564
16506
23735
25053
27728
40692
15439
32504
29987
24482
19642
19161
17728
15533
35642
26499
12899
17554
23742
21359
19632
20574
38519
29156
24443
35566
17441
35199
20598
36643
64658
33577
42115
23789
24757
35415
21092
34027
22419
21902
26983
18047
20491
37659
25357
20425
21382
23665
27344
27930
30620
34305
13854
37307
23239
35158
62829
39235
16954
23846
28228
11875
24622
16986
26229
23765
13106
20936
29472
40052
16440
20704
34805
16887
21714

Answer 1

(a)

Descriptive Statistics: medinc

Variable Mean StDev
medinc 25614 9529

(b)

Descriptive Statistics: Sample 1, Sample 2, Sample 3, Sample 4, Sample 5, ...

Total
Variable Count Mean
Sample 1 40 26819
Sample 2 40 25201
Sample 3 40 24509
Sample 4 40 24152
Sample 5 40 30185
Sample 6 40 26429
Sample 7 40 24607
Sample 8 40 24844
Sample 9 40 25124
Sample 10 40 23946
Sample 11 40 25935
Sample 12 40 25982
Sample 13 40 26271
Sample 14 40 26684
Sample 15 40 24952
Sample 16 40 27075
Sample 17 40 25182
Sample 18 40 23692
Sample 19 40 27289
Sample 20 40 26466

(c)

Descriptive Statistics: Sample means

Total
Variable Count Mean StDev
Sample means 20 25767 1496

(d) mean in part (c)=25767 and the population mean=25614. This difference is occurred since we do not consider all possible combinations of samples of size 40. If we consider all possible combinations i.e. $\binom{849}{40}$ (sampling without replacement)

or 849⁴⁰ (sampling with replacement) then the mean of all possible samples are equal to population mean.

(e)

$Since~Var(\bar{x})=\frac{\sigma^2}{n}~or~\sigma=population~sd=\sqrt{nVar(\bar{x})}~so\\ \widehat{population~sd}=\sqrt{40}* 1496=9461.535$

which is very closed to population sd.