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SIN502S - STATISTICAL INFERENCE 1 - 2ND OPP - JANUARY 2024 |
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nAmlBIA UnlVERSITY
OF SCIEnCEAnDTECHnOLOGY
FacultyofHealthN, atural
ResourceasndApplied
Sciences
Schoool f NaturalandApplied
Sciences
Departmentof Mathematic!">,
Statistics and Actuarial Science
13JacksonKaujeuaStreet
PrivateBag13388
Windhoek
NAMIBIA
T: •264 612072913
E: msas@nust.na
W: www.nust.na
QUALIFICATION: BACHELOR OF SCIENCES IN APPLIED MATHEMATICS AND STATISTICS
QUALIFICATION CODE: 07BAMS
LEVEL:5
COURSE:STATISTICAL INFERENCE 1
DATE: JANUARY 2024
DURATION: 3 HOURS
COURSECODE: SIN502S
SESSION: 2
MARKS: 100
SUPPLEMENTARY/SECONDOPPORTUNITY: EXAMINATION QUESTION PAPER
EXAMINERS: MR E. MWAHI, DR D. NTIRAMPEBA
MODERATOR: DRJ. ONG'ALA
INSTRUCTIONS:
1. Answer all questions on the separate answer sheet.
2. Please write neatly and legibly.
3. Do not use the left side margin of the exam paper. This must be allowed for the examiner.
4. No books, notes and other additional aids are allowed.
5. Mark all answers clearly with their respective question numbers.
PERMISSIBLE MATERIALS:
1. Non-Programmable Calculator
ATTACHEMENTS:
1. T- Table
2. Normal distribution table
3. Chi-square table
4. F-table
5. Manny Whitney U table
This paper consists of 6 pages including this front page.
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QUESTION 1 [20 MARKS]
1.1 What is the standard deviation of a sampling distribution called?
[2]
A. Sampling error
B. Sample error
C. Standard error
D. Simple error
1.2 What does it mean when you calculate a 95% confidence interval?
[2]
A. The process you used will capture the true statistic 95% of the time in the long run
B. You can be "95% confident" that your interval will include the population parameter
C. You can be "5% confident" that your interval will include the population parameter
D. All of the above statements are true
1.3 As a general rule, researchers tend to use __ percent confidence intervals.
[2]
A. 99%
B. 95%
C. 50%
D. none of the above
1.4 Which of the following is the researcher usually interested in supporting when he or
she is engaging in hypothesis testing?
[2]
A. The alternative hypothesis
B. The null hypothesis
C. Both the alternative and null hypothesis
D. Neither the alternative or null hypothesis
1.5 ----
are the values that mark the boundaries of the confidence interval. [2]
A. Confidence intervals
B. Confidence limits
C. Levels of confidence
D. Margin of error
1.6 What is the key question in the field of statistical estimation?
[2]
A. Based on my random sample, what is my estimate of the population parameter?
B. Based on my random sample, what is my estimate of normal distribution?
C. Is the value of my sample statistic unlikely enough for me to reject the null
hypothesis?
D. There is no key question in statistical estimation
1.7 As sample size goes up, what tends to happen to 95% confidence intervals?
[2]
A. They become more precise
B. They become narrower
C.They become wider
D. Both a and b
STATISTICALINFERENCE1 (SIN502S)
2nd Opportunity January 2024
2
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1.8 When the researcher fails to rejects a false null hypothesis, a __ error occurs. [2]
A. Type I
B. Type A
C. Type II
D. Type B
1.9 A student calculates a 90% confidence interval for population mean when population
standard deviation a is unknown and n = 9. The confidence interval is -24.3 cents to
64.3 cents, the sample mean is:
[2]
A.40
B. -24.3
C. 64.3
D.20
1.10 A 95% confidence interval for population proportion is 32.4% to 47.6%, the value of
sample proportion is:
[2]
A.40%
B. 32.4%
C. 47.6%
D. 80%
STATISTICAL INFERENCE 1 (SIN502S)
2nd Opportunity January 2024
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QUESTION 2 [30 MARKS]
2.1 A random sample of employees of a large company was asked the question, "Do you
participate in the company's stock purchase plan?" The answers are shown below.
yes no no yes no no yes yes no no
no yes yes yes no yes no no yes yes
no yes yes no yes yes no yes yes yes
yes no no yes yes yes yes yes no yes
no yes yes no yes yes yes yes yes yes
Use a 90% confidence interval to estimate the proportion of employees who do not
participate in the company's stock purchase plan.
[SJ
2.2 In a packing plant, a machine packs cartons with jars. It is supposed that a new
machine will pack faster on the average than the machine currently used. To test that
hypothesis, the times it takes each machine to pack ten cartons are recorded. The
results in seconds are shown in the following table. The two populations are assumed
to be normally distributed with equal variances.
New 42.1 41.3 42.4 43.2 41.8 41.0 41.8 42.8 42.3 42.7
Old 42.7 43.8 42.5 43.1 44.0 43.6 43.3 43.5 41.7 44.1
2.2.1 Find the pooled variance of the two samples
[6]
2.2.2 Do the data provide sufficient evidence to conclude that, on the average, the
new machine packs faster? Use the 5% level of significance.
[8]
2.2.3 Construct a 99% confidence interval for the difference between the mean time
it takes the new machine to pack ten cartons and the mean time it takes the
present machine to pack ten cartons.
[6]
STATISTICAL INFERENCE1 (SIN502S)
2nd Opportunity January 2024
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2.3 A manufacturer of automobile batteries claims that the distribution of the lengths of
battery life is normally distributed with a mean of 100 hours and a standard deviation
of 12 hours. A sample of 50 batteries from this manufacturer was randomly selected
and it was discovered to have a mean battery life of 95.5 hours. What is the probability
that the sample average life of a battery is between 95 and 100 hours?
[SJ
QUESTION 3 (40 MARKS]
3.1 Ors. Leon Festinger and James M. Carlsmith performed an experiment in cognitive
dissonance ("Cognitive Consequences of Forced Compliance"). In it, they had three
groups: (1) those in Control condition; (2) those in One Dollar condition and (3) those
in Twenty Dollar condition. Copy and complete the partially completed ANOVA table
consistent with the results of their experiment below.
[7]
Source of
Variation
Between
Within
Total
Sum of
square (SS)
...
...
...
Degrees of
freedom (df)
...
57
...
Mean square
{MS)
...
2
...
F-statistic
9
3.2 In preparing a national promotional campaign to raise funds for Orphans and
Vulnerable Children (OVC),the organising charity examined previous records of
donation to establish if age of donor is a factor in the monetary size of the donation
received from a donor. Their records we arranged in the following contingency table:
Size of donation
Age group
A(20-34)
8(35-49)
C(Above 49)
Above N$100
25
40
47
N$50-N$100
69
51
74
Under N$50
36
29
19
Can it be concluded that the age of donor influences the size of the donation to this
charity? Test at 1 % level of significance.
(12]
STATISTICAL INFERENCE 1 (SIN502S)
2nd Opportunity January 2024
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3.3 The following table displays the summated ratings accumulated over a 7-point scale
rated on two brands of Colombian coffee.
A
24
27
19
24
22
26
27
25
22
B
26
27
22
27
25
27
26
27
23
3.3.1 Mention three underlying assumptions of the Mann-Whitney U test.
[3]
3.3.2 Using the Mann-Whitney U test, test at 1% level of significance if there is
evidence of the difference in summated ratings of brand A and brand B. (10]
3.4 An instructor hypothesizes that the standard deviation of the final exam grades in
her statistics class is larger for the male students than it is for the female students.
The data from the final exam for the last semester are shown below. Assume that
exam marks are normally distributed.
Males
Females
n = 16
S = 4.2
n = 18
S = 2.3
Is there enough evidence to support her claim, using the 10% level of significance?[8]
=============END OF EXAMINATION===========
STATISTICAL INFERENCE 1 (SIN502S)
2nd Opportunity January 2024
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APPENDIX C: The Standard Normal Distribution
/~
/
.,,,,/,.,.
'......._____
-==-~------0·- z
z
0.00
0.0 I0.0000
0.1 0.0398
0.2 0.0793
0.3 0.1179
0.01
[0.0040
0.0438
0.0832
0.1217
I. 0.02
lo.0080
1
0.0478
0.0871
0.1255
0.03
I 0.0120
I0.0517
0.0910
0.1293
0.04
0.0160
0.0557
'
0.0948
0.1331
0.05
0.0199
0.0596
0.0987
0.1368
.I 0.06
! 0.0239
I0.0636
I 0.1026
·10.1406
0.07
0.0279
0.0675
0.1064
0.1443
0.08
0.0319
0.0714
0.1103
0.1480
I 0.09
I0.0359
[0.0753
10.1141
10.1517
0.4 0.1554 '0.1591 10.162s 0.1664 0.1700 0.1736 [0.1772 0.1808 0.1844 ,0.1879
0.5 0.1915
0.6 II 0.2257
0.7 I 0.2580
-; ·-
0.8 0.2881
0.9 0.3159
0.1950
0.2291
[0.2611
0.2910
-··
0.3186
[0.1985
lo.2324
[0.26:2.
'0.2939
0.3212
I 0.2019
I 0.2357
I 0.2673.
I0.2967
0.3238
0.2054
0.2389
0.2704
0.2995
0.3264
0.2088
0.2422
0.2734
0.3023
0.3289
10.2123
jo.2454
I0.2764
Io.3051
·10.3315
0.2157
0.2486
0.2794
0.3078
0.3340
0.2190
0.2517
10.2823
0.3106
0.3365
0.2224
I 0.2549
1
I0.2852
I o.3133
Io.3389
1.0 0.3413 0.3438 p.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 .[0.3621
1.1 0.3643
1.2 Io.3849
1.3 I0.4032
1.4 0.4192
1.5 0.4332
1.6 0.4452
1.7 0.4554
1.8 I o.4641
1.9 I 0.4713
2.0 Io.4772
2.1 0.4821
2.2 0.4861
2.3 0.4893
2.4 Io.4918
0.3665
I
j0.3869
j0.3686
·10.3888
[0.4049
0.4207
[o.4066
I
[0.4222
0.4345 0.4357
0.4463 . l 0.4474
0.4564
0.4649
,0.4573
I
[0.4656
10.4719 io.4726
lo.4778
0.4826
0.4864
0.4896
lo.4783
1'0-.-4-83-·0··-
0.4868
I
0.4898
0.4920 lo.4922
0.3708
0.3907
I 0.4032
I 0.4236
I 0.4370
0.4484
0.4582
0.4664
Io.4732
Io.4788
j~.4834
0.4871
0.4901
0.4925
0.3729
0.3925
··--
0.4099
0.4251
0.4382
0.4495
0.4591
0.4671
0.4738
0.4793
0.4838
,0.4875
:o.4904
0.4927
10.3749
I
0.3944
0.4115
0.4265
0.4394
0.4505
0.4599
0.4678
0.4744
0.4798
0.4842
0.4878
0.4906
10.4929
lo.3770
I
lo.igs2
[0.4131
0.4279
Io.4406
I 0.4515
1j0.4608
io.4686
lo.4750
'0.4803
! 0.4846
Io.4881
0.4909
I
I
,0.4931
0.3790
0.3980
0.4147
0.4292
0.4418
0.4525
0.4616
0.4693
0.4756
0.4808
0.4850
0.4884
0.4911
0.4932
0.3810
0.3997
0.4162
0.4306
'0.4429
0.4535
0.4625
0.4699
0.4761
0.4812
10.4854
0.4887
0.4913
0.4934
10.3830
I 1I 0.4015
I 10.4177
Io.4319
1:o.4441
1:0.4545
lo.4633
I,0.4706
0.4767
Io.4817
j 0.:857
I o.4890
110.4916
'I10.4936
2.5 Io.49~.8 j0.4940
2.6 0.4953 - 0-.49·-5-5
2.7 0.4965 0.4966
[o.4941 0.4943 0.4945
10.4-956-- ] o.·4-9-5-7--· -0-.-4-9·5--9-
0.4967 0.4968 0.4969
0.4946
·•·
0.4960
•-·--
0.4970
0.4948
J ~4~61_
Io.4971
0.4949
-0-.-496·2···-·
1 o.4972
0.4951
10.4963
0.4973
•0. .4952
1 o.4964
I10.4974
2.8 0.4974
2.9 J o.4981
3.0 .l,o.-4-9·87
0.4975 0.4976 0.4977
0.4982 10:4982
I
••
0.4983
-10:498?~_:· [o.4987 .. J 0,4988
10.4977
0.4984
0.4988
'0.4978
10.4984
I
0.4989
l•o.4979 I o.4979
lo.4985 0.4985
-·
lo.4989 0.4989
0.4980
0.4986
0.4990
llo.4981
I
10.4986
0.4990
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APPENDIX D: The t-distribution
-~-t(e.dfJ.
df\\p
.l o.4o
0.25
0.10
0.05
0.025
0.01
0.005
1 I 0.324920 11.000000 ] 3.077684 I 6.313752 12.70620 131.82052 63.65674
2
I 0.288675 l0.816497 I11.885618 12,919986 4.30265
116.96456 I 9.92484
3
0.276671 ,0,764892 ] 1.637744 2.353363 3.18245 14,54070
5.84091
4 I 0.210122 lo.740697 ] 1.533206 12,131847 2.77645 113.74695 4.60409
1
5 I 0.26718,1 lo.726687 11.475884 12.015048 2.57058 ·13.36493 4.03214
6
0.264835 0.717558 1.439756 1.943180 2.44691
13.14267
13,70743
7
0.263167 10.711142 11.414924 1.894579 2.36462 12.99795
3.49948
8 I 0.261921 10.106387 [ 1.396815 11.859548 2.30600 12.89646
3.35539
9 I 0.260955 ___lo.102122 _J~:3830~9
1.833113
10
0.260185
11
0.259556
12 _I 0.259033
0.699812 - 11~.3·7-·2·1--•8•·4 -
0.697445 _I 1.363430
1
fo.695483 ··· I 1.356217
1.812461
1.795885
1.782288
12.26216 _J2.82!~4
3.24984
..I 2.22814 - ·12.76377 3.16921
2.20099 1·2.71808 3.10581
2.17881 l2.68100
3.05454
13 I 0.258591
14
0.258213
15 · ···l·o· .257885-
I 16 . 0.257599
lo.693829
I o.692417
-·10.691197
[o.690132
I 1.350171
I
11.345030
J 1.340606
Ijl.336757
1.770933
1.761310
1.753050
[ 1.745884
2.16037
1 2.14479
2.13145
12.11991
12.65031
i'2.62449
] 2.60248
l12.58349
3.01228
2.97684
I 2.94671
2.92078
17
0.257347 0.689195 ru33379 I 1.739607 2.10982 ;12.56693 12.89823
1
18 ' 0.257123 10.688364 I 1.330391 1.734064 2.10092 [ 2.55238
2.87844
19
I0.256923 10.687621 i 1.327728
1.729133
2.09302
[ 2.53948
2.86093
I 20
0.256743 !o.686954 IJi.325341 I 1.724718 2.08596 'J2.s2798
2.84534
I
21
0.256580 I 0.686352 I 1.323188 1.720743 2.07961 12,51765
2.83136
22
0.256432 [o.685805 I 1.321237 1.717144 2.07387 12.50832
2.81876
23 I 0.256297 lo.685306 111.319460 1.713872 2.06866
24 0.256173 0.684850 1.317836 1.710882 2.06390
112.49987
12.49216
2.80734
2.79694
25
l 26
I 27
28
0.256060 ..... [0,684430 11.316345
I0.255955 [o.684043 ... 111.314972
1.708141
1.705618
0.255858
'
-~-0.6-83. 685 ..
11.313.703
1.703288
J I 0.255768 . _ [o.68_3353_ ~.31~527 -~- 1.701131
2.05954
12.05553
2.05183
2.04841
,--- 29 -----1_0:-2ss'68~~--10:683044... \\il.311434
!~--3-0 -- 0.255605 --,o.6s2756-
I inf I 0.253347 'lo.674490
1i1]rn41s
111.281552
[ 1.699127 · -lf?~52i
l 1.697261 12.04227
I 1.644854 1.95996
J 2.43511
II2. .47863
i 2:41266
J 2.46?~-4
712~46202
[2.45726
112.32635
2.78744
2.77871
f 2.77068
! 2.76326
2.75639
2.75000
I
2.57583
0.0005
636.6192
31.5991
12.9240
8.6103
6.8688
5.9588
5.4079
15.0413
4.7809
4.5869
4.4370
4.3178
4.2208
4.1405
4.0728
14.0150
3.9651
3.9216
3.8834
3.8495
3.8193
3.7921
3.7676
3.7454
3.7251
3.7066
3.6896
t3,6739
3.6594
3.6460
3.2905
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APPENDIX E: The Chi-Square Distribution
df\\p .995
1 0.00004
2 0.01003
3 0.07172
4 ,0.20699
.990 I .975
.950
.900 I .750
.500
.250
.100
.050
.025
.010
.005
0.00016 lo.00098 0.00393 I 0.01579 l'.0.10153 0.45494 11.32330 2.70554 3.84146 5.02389 16.63490 17.87944
1
0.02010 1o.o5~~4- _c_~_:_H!._2~_5.2~1J07~ ],o.57536 11.38629 [2.7n~~ __4.60517 15.99146 J 7.37776 9.21034 110.59663
0.11483 I o.215so o.35185 __[o.58437 111.21253 12.36597 4.10834 l?-25139 17.81473 9.34840 11_.34487_li2.83816
I 0.29711 10.48442 0.71072 j 1.06362 [1.92256 I 3.35669 5.38527 17.77944 9.48773 11.14329 13.27670 114.86026
5 o.41174 :o.55430 lo.83121 1.14548 I 1.61031 I 2.67460 14.35146 16.62568 19.23636 11.07050 12.83250 I 15.08627 I 16.74960
I I I 6 10.67573 0.87209 11.23734 rl-.63538 12.20413 13.45460 15.34812 17:8408()' :10:64464 12.59159 114.44938 I 16.81189 I118.54758
I 7 lo.98926 1.23904 I 1.68987 i16735 [2:33311 14.25485 16.34581 19.03715 I12.01704 14.06714116.01276 18.47531 120.27774
s 1.34441 1.64650 12:17973 2.73264 3.48954 [5.07064 7.34412 10.21885 113.36157 15.50731 17.53455 20.09024 121.95495
1
9 1.73493 2.08790 2.70039 3.32511 14.16816 I 5.89883 8.34283 11.38875 114.68366 16.91898 19.02277 21.66599123.58935
10 2.15586
11 2.60322
2.55821
3.05348
3.24697
3.81575
3.94030 14.86518 I 6.73720 9.34182 l12.54886 15.98718 18.30704 20.48318 i 23.20925 I 25.18818
I 4.57481 1?·?7?7B j 17.5~41~ l1?-~4100 l13.70069 17.27501 19.67514 21.92005 I 24.72497 I126.75685
12 13.07382 3.57057 ~A03?9__j ~:_2_2_~0316:3§j~o_-]l8.43842 111.34032 114.84540 [~j.5_:1.93-l~~-L- 02607 '23.33666 26.21697 128.29952
13 3.56503 4.10692 ls.00875-, 5.89186 17.04150 119.29907 12.33976 1 15.98391 119.81193 22.36203 24.73560 n68825 i29.81947
14 4.07467 4.66043 i5.62873 6.57063 17.78953 I 10.16531 113.33927 117.11693 121.06414 23.68479 26.11895 29.14124131.31935
15 4.50092 5.22935 16.26214 17.26094 I18.54676 1111.03654 l14.33886 118.24509 22.30713 _i24.99579 27.48839 30.57791 32.so132
i 16 5_14221 5.81221 [ 6.907~6- 7.96165 19.31224 I[u.91222 115.33850 119.36886 !23.54183 }6.29623 I 2s.84535 31.99993. [34.26719
17 5.69722 6.40776 ]~7:56419 8.67176 110.08519 [12.79193 16.33818 20.48868 [24.76904 27.58711 30.19101 33.40866 [35_71347
1s 6.26480 17.01491 18.23075 9.39046 I 10.86494 r'13.67529 _17_33790 21.60489 125.98942 28.86930 31.52638 34.80531 37.15645
1
j I 19 6.84397 7.63273 _18.90652 ' 10.H701 _1L65091 14.56200 118.33765 l22.71781 27.20357 30.14353 32.85233136.19087 38.58226
I 20 7.43384 8.26040 j 9.59078 I 10.s5o_s1_'I12.~261 ]115.45177 I19.33743 [23.82769 ]128.41198 31.41043 I 34.16961 37.56623 l39.99685
21 8.03365 8.89720 10.23290 111.59131 113.23960'l16.34438 120.33723 1'24.93478 129.61509 32.67057 I 35.47888 38.93217 41.40106
22 8.64272 ·· 19.54249- ·li.a:9a23_2 12.~3801 ] 14.~4149 117.23962 21.33704 · 26.039_27 130.81328 33.92444 36.78071 40.28936 42.79565
23 9.26042 10.19572 11.68855 13.09051 114.84796 lrn,13730 122.33688 127.14134 32.00690 35.17246 38.07563 41.63840 I 44.18128
24 9.88623 10.85636112.40115 13.84843 115.65868I119.03725 123.33673 128.24115 133.19624 36.41503 I 39.36408 42.97982 45.55851
I 25 10.51965 11.52398 [13.119.?2 14.61 '.41 ~6.473~1 [19.93934 24.33659 29.33885_ 134.38159 37.65248 40.64647 44.31410 46.92789
26 11.160-24 (1_~~i81~Ji:8_~~9.?. 1~·~?~~6J 117291~8 J20_34343 I125.33646 130.434~71~5'.56317_ ~8.88514 41.92317 45.6416~_148.28988_
i 27 ln.80759
12.87850 I 14.57338] 16.15140 !118.11390]121:74940 [26:33634 131.528411136.74122
40.11327
43.19451
46.962941 49.64492
1
2s
12.46134
13.564711115.30786 116.92788 ilB.93924
1
'.122.65716r!27.33623 l 32.62049
137.91592
41.33714
144.46079
48.27824 .15o.99338
29 13.12115 114.25645 116.04707 17.70837119.76774 j _23.56659 j28.33613 133.71091 '39.08747 42.55697 45.72229 49.58788 J 52.33562
.I I?~- j 30 13:786~{ J4.9.s3~J~6. 79~77 j l.8:.~92_6~[J:o.59923 124.47761 l29.33603_ 7997j I 40.25602 43.77297 46.97924 I 50.89218 ~3.67196
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1.10 Page 10 |
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F Table for alpha::0.05
11.\\d.M
df2/df1
1
2
3
4
5
1
161.4476
18.5128
10.128
7.7086
6.6079
2
199.5
19
9.5521
6.9443
5.7861
3
215.7073
19.1643
9.2766
6.5914
5.4095
4
224.5832
19.2468
9.1172
6.3882
5.1922
5
230.1619
19.2964
9.0135
6.2561
5.0503
6
5.9874
5.1433
4.7571
4.5337
4.3874
7
5.5914
4.7374
4.3468
4.1203
3.9715
8
5.3177
4.459
4.0662
3.8379
3.6875
9
5.1174
4.2565
3.8625
3.6331
3.4817
10
4.9646
4.1028
3.7083
3.478
3.3258
11
4.8443
3.9823
3.5874
3.3567
3.2039
12
4.7472
3.8853
3.4903
3.2592
3.1059
n
4.6672
3.8056
3.4105
3.1791
3.0254
14
4.6001
3.7389
3.3439
3.1122
2.9582
15
4.5431
3.6823
3.2874
3.0556
2.9013
16
4.494
3.6337
3.2389
3.0069
2.8524
17
4.4513
3.5915
3.1968
2.9647
2.81
18
4.4139
3.5546
3.1599
2.9277
2.7729
19
4.3807
3.5219
3.1274
2.8951
2.7401
20
4.3512
3.4928
3.0984
2.8661
2.7109
21
4.3248
3.4668
3.0725
2.8401
2.6848
22
4.3009
3.4434
3.0491
2.8167
2.6613
23
4.2793
3.4221
3.028
2.7955
2.64
24
4.2597
3.4028
3.0088
2.7763
2.6207
25
4.2417
3.3852
2.9912
2.7587
2.603
26
4.2252
3.369
2.9752
2.7426
2.5868
27
4.21
3.3541
2.9604
2.7278
2.5719
28
4.196
3.3404
2.9467
2.7141
2.5581
29
4.183
3.3277
2.934
2.7014
2.5454
30
4.1709
3.3158
2.9223
2.6896
2.5336
40
4.0847
3.2317
2.8387
2.606
2.4495
60
4.0012
3.1504
2.7581
2.5252
2.3683
120
3.9201
3.0718
2.6802
2.4472
2.2899
inf
3.8415
2.9957
2.6049
2.3719
2.2141
6
233.986
19.3295
8.9406
6.1631
4.9503
7
236.7684
19.3532
8.8867
6.0942
4.8759
8
238.8827
19.371
8.8452
6.041
4.8183
9
240.5433
19.3848
8.8123
5.9988
4.7725
10
241.8817
19.3959
8.7855
5.9644
4.7351
4.2839
3.866
3.5806
3.3738
3.2172
4.2067
3.787
3.5005
3.2927
3.1355
4.1468
3.7257
3.4381
3.2296
3.0717
4.099
3.6767
3.3881
3.1789
3.0204
4.06
3.6365
3.3472
3.1373
2.9782
3.0946
2.9961
2.9153
2.8477
2.7905
3.0123
2.9134
2.8321
2.7642
2.7066
2.948
2.8486
2.7669
2.6987
2.6408
2.8962
2.7964
2. 7144
2.6458
2.5876
2.8536
2.7534
2.671
2.6022
2.5437
2.7413
2.6987
2.6613
2.6283
2.599
2.6572
2.6143
2.5767
2.5435
2.514
2.5911
2.548
2.5102
2.4768
2.4471
2.5377
2.4943
2.4563
2.4227
2.3928
2.4935
2.4499
2.4117
2.3779
2.3479
2.5727
2.5491
2.5277
2.5082
2.4904
2.4876
2.4638
2.4422
2.4226
2.4047
2.4205
2.3965
2.3748
2.3551
2.3371
2.366
2.3419
2.3201
2.3002
2.2821
2.321
2.2967
2.2747
2.2547
2.2365
2.4741
2.4591
2.4453
2.4324
2.4205
2.3883
2.3732
2.3593
2.3463
2.3343
2.3205
2.3053
2.2913
2.2783
2.2662
2.2655
2.2501
2.236
2.2229
2.2107
2.2197
2.2043
2.19
2.1768
2.1646
2.3359
2.2541
2.175
2.0986
2.249
2.1665
2.0868
2.0096
2.1802
2.097
2.0164
1.9384
2.124
2.0401
1.9588
1.8799
2.0772
1.9926
1.9105
1.8307
12
243.906
19.4125
8.7446
5.9117
4.6777
15
245.9499
19.4291
8.7029
5.8578
4.6188
20
248.0131
19.4458
8.6602
5.8025
4.5581
24
249.0518
19.4541
8.6385
5.7744
4.5272
30
250.0951
19.4624
8.6166
5.7459
4.4957
40
251.1432
19.4707
8.5944
5.717
4.4638
60
252.1957
19.4791
8.572
5.6877
4.4314
120
253.2529
19.4874
8.5494
5.6581
4.3985
INF
254.3144
19.4957
8.5264
5.6281
4.365
3.9999
3.5747
3.2839
3.0729
2.913
3.9381
3.5107
3.2184
3.0061
2.845
3.8742
3.4445
3.1503
2.9365
2.774
3.8415
3.4105
3.1152
2.9005
2.7372
3.8082
3.3758
3.0794
2.8637
2.6996
3.7743
3.3404
3.0428
2.8259
2.6609
3.7398
3.3043
3.0053
2.7872
2.6211
3.7047
3.2674
2.9669
2.7475
2.5801
3.6689
3.2298
2.9276
2.7067
2.5379
2.7876
2.6866
2.6037
2.5342
2.4753
2.7186
2.6169
2.5331
2.463
2.4034
2.6464
2.5436
2.4589
2.3879
2.3275
2.609
2.5055
2.4202
2.3487
2.2878
2.5705
2.4663
2.3803
2.3082
2.2468
2.5309
2.4259
2.3392
2.2664
2.2043
2.4901
2.3842
2.2966
2.2229
2.1601
2.448
2.341
2.2524
2.1778
2.1141
2.4045
2.2962
2.2064
2.1307
2.0658
2.4247
2.3807
2.3421
2.308
2.2776
2.3522
2.3077
2.2686
2.2341
2.2033
2.2756
2.2304
2.1906
2.1555
2.1242
2.2354
2.1898
2.1497
2.1141
2.0825
2.1938
2.1477
2.1071
2.0712
2.0391
2.1507
2.104
2.0629
2.0264
1.9938
2.1058
2.0584
2.0166
1.9795
1.9464
2.0589
2.0107
1.9681
1.9302
1.8963
2.0096
1.9604
1.9168
1.878
1.8432
2.2504
2.2258
2.2036
2.1834
2.1649
2.1757
2.1508
2.1282
2.1077
2.0889
2.096
2.0707
2.0476
2.0267
2.0075
2.054
2.0283
2.005
1.9838
1.9643
2.0102
1.9842
1.9605
1.939
1.9192
1.9645
1.938
1.9139
1.892
1.8718
1.9165
1.8894
1.8648
1.8424
1.8217
1.8657
1.838
1.8128
1.7896
1.7684
1.8117
1.7831
1.757
1.733
1.711
2.1479
2.1323
2.1179
2.1045
2.0921
2.0716
2.0558
2.0411
2.0275
2.0148
1.9898
1.9736
1.9586
1.9446
1.9317
1.9464
1.9299
1.9147
1.9005
1.8874
1.901
1.8842
1.8687
1.8543
1.8409
1.8533
1.8361
1.8203
1.8055
1.7918
1.8027
1.7851
1.7689
1.7537
1.7396
1.7488
1.7306
1.7138
1.6981
1.6835
1.6906
1.6717
1.6541
1.6376
1.6223
2.0035
1.9174
1.8337
1.7522
1.9245
1.8364
1.7505
1.6664
1.8389
1.748
1.6587
1.5705
1.7929
1.7001
1.6034
1.5173
1.7444
1.6491
1.5543
1.4591
1.6928
1.5943
1.4952
1.394
1.6373
1.5343
1.429
1.318
1.5766
1.4673
1.3519
1.2214
1.5089
1.3893
1.2539
1
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2 Pages 11-20 |
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2.1 Page 11 |
▲back to top |
Critical Values of the Mann-Whitney U
(Two-Tailed Testing)
n2 ex 3
3
.05
.01
--
--
4
.05 --
.01 --
5
.05
.01
0
--
6
.05
.01
I
--
7
.05
.01
1
--
8
.05
.01
2
--
9
.05 2
.01 0
10
.05
.01
11
.05
.01
3
0.,
.)
0
12
.05
.01
4
1
13
.05
.01
4
1
14
.05
.01
5
1
15
.05
.01
5
2
16
.05
.01
6
2
17
.05
.01
6
2
18
.05
.OJ
7
2
19
.05
.OJ
7
3
20
.05
.01
8
3
n1
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0 0 I 12 2 3 3 4 4 5 5 6 6 7 7 8
0 0 0 0 0 0 0 0 I 1 12 2 2 2 3 3
0 I 2 3 4 4 5 6 7 8 9 10 11 I I 12 13 14
-- 0 0 0 I 1 2 2 3 3 4 5 5 6 6 7 8
I 2 3 5 6 7 8 9 II 12 13 14 15 17 18 19 20
-- 0 I 1 2 3 4 5 6 7 7 8 9 10 11 12 13
2 3 5 6 8 10 11 13 14 16 17 19 21 22 24 25 27
0 I 2 3 4 5 6 7 9 10 11 12 13 15 16 17 18
3
5
6
8
10
12
14
16
18 20
22
24
26
28
30
.,')
.J-
34
0 1 3 4 6 7 9 10 12 13 15 16 18 19 21 22 24
4 6 8 10 13 15 17 19 22 24 26 29 31 34 36 38 41
I 2 4 6 7 9 11 13 15 17 18 20 22 24 26 28 30
4 7 10 12 15 17 20 '_)..), 26 28 31 34 37 39 42 45 48
1 3 5 7 9 11 13 16 18 20 22 24 27 29 31 33 36
5 8 11 14 17 20 23 26 29 33 36 39 42 45 48 52 55
2 4 6 9 11 13 16 18 21 24 26 29 31 34 37 39 42
6
9
13
16
19
').,
-.J
26 30 33 37 40 44 47 51 55 58 62
2 5 7 10 13 16 18 21 24 27 30 33 36 39 42 45 48
7 11 14 18 22 26 29 33 37 41 45 49 53 57 61 65 69
3 6 9 12 15 18 21 24 27 31 34 37 41 44 47 51 54
8 12 16 20 24 28 33 37 41 45 50 54 59 63 67 72 76
3 7 10 13 17 20 24 27 31 34 38 42 45 49 53 56 60
9 13 17 22 26 31 36 40 45 50 55 59 64 67 74 78 83
4 7 I I 15 18 22 26 30 34 38 42 46 50 54 58 63 67
10 14 19 24 29 34 39 44 49 54 59 64 70 75 80 85 90
5 8 12 16 20 . 24 29 33 37 42 46 51 55 60 64 69 73
11 15 21 26 31 37 42 47 53 59 64 70 75 81 86 92 98
5 9 13 18 22 27 31 36 41 45 50 55 60 65 70 74 79
1I 17 22 28 34 39 45 51 57 63 67 75 81 87 93 99 105
6 10 15 19 24 29 34 39 44 49 54 60 65 70 75 81 86
12 18 24 30 36 42 48 55 61 67 74 80 86 93 99 106 112
6 11 16 21 26 31 37 42 47 53 58 64 70 75 81 87 92
13
19
25
.,')
.)_
38 45 52 58 65 72 78 85 92 99 106 113 119
7 12 17 22 28 33 39 45 51 56 63 69 74 81 87 93 99
14 20 ?7 34 41 48 55 62 69 76 83 90 98 105 112 119 127
8 13 I8 24 30 36 42 48 54 60 67 73 79 86 92 99 105