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SIN502S - STATISTICAL INFERENCE 1 - 1ST OPP - NOV 2022 |
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n Am I BI A u n IVER
s I TV
OF SCI En CE Ano
TECH
n OLOGY
Faculty of Health, Natural Resources and Applied Sciences
Department of Mathematics and Statistics
QUALIFICATIONS:BACHELOROF SCIENCESIN APPLIEDMATHEMATICSAND STATISTICS
QUALIFICATION CODE: 07BSAM
LEVEL: 5
COURSE:STATISTICALINFERENCE1
COURSECODE: SIN502S
DATE: NOVEMBER 2022
SESSION: 1
DURATION: 3 HOURS
MARKS: 100
EXAMINER(S)
FIRSTOPPORTUNITYEXAMINATION QUESTION PAPER
MR. EM. MWAHI, DR.D. NTIRAMPEBA
MODERATOR:
DR.J. ONG'ALA
THIS QUESTIONPAPERCONSISTSOF 6 PAGES
(Including this front page)
INSTRUCTIONS
1. Answer all the questions and number your solutions correctly.
2. Question 1 of this question paper entails multiple choice questions with options A to
D. Write down the letter corresponding to the best option for each question.
3. For Question 2, 3 & 4 you are required to show clearly all the steps used in the
calculations.
4. All written work MUST be done in blue or black ink.
5. Untidy/ illegible work will attract no marks.
PERMISSIBLEMATERIALS
1. Non-Programmable Calculator without the cover
ATTACHMENTS
Z-table, t-table, Chi-square table, Mann-Whitney U table and the F-table
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SECTION A [MULTIPLE CHOICE]
Write down the letter corresponding to the best answer for each question.
QUESTION 1 [20 MARKS]
1.1 You take a random sample from some population and form a 96% confidence
interval for the population mean, which quantity is guaranteed to be in the
interval you formed?
[2]
A) 0
B)µ
C) x
D) 0.96
1.2 Increasing the sample size, while holding the confidence level the same, will
do what to the length of your confidence interval?
[2]
A) Make it bigger B) Make it smaller C) It will stay the same
D) Cannot be determined from the given information
1.3 What should be the value of z used in a 92% confidence interval?
[2]
A) 2.70
B) 1.75
C) 1.81
D) 1.89
1.4 Why do we use inferential statistics?
[2]
A) To help explain the outcomes of random phenomena
B) To make informed predictions about parameters we don't know
C) To describe samples that are normal and large enough (n>30)
D) To generate samples of random data for a more reliable analysis
1.5 Statistics and parameters ....
[2]
x A) Are both used to make inferences about
B) Describe the population and the sample, respectively.
C) Describe the sample and the population, respectively.
D) Describe the same group of individuals.
1.6 A waiter believes that his tips from various customers have a slightly right
skewed distribution with a mean of 10 dollars and a standard deviation of 2.50
dollars. What is the probability that the average of 35 customers will be more
than 13 dollars?
[2]
A) Almost 0.5
B. Almost zero
C. 0.1151
D. 0.8849
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1.7 Asample of size 55 is drawn from a slightly skewed distribution. What is the
approximate shape of the sampling distribution?
[2]
A) Skewed Distribution
C) Normal Distribution
B) Binomial Distribution
D) Uniform Distribution
1.8 The null hypothesis Ho: p=0.5 against the alternative Ha: p>0.5 was not
rejected at level alpha=0.1. Nate wants to know what the test will result at
level alpha=0.01. What will be his conclusion?
[2]
A) Reject Ho.
B) Fail to Reject Ho.
C) No conclusion can be made.
D) Reject Ha.
1.9 For a test with the null hypothesis Ho: p = 0.5 vs. the alternative Ha: p > 0.5,
the null hypothesis was rejected at level alpha=0.05. Das wants to perform
the same test at level alpha=0.10. What will be his conclusion?
[2]
A) Reject Ho.
B) Fail to Reject Ho.
C) No conclusion can be made.
D) Reject Ha.
1.10 A null hypothesis was rejected at level alpha=0.1 0.What will be the result of
the test at level alpha=0.05?
[2]
A) Reject Ho
B) Fail to Reject Ho
C) No conclusion can be made
D) Reject Ha
TURN OVER
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SECTION B
QUESTION 2 [31 Marks]
2.1 The lifetime of a light bulb is normally distributed with the mean 3000 hours and
a standard deviation of 696 hours. A simple random sample of 36 bulbs is taken.
(a) What are the expected value, standard deviation, and shape of the
sampling distribution of x?
[3]
(b) What is the probability that the average life time in the sample will be
between 2670.56 and 2809.76 hours?
[5]
(c) How large of a sample needs to be taken to provide a 0.01 probability that
the average lifetime in the sample will be equal to or greater than 3219.24
hours?
[5]
2.2 The personnel department of a large corporation wants to estimate the family
dental expenses of its employees to determine the feasibility of providing dental
insurance plan. A random sample of 10 employees reveals the following family
dental expenses (in N$) for the past year.
110 362 246 85 510 208 173 425 316 179
(a) Find the mean point estimate for the employees' dental expenditure in the
past year.
[3]
(b) Compute a 95% confidence interval for the true population mean in the
employees' dental expenditure.
[5]
2.3 An independent random sample of 200 college football players and 150 college
basketball players in a certain state showed that 65% of football players
received academic tutors while 58% of basketball players received academic
tutors. Construct a 90 percent confidence interval for the difference in the
proportion of football players that received tutors and the proportion of
basketball players that received tutors for the population of this state. [10]
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QUESTION 3 [26 Marks]
3.1 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
n = 18
S =4.2
S = 2.3
(a) Is there enough evidence to support her claim, using the 10% level of
significance?
[8]
(b) What assumptions do you make to justify the use of this test?
[2]
3.2 A dietitian wishes to see if a person's cholesterol level will change if the diet is
supplemented by a certain mineral. Six subjects were pretested, and then they
took the mineral supplement for a 6-week period. The results are shown in the
table below. Assuming that the population between pairs is normally
distributed, can it be concluded that the cholesterol level has been changed?
use alpha=0.1
[12]
Subject 1
2
3
4
5
6
Before 210
235
208
190
172
244
After
190
170
210
188
173
228
3.3 Copy and complete the ANOVA table below:
Source
Between
samples
Within
samples
Total
Degree of
freedom
3
Sum of
squares
216.67
Mean
square
.........
13
... ... ...
. .........
16
1090.47
[4]
F-ratio
..........
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QUESTION 4 [23 Marks]
4.1 Give the underlying assumptions of the Manny-Whitney test
[3]
4.2 A study was conducted on 16 daily cattle. Eight cows were randomly
assigned to a liquid regimen of water only (group 1); the others received liquid
regimen of water with some ingredients (group 2). In addition, each animal
was given 7.5 kg of grain per day and allowed to graze on hay at will.
Although no significant differences were observed between the groups in the
dairy milk-production gauges, such as milk-production and fat content of milk,
the following data on daily hay consumption (in kg/cow) were of interest:
Group1 15.1 8.9
Group2 6.8
7.5
14.8 8.4
13.1 12.8 15.5 15.9
8.6
14.2 8.9
15.1 9.2
9.5
Use these data to test the researcher's hypothesis that there is a difference in
mean hay consumption for the two diets. Use Manny-Whitney test with alpha=
0.05.
[10]
4.3 A public opinion poll surveyed a simple random sample of 1000 voters.
Respondents were classified by gender (male or female) and by voting
preference (Republican, Democrat, or Independent). Results are shown in the
contingency table below.
Gender
Male
Female
Republican
200
250
Voting Preferences
Democrat
150
300
Independent
50
50
Do the men's voting preferences independent of the women's preferences? Use
a 0.05 level of significance.
[15]
----------------------END
OF EXAMINATION-----------------------
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APPENDIX C:. The Standard Normal Distribution
z j[ ·o:oo:i O.oJ 0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
o.o. J:ii.o0~ J'ii:0040 /oiio8o jio120, jo.0160 Tii:0199
0.1 ······1:0:.:o~.i,i0i4J 3::o8.0478 '0.0517 i!o.0557 :·o.0596
·:c··-·--·--
0.2 : 0.0793 0.0832 ·o.0871 '0.0910 l<0.0948 '0.0987
0.023~ :'o.0279 '!ii:o3i9 ::o.0359
'0.0636 · ·o.0675 .'0.0114 · fii:o7s3
: 0.1026 .i0.,~0.il(0;.,4)103 [0.1141
0.3 j 0.1179
ii' ......
0.4 ; 0.1554
0.5.····:;o.i9is
·iii ·1a.~2s7
°:o:!217 :-0.1255 ;0.1293 i'o.1331
... .0.1591 0...1·6··2·8···········0··.·1····6···64 i;o.,1100
,·o.19so·····'i·i·:·1iiiis····:ro:::1019 :ro.2054 l 0.2088
:,ci,~29-1 '0.2324
:a.1406
, 0.1112
,0:14:13
:0.1808
0.2157
::~:;~:;.; Po.1480 [0.1517
:0.1879
!io.2549
.. f'0.2580 0.2611 '0.2642
A.21~1
,'.0.2939 '0.2951 ...Ta:iiiiis j 0.3023 ___]0.3051 , .0,3.0?~i[0.3106 ':o.3133
·'o.:iia6 'o.3212 ··'o.3238 io.3264 i'o.3289 ).3315 o.3340 :-0.:iJiis
;o.j~~l _:o.;;1; .....•.·.o.3438
__lo~~~- _!iiij~ii{i ~3~31 lo.3554 '.0.3577 [0·~5.?.9.-
l 1.1 Lo.3643 :,0.3665 ::ii:3Gaii :fo:3ioa· Ji::iii!i 1ii.:ii::i9 o.3770 :o.3790 ]'o:iiii:ii
········-r········•-······
., 0.3869 :p.3888 '[0,3901 n;;::;;;is ro.3944 ro.3962 )i.39ao ''o.3997
.1,4 .•..'.o.,~!~•~t!-;i:;~;!~~:1::~:;:!:'~:::i~t!~l:,·;:i·:~:::::
•o:.i:i06:.: . -1.5---- i0.4-33-2 . ··o.4345 0.4357- !0.4370- ·::o.4382- [~.4394 -·
1.6 ,i.'Q__.44S2.. -·°-·~6~. _. ___
:0-~ ___l:0:419-~ i0.4505 :0.45~5. -
·j:i~~ i:1 1o:4554
0
'0.4573 fo.4582 :'o.4591 : o.4599 To:i6.ga : ).4616
':0.,4625
1.8 Lo.4641 ·· 'o.4649 ·o.4656
, o.4686 'o.4693 :o.4699
i.?Jt4;;L"?:4119 ......:..::.:.:.:..:.·.,.,......-c.c.:....
..-..,.-.-......cc::'o.4761
,OA?2_(i :0.4732 i0.4738 i 0.4744 , 0.4750 :o.4756
9:~~.J... 2.0 .!.0.4772 .'0.4778 .·.o.4783 ·:.:,o.4788 ..foA793 >o.,i;,91j 3
:0..4808 i:0.4812
]p,~~ 2.1 1:0.4821- 0.4826--).4830:··1~.4834 .. }OA8~~ . ;OA84~
! 0.4~~~ :<o.4854
2.2·····1·~··:['.''!?1.l!:4864 ····0·.·4868:o.4871
a:48ai !l~~7i= 1'ii.4aiii :
·;;:;;884 ,~.4887
i~ii?J.ir~:~9:04- 2.3 ;,,0.4..8..9..3.........~..:.:..i.i.i.~~-:'a.4898 :...
0.4906 i0...4..9..0..9. 0..4..9..1..1.......1.'.<..i.:.4..9..)..3...........
2.4 -,.i0.491--8- , 0.4920 -- ·0.4922 1'0.492s_Jo,49~?.~!0.4929. !0.4931 - 0.4932--][o,4934-
' -~,s_J;ii:4jijf }0~4940 :0.4941 U0.4943--ji:;jii;js-·1o.4946 :,0.49·48-- ··.-0.4949 ······•··0·.4951
iii
.-ii.4955 o:4iisii '0.4957 ::a:4iis!i To.iiiGo i 0.4961 i:o.4962 ....; .
;_; li~:4965 T.0.4966 .0.4967 -!0.4968 jJ0.4969 :0.4970 :0.4971. ! o,:4972 :~.4973 i:0.4974
:;;:~7-.f_ 2.8_ ::o,~~~:::~:~iiii
:;9,4·~··~··~··· 2.9 ·io:4981·:··
;o.4977 :;o.4977 J0.4978 - : 0.49?9-_.. ,0.~?7_?...::~4980 ,. - ·--
'o.4982 :fo.4983 ii0.4984 ....:ii.498. 4:'o.4985 . 10:-498..5'o:498.!.tf0.:~?86
3.o To.4f.~.?..
.. /O..A987
: o.4989 : 0.4989 :o.4989 ;0.4990 -'fo.4990
APPENDIX D: The t-distribution
.-.~..f..~..- :i;~:: :71:it{:·· ::a::::o -::1:~~~~··•·····j;;_~;;l[
.:::;:~lt~:····..·o.·.o.jo.io5~i:~;}··-····
2 :o.288675 ![0.816497__ :11.885618 :!2.919986 ';4.30265 [;6.96456 '!9.92484 !'31.5991
3 ;,0.216611 t!o.764892 i;t.637744 1!2.353363 ;!3.18245 ,fi.5_4070•• ;[5,84091
..·~·1··[~--;;~;~· io:740697 ··i!\\:53;;0~·· ::2.131847 ::1;:·;;645 :,,::3..7. 46-9--5- ,',14.60409
:r:i:35:1~~ 5 · !_o:~6?is1. .Ho,726Eill!. l[1.i7513e.•I :12.015048 '[2.s1058
J4.o3214
=J~~6483s 6
to.i115s·8 - !'i.439!s~·-=::[~~i~-
::2."':~~1=- ·-:r3:1~:i~;~_:).10.?~~-
l:!2.9240
!.8.6103
i,,6.8.6..8..8.........
F~9588
7 :to.263167
,====
:2.36462 2.99795 ::3.49948 \\ 5.4079
8..
...,.,.. .. .......
ll?..:~.6.1..9~1;:.0...106387
.;l!.??Eill.1·~····•·1·.859548·····,"':2~.3~06~0~0='''",2~.~89~64=6=::....1,i;'53..034515339
9
10
1io:2i;o9ss Tii:102n2
i i i~:~~a1 .
H1:iio2ii , 1.833113 !~-~6216 ::2.02144···•· H~:24984
:Jt1,.~?218.'.l...:il:81~4~ 1 .... !2.22814 ::2.76377 ·;3.16927
4.7809
!~ 5869
11 _·1[0259556 J:o.697445 · :k3G3430--!l!.795885
'2.20099 - ::2.71808..... ij3.!0581 -· '.4.4370
12 ]lo.259033 :!o.695483 t:1.356217
13 ·:,0.258591 iio.693829
88 !2.17881
H2.16037
::2.68100
:2.65031
!3.05454 i:4.3178
.....i •... ,....... .
!3.01228 !'4.2208
14 lo.2s8213 t:o.692417 ·;:~-~5030 [!1.161310 i[;:14419 j2.62449 jj2.97684 '4.1405
15 j[0.257885 Lo.691197 i'l.340606 h.753050 '!2.13145 ·2.60248 ,[2.94671 :4.0728
16
io.257599 i'o.690132 :1.336757 ..--:,J.745884 H2.11991 ''2.s8349
'······-····--·--=·-·- ···- ·-----•~ _______,,_ ----'
__ ,, __ --------
!2.92078
''--·---·
17 io.257341 fio.689195.. ][1.333379 . i1.7396o7 ;i2.10932 :ii:sGii93·-..-.·.·..,-
,:
:'o.688364 fJ.330~?1 [!1-~~064 i:2.10092 ?-55;!:l~ ..
: . .... ,
'0.687621 'J.327728 [!J.729133 ·:2.09302 )2.53948 ':2.86093 i 3.8834
;
3.~tt"' 20 1:0.256743 ?~.~;6;~i';:;;~;~fr;i':1241:2.108s596 H2.52798 ':2.84534 :
:it~~::::::·::::::::~.:::::~ ;; '!~;56580 i!0.686352 \\::::::
'J.720743
,-·-----·
====
~==='.
.......',.~.-.?.?.7. .6
;ti:·:.!:[:;:;~~~:;;t;~~~:.1j~;~i;:~~t :j::1} 24 j:0.256173 !io.684850
':2.06390 ''2.49216 ... }2.79694 ·,~_.7454
28 :•0.255768 !io.683353 h.312521
29 Jo:~?.5.?!.!8°.:'..6:83044 
APPENDIX E: The Chi-Square Distribution
. -.
~K
.,::
·•
X_:,!.
)isii·::: :iio· [d~P.iL~.5.::. :9?t-.'.[ii1.~J,·
:L :~00 ..:: ·?s.o .-5.°..?
11
.100
.050
..J: .025 ... :.a..1.0. .005
·-. i:a.:ciocio4Jo~00016·i:~:ioi~!J19:8S·7.>?\\o>:.oa~..319.'503.3454J9l41323j320.7055"43.841\\456.0:23'~'6~.63490-!!787944
]:o:010(003?2olo}i;oso641'o)o~s9:)i.i2o.1.s07752b3.63B62•9j:2.77:21549605!157.991l4ir6niis 119.21J0[3I04..59663
, ?..j.i°-:07].Joi:ii,i~::i~i~:ii1,si_i:iio5·..1}s[.os_:s~:..i..~...............
l6.~5.1...i.3i.:.Ji;:;ii·1i·i·:·31.4:5:1°-Jf1.1.:?4'1~il[~2.:~3.81.~···-
:··1~::::::·::·:·:·:.·.;:,:::':.~~;:,::::•.:·-:i:•::-:·:··:;::::::.:·_·.:[:::::·i:·:-;~·-:-;-:-:·o··;l·:·:::::::·;;
6 j,06.75'70~.8.720.]91~373,,41,635.38)~-.?.:?.~415.436[053481t:284?80[10.64464jil25915?Jii!tl64.4:84i9!.1i3t8.59475B
j[o~~~ji,61.239:.0]i4~:68j9•-8~7.:.1...,~[..~.?~.~3~..51.j:;4.2·5:468.354.5!8[91-??!:~!.5i.~:01f°.~_-[14~6.7.14_j;1_6:01~76.i[1~.:~7.531Ji2027774
. ' Ii_3; 44i4:ii:646;5;02!7.97.H32::i:i2i:ii;:4;:8J9!5:S4:0706!74_3I244Hoi :2jii;8r1s3_:36H15is±o13J!11:.:534}5250:09i_\\:02214:9549.5
--ii-··;:134,9·2J0879;02.1°.0.,.:3~39~s_1·1::~_..:.:......
. .:[i1:iaaf~-,liiii.ii~~·l]i:.119.i°i.i~.i257.~.7ii;!2!1'.:~.~35-:._~5s9Js
--1.0.1.12. 55_8,62.5582,31.24697J'3.-9-4,;043.8065;'.6i',87_3.72.0..... _'_!1_2.54886_:[IS.98718)18::3[02730.240.t9;250:.;42853,18818
i_1J/G0:1{232.05:3-4~8-815T74_s5749:_fs15_.m'8.,7.s94_i:140.34i_i1rn0.0o69f11:2jsa,ff\\9:i6i2-i1s9i:2i offo2s.i:-i2.i91T2675685.
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'..1.3·.1;3,.5,l6i45,0l036'9's2~008.)7is5.891--8:617.04.1,1:590,2, 9.J9i027.33976·1r1sl1:998.83!9!l931!i2li.23•6.27033560J:21.688189.2457.l!29
. ... :4.660J453.6287:63.57~6.3.j7.7~95j}'1[1°.):3136952371]117:1,121~036.414]:~!;[~276?.1!8?5}~'l.j~124Ji:ll.31935
:r:::f:~::::~~=:1l~~:::: .-:i:i::::·::::::::: :::::: ::::1:;;::-::~: :f 1 ......,..... ,_5229365.2621_7~2·-·~··.-;~~-~.7.J.I.:6L_036.:.~[1~?3.81..828~~J5i.[<2>.2.9.3. ?~(3..Jl!'2.2~.:19,?45J~!~3~.3o9.~1~1J:3.2:80132
_:::::~~r::::,:;~9:·;-::~_:: ;:;::
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8.9065'"2io,i:io1r1;:ss0'9i:ii..:5ci;~!1oBo :i:;·1l262s111:iaii:20:i[s:ii!o.14J:[i:si:ii:as~:i1J)<:3Jo;i;!~:iii:sa2iG
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:.2. ~.'.9J-88J6.21.3..?8. .5.6,.[..!32.:.460. 1:13s..~~[31.5j .65.:.1~?.:G0l3l.7.·[·2·3.5367Jj32254. 1.;1[3.5?._:_194612._54°./~[.~16[.3.93.64.o~Jl~...'2.9.7?..~2.J!~.~.:.5.5.~.5.1
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• 26j.•1.1.16(_)1224._119;'1831:554:3:19.05??9:1~67..,128_8!,_1._2_8,0434[235.336jj43640_345[.3?~.5.6.3.1l7llJli5~1!~:44._1-?j'2435.16t~14J:4!
·..21!1i,:ao1s·i1i]2818isi:o;:·5: 73;J;a;isi.iol!la:;;39'i01'7. 494:1026.33[63314.s-2, -8i:4iG1 14;1[2:;2;;_iiiii];4i1949s6i·2i:9]:1;4i4i9.6.4J492
'.tt:~:::::~;;:[:::-~.:,!:;;:: i2.~_rj142613'_41)~3647J.3,J°1.7.5~_166~. 2!8!~1-.89319;2.2.:46.57.!6.:J[23.7.2?:36?.12j0!39j4719.5~i2,j3)34.7.!4.J!'.i.'[444,86:027798.2~);50.9933B
::·~:·::·:·;1;:•·:
~!:::::::·::·::;:::;!:::::::-:,:;::·:.i::::::::.::ii:::::.:·::::
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9 Page 9 |
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F Table for alpha=0.10
df2/dfl
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
40
60
120
Inf
1
39.86346
8.52632
5.53832
4.54477
4.06042
2
49.5
9
5.46238
4.32456
3.77972
3
53.59324
9.16179
5.39077
4.19086
3.61948
4
55.83296
9.24342
5.34264
4.10725
3.5202
5
57.24008
9.29263
5.30916
4.05058
3.45298
6
58.20442
9.32553
5.28473
4.00975
3.40451
7
58.90595
9.34908
5.26619
3.97897
3.3679
8
59.43898
9.36677
5.25167
3.95494
3.33928
9
59.85759
9.38054
5.24
3.93567
3.31628
10
60.19498
9.39157
5.23041
3.91988
3.2974
12
60.70521
9.40813
5.21562
3.89553
3.26824
15
61.22034
9.42471
5.20031
3.87036
3.23801
20
61.74029
9.44131
5.18448
3.84434
3.20665
24
62.00205
9.44962
5.17636
3.83099
3.19052
30
62.26497
9.45793
5.16811
3.81742
3.17408
40
62.52905
9.46624
5.15972
3.80361
3.15732
60
62.79428
9.47456
5.15119
3.78957
3.14023
120
63.06064
9.48289
5.14251
3.77527
3.12279
INF
63.32812
9.49122
5.1337
3.76073
3.105
3.77595
3.58943
3.45792
3.3603
3.28502
3.4633
3.25744
3.11312
3.00645
2.92447
3.28876
3.07407
2.9238
2.81286
2.72767
3.18076
2.96053
2.80643
2.69268
2.60534
3.10751
2.88334
2.72645
2.61061
2.52164
3.05455
2.82739
2.66833
2.55086
2.46058
3.01446
2.78493
2.62413
2.50531
2.41397
2.98304
2.75158
2.58935
2.46941
2.37715
2.95774
2.72468
2.56124
2.44034
2.34731
2.93693
2.70251
2.53804
2.41632
2.3226
2.90472
2.66811
2.50196
2.37888
2.28405
2.87122
2.63223
2.46422
2.33962
2.24351
2.83634
2.59473
2.42464
2.29832
2.20074
2.81834
2.57533
2.4041
2.27683
2.17843
2.79996
2.55546
2.38302
2.25472
2.15543
2.78117
2.5351
2.36136
2.23196
2.13169
2.76195
2.51422
2.3391
2.20849
2.10716
2.74229
2.49279
2.31618
2.18427
2.08176
2.72216
2.47079
2.29257
2.15923
2.05542
3.2252
3.17655
3.13621
3.10221
3.07319
2.85951
2.8068
2.76317
2.72647
2.69517
2.66023
2.60552
2.56027
2.52222
2.48979
2.53619
2.4801
2.43371
2.39469
2.36143
2.45118
2.39402
2.34672
2.30694
2.27302
2.38907
2.33102
2.28298
2.24256
2.20808
2.34157
2.28278
2.2341
2.19313
2.15818
2.304
2.24457.
2.19535
2.1539
2.11853
2.2735
2.21352
2.16382
2.12195
2.08621
2.24823
2.18776
2.13763
2.0954
2.05932
2.20873
2.14744
2.09659
2.05371
2.01707
2.16709
2.10485
2.05316
2.00953
1.97222
2.12305
2.05968
2.00698
1.96245
1.92431
2.10001
2.03599
1.98272
1.93766
1.89904
2.07621
2.01149
1.95757
1.91193
1.87277
2.05161
1.9861
1.93147
1.88516
1.84539
2.02612
1.95973
1.90429
1.85723
1.81676
1.99965
1.93228
1.87591
1.828
1.78672
1.97211
1.90361
1.8462
1.79728
1.75505
3.04811
3.02623
3.00698
2.9899
2.97465
2.66817
2.64464
2.62395
2.60561
2.58925
2.46181
2.43743
2.41601
2.39702
2.38009
2.33274
2.30775
2.28577
2.2663
2.24893
2.24376
2.21825
2.19583
2.17596
2.15823
2.17833
2.15239
2.12958
2.10936
2.09132
2.128
2.10169
2.07854
2.05802
2.0397
2.08798
2.06134
2.03789
2.0171
1.99853
2.05533
2.02839
2.00467
1.98364
1.96485
2.02815
2.00094
1.97698
1.95573
1.93674
1.98539
1.95772
1.93334
1.9117
1.89236
1.93992
1.91169
1.88681
1.86471
1.84494
1.89127
1.86236
1.83685
1.81416
1.79384
1.86556
1.83624
1.81035
1.78731
1.76667
1.83879
1.80901
1.78269
1.75924
1.73822
1.81084
1.78053
1.75371
1.72979
1.70833
1.78156
1.75063
1.72322
1.69876
1.67678
1.75075
1.71909
1.69099
1.66587
1.64326
1.71817
1.68564
1.65671
1.63077
1.60738
2.96096
2.94858
2.93736
2.92712
2.91774
2.57457
2.56131
2.54929
2.53833
2.52831
2.36489
2.35117
2.33873
2.32739
2.31702
2.23334
2.21927
2.20651
2.19488
2.18424
2.14231
2.12794
2.11491
2.10303
2.09216
2.07512
2.0605
2.04723
2.03513
2.02406
2.02325
2.0084
1.99492
1.98263
1.97138
1.98186
1.9668
1.95312
1.94066
1.92925
1.94797
1.93273
1.91888
1.90625
1.89469
1.91967
1.90425
1.89025
1.87748
1.86578
1.87497
1.85925
1.84497
1.83194
1.82
1.82715
1.81106
1.79643
1.78308
1.77083
1.77555
1.75899
1.74392
1.73015
1.71752
1.74807
1.73122
1.71588
1.70185
1.68898
1.71927
1.70208
1.68643
1.6721
1.65895
1.68896
1.67138
1.65535
1.64067
1.62718
1.65691
1.63885
1.62237
1.60726
1.59335
1.62278
1.60415
1.58711
1.57146
1.55703
1.58615
1.56678
1.54903
1.5327
1.5176
2.90913
2.90119
2.89385
2.88703
2.88069
2.5191
2.51061
2.50276
2.49548
2.48872
2.30749
2.29871
2.2906
2.28307
2.27607
2.17447
2.16546
2.15714
2.14941
2.14223
2.08218
2.07298
2.06447
2.05658
2.04925
2.01389
2.00452
1.99585
1.98781
1.98033
1.96104
1.95151
1.9427
1.93452
1.92692
1.91876
1.90909
1.90014
1.89184
1.88412
1.88407
1.87427
1.8652
1.85679
1.84896
1.85503
1.84511
1.83593
1.82741
1.81949
1.80902
1.79889
1.78951
1.78081
1.7727
1.75957
1.74917
1.73954
1.7306
1.72227
1.70589
1.69514
1.68519
1.67593
1.66731
1.67712
1.66616
1.656
1.64655
1.63774
1.64682
1.6356
1.62519
1.61551
1.60648
1.61472
1.6032
1.5925
1.58253
1.57323
1.5805
1.56859
1.55753
1.54721
1.53757
1.54368
1.53129
1.51976
1.50899
1.49891
1.5036
1.49057
1.47841
1.46704
1.45636
2.83535
2.79107
2.74781
2.70554
2.44037
2.39325
2.34734
2.30259
2.22609
2.17741
2.12999
2.0838
2.09095
2.04099
1.9923
1.94486
1.99682
1.94571
1.89587
1.84727
1.92688
1.87472
1.82381
1.77411
1.87252
1.81939
1.76748
1.71672
1.82886
1.77483
1.72196
1.6702
1.7929
1.73802
1.68425
1.63152
1.76269
1.70701
1.65238
1.59872
1.71456
1.65743
1.6012
1.54578
1.66241
1.60337
1.545
1.48714
1.60515
1.54349
1.48207
1.4206
1.57411
1.51072
1.44723
1.38318
1.54108
1.47554
1.40938
1.34187
1.50562
1.43734
1.3676
1.29513
1.46716
1.3952
1.32034
1.23995
1.42476
1.34757
1.26457
1.1686
1.37691
1.29146
1.19256
1
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10 Page 10 |
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'l
Critical Values of the Mann-Whitney U
(Two-Tailed Testing)
n2 a
3
4
5
6
7
8
9
n1
10 11 12 13 14 15 16 17 18 19 20
3
.05 --
.01 --
0
0
0
0
I
0
I
0
2
0
2
0
3
0
3
0
4
I
4
I
5
I
5
2
6
2
6
2
7
2
7
3
8
3
4
.05
.01
--
--
0
--
I
0
2
0
3
0
4
I
4
I
5
2
6
2
7
3
8
3
9
4
10 I I 11 12 13 14
5 5 6 6 78
5
.05 0
.01 --
1
--
2
0
3
I
5
I
6
2
7
3
8
4
9 11 12 13 14 15 17 18 19 20
5 6 7 7 8 9 10 11 12 13
6
.05
.01
I
--
2
0
3
I
5
2
6
3
8
4
10 I I 13 14 16 17 19 21 22 24 25 27
5 6 7 9 10 11 12 13 15 16 17 18
7
.05
.01
I
--
3
0
5
I
6
3
8
4
IO 12 14 16 18 20 22 24 26 28 30 32 34
6 7 9 IO 12 13 15 16 18 19 21 22 24
8
.05
.01
2
--
4
I
6
2
8
4
lO 13 15 17 19 22 24 26 29 31 34 36 38 41
6 7 9 11 13 15 17 18 20 22 24 26 28 30
9
.05
.01
2
0
4
I
7
3
10 12 15 17 20 23 26 28 31 34 37 39 42 45 48
5 7 9 11 13 16 18 20 22 24 27 29 31 33 36
10
.05
.01
3
0
5
2
8 11 14 17 20 23 26 29 33 36 39 42 45 48 52 55
4 6 9 11 13 16 18 21 24 26 29 31 34 37 39 42
11
.05
.01
3
0
6
2
9
5
13 16 19 23 26 30 33 37 40 44 47 51 55 58 62
7 10 13 16 18 21 24 27 30 33 36 39 42 45 48
12
.05
.01
4
I
7
3
11 14 18 22 26 29 33 37 41 45 49 53 57 61 65 69
6 9 12 15 18 21 24 27 31 34 37 41 44 47 51 54
13
.05
.01
4
I
8
3
12 16 20 24 28 33 37 41 45 50 54 59 63 67 72 76
7 IO 13 17 20 24 27 31 34 38 42 45 49 53 56 60
14
.05
.01
5
I
9
4
13 17 22 26 31 36 40 45 50 55 59 64 67 74 78 83
7 11 15 18 22 26 30 34 38 42 46 50 54 58 63 67
15
.05
.01
5
2
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
16
.05
.01
6
2
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
17
.05
.01
6
2
11 17 22 28 34 39 45 51 57 63 67 75 81 87 93 99 l05
6 10 15 19 24 29 34 39 44 49 54 60 65 70 75 81 86
18
.05
.01
7
2
12 18 24 30 36 42 48 55 61 67 74 80 86 93 99 106 I 12
6 11 16 21 26 31 37 42 47 53 58 64 70 75 81 87 92
19
.05
.01
7
3
13 19 25 32 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
20
.05
.01
8
3
14 20 27 34 41 48 55 62 69 76 83 90 98 105 112 I 19 127
8 13 18 24 30 36 42 48 54 60 67 73 79 86 92 99 105