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SIN502S - STATISTICAL INFERENCE 1 - 2ND OPP - JANUARY 2025 |
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p
nAmlBIA UnlVERSITY
OF SCIEnCE
FacultoyfHealthN, atural
ResourcaensdApplied
Sciences
Schoool f NaturalandApplied
Sciences
Departmentof Mathematics,
StatisticsandActuariaSl cience
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 2025
DURATION: 3 HOURS
COURSECODE: SIN502S
SESSION: 2
MARKS: 100
SUPPLEMENTARY/SECOND OPPORTUNITY: EXAMINATION QUESTION PAPER
INSTRUCTIONS:
EXAMINERS: MR E. MWAHI
MODERATOR: DRD. NTIRAMPEBA
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. Z-table
3. Chi-square table
4. F-table·
5. U-table
This paper consists of 5 pages including this front page.
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QUESTION 1
[20 MARKS]
Write down the letter corresponding to the best answer for each question.
1.1 When the population is divided into mutually exclusive sets, and then a simple random
sample is drawn from each set, this is called:
[2]
A. Simple random sampling.
B. Stratified random sampling.
C. Cluster random sampling.
D. Systematic random sampling.
1.2 A marketing research firm divides the population of a state into geographic areas, and
randomly selects some of the areas and takes a simple random sample of each
selected area. This is an example of a
[2]
A. Cluster random sample
B. Systematic random sample
C. Simple random sample
D. Stratified random sample.
1.3 The use of the laws of probability to make inferences and draw statistical conclusions
about populations based on sample data is referred to as____
[2]
A. Descriptive statistics
B. Inferential statistics
C. Sample statistics
D. Population statistics
1.4 A manufacturer of contact lenses is studying the curvature of the lenses it sells. In
particular, the last 500 lenses sold had an average curvature of 0.5.
The population is
[2]
A. The 500 lenses.
B. 0.5.
C. The lenses sold today.
D. All the lenses sold by the manufacturer.
1.5 A political scientist is studying voters in California. It is appropriate for him to use a
mean to describe:
[2]
A. The age of a typical voter.
B. The party affiliation of a typical voter.
C. The sex of a typical voter.
D. The county of residence of a typical voter.
1.6 A researcher is studying students' behaviour in colleges in California. She takes a
sample of 400 students from 10 colleges. The average age of all college students in
California is?
[2]
A. A statistic.
B. A parameter.
C. The median.
D. A population.
STATISTICALINFERENCE1 (SIN502S)
2NDOPPORTUNITYJANUARY2025
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1.7 The standard deviation of a normal population is 10. You take a sample of 25 items
from this population and compute a 95% confidence interval. To compute the
confidence interval, you will use
[2]
A. The t table because the degrees of freedom will be 24.
B.The t table because the sample standard deviation is known.
C. The z table because the population standard deviation is known.
D. The z table because the sample size is small.
1.8 If in a random sample of 400 items, 88 are found to be defective. If the null hypothesis
is that 20% of the items in the population are defective, what is the value of the test
statistic?
[2]
A. 0.02
B. 1
C. 0.9656
D. 1.03
1.9 A 92% 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%
1.10 In a simple random survey of 89 teachers of high school AP Statistics, 73 said that it
was the most satisfying, most enjoyable course they had ever taught. Establish a 98%
confidence interval estimate of the proportion of all high school AP Statistics
teachers who feel this way.
[2]
A. 0.820 ± 0.004
B. 0.820 ± 0.041
C. 0.820 ± 0.084
D. 0.820 ± 0.095
QUESTION 2
[44 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 is the expected value, standard deviation, and shape of the sampling distribution
of x?
[3]
(b) What is the probability that the average lifetime 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]
STATISTICALINFERENCE1 (SIN502S)
2NDOPPORTUNITYJANUARY2025
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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 A study was conducted to investigate the effectiveness of hypnotism in reducing
pain. Results for randomly selected subjects are shown in the table.
Before 6.6
6.5
9.0
10.3 11.3
8.1
6.3
11.6
After
6.8
2.4
7.4
8.5
8.1
6.1
3.4
2.0
At a 5% level of significance, from the sample data, is there sufficient evidence to
conclude that the sensory measurements, on average, are lower after hypnotism?
Conduct an appropriate hypothesis test.
[10]
2.4 With individual lines at its various windows, a post-office is interested in the standard
deviation for normally distributed waiting times for customers on Friday. The post-
office experiments with a single main waiting line and find that for a random sample
of 25 customers, the waiting times for customers have a variance of 12.25 minutes.
(a) With a significance level of 5%, construct a confidence interval estimate for the
variance waiting times of all customers at this post-office on a Friday. [6]
(b) Assuming that the estimated population variance at this post-office is 51.84
minutes, is there evidence at 1% level of significance to conclude that a single
main waiting line causes lower variation among waiting times?
[7]
STATISTICALINFERENCE1 (SIN502S)
2NDOPPORTUNITYJANUARY2025
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QUESTION 3
[36 MARKS]
3.1 The contingency table below shows a random sample of 500 U. S adults who were
questioned regarding their political affiliation and opinion on a tax reform bill.
Political Affiliation
Democrat
Republican
Favor
138
64
Opinion on Tax Reform
Indifferent
83
67
Oppose
64
84
Test if the political affiliation and their opinion on a tax reform bill are dependent at
a 5% level of significance.
[13]
3.2 Someone has told you that men are better in abstract reasoning than women. You are
sceptical, so you decide to test this idea. You randomly select eight adult men and
eight adult women living in your hometown and administer an abstract reasoning test.
A higher score reflects better abstract reasoning abilities. You obtain the following
scores:
Men
70
86
60
92
82
50
74
94
Women 81
80
50
95
93
65
90
75
At the 5% level of significance, use the Mann-Whitney U test to test if there is a
difference in scores between the two groups.
[8]
3.3 Students were given different drug treatments before revising for their exams. Some
were given a memory drug, some a placebo drug and some no treatment. The exam
scores{%) are shown below for the three different groups:
Memory drug
70
77
83
Placebo
37
43
50
No treatment
30
10
17
Test the hypothesis that the treatments had different effects. Use alpha = 0.05. [15]
STATISTICALINFERENCE1 (SIN502S)
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APPENDIX E: The Chi-Square Distribution
/
~i:;=...-.----,; .995
i 1 !to.00004
II.. .990 1.._1 .975
110.00016 ilo.00098
.950 I,·.:._
1_1, .900
ii .750 !,·1 .500
.250
l_jo.00393 ·10.01579 ![0.10153 _Jio.45494 !!1.32330
ii '1! .100
.050 iii. .025
il2.70554 _jJ3.84146 il5.02389
:!.:_I .010
.005
!16.63490 !!7.87944
2 0.01003 !0.02010 I 0.05064 1·10.10259 !0.21072 llo.57536 111.38629 !!2.77259 1]4.60517 115.99146 117.37776 119.21034 !!10.59663
3 ! o.07172 lo.11483 ilo.21580 Jio.35185 ijo.58437 !!1.21253 !J2.36597 i[4.10834 116.25139 Jj7.81473 j!9.34840 !!11.34487112.83816
I 4 j!o.20699 110.29711 i)o.48442 Jjo.71072 111.06362 jj1.92255··113.35669 j/5.38527 jJ7.77944 I 9.48773 jj11.143291113.27670 i 14.86026
T 5 i[o.41174 llo.55430 ilo.83121 il1.14548 111.61031 [i2.67460 !14.35146 !!6.62568 IJ9.23636 i 11.07050 i!12.83250lii5.086271116.74960
i 6 Io.67573 10.87209 11.23734 111.63538 112.20413 !!3.45460 !15.34812 117.84080 1i10.64464 12.59159 li14.44938 !!16.81189 ih8.54758
r-·7·1o.98926 111.23904 i 1.68987 ·11z.16735 112.83311 >:4.25485 116.34581 119.03715 ii12.01704 li14.06714 li16.01276 jh8.47531 bo.27774
I a 111.34441 11.64650 112.17973 li2.73264 !3.48954 i!5.07064 117.34412 ih~--;-~~~~-[!13.36157J15.50731 li17.534551120.090241!21.95495
I::~~"]11.73493 112.08790 112.70039 1:13.32511 j4.16816 !!5.89883 1:18.34283 ihl.38875 !TDi:68366·1116.91898r]19.022771121.66599ii23.58935
+-1OTi2.15586 Ti2.55821 ij3.24697 jJ3.9403o li4.86518 116.73720 !J9.34182 i\\12.54886 jl15.98718 Jj18.30704 jJ20.48318 1!23.20925 !125.18818
-1--11/i2.60322 l!3.05348 !13,81575 114.57481 115.57778 '::7,58414 :il0.34100 i:113.70069ff11·:21s-oTi.r1·9:6-7514-r.i21.9200ii524.72497 !:i26.75685
J j 12 3.07382 13.57057 114.40379 115.22603 116.30380 !!8.43842 li11.34032 li14.84540 !118.54935[b.02607 \\[23.33666 i\\26.21697 1!28.29952
r----i-3-i3.56503 14.10692 il5.00875 !15.89186 117.04150 ·9.299oi·--n12.33976 ii15.98391 ii19.81193-ll22.36203 li24.7356o 1127.68825i 29.81947
r 14 ! 4.07467 !4,66043 li5.62873 116,57063 li;,;~~53···11~-o~~~~~-~-;113.339ii2l?7.11693 ii21.06414 ll23.6847~--]i26.11895i)29.14124 j 31.31935
·:-.x~] 4.60092 15.22935 116.26214 !17.26094118,54676 !J~l.03654 li14.33886 ih~:~4.?°.?Jp2.30713 il24.99579 li27.48839 fl30.57791 ii32.80132
l I I 16 115.14221 115.81221 l\\6.90766 j 7.96165 _Jl9.31224 1111.912221!15.338501119.36886!!23.54183 26.29623 /!28.84535 !131.99993 34.26719
I i I 17 1!5.69722 IJ6.40776 1]7.56419 8.67176 IJ10.08519 i!12.79193-ji16.33818 !]~?.:4.~~~~.JE4.:?.~?°._4.Jl27.587(3101.19101 ii33.40866 35.71847
i 1a 116.26480 17.01491 118.23075 1!9.39046 !10.86494113.675291!17.33790 1121.604891!25.98942 28.869301131.52638 [134.80531!37.15645
L~9Jl6.84397 17,63273 ! 8.90652 110.11701 El.65091 i 14.562°.°.jl18,33765 i!22-:1iiai"li27.20357 L~?.:~4.~?~J!32.85233il36.19087: 38.58226
I J~[!?:_4.3384 18.26040 9.59078 ! 10.85081 !12.44261 ii15.45177 il19.33743 jj23.82769I28.411981131.41043 ij34.16961 !!37.56623139.99685
i i i I , __i_l8.03365Jl_8.89720 10.28290 11.59131 113.23960Jj1G.3443ajl20.33723 jj24.93478 29.6~?.?.~Jl32.67057 jj35.47888 \\138.93217 41.40106
i 22 !!8.64272 ]19.54249 10.98232 12.33801 i14.041491E7.:~.3~6.~Jl2u3704 ll26.03927 li3o.81328 133.92444 !136.78071ii40.289361142.79565
r~;! i 9.26042 110.19572 11.68855 13.09051 \\14.84796 ih8.13730··1122.33688li27.14134 32.00690 : 35.17246 ih8.07563 [141.63840!144.18128
L~-~j 9.88623 j10.85636 IJ!~:4.?.~~?.i13.84843 !15.65868l!1~--~;;;~-!i23_335731128.24115 il33.19624 35.41503 !j39.36408 i]~;:;;~~;!!45.55851
l 25 110.51965 111.5239811n11972 114.61141 116.47341 1119.93934r124.336591129.338851134.38159 37.65248!140.646471144.314101146.92789
}JD11.16024 '.112.19815il13.84390 ! 15.37916 i~;:;~~~~-ji20.8434;]125.33646 jj;~:~;~~;jh5.56317 38.88514jj41.923171145.641681148.28988
i ~?.Jl11.807591112.8785011.~~:??~~~J116.15140 118.113901!21.74940j/26.336341131.5284111~6..
i 40.113271143.194511146.96294 49.64492
J ~~jJ2.46134 li13.56471 j 15.30786 !116.92788 _!18.93924iJ~~:6.??.~6.JJ27.3362l3i32.62049 1137.91592 41.33714 !44.46079 il48.27824 l!5o.99338
• ~~_Ji13.12115 ll14.25645 ij~_6.:?,470j717.70837 !19.767741123.56659 !128.33613!133.71091jl39.08747 42.55697 !45.722291149.58788 ! 52.33562
77J 77 [J1::7.86.7.=Ji14:9._5.:46.Ji_1.6.7901.8.49.=_6.6._0!20:599.::J!24.641._l!~9..33.603
APPENDIX D: The t-distribution
2 110.288675 llo.816497 !11.885618 !!2.919986 114.30265
!!9.92484 !!31.5991 u
i:===3===;,lc::o=.2=76===6==:7;!='7.11=!6=40==8. =9=2!==1::.1=:6;:3=7=7=44=·=·······::::1:.!1=:·3t=_112=8.32==54==353===6n=131===41=1_ii==554==.08=74=0=09=1===t!i!i==2=.9=2
1
4
!0.270722 i!o.740697 lii.533206 !!2.131847 112.77645 113.74695 114.60409 118.6103 i1
s
10.267181 110.726687 !!1.475884 ::2.015048 112.57058 113.36493 114.03214 li6.8688
il
1
6
lo.264835 l!o.717558 ![1.439756 iJ.943180 !12.44691 !b.14267 !b.70743 1!5.9588
l. 1
IIR=====1i=' ====±i====r=====n=====t'r'====:====1'.1:=====t=====±
7 110.263167 tlo.711142 1:1.414924 !11.894579 !12.36462 112.99795 1!3.49948 ::5.4079
ii
1
II!========s::::i+Jt=o==·=2=6=1:9:2=1====:110=_=70=6=3=87=::::i:;:!1:i=1;:..=3i:29:.638:o:1:6:5=:0::=10:1:::1:.1:8:l=::5::=l=92:5=:.4=88::9: 6:4:6:::::;!3:.=3=5:5:3=9
9
1
l!o.260955 lio.702722 \\!1.333029 !!1.333113 ::2.26216
12.82144
113.24984 !~:?~?.-..~.J. ]
.t===10===1'1t::::o=.2=60=1=8=5====;iioil.639792811824 iil.812461 ii2.22814 12.76377 113.16927 A.5869
ii
I U 11 lo.259556 l:o==j .6=97=4=4=!=51=.:=::3,6=3=4=310==7.=··9··5==·8··==8:5l:==.2i:=i:=:!:.2:;=j 00=9=9==±:2=_==7:1::=:;8ib=.01=05881
14.4370
!]
1·;::::! =1=2===110.259033 llo.695483 111.356217 i\\.782288 /'2.17881
!2.68100
3.05454
!4.3178
jj
l:::i==13====t!:i1=0=.2=58=5=9=1==!\\o.69ii3i.832590171 111.770933 ::2.16037 :2.65031 113.01228 !4.2208
!I
===::::;
t::::====tt====,====::1··'-i-
;·-1
i! ==1=4==+====t:=o.=69=2=4=17==i:: '=1=.3=4=5=03=i=0i7=l=::•6;:1=3=1=0=:::it:i2=·=144=79=:::=±:=:21i=2.6==.92=7464=89=j4=~=±=::::.=::!;'~:i==?=5=
15 !io.257885 io.691197 !!~.:~.~?606 !!1.753050 ii2.13145 ::2.60248 ih.94671 !!4.0728
'
i1===1=6110.257599
!r===1::7::::::,::I
i!o.690132 l!i.336757 !!1·=.7=4=58=8=4===r1!2.111929.158349 !12.92078 1!4.0150
ii 1
::1.333379 1.739607 it:12=.1=09=8=!2==_:=::5:6:!=6=9=3===1':::::::121==.38.==99==8625==3l==1==
18 jjo.257123 !lo.688364
1.734064 _!'2.10092 ::2.55238 ib.87844 !t=!3.=92=1=6===
19 i\\o.256923 llo.687621 111.327728 l!i.729133 !2.09302 ::2.53948 !!2.86093 !!3.8834
.
20 iio.256743 iio.686954 \\ii.325341 ······i·!i.724718 !2.08596 ::2.52798 (l::::12=.8=45=3=4==!13.849j5j
21 ilo.256580 0.686352 jji.323188 !11.720743
==i.t::::===· =:it=====•t=====i
22 ti0.256432 J0.685805 !11.32.1237 ill.717144
ll 23 !!o.256297 ·o.685306 !11.319460 111.713872
I=] ===2=4,t=.
=====t~,====·'•i=.
110.256173 10.684850
====:,======..
111.317836 !1.710882
112.07961
't::=::=' =c=j''
!!2.07387
lb.51765
!12.50832
ii2.83136
,,
112.81876
rt::::13=.8=19=3==
,,
l._l
113.7921 jj
!!2.06866 j2.49987 112.80734 ii3.7676
. .....:.:...;..,.====::::::tt======,i=====i
!12.06390 12.49216 ::2.79694 ::3_7454
2s lio.256060 i!o.684430 !!1.316345 ,i.708141 2.05954 ib.48511
26 iio.255955 it::::io=.6=84=0=4=3=::;-r.ti=11.=73.=10==45=69==17=281===2::.:=;:1:0tj=::5:5! =5=3'2=.:4::7:8;:63
112.78744 113.7251
i!=1:2=7.=78=7=1=::::7:;0it==6j3==6.====!
27 lio.255858 !o.683685 111.313703 11.703288
ti=====''·····-·····=····=···=·=n·=·=-==·=· =1"-•=······==·=··=;:;=====,
2s :i0.255768 ;o.683353 !i1.312527 1,1.701131
ib.05183
1,2.04841
!!2.47266
::2.46714
112.77068
112.76326
3.6896
,3.6739
29 !jo.255684 10.683044 111.311434 111.699127 112.04523 !h.46202 112.75639 !t::::!3=.6=59=4==:
=====i=====:it=====,t=====+i===:::::::::j'~'
1.1. 30 ! 0.255605 10.682756 ih.310415 ih.697261
•+=====;:,=====+.=::::::::..=.==::::+ ====:::+======,
===:::::;t·=====ti:::·
ib.04227 ih.45726
====+!=====+
ib.75000 1\\3.6460
!i
i1±1:::::=in=f===•1d•10.:.=:.25..:3.-.3.-.:4.:=.:7.·:·.=· ··-=·!·=o··.6··=7·4·==4=·9l·•0tt!::::::1::1=1=.2=8±1t5''.5::2:::
111.644854 111.95996 !!2.32635 !l2.s7583 ii3.29o5
.._...:.-.:..:=.=::::::i't::i-====t=····=····· =······=··..:·.:·..:.·.:.:-.::=.:-=:·.=-=::·:·:·:=::·::·!·=··='t=t =i
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F Table for alpha=0.05
.}\\_'
t·
F '!.05 df1cn1.
dl2/dfl
1
1
161.4476
2
18.5128
3
10.128
4
7.7086
5
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
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
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
6
5.9874
5.1433
4.7571
4.5337
4.3874
4.2839
4.2067
4.1468
4.099
4.06
3.9999
3.9381
3.8742
3.8415
3.8082
3.7743
3.7398
3.7047
3.6689
7
5.5914
4.7374
4.3468
4.1203
3.9715
3.866
3.787
3.7257
3.6767
3.6365
3.5747
3.5107
3.4445
3.4105
3.3758
3.3404
3.3043
3.2674
3.2298
8
5.3177
4.459
4.0662
3.8379
3.6875
3.5806
3.5005
3.4381
3.3881
3.3472
3.2839
3.2184
3.1503
3.1152
3.0794
3.0428
3.0053
2.9669
2.9276
9
5.1174
4.2565
3.8625
3.6331
3.4817
3.3738
3.2927
3.2296
3.1789
3.1373
3.0729
3.0061
2.9365
2.9005
2.8637
2.8259
2.7872
2.7475
2.7067
10
4.9646
4.1028
3.7083
3.478
3.3258
3.2172
3.1355
3.0717
3.0204
2.9782
2.913
2.845
2.774
2.7372
2.6996
2.6609
2.6211
2.5801
2.5379
11
4.8443
3.9823
3.5874
3.3567
3.2039
3.0946
3.0123
2.948
2.8962
2.8536
2.7876
2.7186
2.6464
2.609
2.5705
2.5309
2.4901
2.448 2.4045
12
4.7472
3.8853
3.4903
3.2592
3.1059
2.9961
2.9134
2.8486
2.7964
2.7534
2.6866
2.6169
2.5436
2.5055
2.4663
2.4259
2.3842
2.341 2.2962
13
4.6672
3.8056
3.4105
3.1791
3.0254
2.9153
2.8321
2.7669
2.7144
2.671
2.6037
2.5331
2.4589
2.4202
2.3803
2.3392
2.2966
2.2524
2.2064
14
4.6001
3.7389
3.3439
3.1122
2.9582
2.8477
2.7642
2.6987
2.6458
2.6022
2.5342
2.463
2.3879
2.3487
2.3082
2.2664
2.2229
2.1778
2.1307
15
4.5431
3.6823
3.2874
3.0556
2.9013
2.7905
2.7066
2.6408
2.5876
2.5437
2.4753
2.4034
2.3275
2.2878
2.2468
2.2043
2.1601
2.1141
2.0658
16
4.494
3.6337
3.2389
3.0069
2.8524
2.7413
2.6572
2.5911
2.5377
2.4935
2.4247
2.3522
2.2756
2.2354
2.1938
2.1507
2.1058
2.0589
2.0096
17
4.4513
3.5915
3.1968
2.9647
2.81 2.6987 2.6143
2.548
2.4943
2.4499
2.3807
2.3077
2.2304
2.1898
2.1477
2.104
2.0584
2.0107
1.9604
18
4.4139
3.5546
3.1599
2.9277
2.7729
2.6613
2.5767
2.5102
2.4563
2.4117
2.3421
2.2686
2.1906
2.1497
2.1071
2.0629
2.0166
1.9681
1.9168
19
4.3807
3.5219
3.1274
2.8951
2.7401
2.6283
2.5435
2.4768
2.4227
2.3779
2.308
2.2341
2.1555
2.1141
2.0712
2.0264
1.9795
1.9302
1.878
20
4.3512
3.4928
3.0984
2.8661
2.7109
2.599
2.514
2.4471
2.3928
2.3479
2.2776
2.2033
2.1242
2.0825
2.0391
1.9938
1.9464
1.8963
1.8432
21
4.3248
3.4668
3.0725
2.8401
2.6848
2.5727
2.4876
2.4205
2.366
2.321
2.2504
2.1757
2.096
2.054
2.0102
1.9645
1.9165
1.8657
1.8117
22
4.3009
3.4434
3.0491
2.8167
2.6613
2.5491
2.4638
2.3965
2.3419
2.2967
2.2258
2.1508
2.0707
2.0283
1.9842
1.938 1.8894
1.838
1.7831
23
4.2793
3.4221
3.028
2.7955
2.64
2.5277
2.4422
2.3748
2.3201
2.2747
2.2036
2.1282
2.0476
2.005
1.9605
1.9139
1.8648
1.8128
1.757
24
4.2597
3.4028
3.0088
2.7763
2.6207
2.5082
2.4226
2.3551
2.3002
2.2547
2.1834
2.1077
2.0267
1.9838
1.939
1.892
1.8424
1.7896
1.733
25
4.2417
3.3852
2.9912
2.7587
2.603
2.4904
2.4047
2.3371
2.2821
2.2365
2.1649
2.0889
2.0075
1.9643
1.9192
1.8718
1.8217
1.7684
1.711
26
4.2252
3.369
2.9752
2.7426
2.5868
2.4741
2.3883
2.3205
2.2655
2.2197
2.1479
2.0716
1.9898
1.9464
1.901
1.8533
1.8027
1.7488
1.6906
27
4.21
3.3541
2.9604
2.7278
2.5719
2.4591
2.3732
2.3053
2.2501
2.2043
2.1323
2.0558
1.9736
1.9299
1.8842
1.8361
1.7851
1.7306
1.6717
28
4.196
3.3404
2.9467
2.7141
2.5581
2.4453
2.3593
2.2913
2.236
2.19
2.1179
2.0411
1.9586
1.9147
1.8687
1.8203
1.7689
1.7138
1.6541
29
4.183
3.3277
2.934
2.7014
2.5454
2.4324
2.3463
2.2783
2.2229
2.1768
2.1045
2.0275
1.9446
1.9005
1.8543
1.8055
1.7537
1.6981
1.6376
30
4.1709
3.3158
2.9223
2.6896
2.5336
2.4205
2.3343
2.2662
2.2107
2.1646
2.0921
2.0148
1.9317
1.8874
1.8409
1.7918
1.7396
1.6835
1.6223
40
4.0847
3.2317
2.8387
2.606
2.4495
2.3359
2.249
2.1802
2.124
2.0772
2.0035
1.9245
1.8389
1.7929
1.7444
1.6928
1.6373
1.5766
1.5089
60
4.0012
3.1504
2.7581
2.5252
2.3683
2.2541
2.1665
2.097
2.0401
1.9926
1.9174
1.8364
1.748
1.7001
1.6491
1.5943
1.5343
1.4673
1.3893
120
3.9201
3.0718
2.6802
2.4472
2.2899
2.175
2.0868
2.0164
1.9588
1.9105
1.8337
1.7505
1.6587
1.6084
1.5543
1.4952
1.429
1.3519
1.2539
inf
3.8415
2.9957
2.6049
2.3719
2.2141
2.0986
2.0096
1.9384
1.8799
1.8307
1.7522
1.6664
1.5705
1.5173
1.4591
1.394
1.318 1.2214
1
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9 Page 9 |
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Critical Values of the Mann-Whitney U
(Two-Tailed Testing)
n2 a. 3
3
.05
.01
--
--
4
.05
.01
--
--
5
.05
.01
0
--
6
.05 I
.01 --
7
.05 I
.01 --
8
.05
.01
2
--
9
.05
.01
2
0
10
.05
.01
3
0
11
.05
.01
3
0
12
.05
.01
4
I
13 .05 4
.01 I
14
.05
.01
5
1
15
.05
.01
5
2
16
.05
.01
6
2
17
.05
.01
6
2
18
.05
.01
7
2
19
.05
.01
7
3
20
.05
.01
8
3
11I
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
00 I I 2233445566778
00000000 I I I 222233
0 1 2 3 4 4 5 6 7 8 9 10 11 11 12 13 14
-- 0 0 0 I I 2 2 3 3 4 5 5 6 6 7 8
I 2 3 5 6 7 8 9 11 12 13 14 15 17 18 19 20
-- 0 I I 2 3 4 5 6 7 7 8 9 10 11 12 13
2 3 5 6 8 10 II 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 32 34
0 I 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 23 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 23 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 11 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
11 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 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
14 20 27 34 41 48 55 62 69 76 83 90 98 105 112 119 127
8 13 18 24 30 36 42 48 54 60 67 73 79 86 92 99 105
|
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10 Page 10 |
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APPENDIX C: The Standard Normal Distribution
..____,_0 __z_ .
I., z
,: [I ii o.oo
:1
11 0.01
1··1·. 0.02
jili o.o3 ... :1 o.o4
o.o5
0.06 !,:·I o.o7
0.08
o.o9
J o.o IIl=o=.o=oo=o=::::;it:lo==·jo=·o=4o=1o==!o2==ll.00,!==i:=1:4;o=:l:.;o6=:lI=0.00==81==o9=9=P::I:;!:it=oio.==0.0==2'2=73o==.9=9=0=:3:=µ=;'1=9=:ttlo=.
1 I 0._1 0._3098 l.l.!..i Jlo.0438 101.0478 \\0.0517 iio.0557 i!g:9596J!o.0636 l0.0675 0_0714 ,lo0. 753
0 2 ;00793 0.0832 ,1t0871 1:0.0910 I!o.0948 !t=!o=.o=98=7==iit=!o=.1=02=6==l1ii=:J.:1:=:01=4.=11=
_I o.3 1110.11791:1.0.1217 10.125_5_Ji?.:~293!lo.1331110.1368
0=·=14=8=0=1+!,lo
I -1=,
o.4 lio.1554 1!0.159110.1628 lio.1664 110.1700!lo.1736
=oo=.6.5=I=01.2!2:;5::7:::i·!ol·o.=.212=991111·:=110·15.01=.92:8-3:5:2:;4II1.i110io1..22030.1519795l0·!i.0o..22035849
lJo.2088
llo.2422
llo.1772
!
llo.2454
Jto.1808
!o.2486
1 o.1844
1o.2517
jlo.1879
·'jo.2549
o.7 Io.2580 !0.2611 lo.2642 jJo.2673lio.2704 !fo.2734llo.2764 !o.2794 ilo.2323 jlo.2352
0.8 I0.2881
Jo:293l9!o".2967 110.2995i'io.3023iio.3051 llo.3078 l'o.3106 IJo.3133
I o.9 Io.3159
~i ==1.=0i:l=lo=.=3~41,.3,
l_lo.3186!o.3212
'
llo.3238
.
J;jro-.3264
.:·l=1·.=0=.3=2=3=9lilio:L.3i•3•1=5··j:J.o·=.=3==3==4=t0i=l1li•=ti.====o:,==--.;=3,i=· 36=5==i!:i
l!0.3438 uo.3461 !Jo.3485!,0.3508 ilo.3531 !\\o.3554 u0.3577 101.3599 110.3621
-::j::1:l:=.1I:::1o.=3=64=3=::::i=!o=.3=6=fi6o5.3=7:0::81[!:Joo..33762896llo.3749 1lo.3770 1.·. •.1.=llo
I 1.2 12;1 o.3849 jo.3869 101.3888.Jio.3907 lio.3925 !io.3944llo.3962 ·..0i.
llo.3997 1·0! .4015
:j::j =1=.3===jr;::lo=J.o4=.04==302=1=4==t9:;=:!::;11:0io==_.=44=00==86==26==+::jJl:oi=:I.=40,=09.4=193=1=1q]0:.4;!1o4=7.=Il4o1.4=116=2511=i1 0=.=41i_77
1=1 =1.=4===ilt::!o.=4=19=2==1j·,:+=lo=.4=j2jo=.0472=5=1llt!+o=.4!2o6=5.4i=io2.4=2729=:!:j:o::.4jl2l.962.4i2io3.64306 llo.4319
_I 1.5 lio.4332 ilo.4345 iio.4357 lio.4370 iio.4382 l[0.4394i!o.4406 ija:44i"s"lio.4429 !jo.4441
I 1.6 llo.4452 ilo.4463 llo.4474 !jo.4484 l[o.4495!!o.4505i!o.4515 llo.4525 !Jo.45351Jo.4545
_11,:t=:::1=.7=:::;110.45!!5o4.4564llo.4573 i!o:458.2Tlo.4591ilo.4599 ![0.4608llo.4616 jl0.4625 llo.4633
+==I =1.=8====tlio.4il6o4.41649 llo.4656Jlo.4664 ·lia:4671l!o.4678i[o.4686 i'o.4693 llo.4699 !io.4706
I 1.9 !fo.4713 Jlo.4719-']!o.4726!\\o.4732jlo.4738
2.0 110.4772ii0.4778 j/o.4783 jio.4788 ilo.4793 i!o.4798l!o.4803 I0.4808 llo.4812 llo.4817
I 2.1 l\\0.4821 i:lo.4826!lo.483-0--flo.4834i!o.4838iio.4842 i!o.4846 l]o.4850 lo.4854 lio.4857
:l=I2=.2==ii;:::io=.4=86=1===1t:10i=o.=4.=48===8:r6=.417=i=:=[1o1==1A=i=t=:1:80==.47.84=7=58li=o6.488=8=1=!t!10=.4884 I10.4887Ilo.4890
I 2.3 jJo.4893 J!o.4896l!o.4898 l!o.4901 llo.4904 0.4906 iio.4909 ilo.4911 II0.4913jj0.4916
I 2.4 !io.4918 llo.4920 \\lo.4922 lt::io=.4=92=5==lit==·l4o==9.42=92==::7:;=i,=tli=•.ot=i~'0~=1=~,3o==·.·=-~4=9--=--3==:64-=-:=_:=:_·=:=ij==:
I 2.5 l!o.4938 [lo.4940
I 2.6 110.4953Jjo.4955 l:io.4956Jig:~~_5_?J_.l_0_.4959 110.4960o.4961 ilo.4962 lo.4963 l[o.4964
I 2.7 !!o.4965 IJo.4966llo.4967 li0.4968 ilo.4969 !lo.4970 0.4971 l0.4972 I0.4973 lo.4974
I 2.8 j!o.4974 ilo.4975 !lo.4976 lio:4977"--!!o.4977·!o.4978 o.4979 '0.4979 ilo.4980 110.4981
I 2.9 Ilo.4981 I10·:4·9·8··2·········11]0.4982
===+ll=o.=49=8=5=:t•l=o=.4=9
I _.J_l?~~~~~-----!]?~4)?9:8~7~87 1\\0.4988 ,i0.4988 iio.4989 0.4989 Jjo.4989l,?:~9.9.J?!_o.4990.._.