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SIN502S - STATISTICAL INFERENCE 1 - 1ST OPP - NOVEMBER 2024 |
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r
nAm I BIA un IVERSITY
OF SCIEnCEAnOTECHnDLOGY
FacultyofHealthN, atural
ResourceasndApplied
Sciences
Schoool f NaturalandApplied
Sciences
Departmentof Mathematics,
StatisticsandActuarialScience
13JacksonKaujeuaStreet
Private Bag13388
Windhoek
NAMIBIA
T: +264612072913
E: msas@nust.na
W: www.nust.na
QUALIFICATION: BACHELOR OF SCIENCES IN APPLIED MATHEMATICS AND STATISTICS
QUALIFICATIONCODE: 07BAMS
LEVEL:5
COURSE:STATISTICAL INFERENCE 1
DATE: NOVEMBER 2024
DURATION: 3 HOURS
COURSECODE: SIN502S
SESSION: 1
MARKS: 100
FIRST OPPORTUNITY: EXAMINATION QUESTION PAPER
INSTRUCTIONS:
EXAMINERS: MR E. MWAHI
MODERATOR: DR D. 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
[18 MARKS]
Write down the letter corresponding to the best answer for each question.
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.O
B. µ
C. x
D. 0.96
1.2 What will happen to the length of the confidence interval if the sample size is
increased but the confidence level is kept constant?
[2]
A) it becomes bigger B) it becomes 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]
A. Are both used to make inferences about x
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 sample 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) Chi-square Distribution
D) Uniform Distribution
STATISTICALINFERENCE1 (SIN502S)
15T OPPORTUNITYNOVEMBER2024
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1.7 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.
1.8 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.
1.9 A null hypothesis was rejected at level alpha=0.10.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
STATISTICALINFERENCE1 (SIN502S)
15T OPPORTUNITYNOVEMBER2024
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QUESTION 2
(40 MARKS]
2.1 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.
2.1.1 What is the probability that the average of 35 customers will be more than 11
dollars?
[4]
2.1.2 What is the probability that the average of 25 customers will be between 8.5
dollars and 11 dollars?
[5]
2.2 In a study of the relationship of the shape of a tablet to its dissolution time, 6 disk-
shaped ibuprofen tablets and 8 oval-shaped ibuprofen tablets were dissolved in
water. The two population variances are assumed to be nearly equal. The dissolve
times, in seconds, were as follows:
Disk: 269.0
249.3
255.2
252.7
247.0
261.6
Oval: 268.8
260.0
273.5
253.9
278.5
289.4
261.6
280.2
(a) Estimate and interpret the true population mean difference in dissolve times
between the two shapes of a tablet with the 5% level of significance.
[11]
(b) Can we conclude that the mean in dissolve times differ between the two shapes?
Use alpha = 0.01.
[8]
QUESTION 3
(42 MARKS]
3.1 We want to know whether boys are more likely to enrol for science at NUST. Below
is the table documenting the percentage of boys and girls who study science and
humanities:
Gender
Field of study
Science
Humanities
Boys
46
71
Girls
47
83
Can it be concluded that there exist a relationship between gender and the field of
study? Test at 10 % level of significance.
[12]
STATISTICALINFERENCE1 (SIN502S)
151 OPPORTUNITYNOVEMBER2024
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3.2 Does physical exercise alleviate depression? We find some depressed people and
check that they are all equivalently depressed to begin with. Then we allocate each
person randomly to one of two groups: 20 minutes of jogging per day; or 60 minutes
of jogging per day. At the end of a month, we ask each participant to rate how
depressed they now feel on a Likert scale that runs from 1 ("totally miserable")
through to 100 (ecstatically happy"). Ratings were recorded in the table below:
Jogging for 20 22
27
39
29
46
48 49
minutes
Jogging for 60 59
66
38
49
56
60 56
minutes
(a) Use the Mann-Whitney U test to test if there is a difference in ratings
between the two groups. Use alpha = 0.01
[12]
(b) Suppose the data meet the requirements for a parametric test, what
parametric test can be used instead of the Mann-Whitney U test?
[2]
3.3 The Mozart effect refers to a boost of average performance on tests for elementary
school students if the students listen to Mozart's chamber music for a period
immediately before the test. To attempt to test whether the Mozart effect exists, an
elementary school teacher conducted an experiment by dividing her third-grade class
of 6 students into three groups of 2. The first group was given an end-of-grade test
without music; the second group listened to Mozart's chamber music for 10 minutes;
and the third groups listened to Mozart's chamber music for 20 minutes before the
test. The scores of the six students are given below:
Groupl
80
63
Group 2 Group 3
79
73
73
82
(a) Using the ANOVA F-test at a=0.10, is there sufficient evidence in the data to
suggest that the Mozart effect exists?
[12]
(b) Based on your results in (a), is it justifiable to do a post hoc test?
Give a reason to support your answer.
[2]
STATISTICALINFERENCE1 (SIN502S)
15T OPPORTUNITYNOVEMBER2024
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APPENDIX C: The Standard Normal Distribution
+I=I =zo=.=o =1li=:olo~.0=i0.o0==0=+ou!ot==.!00=04·=00lii1=o=o.0==0+.80li0=l2li=oo.=0=.1+.20j0=1113I=!=0o-.:=0t-.10+6==04!0=.=0e1=9=9o1.1=00.=50=2·=3=09!r06-l.=i0==2!t7-9!=I=0o.0=3.01=9 711=0=..0+3=5=9o=.0
:.t:==o=.1==+_1=__t_=__io_=__.o_=39=8=
0.2 j!o.0793!io.0332 lo.0871 llo.0910110.0948!io.0987 0.1026 I0.1064 0.1103 !!0.1141
I o.3 lio.1179 110.1217110.125J510.1293ilo.1331!!0.1368 0.1406
:t=I=o=.4==+1t=io=.1=55=4i=o::.:=t!1:t6:=:!!02o==·8.=1==1::65:11=t9:l=6:o:=1I =4.1==:t;71:=I;:00==-!:t=!o=.1=7=36=1=10+=t=·=o=18.1==47=74==2:=ri1+-1
0.5 iio.1915 !.Jo.1950jio.1985Jio.2019Jlo.2054 i!o.2088 10.2123 0.2157 0.2190 l!o.2224
0.6 IJ;0.2257 il!0.2291 j0.2324 11_1,,,0.2357u!I o.2389!.' J0.2422!0.2454!. 0.2486 0.2517 ''
:i.=t:==oo==..87=o==t1l."9j1oi==0.2:=.:25:=!88:=810=1iil=.!;0io1i..022=96.11301=I11il110io=5..o22=.96=934=3922::=ll:ti!:=o1to1=.=.22i=2169==76:=037:=:rilt:!t.:too3:=_:..~=j22=_j=19o7_=__89=0_=_§=54_=_4il6!_=l!=µ=oo3_=:.:.2:2:3;!:=t7o0t=i33=2==43.=38'o=0==..2=3i2i=n0:7:8=56191=41==oii:ol==:o;.}=:2.i3=t7:=0:9:7=:4o8==lll=_l~.o3rj!+o.=o3.-.321=31803==268=53l1i9ii~=_o-o=.3.r21·8o35.33234o
t===l.=o===ilio.3413 !i=I0.=34=3=8=:tr;:=io===.i3ll=o4.3=468=51 lio.3508 :i=:o=.3=53=1=:::;·;1=00..=33557=75=4=l~;!~-J0.3621
1.1 iio.3643 !J0.3665Jio.3686Jio.3708iio.3729l!o.3749 !o.3770ilo.3790llo.3810lio.3830
1.2 :10.3849 !lo.3869l!o.3888llo.3907l;o.3925/io.3944llo.3962!lo.3980jjo.3997110.4015
:.t:==1=.3=::;t:t::io=.4=031==20=.=::4:;0:1i=ot6:=:=1.460==t0:.-+4=i1=180ti0==42.=4=9=0=9ii9, ==;i·t::t=io:.41i=:1ii==1o=.:54::=1=4:=7,:,:=:1n1i:0;:::.::i4=1=o:=6:.=2:
:t::I
!io.4265iio.4279!0.4292 l'jo.4306!I0.4319
I 1.5 llo.4332 1·10.434j5io.4357lio.4370i!o:43_8_2_i;_-_!o_.=43=9=4==!1;"·+=i,_o=.4=4=06==::::t:i;l_l
·1 1.6 1-lo.4452 l1110._.4463--Ji,o.44711_i,o.448114i,o.4495i_l;a__._=_4=50_1=05==.4:i=:t5:__=_15==l,!,0=·=45=1:,30=5.4==±5;=:i4, =
'_I 1.7 llo.4554 ilo.4564ljo.4573iio.4582jlo:4591!;-io=.4=59=9==n!!=o.=4lo6.=406=285=i:!:o:;.[4_6o3.34616
I 1.8 iio.4641 !,llo.464!9lo.4656Jio.4664Jlo.4671iio.4678iio.4686i0.4693 1]o.4699 1io.4706
I 1.9 l!o.4713qo.4719-[lo.4726l!o.473-2ilo.4738!lo.4744f!o.4750ilo.4756I0.4761 [o-.-4767
:l=I =2.=o===ilio.417:7120.477l8io.478'3]Jo:478-1810.4793iio.4798i0.4803 l:io.48080.4812 llo.4817
_t=::2=.j=1==110.43i2lo1.4826j!o.4830llo.4834!io.4838ilo.4842i0.4846 !0.4850 1/o.4854lio.4857
I 2.2 Iio.4861 i;,:ilo.48·6,4to:4868ll,,~---481117, 10.487i5io.4878iio.4881i0.4884 I0.4887 Iio.4890
+=[
t=jo.=4=89=3==ilt==ojo.==4.48=io89=9=.846===:t9:=tt=++0jI,1=oI1=.4=9=04==lr:t-:io=.4=9=06=0::=:t·!=t=49!o=i=o1.=4.3=4=9=~09=='-=t±9i1===
I 2.4 i!o.4918!io.4920l/0.4922l[~:~2-~~--_Jio.4!9o2.47929 0.4931 i[o.4932110.4934ito.4936
I 2.5 IJ0.4938ilo.4940j_ij'.o.49,4_1i_0.494l_t=3_i,o.=4=94=5=::itj=o.=49=4=6=1:t~io=.4=9=4===8,i==!,
r--2.6·---110.495!3lo.4955lio.4956;10.4957ll0.4959I0.4960 l!o.4961ir.4962 io.4963 lio.4964
-1 2.7 llo.4965 ijo.4966lio.4967T!~-~968ilo.4969I0.4970 !!o.4971!0.4972 lo.4973 lio:4974
-C~IC_!-:_1 0.4.974 !!o.4975llo.4976Ijo.4977 jjo.4977lt=io=.4=97=8=::10ti=;i.=4o==.9=tj=4i=79=97=9=:1lt
0 4981
1
i_lo.4982jjo.4982l!o.4983ilo.4984jjo.4984!io.4985llo.4985lio.4986jjo.4986
I 3.o l+=!o=.4=1f:9::8:_:=!_?o7==__:=:=.:=1;-i_-,-+:~-=1=-L-=-,=.-:-:~::8!=~-=-=~=~=--~:7:::1:=o=.=4=9:8:8:::110:.4:9:3=3====~
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Critical Values of the Mann-Whitney U
(Two-Tailed Testing)
llz a 3
3
.05
.01
--
--
4
.05
.01
--
--
5
.05
.01
0
--
6
."05 I
.01 --
7
.05 1
.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
.01
4
I
14
.05
.01
5
I
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
n,
4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20
0 0 11223344 55 66778
0 00 00000 11122 2233
0 1 2 3 4 4 5 6 7 8 9 10 11 11 12 13 14
-- 0 0 0 1 1 2 2 3 3 4 5 5 6 6 7 8
1 2 3 5 6 7 8 9 11 12 13 14 15 17 18 19 20
-- 0 1 1 2 3 4 5 6 7 7 8 9 10 11 12 13
2 3 5 6 8 IO II 13 14 16 17 19 21 22 24 25 27
0 I 2 3 4 5 6 7 9 10 I I 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 II 13 15 17 18 20 22 24 26 28 30
4 7 IO 12 15 17 20 23 26 28 31 34 37 39 42 45 48
1 3 5 7 9 I 1 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 IO 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 I I 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 l 19
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 I 19 127
8 13 18 24 30 36 42 48 54 60 67 73 79 86 92 99 105
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APPENDIX D: The t-distribution
ii 0.40
•
0
02
ii
5
0 5 :i
;:==d=~=P==il.,;::io=.3=2=49=2=o=+!i=1.=o=o:=·4:==:=;:o;=:6;=:;.3:i=l31=°.=o3=·7;:=..:.=.:·:=:~82==:=2~ti=.i=!1:2c=::.:7l3==1=.8=:~=o:=2===4+=1~,11=::6:i36===.3=1=::56==°:==·:=.=~=: i!.:,
2 llo.288675 i!o.816497 111.885618 2.919986 i:4.30265 116.96456 119.92484 jj31,5991 1
====3==ttj 1=0.=27=6=6=717==:6:±4j=:t8==i±o9i=t=.2h=.6=3=7=744==H=2.=35=3=3=63==;jj!=3=.1=82=4=51=5=!.8r:==4!4==0.=59=14===o:7:±=jo:!===1=r2i±~=ii.9=2
4 !io.270722 llo.740697 111.533206 2.131847 !!2.77645 1!3.74695 4.60409 /!8.6103
I/
+====+!=· ===:i-1====:=::::i:=========t-:====•it:·
===:it=" ===:tt=======r-1:1
6 1.10.264835 ilo.717558 111.439756 111.943180 112.44591 :!3.14267 113.70743 115.9588 1!
7 110.263167 lio.n1142 111.414924 !11.894579 li2.36462 !l2.99795 !13.49948 !l5.4079
!I
8==la i=o.=2=61==/9::=:02=7.1=0=6=38=7i1=.3=9! 6815 li1.859548 112.30600 j :=2=.8=96=4=!63==~·='r3. 5=5=39==:::;!,;=l5=.0=4=
9 ==sl'.;=::i?=·2=60=9=5=5=,:;.to=.=70=2=7=22=i=·=ili!11=.3.8=33032=9 1=1=3=::;i;:l2=·=26=2=1=6==:;!!2.82144
10 !!0.260185 i o.699812 iii.372184 !!1.812451 !12.22814 !12.76377
11 !lo.259556 !o.697445 :1_-i_3-63-___-4__---3-;0_l!t=1=.7=95=8=85=::i!b.2101029.791808
ii 12=:::::t,.o;:=::.:2,=59=.0..=..t3.:=ro-=3=.=69=5=4=l813.3=5::6:2ji17 1.782288 Ii2.17881 ii2.68100
ih.15927 ii4.5869
i•i3.10581 114.4370
i,,i
i=i3=.0=54=5=l4==i4:=i.l3=17=8===i
13===1.
1
=0::=·=:2::.·:5::::,::8:.::;5:9:=1 o=.6=93=8=2=9=~!;:i1=·=35=0=1=n=1=1:2:.:1;i5l10.37770933!2.65031
113.01228 1!4.2208
[j
1
14 !lo.258213 ilo.692417 iJ1.345030 111.751310 !!2.14479 !12.62449 2.97684
.1405
15==i:;::::io=7.2==85=8=5=:;1'10=_=69=1l=.13==947==0==,6:t0:!=I16.=7:5;3! 050 112.13145- -------rb.60248 !l:::[2=.9=46=7=·1==0=7t=:t2i =8===tll
16== 1;:::10=.2=5..7.=._=.=._5.i=!i9:t=o9==_=59=0=11.3=33627=5=7iii i'i1==7.=45=8=8=142==:r_.i="11==9i=ii92=.518349
ii2.92078
I]
I 17 lo.257347 iio.689195 i't1:=.i3=33=3=791==7=.=l!3t9==i 6=0=7='"ir-t:12==·=1=0:9=8:2:::::1,J~t=~==.=5=6=6:9:3::::i!.tj2=
18
i.:.....
19
ilo.257123 iio.688364 · 1 1.330391
i llo.256923 !!o.687621 1.327728
il.734064 i!2.10092 !12.55238 2.87844 !!3.9216
11
;-1.-729i33 i;:::::[;=.0=9=31i02==2_===5=3;=:i9=--4=8==++=l2=.8=6=09=3==;1ii.i3.
20 jio.256743 i!o.686954 !!1.325341 ·---:=[1=.7=24===7ti=b1.0=88596 !!2.52798 IJ2.84534 ~·t=i3=.84=9=5===
21 ljo.256580 ;lo.686352 li1.323188 !!1.720743 112.07961 !12.51765 !12.83136 !!3.8193
!,.!,
.
22
lh.717144 ii2.07387 !1;::::2=.5=08=3=2=::::;i;::::!2=.8=18=7=6=::::;
=::i:i=====::i,~.====:;:;:
..====::i ..f"====+====--:::;•H+•;:=i ====.+ ====,;:;::, ====
;I
23 l!0.256297 !!o.685306 li1.319460 ll1.n3872 !i2.06866 112.49987 1!2.80734 !13.7676 \\!
24==±1.;::::10=.2=56=1=7=3=:;:1t10=.=68=4=8=5i0il=.7=1=0l8ii812.31-7-8-13;6:::i2=.0=6=39=0==i"i2=_=49=2=1=6===1j;=:j2=.7=96=9=
25 ilo.256060 ::o.684430 'll.316345 ii1.708141
26=::::::!t=lo=.2=55=9=5=5==tit:lo=.=16=8.3==41==04=94=37==1=2.i7=i=0:i5618
27 !lo.255858 !!o.683685 1.313703 !1.703288
lt=l2=.0=5=95=4==:iltl2=·=48=5=1=1=::::1;i.!123.7.78275414 IJ
ib.05553 112.47863 jt:=7j2==7.8=7=1=::::7;i0t:==6b==6.==11
ii !!2.05183 112.47266 !!2.77068 113.6896
28
29 llo.255684 (o.683044 i!l.311434 jl.699127 !!2.04523 li2.45202 !12.75639 !!3.6594
II::::::::310r!=.,;:::lo:.2:5:55:o:5:::11:o=.=6=8=!21:=7.:35=:51:=:n0:=:141=5=::;,l:=i1=.6=97=2=6===1i:=b:1:t=l2==.4.==054==722==26==7=7:5:0:i0ib0. ib.6460
I±!:::::::i=nf==='±!:l:o:.:2::5::3::3=4=7=::::!tiio=.=6=7=44=9·+0
111.281552 ,h.644854 --·--1=1=.9=59=9=6==;:r:!12=.3=2=6=35==rji]!23..527950853
.:.:.:·.:=::=: =::::!:t::====1-i:=====t====::it=====i±:====
!I
!i
..:..:.:.!.:) :::
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F Table for alpha=0.10
~A,, ···
·I' :· ,, .
'·
.Fi.10.a1.1df2J
df2/dfl
1
2
3
4
5
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
6
3.77595 3.4633 3.28876 3.18076 3.10751 3.05455 3.01446 2.98304 2.95774 2.93693 2.90472 2.87122 2.83634 2.81834 2.79996 2.78117 2.76195 2.74229 2.72216
7
3.58943 3.25744 3.07407 2.96053 2.88334 2.82739 2.78493 2.75158 2.72468 2.70251 2.66811 2.63223 2.59473 2.57533 2.55546
2.5351 2.51422 2.49279 2.47079
8
3.45792 3.11312
2.9238 2.80643 2.72645 2.66833 2.62413 2.58935 2.56124 2.53804 2.50196 2.46422 2.42464
2.4041 2.38302 2.36136
2.3391 2.31618 2.29257
9
3.3603 3.00645 2.81286 2.69268 2.61061 2.55086 2.50531 2.46941 2.44034 2.41632 2.37888 2.33962 2.29832 2.27683 2.25472 2.23196 2.20849 2.18427 2.15923
10
3.28502 2.92447 2.72767 2.60534 2.52164 2.46058 2.41397 2.37715 2.34731
2.3226 2.28405 2.24351 2.20074 2.17843 2.15543 2.13169 2.10716 2.08176 2.05542
11
3.2252 2.85951 2.66023 2.53619 2.45118 2.38907 2.34157
2.304 2.2735 2.24823 2.20873 2.16709 2.12305 2.10001 2.07621 2.05161 2.02612 1.99965 1.97211
12
3.17655
2.8068 2.60552
2.4801 2.39402 2.33102 2.28278 2.24457 2.21352 2.18776 2.14744 2.10485 2.05968 2.03599 2.01149
1.9861 1.95973 1.93228 1.90361
13
3.13621 2.76317 2.56027 2.43371 2.34672 2.28298
2.2341 2.19535 2.16382 2.13763 2.09659 2.05316 2.00698 1.98272 1.95757 1.93147 1.90429 1.87591
1.8462
14
3.10221 2.72647 2.52222 2.39469 2.30694 2.24256 2.19313
2.1539 2.12195
2.0954 2.05371 2.00953 1.96245 1.93766 1.91193 1.88516 1.85723
1.828 1.79728
15
3.07319 2.69517 2.48979 2.36143 2.27302 2.20808 2.15818 2.11853 2.08621 2.05932 2.01707 1.97222 1.92431 1.89904 1.87277 1.84539 1.81676 1.78672 1.75505
16
3.04811 2.66817 2.46181 2.33274 2.24376 2.17833
2.128 2.08798 2.05533 2.02815 1.98539 1.93992 1.89127 1.86556 1.83879 1.81084 1.78156 1.75075 1.71817
17
3.02623 2.64464 2.43743 2.30775 2.21825 2.15239 2.10169 2.06134 2.02839 2.00094 1.95772 1.91169 1.86236 1.83624 1.80901 1.78053 1.75063 1.71909 1.68564
18
3.00698 2.62395 2.41601 2.28577 2.19583 2.12958 2.07854 2.03789 2.00467 1.97698 1.93334 1.88681 1.83685 1.81035 1.78269 1.75371 1.72322 1.69099 1.65671
19
2.9899 2.60561 2.39702
2.2663 2.17596 2.10936 2.05802
2.0171 1.98364 1.95573
1.9117 1.86471 1.81416 1.78731 1.75924 1.72979 1.69876 1.66587 1.63077
20
2.97465 2.58925 2.38009 2.24893 2.15823 2.09132
2.0397 1.99853 1.96485 1.93674 1.89236 1.84494 1.79384 1.76667 1.73822 1.70833 1.67678 1.64326 1.60738
21
2.96096 2.57457 2.36489 2.23334 2.14231 2.07512 2.02325 1.98186 1.94797 1.91967 1.87497 1.82715 1.77555 1.74807 1.71927 1.68896 1.65691 1.62278 1.58615
22
2.94858 2.56131 2.35117 2.21927 2.12794
2.0605
2.0084
1.9668 1.93273 1.90425 1.85925 1.81106 1.75899 1.73122 1.70208 1.67138 1.63885 1.60415 1.56678
23
2.93736 2.54929 2.33873 2.20651 2.11491 2.04723 1.99492 1.95312 1.91888 1.89025 1.84497 1.79643 1.74392 1.71588 1.68643 1.65535 1.62237 1.58711 1.54903
24
2.92712 2.53833 2.32739 2.19488 2.10303 2.03513 1.98263 1.94066 1.90625 1.87748 1.83194 1.78308 1.73015 1.70185
1.6721 1.64067 1.60726 1.57146
1.5327
25
2.91774 2.52831 2.31702 2.18424 2.09216 2.02406 1.97138 1.92925 1.89469 1.86578
1.82 1.77083 1.71752 1.68898 1.65895 1.62718 1.59335 1.55703
1.5176
26
2.90913
2.5191 2.30749 2.17447 2.08218 2.01389 1.96104 1.91876 1.88407 1.85503 1.80902 1.75957 1.70589 1.67712 1.64682 1.61472
1.5805 1.54368
1.5036
27
2.90119 2.51061 2.29871 2.16546 2.07298 2.00452 1.95151 1.90909 1.87427 1.84511 1.79889 1.74917 1.69514 1.66616
1.6356
1.6032 1.56859 1.53129 1.49057
28
2.89385 2.50276
2.2906 2.15714 2.06447 1.99585
1.9427 1.90014
1.8652 1.83593 1.78951 1.73954 1.68519
1.656 1.62519
1.5925 1.55753 1.51976 1.47841
29
2.88703 2.49548 2.28307 2.14941 2.05658 1.98781 1.93452 1.89184 1.85679 1.82741 1.78081
1.7306 1.67593 1.64655 1.61551 1.58253 1.54721 1.50899 1.46704
30
2.88069 2.48872 2.27607 2.14223 2.04925 1.98033 1.92692 1.88412 1.84896 1.81949
1.7727 1.72227 1.66731 1.63774 1.60648 1.57323 1.53757 1.49891 1.45636
40
2.83535 2.44037 2.22609 2.09095 1.99682 1.92688 1.87252 1.82886
1.7929 1.76269 1.71456 1.66241 1.60515 1.57411 1.54108 1.50562 1.46716 1.42476 1.37691
60
2.79107 2.39325 2.17741 2.04099 1.94571 1.87472 1.81939 1.77483 1.73802 1.70701 1.65743 1.60337 1.54349 1.51072 1.47554 1.43734
1.3952 1.34757 1.29146
120
2.74781 2.34734 2.12999
1.9923 1.89587 1.82381 1.76748 1.72196 1.68425 1.65238
1.6012
1.545 1.48207 1.44723 1.40938
1.3676 1.32034 1.26457 1.19256
inf
2.70554 2.30259
2.0838 1.94486 1.84727 1.77411 1.71672
1.6702 1.63152 1.59872 1.54578 1.48714
1.4206 1.38318 1.34187 1.29513 1.23995
1.1686
1
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