POLYTECHNIC OF NAMIBIA SCHOOL OF HEALTH AND APPLIED SCIENCES

DEPARTMENT of MATHEMATICS AND STATISTICS

QUALIFICATIONS:

1. BACHELOR OF INFORMATION TECHNOLOGY (Business Computing) 2. BACHELOR OF INFORMATION TECHNOLOGY (Software Development)

COURSE NAME: APPLIED STATISTICS & PROBABILITIES for IT

COURSE CODE: ASP61 OS

DATE:

DURATION:

MARKS:

JUNE 2015

3 HOURS

100

1st OPPORTUNITY EXAMINATION PAPER

EXAMINER:

MODERATOR:

INSTRUCTIONS:

MR. ROUX, A.J

DR. AJIBOLA, IKO

a. Answer all the questions in the booklet provided b. Show clearly all the steps used in the calculations c. All written work MUST be done in blue or black ink and sketches must be

done in pencils.

PERMISSABLE MATERIAL 1. Calculator

APPENDIX: 1) Formulae Sheet 2) Statistical Tables

This ｰ｡ｾｊ･ｲ＠ consist of 6 pages including this cover

1

ﾷｾ＠ '

QUESTION 1 [201

1.1 Which of the following measures of central tendency can reliably be used when dataset

has outliers?

a) Mean b) Median c) Mode d) All the above [2]

1.2 A sample is

a) An experiment in the population

b) A subset of the population

c) A variable in the population

d) An outcome of the population [2]

1.3 A parameter refers to

a) Calculation made from the population

b) A measurement that is made from the population

c) A value observed in the experiment

d) All of the above

[2]

1.4 Weight is a ____ variable

a) Continuous b) Discrete c) Ordinal d) Interval [2]

1.5 Researchers do sampling because of all of the following reasons except

a) Reduce cost

b) Can be done in a shorter time frame

c) Sampling is interesting

d) Easy to manage due to manageable logistics requirements [2]

1.6 Event A and B are said to be collectively exhaustive in statistics if

a) A and B are mutually exclusive

b) The union of A and B equals the sample space

c) The union of A and B is an empty set

d) The intersection of A and B is the same as the union [2]

2

1. 7 Which of the following is NOT a possible probability

65 a)-1oo b) 1.16 c) 0 d) All of the above [2]

1.8 A student is chosen at random from a class of28 girls and 12 boys. What is the

probability that the student is NOT a boy

3 a)-

10 b) 28

12 c) 28 d) 2.

10 [2]

1.9 On a multiple choice test, each question has 4 possible answers. If you make a random

guess on the first question, what is the probability that you are correct

a) 4 b) 0 c)0.25 d) 100 [2]

1.10 A 6-sided die is rolled. What is the probability of rolling a 3 or a 6

a) 12 b)! 6

QUESTION 2 [101

1 c)-

3 d) 0 [2]

The data below shows scores in ASP61 OS for a random sample of 7 students in a class test.

86, 72, 23, 56, 62,

Use the data provided to find the following:

2.1 The average score

a) 64 b) 63.4

2.2 The modal scores

a) 86 b) no mode

2.3 The median scores

94, 48

c) 100

c) 23

a) 72 b) 62 c) no median

3

d) none of the provided

[1]

d) none of the provided

[1]

d) none of the provided

[2]

2.4 The range of the scores

a) 72 b) 73

2.5 The first quartile of the scores

a) 62 b) 48

2.6 The third quartile of the scores

a) 88 b) 94

2.7 The inter-quartile range for the scores

a) 0 b)38

QUESTION 3 [15]

c) 38

c) 71

c) 62

c)17

[2]

d) none of the provided

[1]

d) none of the provided

[2]

d) none of the provided

[2]

d) none of the provided

[1]

A switchboard operator receives on average 6 calls per minute. What is the probability that for the next minute:

3.1) No calls will be received 3.2) Exactly 4 calls will be received 3.3) At least three calls will be received

Question 4 [15]

[4] [4] [7]

It is known that 90% of drivers are wearing seat belts. Thus, in a random sample of 25 drivers, what is the probability that:

4.1) Exactly 20 of them will wear their seat belts 4.2) At least twenty of them will wear their seat belts 4.3) At most twenty four of them will wear their seat belts

4

[3] [6] [6]

ＳＮｊｾＺｬ･｣ｴ＠ the suitable critical value for the decision rule (3)

a) ± 1.645 b) ± 1. 96 c) ± 2.575 d) None of the above

3.4) Compute the sample statistic (3)

a) M 0.125 b) M 2. 225 c) -1.024 d) None of the above

3.5) What is the decision (3)

a) Reject Ho

b) Do not reject H o .... /• J

,•'''

,/

c) None of ｴｨｾ＠ .. above

lrnn..cun that 90% of drivers are wearing seat belts. Thus, in a random sample o s, what is the probability that:

4.1) Exactly 20 of them will wear their seat belts · ·····----... 4.2) At least twenty of them will wear their seat belts

ＭＭＧＴＺＳｽＭＭｾＮｾＭｾｴｾｯｳｴ＠ twenty four of!Jle!!! ｷ｟ｩＡＮＡｽｙｾＮｲｵＺ＠ . .th.e.ir .. .s.eat.belts ...... ﾷＭＭﾷﾷｾＭ ... --.... ｾＭ .............. ---·--·----... -

QUESTION 5 [10]

A factory production line is manufacturing bolts using three machines, A, B and C. Of the

total output, machine A is responsible for 25%, machine B for 35% and machine C for the

rest. It is known from previous experience with the machines that 5o/o of the output from

machine A is defective, 4% from machine Band 2% from machine C.

5.1) What is the probability that a randomly selected bolt is defective?

5

[7]

5 .2) A bolt is chosen at random from the production line and found to be defective. What

is the probability that it came from machine B? [3]

QUESTION 6 [301

The daily study time for high school learners for the coming final examination period can be approximated by a normal distribution with a mean time of 100 minutes per day, and a standard deviation of 1 0 minutes.

6.1) Determine the probability that the maximum daily study time will be between 100 minutes and 125 minutes ( 5)

6.2) Determine the probability that the maximum daily study time will be between 94 minutes and 108 minutes (8)

6.3) Determine the probability that a given day's maximum study time will be exceed 87 minutes (5)

6.4) A chief electrical engineer at a small coal driven power station has accumulated the following information on quarterly electricity generation at his plant (in MegaWatts):

' 2005 2006 : 2007 i 2008

-----------

Jan-Mar 21 35 39 78

ｾｐｾ］ＡｾＮｾＮＭ 42 54 82 114

ｊｵｩＭｾｾ｟ｐｴ＠ ... 60 91 136 160 Oct-Dec 12 14 28 40

ＭＭｾｾＧ＠ ---- .. ＭｾＭＭｾＭ

Find the typical median seasonal indexes (12)

Xxxxxxxxxxxxxxxxxxxxxx END OF EXAMINATION PAPER xxxxxxxxxxxxxxxxxx

6

STATISTICAL FORMULAE SHEETS & TABLES

APPENDIX A: Formulae Sheet

APPENDIX B: Additional Formulae Sheet

APPENDIX C: The Standard Normal Distribution

APPENDIX D: Areas in the tail of the Standard Normal Distribution

APPENDIX E: The t-distribution

APPENDIX F: The Chi-Square Distribution

APPENDIX A

Population mean, raw data

Sample mean, raw data

n

Weighted mean

Geometric mean

Geometric mean rate of increase

GM Value at end of period

Value at start of period

Sample mean grouped data

X = }:fx n

Median of grouped data

- 1.0

T -CF Median = L + (Class width)

f

Mean deviation

"' I X-X I MD = LJ

n

Linear regression equation

Y = a+ qX

Sample variance for raw data

n-1

Sample variance, raw data computational form

I;X2 - <Dt>z g2 = n

n-1

Sample standard deviation, raw data

_EX2 - CVCJ2 S = '\I r.

ｾ＠ n -1

Sample standarddeviation, grouped data

Coefficient of variation

s CV= {lOO)

X

Location of percentile p

Lp = (n+l) 100

Pearson' s Correlation coefficient

n (l:XY) - (}:X) (l: Y) r =

Correlation test of hypothesis

t = ｲｾ＠

Population standard deviation for raw data

". ｾ＠ r.cx ｾ＠ <'l'

Population variance for raw data

E<X- f .. L)2 az =

N

Slope of regression line

b = n (l:XY) - {2:X) (2:Y)

Intercept of a regression line

a

The Range

Range

2:Y --b

n ( r:)

highest - lowest

APPENDIX B: ADDITIONAL FORMULAE

Mode = L + ( d1 J x c dl +dz

. . Q jn position 1 = -4

. . p jn position ,. = -. 100

P(AIB)= P(AnB) P(B)

X-J.l z=--

a

zcalc = xi -Xz

sz sz _1 +_l

ｾ＠ nz

p-:c z=--====

ｾｩｲＨｬＺ＠ ir)

P= A (l+if

value

value ( jn -F))xc

P. = L + ....:_1_0_0 _ __:;___ 1 fpj

X-J.l zcalc ］｡Ｏｾ＠

xi-Xz

(calc = ---;================(===J (n ＭＱＩｳｾ＠ + (n -1)si __!__ + __!__

ｾＫｮ Ｒ ＭＱ＠ ｾ＠ n2

PV = P(1+if (1+ JY

r = (1 + i)m -1

q =1- p

D = B(l-iY

APPENDIX C: The Standard Normal Distribution

z I o.oo 1 o.o1 o.o2 o.o3 I 0.04 I 0.05 I 0.06 I 0.07 I 0.08 I 0.09 I

o.o 1 o.oooo 1 o.oo4o 1 o.oo8o 1 o.o 120 jo.0160 jo.0199 10.0239 jo.0279 1 o.o319 lo.o359

0.1 jo.0398 jo.0438 lo.0478_ jo.0517 _ jo.o557 .. _,0.0596 jo.o636 jo.0675 jo.o714 jo.o753

0.2 lo.0793 jo.o832 lo.0871 _jo.0910 __ jo.0948 jo.o987 jo.1o26 jo.I064 jo.1103 jo.1141

o.3 lo.1179 !o.1211 lo.I255 lo.1293 10.1331 jo.1368 10.1406 lo.1443 1 o.I48o lo.1517

o.4 1 o.I554 1 o.I591 lo.I628 lo.I664 1 o.11oo 10.1736 jo.1112 jo.I808 lo.I844-jo.1879

1 o.5 I 0.1915 - I 0.1950 I 0.1985 ｪｯＮｾＰＱＹ＠ jo.2054 _10.2088 10.2123 lo.2157 10.2190 10.2224

ｬｯＮＶＭｦｄＮｭＷＭｬｯＮＲＲｾＱｯＮＲＳＲＴ＠ jo.2357 jo.2389 lo.2422 10.2454 jo.2486 lo.2517 lo.2549

1 o.1 lo.258o jo.26u jo.2642 lo.2673 10.2704 jo.2734 lo.2764 lo.2794 1 o.2823 10.2852

I o.8 jo.2881 jo.2910 ＱｾＮＲＹＳＹ＠ u jo.2967 _ ＱｾＮＲＹＹＵ＠ 10.3023 jo.3051 10.3078 ｬｯｾＳＱＰＶ＠ 10.3133

I o.9 f0.3159 joJ186 _lo.3212 loJ238 joJ264 10.3289 10.3315 jo.3340 loJ365 10.3389

I 1.0 loJ413 loJ438 jo.3461 loJ485 loJ508 10.3531 jo.3554 jo.3577 r

loJ621 1 oJ599

1 1.1 10.3643 10.3665 10.3686 10.3708 loJ729 10.3749 jo.3770 joJ790 r----rm-0.3810 0.3830

I 1.2 loJ849 joJ869 !o.3888 lo.3907 lo.3925 10.3944 lo.3962 10.3980 10.3997 lo.4015

1 1.3 10.4032 1 o.4o49 1 o.4o66 1 o.4082 10.4099 10.4115 fo.4131 10.4147 !0.4162 1 o.4177

I 1.4 f0Ai92-lo.4207 lo.4222 jo.4236 10.4251 10.4265 jo.4279 10.4292 jo.4306 ioA3i9-

I 1.5 'jo.4332_ _lo.4345 ｪｯＮｾＳＵＷ＠ lo.4370 10.4382 Ｑ｟ＰＮｾＳＹＴ｟Ｍ lo.4406 10.4418 lo.4429 10.4441

I 1.6 joA452 jo.4463 jo.4474 lo.4484 lo.4495 lo.4505 lo.4515 lo.4525 10.4535 jo.4545

I 1.7 lo.4554 jo.4564 jo.4573 lo.4582 lo.4591 lo.4599 lo.4608 10.4616 10.4625 1 o.4633

I 1.8 lo.4641 lo.4649 lo.4656 jo.4664 I I I I 1.9 lo.4713 jo.4719 .,0.4726 !o.4732 lo.4738 jo.4744 r--lo.-47-5o-r--jo.-47-56-lr--o.-47-61-1.-o.-47-67-!

I 2.0 jo.4772 lo.4778 lo.4783 lo.4788 lo.4793 lo.4798 lo.4803 lo.4808 jo.4812 ro.-4817 I I 2.1 10.4821 lo.4826 jo.4830 10.4834 lr--o.-48_3_8 ＭｲｾＰＭＮＴＭＸＴＭＲＭＱＰＮＴＸＴＶ＠ 10.4850 10.4854 ｛ｏａｾｾ＠

0.4671 0.4678 0.4686 0.4693 0.4699 0.4706

[ 2.2 f0As61-lo.4864 jo.4868 joASn-lo.4875 lo.4878 ....-jo.-48_8_1 -r-lo-.4-88_4_ !o.4887 jo.4890 j

! 2.3 lo.4893 joA896 lo.4898 io.4901 l.-o.-49-04--·Io_.4_9_06-Io.4909 j0.4911 jo.4913 fo.4916-l 1 2.4 ｾＮＴＹＱＸ＠ lo.4920 lo.4922 jo.4925 lo.4927 jo.4929 r--lo.-49_3_1 -r-jo-.4-93_2_1r--o.-49-34-j0.4936

1 2.5 I 0.4938 1 o.494o [OA941 10.4943 1 ｯＮｾＹＴＵ＠ 10.4946 10.4948 1 0.4949 ｦｑＮＴＹｾ＠ 1 0.49521

1 2.6 10.4953 10.4955 1 0.495_? 10.4957 10.4959 _ 1 ｾＮＴＹＶＰ＠ 10.4961 1 0.4962 10.4963 10.4964 I 2.1 joA965 [OA966 lo.4967 ｪｯＮｾＹＶＸ＠ lo.4969 lo.497o lo.4971 lo.49?2 fo.4973 fD.4974-2.8 10.4974 10.4975 10.4976 jo.4977 .---10-.49_7_7 -rlo-_.4-9-78-,0.4979 10.4979 ｲｯｾＴＹＸＰ＠ jo.4981

I _ 2.9 jo.4981 lo.4982 lo.4982 ｬｯＮｾＹＸＳ＠ _ .---jo.-49-84 _____ -,_lo--.4-98-4-rl-o.4-9-85-I0.4985 lo.4986 _ lo.4986 I j3Jl[OA98-7fOA987-fOA987-Io.4988 lo.4988 IQ.4989 jo.4989 jo.4989 fo.499o--lo..499o Ｍｾ＠

APPENDIX D

AREAS IN THE TAIL OF THE NORMAL DISTRIBOTION

z .00 .01 .02 .03 .04

0.0 .5000 .4960 .4920 .4880 .4840 0.1 .4602 .4562 .4522 .4483 .4443 ,. '} .4207 .4168 .4129 .4090 .4052 v . .;;..,

u.3 .3821 .3183 .3745 .3707 .3669 0.4 .3446 .3409 .3372 .3336 .3300 0.5 .3085 .3050 .3015 .2981 .2946 0.6 .2743 .2709 .2676 .2643 .2611 0.7 .2420 .2389 .2358 .2327 .2296 0.8 .2119 .2090 .2061 .2033 .2005 0.9 .1841 .1814 .1788 .1762 .1736 1.0 .1587 .1562 .1539 .1515 .1492 1.1 .1357 .1335 .1314 .1292 .1271 1.2 .1151 .1131 .1112 .1093 .1075 1.3 .0968 .0951 .0934 .0918 .0901 1.4 .0808 .0793 .0778 .0764 .0749 1.5 .0668 .0655 .0643 .0630 .0618 1.6 .0548 .0537 .0526 .0516 .0505 1.7 .0446 .0436 .0427 .0418 .0409 1.8 .0359 .0351 .0344 .0336 .0329 1.9 .0287 .0281 .0274 .0268 .0262 2.0 .0228 .0222 .0217 .0212 .0207 2.1 .0179 .0174 .0170 .0166 .0162 2.2 .0139 .0136 .0132 .0129 .0125 2.3 .0107 .0104 .0102 .0099 .0096 2.4 .0082 .0080 .0078 .0075 .0073 ? ,-_,:::> .0062 .0060 .0059 .0057 .0055 2.6 .0047 .0045 .0044 .0043 .0041 2.7 .0035 .0034 .0033 .0032 .0031 2.8 .0026 .0025 .0024 .0023 .0023 2.9 .0019 .0018 .0018 .0017 .0016 3.0 .0014 .0013 .0013 .0012 .0012

.05 .06 .07 .08 .09

.4801 .4761 .4721 .4681 .4641

.4404 .4364 .4325 .4286 .4247

.4013 .3974 .3936 .3897 .3859

.3632 .3594 .3557 .3520 .3483

.3264 .3228 .3192 .3156 .3121

.2912 .2877 .2843 .2810 .2776

.2578 .2546 .2514 .2483 .2451

.2266 .2236 .2206. .2177 .2148

.1977 .1949 .1922 .1894 .1867

.1711 .1685 .1660 .1635 .1611

.1469 .1446 .1423 .1401 .1379

.1251 .1230 .1210 .1190 .1170

.1056 .1038 .1020 .1003 .0985

.0885 .0869 .0853 .0838 .0823

.0735 .0721 .0708 .0694 .0681

.0606 .0594 .0582 .0571 .0559

.0495 .0485 .0475 .0465 .0455

.0401 .0392 .0384 .0375 .0367

.0322 .0314 .0307 .0301 .0294

.0256 .0250 .0244 .0239 .0233

.0202 .0197 .0192 .0188 .0183

.0158 :0154 .0150 .0146 .0143

.0122 .0119 .0116 .0133 .0110

.0094 .0091 .0089 .0087 .0084

.0071 .0069 .0068 .0066 .0064

.0054 .0052 .0051 .0049 .0048

.0040 .0039 .0038 .0037 .0036

.0030 .0029 .0028 .0027 .0026

.0022 .0021 .0021 .0020 .0019

.0016 .0015 .0015 .0014 .0014

.0011 .0011 .0011 .0010 .0010

APPENDIX E : The t-distribution

I dt\p I 0.40 0.25 0.10 0.05 0.025 0.01 0.005 0.0005 I

I

I 1 1 o.32492o 11.000000 13.077684 16.313752 112.70620 131.82052 163.65674 1636.6192

I 2 1 o.288675 1 o.816497 11.885618 12.919986 14.30265 16.96456 _,9.92484.- 131.5991

I 3 1 o.276671 1 o.764892 11.637744 12.353363 ｾ＠13.18245 ,4.54070 _15.84091 112.9240

I 4 10.210122 1 o.740697 11.533206 12.131847 ,2.77645 ,3.74695 _,4.60409 18.6103

I 5 jo267'i"81 10.726687 11.475884 12.015048 12.5-7058 13.36493 ,4.03214 16.8688 I

I 6 1 o.264835 1 o.717558 11.439756 11.943180 12.44691 13.14267 ,3.70743 j5.9588 --

i 7 1 o.263167 1 o.711142 11.414924 11.894579 12.36462 f2.99795 13.49948 15.4079 1

r-. 8 1 o.261921 1 o.706387 1_1.396815 11.859548 . 12.30600 ,2.89646 13.35539 5.0413 I

I 9 1 o.260955 10.102122 11.383029 11.833113 12.26216 12.82144 13.24984 4.7809

I 10 1 o.260185 1 o.699812 11.372184 11.812461 12.22814 12.76377 13.16927 4.5869

I 11 1 o.259556 1 o.697445 11.363430 11.795885 12.20099 12.71808 13.10581 4.4370 I

I 12 I 1 o.695483 11.356217 11.782288 12.17881 12.68100 ,3.05454 4.3178

I I 0.259033

I 13 1 o.258591 1 o.693829 11.350171 11.770933 ,2.16037 12.65031 13.01228 4.2208 I

I 14 1 o.258213 1 o.692417 11.345030 11.761310 12.14479 12.62449 ,2.97684 f4.1405

I 15 1 o.257885 1 o.691197 11.340606 11.753050 ,2.13145 12.60248 !2.94671 14.0728

I 16 1 o.257599 1 o.690132 11.336757 11.745884 ,2.11991 12.58349 ,2.92078 !4.0150

I 17 10257347 1 o.689195 11.333379 11.739607 12.10982 ＬＲＮＵＶＶｾＱＲＮＸＹＸＲＳ＠ 13.9651

I 18 1 o.257123 1 o.688364 11.330391 11.734064 12.10092 r 2.55238 12.87844 f3.9216

119 1 o.256923 1 o.687621 11.327728 11.729133 ,2.09302 ! 2.53948 12.86093 13.8834 I I 20 1 o.256743 jo.686954 11.325341 11.724718 12.08596 12.52798 12.84534 ,3.8495--

I 21 1-0.256580 1 o.686352 11.323188 11.720743 ,2.07961 12.51765 12.83136 13.8193

I 22 1 o.256432 1 o.685805 11.321237 11.717144 ,2.07387 12.50832 ,2.81876 r,7921

I 23 1 o.256297 __ 1 o.685306 _11.319460 11.713872 12.06866 12.49987 -- 12.80734 13.7676-

I 24 1 o.256173 1 o.68485o 11.317836 11.710882 12.06390 12.49216 12.79694 f3.7454

125 10.256060 1 o.68443o 11.316345 11.708141 12.05954 12.48511 12.78744 13.7251

I 26 1 o.255955 1 o.684043 ＬＱＮＳＱｾＹＷＲ＠ 11.705618 ,2.05553 12.47863 ,2.77871 13.7066 I

r-v·-1 0.255858 10.683685 11.313703 11.703288 12.05183 12.47266 ,2.77068 j3.6896 ---.

I 28 1 o.255768 1 o.683353 11.312527 fl.701131 ,2.0484_1 __ 12.46714 ＱＲＮＷＶＭＭｾＭＭ 6326 3.6739

I 29 10.255684 f0.683044 11.311434 11.699127 12.04523 12.46202 f2.75639 ,3.6594

I 30 10.255605 1 o.682756 11.310415 11.697261 12.04227 f2.457M-12.75ooo 13.6460

l inf 1 o.253347 1 o.67449o 11.281552 11.644854 ,1.95996 12.32635 ,2.57583 [ 3.2905 J

APPENDIX F: The Chi-Square Distribution

1 df\p 1 .995 .990 .975 .950 .9oo .750 .5oo .250 .1oo .o5o .025 .010 .oo5

ltl o.oooo4 I o.ooo16 I o.ooo98 I o.oo393 I o.01579 I 0.10153 I 0.45494 11.32330 12.70554 13.84146 15.02389 16.63490 17.87944 I 121 o.o1oo3 1 o.o2o1o 1 o.o5o64 1 ｯＺｾｯＲＵＹ＠ 10.21012 1 o.57536 ＱＱＮＳＸＶＲｾ＠ 12.77259 14.60517 15.99146 ＱＷＮＳＷｾＷＶ＠ 19.21034 110.59663

j3Jo.o7I72 jo.11483 lo.21580 lo.35185 jo.s8437 !1.21253 12.36597 14.10834 16.25139 17.81473 19.34840 111.34487112.838161

141 o.20699 1 o.29711 fOA8442 1 o.no12 11.06362 11.92256 13.35669 15.38527 !7.77944 19.48773 111.14329 113.27670 f14.86026 fSI 0.41174 10.55430 jD.83121 jt14548 11.61031 j2.6746o 14.35146 f"6.62568 19.23636 111.07050 112.83250 115.08627116.74960

j61 0.67573 I 0.87209 11.23734 11.63538 12.20413 ,3.45460 15.34812 17.84080 110.64464112.59159114.44938 116.81189,18.54758 I

i 1 1 o.98926 j1.23904Tl.68987-f2.16735 12.83311 [4.25485 16.34581 f9.03715-112.o17o4jt4.06714116.o1276118.47531 !2o.21174 I fslt.34441 11.64650 12.17973 12.73264 13.48954 f5.07064 j7.344i2-j10.21885 113.36157115.50731 jt7.5345512o.o9024121.95495

19-11.73493 12.08790 j2.70039 13.32511 .14.16816 15.89883 ＱＸＮＳｾＲＸＳ＠ 111.38875 r14.68366,16.91898119.02277121.66599123.58935

Jloj2.15586 12.55821 13.24697 13.94030 14.86518 ,6.73720 ,9.34182 112.54886 115.98718118.30704120.48318 123.20925 125.18818

llll2.60322 13.05348 13.81575 14.57481 ,5.57778 17.58414 110.34100 113.70069117.27501 119.67514121.92005,24.72497126.756851

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