J

nAm I Bl A UnIVERSITY OF SCIEnCE AnD TECHnOLOGY

FACUL TV OF HEALTH AND APPLIED SCIENCES

Department of Mathematics and Statistics

QUALIFICATION: BACHELOR OF SCIENCES (APPLIED MATHEMATICS AND STATISTICS)

BACHELOR OF SCIENCE (TOWN AND REGIONAL PLANNING)

QUALIFICATION CODE: 07BAMS and 07BTRP LEVEL: 4

COURSE: INTRODUCTION TO APPLIED COURSE CODE: IASSOlS

STATISTICS

DATE: JUNE 2016

DURATION: 3 HOURS

EXAMINER(S)

MODERATOR:

SESSION:

MARKS: 100

FIRST OPPORTUNITY EXAMINATION QUESTION PAPER

MR ROUX, A.J

MR NTIRAMPEBA, D

INSTRUCTIONS 1. Answer ALL the questions in the booklet provided.

2. Show clearly all the steps used in the calculations.

3. All written work must be done in blue or black ink and sketches must

be done in pencil.

PERMISSIBLE MATERIALS

1. Non-programmable calculator without a cover.

THIS QUESTION PAPER CONSISTS OF 5 PAGES (Including this front page)

Attachments Z-table; 2) formula sheet; 3) t-table 4) 1 x A4 Graph Paper

QUESTION 1 [32]

1.1 State whether each of the following variables is qualitative or quantitative and

indicate its measurement scale.

1.1.1 Age (2}

1.1.2 Gender (2}

1.1.3 Class rank (2}

1.1.4 Make of car (2}

1.1.5 Number of people favouring a particular soft drink (2}

1.2 For each of the following random variables, indicate if the data type is discrete or

continuous

1.2.1 The number of accidents per hour (2}

1.2.2 The temperature at sunset (2}

1.2.3 The amount of money in my pocket. (2)

1.2.4 The fuel used by a taxi to travel to the coast (2}

1.2.5 The size categories for shoes (2}

1.3 Write down ON LV the letter corresponding to your choice next to the question number.

1.3.1 Events A and Bare said to be mutually exclusive in statistics if: (2}

a) A and Bare independent events b) The intersection of A and B is an empty set

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

1.3.2 Events A and B are said to be collectively exhaustive in statistics if: (2}

a) A and Bare 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

1

1.3.3 What is a sample space? {2)

a) The sample from a national survey b) The space between events A and B c) The set of all possible outcomes of an experiment d) None of the above

1.3.4 A subjective probability is a probability (2)

a) Between 0 and 1 b) Very close to zero c) Based on individual judgement or assessment d) None of the above

1.3.5 A student is chosen at random from a class of 16 girls and 14 boys. What is the

probability that the student chosen is a girl? (2)

a) ｾＵ＠

b) /{5 c) 0.35 d) 0 e) None

1.3.6 A glass jar contains 5 red, 3 blue and 2 green jelly beans. If a jelly bean is chosen at

random from the jar, what is the probability that it is red or green? {2)

a) ｾ＠

b) Xo c) /{o d) 0.35

e) None

2

QUESTION 2 [33]

The data below represents the cost of electricity (in N$) during the month of June, 2003 for

a random sample of 40 two- bedroom apartments in Windhoek

250 600 553 295 210 389 400 625 850 723

157 423 300 239 487 535 762 532 672 678

522 435 628 456 239 863 764 433 677 245

342 296 456 586 349 421 568 825 924 598

2.1 Summarize the data in a frequency distribution with classes of equal width of 100

rand, starting at N$100 - < N$200 ; N$200 - < N$300 ; ext ..

2.2 Use the data obtained in 2.1 to draw a histogram and a polygon

2.3 Use the grouped data set produced in 2.1 to calculate and interpret the

2.3.1 mean

2.3.2 median

2.3.3 mode

(6)

( 6 + 4 = 10)

(5)

(5)

(5)

2.3.4 Based upon your measures of central tendencies calculated, comment of the shape

of the distribution

QUESTION 3 [20]

3.1 Suppose that the following contingency table was set up:

A B

What is the probability of:

3.1.1 Event A

3.1.2 Event A and C

3.1.3 Event A and B

c 10 25

3

D

30 35

(2)

(1)

(1)

(1)

I . ' •

3.1.4 Event B or D (2)

3.1.5 Event CorD (2)

3.1.6 P(A/D) (2)

3.2) A local ambulance service handles 0 to 5 service calls on any given day. The probability distribution for the number of service calls is as follows

Number of service calls (x) Probability, p(x)

0 0.10 1 0.15 2 0.30 3 0.20 4 0.15 5 0.10

3.2.1 Find P ( 1 :::; x :::; 3)

3.2.2 What is the expected number of service calls?

3.2.3 What is the variance in the number of service calls?

3.2.4 What is the standard deviation?

3.2.5 What is the coefficient of variation in the number of service calls

QUESTION 4 [15]

{2)

(2)

(4)

(1)

(2)

The table below shows the annual rainfall (x 100 mm) recorded during the last decade at the Goabeb Research Station in the Namib Desert

2004 2005 2006 2007 2008 2009 2010 2011 2012 2013

3.0 4.2 4.8 3.7 3.4 4.3 5.6 4.4 3.8 4.1

4.1 Construct a scatter plot (4)

4.2 Determine the least squares trend line equation, using the sequential coding method with 2004 = 1 . (7)

4.3 Use the trend line equation obtained in 4.1 to estimate rainfall for 2002 and 2017 (4)

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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

APIPlENUI X A

Population mean, ra>< data

I;x

N

Sample mean, ｲ｡ ｾｲ＠ data

n

ｩｾ･ｩｧｨ＠ ted mean

Xw =

Geometric mean

Geometric mean rate of increase

GM - n Value at end of period

Valu e at start of period

Sample mean grouped data

n

Median of grouped data

- 1 . 0

z.Iedian = L +

ｾ＠ -CF 2

f (Class ｾ＼ｩ､ｴｨＩ＠

Mean deviation

MD= I: I X-X

n

Linear regression equat ion

Y = a+ h X

Sample variance for ;:-a11 data

'\' - 2

6 (X -X)

n-1

Sample variance , ra1< data c omputational form

2.:x2 _ <L:>=l' n

n- 1

Sample standard deviation, ra"rdata

I;X2 - (l;Xl'

S - \J n

ｾ＠ n- 1

Sample standard deviation, grouped data

Coefficient of variation

cv = s X

(100)

Location of percentile

Lp = (n + 1) p

100

Pearson' s Correlation coefficient

n (I;XY)- (I;X) (I;Y) r

Correlation test of hypothesis

t =

Population standard deviation for ra1< data

Population variance for ra1< data

L (X -

N

Slope of regression line

b = n (I;XY) - (I;X ) (I;Y)

Intercept: of a regression line

a

The Range

Range highest .. lmvest

APPENDIX B: ADDITIONAL FORMULAE

0 0 Q . __ jn pOSitiOn J 4

position P1 = jn

100

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

X-JL z=--

7 -ｾｭｬ｣Ｍ

CJ

xl -x2 s2 s2 ｾＫ｟｟Ｑ｟＠

nl n2

p-re z =---==== J rc(l: rc)

P= A (1 + i)"

value

value ( )!!___-F)) x c

p = L +_,__I 0_0_--=--J ft ,

)

P(x) = n! rex (1-rc)"-x x!(n- x)!

X-JL z calc = () / ..[;;.

PV = P(l + i)" (1 + j)"

r = (1 + i) 111 -1

xe-). P(x)= --

x!

q = 1- p

D = B(l- i)"

APPENDIX C: The Standard Normal Distribution

0 z

1 z o.oo 1 o.o1 1 o.o2 1 o.o3 1 o.o4 1 o.o5 1 o.o6 o.o7 1 o.o8 0.09

I o.o lo.oooo [Q.004o-l o.oo8o fD.OI2o--l o.oi60 f0.0199-l o.o239 o.0279 lo.o319 0.0359

I 0.1 ｛ｑＮＰＳｾ Ｑ ｯＮＰＴＳＸ＠ jo.o478 lo.o517 [D.0557 lo.0596 lo.0636 o.0675 lo.0714 0.0753

I 0.2 lo.o793 lo.o832 lo.o871 lo.o9IO ｉｑＮｏ ｾ ｉ ｯ Ｎ ｯＹＸＷ＠ 10.1026 o.I064 10.1103 0.1141

I o.3 lo.ll79 lo.1211 lo.l255 lo.1293 lo.l331 lo. l368 lo.I406 0.1443 lo. I480 0.1517

I 0.4 lo. l554 lo.l591 lo.l628 I0.1664 lo.I700 [Q.l 736 lo.1n2 0.1808 lo.l844 0.1879

I o.5 lo. l915 ｛ｄＮｉＹｳｯＭＭＭ ｬ ｯ Ｎ ｬＹ ｾ Ｑ ｯＮＲｯｬＹ＠ lo.2054 lo.2088 j0.2123 o.2157 lo.2190 IQ.2224

I o.6 lo.2257 lo.2291 ｛ＰＲＳｾ Ｑ ＰＮＲＳＵＷ＠ fo.2389 lo.2422 lo.2454 o.2486 lo.2517 10.2549

I 0.7 lo.2580 [02611 lo.2642 lo.2673 lo.2704 lo.2734 lo.2764 jo2794-l o.2823 lo.2852 -

I o.8 lo.2881 lo.29IO ·1 0.2939 lo.2967 10.2995 lo.3o23 lo.3051 lo.3078 lo.3I06 lo.3133

I o.9 lo.3159 lo.3186 lo.3212 [03238-fo.3264 lo.3289 ｛ＰＳｾ Ｑ ｯＮＳＳＴｯ＠ jo.3365

I t.o l o.3413 lo.3438 jo34-61-l o.3485 -1 0.3508 lo.3531 lo.3554 lo.3577 -J o.3599 lo.3621 I ro.-33 89

I 1.1 lo.3643 lo.3665 lo.3686 lo.3708 10.3729 lo.3749 fD.mo- jo.3790 ·l o.38IO [0.383-o -j

I 1.2 lo.3849 lo.3869 lo.3888 lo.3907 lo.3925 [QJ944 1o.3962 lo.3980 lo.3997 Jo.40I5

1 1.3 1 o.4o32 10.4049 10.4066 1 o.4o82 1 o.4o99 10.4115 10.4131 10.4147 10.4162 r 0.4177

I 1.4 l o.4192 lo.4207 lo.4222 lo.4236 lo.4251 lo.4265 lo.4279 [OA292-I o.4306 ｉ ｄＮＴＳ Ｍ ｾ＠

I 1.5 lo.4332 Jo.4345 jo.4357 lo.4370 lo.4382 lo.4394 lo.4406 0.4418 lo.4429 lo.4441

I t.6 lo.4452 Jo.4463 lo.4474 lo.4484 ｪｯ ａＴ ｾ ｦｄＮＴＵｯ Ｍ Ｕ＠ Ｍｾ ＰＮＴＵＱＵ＠ 0.4525 ｊｇＮ Ｔ ｾ ｊ ｯＮＴＵＴＵ＠

I 1.1 lo.4554 Jo.4564 lo.4573 lo.4582 Jo.4591 lo.4599 lo.4608 0.4616 lo.4625 [0.4633 1 1 1.8 10.4641 10.4649 10.4656 10.4664 10.4671 10.4678 10.4686 0.4693 10.4699 fo.4 706 I I 1.9 lo.4713 lo.4719 lo.4726 lo.4732 lo.4738 lo.4744 lo.4750 0.4756 lo.4761 lo.4767

I 2.0 l o.4772 lo.4778 lo.4783 lo.4788 lo.4793 lo.4798 lo.4803 0.4808 lo.4812 lo.4817

I 2.1 lo.4821 lo.4826 lo.483o j0.4834- lo.4838 10.4842 IQ.4846 0.4850 lo.4854 10.4857

l 2.2 [0.4861 "1 0.4864 10.4868 10.4871 lo.4875 lo.4878-l o.4881 0.4884 fOA881fOA890 I 2.3 - [OA893-I o.4896 IQ.4898 lo.4--9-0I- lo.4904 lo.4906- [0:4909 ·--[0.4911 Ｍｾ ＰＮＴＹＱＳ＠ jo.4916

I 2.4 lo.4918 IQ.4920 lo.4922 lo.4925 lo.4927 lo.4929--l o.4931 lo.4932 [o.4934 10.4936

I 2.5 l o.4938 lo.4940 lo.4941 lo.4943 lo.4945 lo.4946 10.4948 10.4949 lo.4951 lo.4952

1 2.6 -I 0.4953 10.4955 10.4956 10.4957 lo.4959 lo.4960 lo.4961 10.4962 lo.4963 [OA%4 Ｑ ｾ Ｑ ＰＮＴＹＶＵ＠ jo.4966 lo.4967 lo.4968 lo.4969 lo.4970 lo.4971 10.4972 lo.4973 10.4974

I 2.8 lo.4974 joA975- jo:4976·--lo.4977 - fo.4977 ｦＰａｭＭ ｪ ｯＮＴＹ ｾ｛Ｐａ ＹＷＹ＠ lo.4980 lo.4981

1 2.9 10.4981 10.4982 10.4982 10.4983 lo.4984 lo.4984 10.4985 10.4985 lo.4986 lo.4986

1 3.o 10.4987 10.4987 10.4987 10.4988 lo.4988 lo.4989 lo.4989 ｜ｯａｾ ｉ ｯＮＴＹＹＰ＠ l o.4990 ..

--

®J_ J e.g., for z ｾＭ 1".34, refer to the -1.3 APPENDI X D: Areas in the tail of the Standard Normal Distribution

row and the 0.04 column to find the cumulative area, 0.0901.

ｾ＠_/ ' ! ..___

0

z 0.00 O.Q1 0.02 0.03 0.04 0.05 0.06 0.07 o.os 0.09

-3.0 0.0013 6.0013 0.0013 0.0012 0.0012 0.00 11 0.001 1 0.00 11 0.0010 0.0010

-2.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.00"14 0.0014

-2.8 0.0026 0.002 5 0.0024 0.0023 0.0023 0.0022 Ｐ ｾ ＰＰＲ Ｑ＠ 0.002 1 0.0020 0.0019

- 2.7 0.0035 0.0034 0.0033 0.0032 0.003 1 0.0030 0.0029 0.0028 0.0027 0.0026

-2.6 0.0047 0.0045 0.0044 00043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036

-2.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 O.i)049 0,0048

- 2.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.007 1 0.0069 0.0068 0.0066 0.0064

-2.3 0.0107 0.0104 0.0102 1}0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084

-2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0 11 9 0.0 11 6 0.011 3 0.011 0

-2.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0 154 0.0150 0.0146 0.0143

-2.0 0.0228 0.0222 0.02 17 0.02 12 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183

-1.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233

-1.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 o:o3 14 0.0307 0.0301 0.0294

-1.7 0.0446 0.0436 0.0427 0.04 18 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367

-1.6 0.0548 0,0537 0.0526 0.0516 0.0505 0.0495 0.0485 0:0475 0.0465 0.0455

-1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.06()6 0.0594 0.0582 0.057 1 0.0559

- 1.4 0.0808 0.0793 0.0778 1}0764 0.0749 0.0735 0.072 1 0.0708 0.0694 0.0681

-1.3 0.0968 0.095 1 0,0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823

-1.2 . 0 11-51. 0 .11 31 . 0.11 12 . . 0. 1093 0.1075 O.J 056 · _.0 .. 1038. 0. 1020 0.10_03 0.0985

-1.1 () 1357 0 .1335 0.1314 0. 1292 0.1271 0.125 1 0. 1230 0. 1210 0. 11 90 0. 1170

-1.0 0.1587 0.1562 0. 1539 0. 1515 0.1492 0.1469 0. 1446 0. 1423 0.1 401 0 . . 1379

- 0.9 0.1841 0. 1814 0 1788 0. 1762 0. 1736 0.17 11 0. 1685 0. 1660 0. 1635 0. 1611

-0.8 0.211 9 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0. 1922 0. 1894 0. 1867

-0.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2 177 0.2148

- 0.6 0.2743 0.2709 0.2676 0.2643 0.26 11 0.2578 0.2546 0.25 14 0.2483 0.245 1

-0.5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776

-0.4 0.3446 0 .3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3 121

-0.3 0.382 1 0.3783 0.3745 0.3707 0.3669 03632 0.3594 0.3557 0.3520 0.3483

-0.2 0.4207 0.4 168 0.4 129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859

-0.1 0.4602 0.4562 0.4522 0.4483 ｏＮｴｾＴＴＳ＠ 0.4404 0.4364 0.4325 0.4286 0.4247

-0.0 0.5000 0,4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.472 1 0.4681 0.4641

Sourc e C ｵｭｵｬｾｴｲﾷＮｲ･＠ ｳｴ ｡ｮ､ｾｲ ､＠ nor rt1al ｰｩ ｯ｢｡｢､ｲｴＱＺｾ＠ gen'2 r3tE:d by f·.:l in.t ab, thE: n rour1dEd to fo ur ､ＧＲ｣ ｩ ｲｮｾ Ａ＠ pi.?Jc:s

z

0.0

0.1 0.2

0.3

0.4 .. ' 0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2 1.3

1.4 1.5

1.6

1.7

1.8

1.9

2.0

2.1 2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3.0

e .. g., for z = 1.34, refer to the 1.3 row and the 0.04 column to

find the cumulative area, 0.9099.

0.00

0.5000

0.5398 0.5793

0.6179 0.6554

0.6915

0.7257

0.7580 0.7881

0.8159 0.8413

0.8643

0.8849 0.9032

0.9192 0.9332

0.9452

0.9554 0.964 1

0.9713

0.9772

0.982 1

0.9861

0.9893

0.9918

0.9938

0.9953 0.9965

0.9974

0.9981

0.9987

0

0.01

0.5040

0.5438 0.5832

0.62 17

0.6591 0.69.50-

0.7291

0.7611 0.7910

0.8186 0.8438

0.8665

0.8869

0.9049

0.9207 0.9345

0.9463

0.9564

0.9649 0.971 9

0.9778

0.9826 0.9864

0.9896

0.9920

0.9940

0.9955

0.9966

0.9975

0.9982 0.9987

z

0.02

0.5080

0.5478

0.5871

0.6255 0.6628

0.6985

0.7324

0.7642 0.7939

0.82 12

0.8461

0.8686

0.8888

0.9066

0.9222 0.9357

0.9474

0.9573 0.9656 0.9726

0.9783

0.9830

0.9868

0.9898 0.9922

0.9941

0.9956 0.9967

0.9976

0.9982

0.9987

0.03

0.5 120

0.55 17 0.5910

0.6293 0.6664 0.7019

0.7357

0.7673 0.7967

0.8238

0.8485

0.8708

0.8907

0.9082

6.9236 0.9370

0.9484

0.9582 0.9664 0.9732

0.9788

0.9834 0.987 1

0.9901

0.9925

0.9943

0.9957

0.9968

0.9977

0.9983

0.9988

APPENDIX 0: Ar eas in the tail of the Standard ;'\lon na! DistJ·ibution

0.04

0.5 160

0.55 57

0.5948

0.6331

0.6700 0,7054

0.7389

0.7704 0.7995

0.8264

0.8508

0.8729

0.8925

0.9099

0.925 1 0.9382

0.9495

0.9591 0.967 1

0.9738

0.9793

0.9838

0.9875

0.9904

0.9927

0.9945

0.9959 0.9969

0.9977

0.9984 0.9988

0.05

0.5199

0.5596 0.5987

0.6368 0.6736

0.7088

0.7422

0.7734 0.8023

0.8289

0.853 1

0.8749

0.8944

0.9115

0.9265 0.9394

0.9505

0.9599 0.9678 0.9744

0.9798

0.9842

0.9878

0.9906

0.9929

0.9946

0.9960 0.9970

0.9978

0.9984

0.9989

0.06

0.5239

0.5636

0.6026

0.6406 0.6772

0.7 123

0.7454

0.7764 0.805 1

0.8315

0.8554

0.8770

0.8962

0.913 1

0.9279 0.9406

0.95 15

0.9608 0.9686

0.9750

0.9803

0.9846 0.9881

0.9909

0.9931

0.9948

0.9961

0.9971

0.9979

0.9985

0.9989

0.07

0.5279

0.5675 0.6064

0.6443 0.6808

0.7 157

0.7486

0.7794 0.8078

0.8340

0.8577

0.8790

0.8980

0.9147

0.9292 0.9418

0.9525

0.9616 0.9693

0.9756

0.9808

0.9850

0.9884

0.9911

0.9932

0.9949

0.9962

0.9972

0.9979

0.9985

0.9989

0.08

0.53 19

0.57 14

0.6103

0.6480 0.6844

0.7 190

0.75 17

0.7823 0.8106

0.8365

0.8599

0.88 10

0.8997

0.9162

0.9306 0.9429

09535

0.9625

0.9699 0.9761

0.9812

0.9854

0.9887

0.9913

0.9934

0.995 1

0.9963

0.9973

0.9980 0.9986

0.9990

0.09

0.5359

0.5753 0.6141

0.65 17 I) 6879

0.7224

0.7549

0.7852 0.8133

0.8389

0.862 1

0.8830

0.9015

0.9177

0.93 19 0.9441

0.9545

0.9633 0.9706

0.9767

0.9817

0.9857

0.9890

0.9916

0.9936

0.9952

0.9964

0.9974

0.9981 0.9986

0.9990

APPENDIX E: The t-distribution

r---... . .... -"'... ···• ....... .

/.. \ ,.· ·· .. .. '·.,,

--/ __ ,.......... ""'··-

../"" "'"-·---

t (p,df)

I df\p 0.40 I 0.25 0.10 .----0-.0-5 - 0.025 0.01 0.005 I 0.0005

1 1 10.324920 fi .oooooo 13.077684 16.313752 l 12.1062o 131.82052 163.65674 1636.6192

ｾｺ Ｍ ｬ ｯＮＲＸＸＶＷＵ＠ 19.816497 ＱＱＮＸｳＵｾｦＲＮＹ ｾＹＹＸＶ＠ 14.30265 16.96456 ·1 9.92484 131.5991

I 3 lo.276671 l o.764892 11.637744 12.353363 13.18245 14.54070 15.84091 112.9240

I 4 lo.210122 l o.740697 1!.5D2o6 12.131847 ｦＲＮＷＷＶｾ Ｍ ｪＳ ＮＷＴＶＹＵ＠ 14.60409 [8.6-10-3 -

I 5 l o.267181 l o.726687 1'-475884 12.015048 12.57058 13.36493 14.03214 16.8688

16- IQ264rn-·l o.717558 r 1.439756 11.943180 j2A4691 13.14267 13.70743 15.9588

I 1 lo.263167 jo.711142 11.414924 l '-894579 12.36462 f2.9sn95 13.49948 15.40-79--

1 8 l o.261921 l o.706387 11.396815 11.859548 12.30600 12.89646 13.35539 15.041-3 -

1 9 1 o.260955 10.102122 l '-383029 1'-833113 12.26216 12.82144 13.24984 ｦＴＮＷＸｾ Ｍ

l 10 fD.260Jss-l o.699812 j t.372184 11.812461 12.22814 12.76377 f3.16927 14.5869

I II l o.259556 l o.697445 11.363430 11.795885 ｦＲＮＲＰＰＹ Ｙ Ｍ｛ ｾＷＱＸＰＸ ＭＭＱ Ｓ Ｎ ＱＰＵＸＱ＠ [4.4370

1 12 l o.259033 lo.695483 l u56217 l '-782288 12.17881 !2.681oo j3w4-14.3178

I 13 lo.258591 ＱＰＮＶＹＳｾ Ｑ ＱＮＳＵＰＱＷＱ＠ fl. 77o933 12.16037 !2.65031 13.01228 142-2-08--

1 I4 fo.258213 l o.692417 [ 1.345030 TI:761310 -12.14479 /2:62449 12.97684 14.1405 -

1 15 o.257885 1 o.691197 l u4o6o6 f1.753oso--12.13145 12.60248 12.94671 14.0728

1 I6 o.257599 [0.69013211.336757 11.745884 12.11991 12.58349 12.92078 14.0150

1 11 o.25734 1 1 o.689195 11.333379 11.739607 12.10982 12.56693 12.89823 13.9651-

1 18 o.257123 1 o.688364 11.330391 l '-734064 12.10o92 12.55238 12.87844 13.9216

1 I9 o.256923 1 o.687621 11.327728 l '-729133 12.09302 12.53948 ＱＲＮＸＶＰＹ ｾ＠ 13.8834

1 20 0.256743 1 o.686954 1'-325341 11.724718 12.08596 12.52798 12.84534 13.8495

I 2I o.25658o l o.686352 fl .323188 11.720743 12.07961 j2.51765 12.83136 13.8193 ·--·

I 22 0.256432 l o.685805 11.321237 11.717144 12.07387 12.50832 12.81876- 13.7921

1 23 fo.256297 l o.685306 11.319460 11.713872 12.06866 12.49987 f2.80TI-4 - ,3.7676

I 24 1 0.256173 1 o.68485o f1.3l7836 I'-110882 12.06390 12.49216 12.79694 13.7454

I 25 l o.25606o l o.684430 11.316345 11.708141 12.05954 12.48511 12.78744 13.7251

1 26 o.255955 o.684043 11.314972 l uo5618 ｛ＲＮＰＵＵＵ ｾ Ｑ ＲＮＴＷＸＶＳ＠ f2:778_1_1 - [3.7066-

1 21 o.255858 o.683685 1'-313703 l '-703288 12.05183 12.47266 12.77068 13.6896--

1 28 o.255768 o.683353 l '-312527 11.101131 12.04841 12.46714 12.76326 13.6739

1 29 o.255684 o.683044 11.311434 l '-699127 12.04523 12.46202 12.75639 13.6594 ·-

1 30 o.255605 o.682756 l 1.31 0415 l '-697261 [2.o4227 12.45726 12.75ooo 13 6 o . 46

I inf 0.253347 0.674490 11.281552 11.644854 11.95996 12.32635 [2.57583 13.2905