I.INIVERSITI SAINS MALAYSIA Final Examination

(1)

I.INIVERSITI SAINS MALAYSIA Final Examination

Academic Session 200812009

April2009

JIM 104 - Introduction To Statistics [Pengantar Statistikl

Duration: 3 hours

lMasa:

³

jaml

Please ensure that this examination paper contains THIRTY TWO printed pages before you begin the examination.

Answer FOUR questions only. You may answer either in Bahasa Malaysia or in English.

Read the instructions carefully before answering.

Each question is worth 100 marks.

[Sila pastiknn bahawa kertas peperilcsaan ini mengandungi TIGA PULUH DUA muka surat yang bercetak sebelum anda memulakan peperiksaan ini.

Jawab EMPAT soalan sahaja. Anda dibenarkan menjawab sama ada dalam Bahasa Malaysia atau Bahasa Inggeris.

Baca arahan dengan teliti sebelum anda menjawab soalan.

Setiap soalan diperuntukkan 100 marknh.J

aar

29

al

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-2- [JrM 104]

1. Given the following data:

25 24 25 24 25 23 2s t9 32 23 22 24 26 25 23 28 25 25 26 27 22 28 24 23 24 2t 25 22 29 23 (a) Find

(i) median, (iD mode.

(40 marks)

(b) Construct a frequency distribution. Use 5 classes.

(30 marks)

(c) From the distribution, find

(i) its mean,

(ii) its standard deviation.

(30 marks)

2. (a) The average age of accountants at Three Rivers Corp is 26 yearc, with a standard deviation of 6 years. The average salary of the accountants is RM31,000, with a standard deviation of RM4,000. Compare the variations of

age and income.

(20 marks)

(b) Which of the following exam scores has a better relative position?

X, a score of 42 on exam with mean: 39 and standard deviation:4.

{ a score of 76 on an exam with mean : 71 and standard deviation : 3.

(20 marks)

...3t-

(3)

J.

-3- [JrM 104]

(c) 5lo/o of families had no children, 2}Yohad one child, l9Yohad two children, 7Yo had three children and 3Yo had four or more children. If a familv is selected at random, find the probability that the family has:

(i) two or three children,

(ii) more than one child,

(iii) less than three children.

Based on the answer in parts (i), (ii) and (iii), which is most likely to occur?

Explain why?

(20 marks)

(d) In statistics class there are 18 juniors and 10 seniors. Six of the seniors are females, and 12 of the juniors are males. If a student is selected at random, find the probability of selecting the following:

(i)

^a

junior or a female,

(iD a senior or a female,

(iii)

^a

junior or a senior.

(20 marks)

(e) 70.3% of females ages 20 to 24 have never been married. Choose 5 females in this age category at random. Find the probability that

(i) none has never been married,

(iD at least one has been married.

(20 marks)

(a) A student takes a 20 question, multiple choice exam with five choices for each question and guesses on each question. Find the probability of guessing at least 15 out of 20 correctlv.

(25 marks)

(b) If 2% of the batteries manufactured by a company are defective, find the probability that in a sample of 144 batteries, 3 are defective ones.

(25 marks)

(c) The average salary for first year teachers is RM27,989. If the distribution is approximately normal with o : RM3,250, find the probability that a

randomly selected first year teacher earns,

(i) between RM20,000 and RM30,000 a year,

(ii) less than RM20,000

a

year.

(25 marks) ..4/-

,L.l

'gl

(4)

4.

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1041

(d) A researcher is interested in estimating the average monthly salary of sports reporters in a large city. He wants to be 90%o confident that his estimate is

conect. If the standard deviation is RMl,100, find the sample size needed to get the desired information and to be accurate to within RMl50.

(25 marks)

(a) The heights of 28 police officers were measured. The standard deviation of

the sample was 1.83 inches. Find the 95% confidence interval of the standard deviation of heights of the officers.

(25 marks)

(b) The average salary for public school teachers for a specific year was reported

to be RM39,385. A random sample of 50 public school teachers in a

particular state had a mean of RM41,680 and a standard deviation of RM5,975. Is there sufficient evidence with a : 0.05 to conclude that the

mean salary differs ffom RM39,385?

(25 marks)

(c) In what ways is the /-distribution similar to the standard normal distribution?

In what ways is the /-distribution different from the standard normal distribution?

(25 marks)

(d) A researcher claims that the standard deviation of the ages of cats is smaller than the standard deviation of the ages of dogs owned by families in a large

city. A randomly selected sample of 29 cats has a standard deviation of 2.7 years, and a random sample of 16 dogs has a standard deviation of 3.5 years.

Is the researcher correct? Use a : 0.05. If there is a difference, suggest a reason for the difference.

(25 marks)

(a) The proportion of students in private schools is around 11%. A random sample of 450 students from a wide geographic area indicated that 55 attended private schools. Estimate the true proportion of students attending private schools with91% confidence. How does your estimate compare to llYo?

(30 marks)

(b) Find the 95Yo confidence interval for the variance and standard deviation for the time it takes a customer to place a telephone order with a large catalogue company if a sample of 23 telephone orders has a standard deviation of 3.8 minutes. Assume the variable is normallv distributed.

(20 marks)

5.

...)/-

(5)

-5- urM

¹⁰⁴¹

(c) A manager states that in his factory, the average number of days per year missed by the employees due to illness is less than the national average of 10.

The following data show the number of days missed by 40 employees last

year. Is there sufficient evidence to believe the manager's statement at a:

0.05? (use s to estimate o). Use the P-value method.

(30 marks)

(d) Two groups of students are given a problem-solving test and the results are compared. Find the 90Yo confidence interval of the true difference in means.

Mathematics majors Computer Science majors xr = 83'6

\=4'3

\ =36

; -10

¹

*2 - t /.L

sz = 3'8 nz =36

(20 marks)

0 6 t23 3 5 4

¹

39607634

7 4 7 I 0 8 123 2510515325

3 11 8 2 2 4 I 9

13i

...61-

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I

-6- [JrM 104]

1. Diberi data berikut:

25 24 25 24 25 23 25 19 32 23 22 24 26 25 23 28 25 25 26 27 22 28 24 23 24 21 2s 22 29 23

(a) Cari

(D median,

(ii) mod.

(40 markah)

(b) Bina taburan kekerapan bagi data di atas. Gunakan 5 kelas.

(30 markah)

(c) Daripada taburan, cari

(D min,

(ii) sisihan piawai.

(30 markah)

2. (a) Purata umur akauntan di syarikat Three Rivers adalah 26 tahun dengan sisihan

piawai 6 tahun. Purata gaji akauntan adalah RM31,000, dengan sisihan piawai RM4,000. Bandingkan variasi bagi umur dan pendapatan.

(20 markah)

(b) Markah peperiksaan yang mana mempunyai kedudukan relatif yang lebih baik?

X, markah peperiksaan 42 denganmin: 39 dan sisihan piawai:4 I, markah peperiksaan 76 denganmin : 71 dan sisihan piawai : 3.

(20 markah)

...7 l-

(7)

3.

-7 - FrM

¹⁰⁴¹

(c) 5LYo keluarga tidak mempunyai anak, 20o/o mempunyai seorang anak, l9o/o

mempunyai 2 orang anak,

To/o

mempunyai 3 orang anak dan 3Yo memExryai 4

orang anak atau lebih. Jika satu keluarga dipilih secara rawak, cari kebarangkalian bahawa keluarga itu mempunyai

^:

(i) dua atau tiga orang anak,

(ii) lebih dari seorang anak,

(iii) kurang dari tiga orang anak.

Berasaskan jawapan dalam bahagian (i), (ii) dan (iii), yang mana paling kerap berlaku? Jelaskan mengapa?

(20 markah)

(d) Dalam kelas statistik terdapat 18 orang junior dan 10 orang senior. Enam orang dari senior adalah perempuan dan 72 orang dari junior adalah lelaki.

Jika seorang pelajar dipilih secara rawak, cari kebarangkalian bahawa:

(i) seorang junior atau seorang perempuan yang terpilih,

(ii) seorang senior atau seorang perempuan yang terpilih,

(iii) seorang junior atau seorang senior yang terpilih.

(20 markah)

(e) 70.3% daripada wanita yang berumur di antara 20 tahun dan 24 tahun tidak pemah berkahwin. 5 orang wanita pada kategori umur tersebut dipilih secara rawak. Cari kebarangkalian bahawa:

(i) tiada seorang pun yang pernah berkahwin,

(ii) paling kurang seorang pernah berkahwin.

(20 markah)

(a) Seorang pelajar mengambil peperiksaan yang mengandungi 20 soalan

objektif, yang mempunyai 5 pilihan jawapan untuk setiap soalan dan meneka jawapan setiap soalan. Dapatkan kebarangkalian ia memperolehi 15 jawapan yang betul.

(25 markah)

(b) Jika 2% daripada bateri yang dibuat oleh sebuah syarikat adalah rosak, dapatkan kebarangkalian bahawa dalam satu sampel 144 bateri, terdapat

3

bateri yang rosak.

(25 markah)

35

...8/-

(8)

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(c) Gaji purata guru tahun pertama adalah RM27,989. Jika taburan adalah hampir normal dengano : RM3,250, hitung kebarangkalian bahawa guru tahun pertama yang terpilih secara rawak mendapat gaji

(i) antaraRM20,000 dan RM30,000 setahun,

(ii) kurang daripada RM20,000 setahun.

(25 markah)

(d) Seorang penyelidik ingin mengetahui purata gaji bulanan pemberita sukan di sebuah bandar besar. Dia menginginkan keyakinan 9AYobahawa arnggarcffrya adalah betul. Jika sisihan piawai adalah RMl,100, dapatkan saiz sampel yang diperlukan untuk mendapatkan maklumat yang diinginkan dan ketepatan dalam lingkungan RM150.

(25 markah)

(a) Ketinggian 28 pegawai polis diukur, sisihan piawai dari sampel adalah 1.83

inci. Dapatkan selang keyakinan 95% daripada sisihan piawai ketinggian pegawai polis.

(25 markah)

(b) Purata gaji untuk guru sekolah untuk tahun tertentu adalah dilaporkan sebanyak RM39,385. Sampel rawak 50 guru sekolah di negeri tertentu mempunyai purata RM41,680 dan sisihan piawai RM5,975. Adakah bukti yang cukup pada aras a: 0.05 untuk menyimpulkan bahawa pwata gaji berbeza daripada RM3 9,3

^{8 5}^?

(25 markah)

(c) Apakah kesamaan antara taburan / dengan taburan normal piawai? Apakah perbezaan antarataburan / dengan taburan normal piawai?

(25 markah)

(d) Seorang penyelidik mendakwa bahawa sisihan piawai umur kucing adalah lebih kecil daripada sisihan piawai umur anjing yang dimiliki oleh keluarga di bandar besar. Satu sampel rawak 29 ekor kucing mempunyai sisihan piawai 2.7 tahun dan satu sampel rawak 16 ekor anjing mempunyai sisihan piawai 3.5

tahun. Adakah dakwaan penyelidik tersebut betul? Gunakan a: 0.05. Jika terdapat perbezaan, berikan alasan untuk perbezaan tersebut.

(25 markah)

...91-

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-9- [JrM 104]

5. (a) Kadar pelajar yang menghadiri sekolah persendirian adalah sekitar 11%.

Sampel rawak 450 pelajar dari kawasan geografi yang luas menunjukkan bahawa 55 pelajar menghadiri sekolah persendirian. Anggarkan kadar sebenar pelajar yang menghadiri sekolah persendirian dengan keyakinan 95%.

Bagaimana hasil anggaran anda dibandingkan dengan IlYo.

(30 markah)

(b) Dapatkan selang keyakinan 95% bagi varians dan sisihan piawai untuk masa yang diperlukan oleh pelanggan membuat tempahan barangan melalui telefon dari katalog sebuah syarikat besar jika satu sampel 23 tempahan mempunyai sisihan piawai 3.8 minit. Anggap pembolehubah bertaburan secara normal.

(20 markah)

(c) Seorang pengurus menyatakan bahawa purata ketidak hadiran para pekerja dalam setahun adalah kurang daripada purata nasional sebanyak 10 hari. Data

berikut menunjukkan bilangan hari ketidak hadiran 40 orang pekerja pada tahun lepas. Adakah cukup bukti untuk mempercayai pernyataan pengurus tersebut pada a: 0.05? (gunakan s untuk menganggar o). Gunakan kaedah nilai-P.

(30 markah)

(d) Dua kumpulan pelajar diberikan ujian soal jawab dan keputusannya dibandingkan. Dapatkan selang keyakinan 90Yo perbezaanmin sebenar.

Major Matematik Major Sains Komputer

t :83.6

sr :4'3

\=36

7z:79'2

sz = 3'8 nz =36

(20 markah)

0 6 r23 3 5 4 I 39607634

7 4 7 r 0 8 t2

³

2510515325

3 11 8 2 2 4 r 9

37

...r0t-

(10)

-10-

Important Formulas

Chapter 3 Data Description

Mean for individual data: X - Ix z-/

n

- l,"f 'x.

Mean for grouped data: X =

T

Standard deviation for a sample:

ll ffi x' -lt,t x\" lnl

IL" l-\L" t t'" )

I n-I

Standard deviation for grouped data:

l>r *:-l;.r x^f hl

--' +

I n-l

Range rule of thumb: s

^,

ry

4 Median for grouped data:

MD=@+t(w)+L_

where

n : sum offrequencies

c/ : cumulative frequency of class immediately preceding the median class

w : width of median class

f : frequency of median class

L^= lower boundary of median class Chapter 4 Probability of Counting Rules Addition rule I (mutually exclusive events):

P(A or B)= P(A)+ r(r)

Addition ruIe2 (events not mutually exclusive):

P(A or ,B) = P(A)+ r(a) - P(A and B) Multiplication rule I (independent events):

r('t and a)= r('a)'P(B)

prM

1041

...rU-

(11)

- 11- IJIM 104]

Multiplication rule 2 (dependent events):

P(A and B)= P(t).r(alt)

conditional probabilit y: r(al,e)='(o T1, u)

P(A)

Complementary events: P(E)=I- P(E)

Fundamental counting rule: Total number of outcomes of a sequence when each event has a different number of possibilities: k, 'kr. kr'

^.

.k,

Permutation rule: Number of permutations of n objects taking r at

a

time is

P- nt' n-' (n-r)t

Combination rule: Number of combinations of r objects selected from n objects is

7'11.

n-r / \.

ln-r)lr!

Chapter 5 Discrete Probability Distributions

Mean for a probability distribution: p =>,lX P(X)]

Variance and standard deviation for a probability distribution:

o' =Zlx' .r(x)]- a'

o' = {I[ x'z .r(x)f- r'

Expectation: n (x) =ZlX

^.

P (X)l

Binomial probability : f (X\

^'

pr .q''r

\ " (n- X)lXl '

Mean for binomial distribution: p = n. p

Variance and standard deviation for the binomial distribution:

o2=n.p.Q o=Jiv.q

E9

...12/-

(12)

-r2- [JrM 104]

Multinomial probability:

P(X) \ / xrlxzt'x31"'x ol " '

Poissonprobability : p(x;1\=u:!,. ,X! where X = 0, 1,2, ...

Hypergeometric probability: P(X)=+Y

Chapter 6 The Normal Distribution Standardscore:, =X - F o, X -X

o,s

Mean of sample means: pt = lt

Standard error of the mean: o1=$

!n

Central limit theorem formula , , -- F

- olJi ,-y

Chapter 7 Confidence Intervals and Sample Size z confidence interval for means:

v - (ot '-)

A -.at2r --

|

< p< F + ro,rl #

^r

\tln 1 \{n

⁷

/ confidence interval for means:

f "\ f .)

X -to,rl +

^I

< p<X +to,rl -

^r

\tln 1

, ,U:,

Sample size for means: n =[ \E fu-e I ) where E is the maximum error of estimate

Confidence interval for a proportion:

F; l-

^ | tlPl 'Pq

it -Q",r) l'z < p < b * (t,,r) l-

...t3/-

(13)

-13-

Sample size for a proportion: , = bA(+\' vE)

where h =: X and A =7- b

n

Confidence interval for variance:

(n-1)s'? , (r-1)r'

-\v

- ,

X';et, It"r,

Confidence interval for standard deviation:

Chi-square test for a single variancp: ,' -('-1-)t' (d.f.: n-l) 6-

Chapter 8 Hypothesis Testing

z test

^'.

z =" X-u t

^r----L

for any value n. If n < 30, population must be normally distributed.

olt.ln

7=J7[ V-a for o unknown and n>30 ,r/vn

t test:

^y

=--fr X-u for n130 (d.f. : n- l)

s/Vn

z test for proportions: z = b-p .tp;F

IJIM 104]

r;-fi

l*<o<

I rign,

41

...t4/-

(14)

-14- urM

1041

Chapter 9 Testing the Difference Between Two Means, Two Variances and Two Proportions

Ztest for comparing two

_ (N,- X,)-(n

(d.f. : the smaller

or7 n1

- | or

ⁿ²

- l)

means (independent samples)

_;

- pr)

Formula for the confidence interval for difference of two means (large samples):

?-+

Ioi , oi

,rl-T-

l\

^nz

f t -z

- xr\* Zotz^lL*5 lry

ⁿ²

nr>30.

1

Ftest for comparing two variances: F =+

S;

where s,t is the larger variance and d.f.N. = 4,-1, d.f.D = nz-I

/ test for comparing two means (independent samples, variances not equal):

, (N,_ N,)_(p,_ p,) lt? . t1

r- ^-l-...-

!n,

^nz

Note:

(x'

sf

...15/-

(15)

Formula for the confidence interval for difference of two means (small independent samples, variance unequal)

:

(x, - x,1-r",,

^ffi \rh n2 < pt - pz .(x, - x,y * r",, ^p*t \r4

ⁿ²

-i5-

(d.f. : smaller of n1- I and n2- l)

t test for comparing two means (independent samples, variances equal):

(d.f.:u+nz-2)

Formula for the confidence interval for difference of two means (small independent samples, variances equal)

^:

lJrM i04l

li;\ M

(X,-Xr)-rotrrl-

I nt+nr-2

/;-.-\ M

(,\i-Xr)+totrtl-

Y nt+nr-2

f]

¹

lnt t-+-

ⁿ²

t-+- n1

\ry

^nz

and d.f. : ltr * nz-2.

/ test for comparing two means for dependent samples:

D-u^ t/)

t-- ,-D where D-"" and

soltln n

(a.r. = n -t)

>Dz -l{>4'l"f

43

...16/-

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-16- [JrM 104]

Formula for confidence interval for the mean of the difference for dependent samples:

D-ro,r+. po.D*t",r$

in {n

(d.f.:n-l)

/ test for comparing two proportions:

(b,- fr,)-(p,- p,)

- -

\' L 1

' t

l-( t 1 )

^lpql -+-

^|

\- -\tq nr)

X,+X, ^ X,

Wnefe P =- nr+nz Pt=- ry

q:l- p hr=L

n2

Formula for the confidence interval for the difference of two proportions:

,^@J4 l pt- p, <(i,- b,)* r,,^ W*P,qt

'! ,\ - ,b \/l-lzz -\l't-P21t"",'! ,\ -

",

...r7 /-

(17)

-17 - urM

¹⁰⁴¹

0.ti

{}.9

_0.95

Ihe

Binomial Disiribution

0.05

0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.901

^0.8

l0

0.09.5

0. t80

0.002

0.0 t0

0.857

0.729

0.1-i-5

0.243

0.007

^0.017

0.00 i

0.73.s

0.-51r

0.2-32

0.3.s4 0.03

t

^0.09ti

0.002

0.0 t-5 0.(x) I

0.6.10

0.190

0.320

^0.41t)

0.0.10

0.090 0.5

r2

^0.343

0.381

^0.441

0.096

^0.^r89

0.008

^0.027

0.360

^0.250

0..+80

^0.500

0.

r60

^0.250

0.

t60

^0.090

0.180

^0.110

61..360

^0.190

0.026

^0.008

0.

r54

^0.076

0.3.16

0.265

0.346

^0.4^t2

0.130

^0.?40

0.0I

0

^0.(x)2

0.077

^0.028

0.?30

^0.^r32

0.346

^0.309

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^0.360

0.078 0.l6ti 0.004

^0.00I

0.037

^0.010

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^0.060

0.276

^{0. r8-5}

0.31

|

^0.324

0.

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0.047

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0.

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^0.3r8

0.13

r

^0.?17

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^0.082

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^0.00I

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0.1.12

^0.^t36

0.179

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0.109

^0.196

0.090

^0.^r98

0.0

r7

^0.0-58

f).0-l()

0.0I()

0.310 0

IlJ0

().(r-l()

().li _I₀

0.001 0.09.s 0.901

0.815

^0.656

0.1

71

^0.2()2

0.014

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0.(X):l

0.171

0.-590

0.20,+

^0.318

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^0.073

0.(x)

I

^0.(x)t{

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^0.^rti5

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0.(x)r

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^0.082

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^0.227

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0,

r68

^0.0.58

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r

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|

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|

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^0.164

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^0.273

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0.066 0.172 0.251

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0.064

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0.302

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0.001 0.006

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0.088

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0.268

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0.002 0.010

0.039

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0.11

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0.221

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0.295

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0.236

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0.086

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0.069

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0.0q1 0.008 0.061 0.2( 9 0.630 0.

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0.302

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0.006

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0.236

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0.295

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0.039 0.t32 0.010

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0.002

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0.004 0.001

0.069

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0.206

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0.283

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0.236

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0.133

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0.053

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0.0 r

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0.00

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0.006

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0.040

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^0.044

0.2r5

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0.251

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0.20t

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0. r

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0.042

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0.01

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0.001 0.004

0.027

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0,089

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0.t'77

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0.236

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0.22t

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0.t47

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0.070

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0.0 t7 0.099 0.341 0.540

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(19)

-19- [JrM 104]

0.8 0.9 0.4 0.7

0.2 0.3 0.05 0.1

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0.351

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0.111

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0.1'19

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0.268

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0.069

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0.086

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0.007 0.001

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0.t32

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0.231

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0.188

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0.103

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0.043

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0.014

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0.003

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0.001

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0.002

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0.9-87

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0.157

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0.209

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0.209

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0.087

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0.035

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0.010

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0.006

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0.t22

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0.183

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0.209

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0.183

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0.122

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0.061

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0.002 0.007 0.024 0.061 0.118 0.177 0.207 0.186 0.127 0.063 0.022 0.005 0.092

0.042 0.014 0.003

:

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0.367

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0.207 0207 0.r57 0.092 0.041 0.014 0.003 0.001

0.005 0.022 0.063 0.r27

0.1 86 0.201

0.474 0.007 0.002

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0.001 0.006 0.023 0.069 0.154 0.246 0.268 0.179 0.055

0.488

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0.366

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0.135

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0.001 0.007 0.023 0.062 0.126

0.i96

0.229

0.r94

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0.002 0.009 0.032 0.086 0.172 0.250 0.250 0.154 0.o44

0.001 0.003 0.014 0.043 0.103 0.188 0.250 0.231 0.132 0.035

0.00r 0.008

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I z

3 4

5 o 7 8 9 10 11 12 13 14 15

0.177

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0.343

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0.206

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47

...20t-

(20)

0.440

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0.371

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0.146

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0.036

0.142

0.006

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0.001

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0.003

0.418

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0.3.14

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0.158

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0.041

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0.008

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0.001

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0.004 0.001

0.028

^0.003

0.113

^0.023

0.211.

^4.073

0.246

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0.200

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0.t20

^0.210

0.055 0.i65 0.020

^0.101

0.006

^0.049

0.001

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0.006 0.001

0.w3

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0.096

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0.191

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0.239

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0.209

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0.136

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0.068

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0.027

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0.008

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0.002

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0.009 0.003 0.001

0.001 0.004

0.014

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0.039

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0.084

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0.142

^o.o+s

'0.189

0.101

0.198

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4.L62

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0.101

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0.047

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0.015

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0.003

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0.003

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¹⁰⁴¹

0.001 0.006 0.020

0.055

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0.120 0.014

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0.200 0.051

0.006

0.246 0.t42

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0.211 0.215

0.L46

0.113 0.329

^0.37

|

0.028 0.185

^0.440

0.002 0.008

0.027

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0.068

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0.136 0.017

0.00r

0.209 0.060

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0.239 0.156 0.04r

0.191 0.280

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0.096 0.315

^0.374

0.023 0.167

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0.9 0.3 0.7

0.003 0.015 0.047 0.101 0.162 0.198

0.i89 0.t42

0.084 0.039 0.014 0.004 0.001

0.002 0.009 0.028 0.067 0.122 0.t75 0.196 0.175

0.t22

0.067 0.028 0.009 0.002

8 9 10 11 L2

t3

L4 15 16 17

0.002 0.010 0.034 0.080 0.138 0.184 0.i93 0.161 0.107 0.057 0.024 0.008 0.002

0.001 0.005 0.018 0.047 0.094 0.148 0.185

u.l6f

0.148 0.094 0.Q47 0.018 0.005 0.001

0.002

0.008

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0.024

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0.057

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0.107

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0.161

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0.193

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0.184

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0.138

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0.080

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0.034

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0.002

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0.002 0

I

aL 3 .+

5 6 7 8 9 10 11 12 IJ

1A 15 16

...21/-

(21)

0.001

0.377 0.135

^0.014

0.377 0.285

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0.1.79 0.285

^0.^r54

0.053 0.180

^0.218

0.01r 0.080

^0.218

0.002 0.027

^0.164

0.007

^0.095

0.001

^0.044

0.017 0.005 0.001

0.150

^0.018

0.300

^0.081

0.284

^0.172

0.168

^0.230

0.070

^0.215

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0.15r

0.005

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0.001

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0.012 0.003 0.001

0.002

0.013

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0.046

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0.105

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0.168

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0.202

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0.187 0.i66 0.138

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0.081

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0.039

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0.015

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0.005

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0.001

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0.004 0.001

0.001

0.009

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0.036

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0.087

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0.149

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0.t92

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0.192

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0.153

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0.098

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0.051

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0.022

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0.008

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0.002

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0.001

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0.002 0.001

0.001 0.003

0.012

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0.033

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0.071

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0.t21

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0.t67

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0.185

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0.167

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0.121

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0.071

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0.033

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0.003

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lJrM

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0.95

0.001 0.00s

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0.300

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0.1s0

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-21 -

0.7 0.8 0.1

0.397 0.376 0.168 0.047

0

1

2 J 4 5 o 7 8 9 10 11 12 1J

t4

15 16

l7

18

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0.002

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0.t44

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0.096

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0.052

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0.022

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0.007

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0.002

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0.001 0.005 0.015 0.039 0.081 0.138 0.1 87 a.202 0.168

0.i05

0.046 0.013 0.002

0.001 0.003

0.0r2

0.035 0.082 0.151

0.2t5

0.230 0.172 0.081 0.018

0.001 0.005 0.071 0.044 0.095 0.164 0.218 0.218

0.1 54 0.068 0.014 0.001

0.002 0.008 0.022 0.051 0.098 0.1 53 n

lQ)

n 10t

0.r49

0.087 0.036 0.009 0.001

0.001 0.007 0.02'7 0.080

0.1 80 0.285 0.285

U, -IJ)

0.002 0.011 0.053 0.179 0.377 0.377 0

1

2 J 4 5 6 7 8 9 10 11 12 IJ 1A 15 16

t7

18

49

...22/-

(22)

-22- [JrM 104]

0.002 0.009

0.032

^a.002

0.090

^0.013

0.190

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0.285 0.I89 0.2'70

^0.377

0.122

^0.358

0;t

0.9 0.1 0.2

n

1

z

a

4 5 6 7 8 9 10

1l t2

IJ IA 15

l6 l7 t8

19 20

0.358 0.122 0.0i2

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0.377 0.2'70 0.058

^0.007

0.189 0.285 0.t37

^0.028

0.060 0.190 0.205

^0.072

0.013 -^T.090- 0.218

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a.002 0.032 0.1'.75 0-l't9 0.009 0.109

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0.002 0.055

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0.022

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0.007

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a.002

^0.031

0.012 0.004 0.001

0.003

0.0

r.2

^0.001

0.035

^0.005

0.075

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0.t24

^0.037

0.166

^0.074

0.180

^0.120

0.160

^0.160

0.1t7

^0.176

0.07i

^0.160

0.035

^0.120

0.015

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0.005

^0.037

0.001

^0.015

0.005 0.001

0.001 0.005 0.015 0.035 0.071 0.117 0.160 0.180 0.166

0.r24

0.075 0.035 0.012 0.003

0.001 0.004 0.012 0.031 0.065 0.1

t4

0.164

0.r92

0.1'79 0.130 0.072 0.028 0.007 0.001

0'002-

0.007 0.022 0.055 0.109 0.175 0.218 ^- 0.205 0.137 0.0s8 0.012

Nore: All values of 0.0005 or less are omitted.

Souce.. John E. Freund. Modent Eletnentary Srarlrrrcr, 8th ^ed.,@ 1992. Reprinted by permission of Prentice-Htll, Inc.. Upper Saddle River' N'J'

...zJt-

(23)

-23 - lJrM

¹⁰⁴¹

.4066 .3659 .1647 .0494 .011I .0020 .0003 .0000 The Poisson Distribution

.9048 .0905 .0045 .0002 .0000 .o000 .0000 .0000

.3329 .3662

.20t4

.0738 .0203 .0045 .0008 .0001 .0000 .0000

.1225 .2572 .2'700 .1 890 .0992 .0417 .0146 .0044 .001I .0003 .0001 .0000

.8r87 .1631 .0164 .0011 .0001 .0000 .0000 .0000

.3012 .3614 .2169 .0867 .0260 .0062 .0012 .0002 .0000 .0000

.il08

.2438 .2681 .1966 .1082 .0476 .0174 .0055 .0015 .0004 .0001 .0000 .0000

.7408 .2222 .0333 .0033 .0003 .0000

.O /UJ .2681 .U]JO .0072 .0007 .0001

.6065 .3033 .0758 .0126 .0016 .0002 .0000 .0000

.zz)

I

.3347 .251 0 .1255 .0471 .0r41 .0035 .0008 .0001 .0000

.0821 .2052 .2565 .2138

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51

...24t-

(24)

-24 - lJrM 1041

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(26)

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-26 - [JrM 104]

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.0001 .000s .0025 .008 ^r .0201 .0398 .0656 .0928 .l148

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(28)

-28 - [JrM 104]

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