Buku Rumus Sifir Matematik 2019

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PENDAHULUAN PENDAHULUAN

Falsafah Pendidikan Kebangsaan (FPK) dan Falsafah Pendidikan Guru Falsafah Pendidikan Kebangsaan (FPK) dan Falsafah Pendidikan Guru (FPG) menekankan kepada perkembangan potensi individu guru secara (FPG) menekankan kepada perkembangan potensi individu guru secara seimbang dan bersepadu dari segi jasmani, emosi, rohani dan intelek. Hasrat seimbang dan bersepadu dari segi jasmani, emosi, rohani dan intelek. Hasrat ini dapat dicapai menerusi pelaksanaan kurikulum pendidikan guru yang ini dapat dicapai menerusi pelaksanaan kurikulum pendidikan guru yang menyepadukan aspek-aspek pengetahuan, kemahiran ikhtisas dan amalan menyepadukan aspek-aspek pengetahuan, kemahiran ikhtisas dan amalan nilai keguruan.

nilai keguruan.

Berasaskan kepada FPK dan FPG, matematik telah menjadi bidang Berasaskan kepada FPK dan FPG, matematik telah menjadi bidang ilmu yang dapat melatih minda supaya berfikir secara mantik dan bersistem ilmu yang dapat melatih minda supaya berfikir secara mantik dan bersistem dalam menyelesaikan masalah dan membuat keputusan. Sejajar dengan itu, dalam menyelesaikan masalah dan membuat keputusan. Sejajar dengan itu, buku Rumus Matematik dan Sifir Statistik ini telah diolah dan disusun semula buku Rumus Matematik dan Sifir Statistik ini telah diolah dan disusun semula dengan mengambil kira kesinambungan yang berterusan mengikut dengan mengambil kira kesinambungan yang berterusan mengikut kursus-kursus yang ditawarkan oleh Institut Pendidikan Guru Malaysia (IPGM). Buku kursus yang ditawarkan oleh Institut Pendidikan Guru Malaysia (IPGM). Buku ini diharapkan dapat menjadi bahan rujukan dan membantu siswa pendidik ini diharapkan dapat menjadi bahan rujukan dan membantu siswa pendidik dalam menyelesaikan masalah matematik menerusi rumus-rumus penting dalam menyelesaikan masalah matematik menerusi rumus-rumus penting dan sifir statistik yang disediakan.

dan sifir statistik yang disediakan.

Buku ini dibahagikan kepada enam tajuk utama iaitu Algebra, Buku ini dibahagikan kepada enam tajuk utama iaitu Algebra, Geometri, Trigonometri, Kalkulus, Statistik dan Matematik Kewangan. Setiap Geometri, Trigonometri, Kalkulus, Statistik dan Matematik Kewangan. Setiap tajuk pula dibahagikan kepada topik-topik kecil supaya proses pembelajaran tajuk pula dibahagikan kepada topik-topik kecil supaya proses pembelajaran dapat dilakukan dengan lebih mudah, teratur dan

dapat dilakukan dengan lebih mudah, teratur dan bermakna. Semoga buku inibermakna. Semoga buku ini dapat memberi manfaat kepada siswa pendidik di institut

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AHL

AHL I I PANEL PANEL PENYEMAKPENYEMAK AN SEMULAN SEMUL AA

1.

1. Dr. Dr. Jong Jong Cherng Cherng MeeiMeei IPG Kampus Pulau Pinang IPG Kampus Pulau Pinang Pulau Pinang

Pulau Pinang 2.

2. Dr. Dr. Teong Teong Mee Mee MeeMee

IPG Kampus Pulau Pinang IPG Kampus Pulau Pinang Pulau Pinang

Pulau Pinang 3.

3. Dr. Dr. Lam Lam Kah Kah KeiKei

IPG Kampus Tengku Ampuan Afzan IPG Kampus Tengku Ampuan Afzan Kuala Lipis, Pahang

Kuala Lipis, Pahang 4.

4. Dr. Dr. Ng Ng Kok Kok FuFu

IPG Kampus Sultan Abdul Halim IPG Kampus Sultan Abdul Halim Sungai Petani, Kedah

Sungai Petani, Kedah 5.

5. Datin Datin Hjh. Hjh. Zaitun Zaitun binti binti OthmanOthman IPG Kampus Pendidikan Islam IPG Kampus Pendidikan Islam Bangi, Selangor

Bangi, Selangor 6.

6. Pn. Pn. Manisah Manisah Mohd Mohd ShahShah

IPG Kampus Pendidkan Teknik IPG Kampus Pendidkan Teknik Nilai, Negeri Sembilan

Nilai, Negeri Sembilan 7.

7. Pn. Pn. Lee Lee Yen Yen TingTing

Institut Pendidikan Guru Malaysia Institut Pendidikan Guru Malaysia Kementerian Pendidikan Malaysia Kementerian Pendidikan Malaysia

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KANDUNGAN KANDUNGAN CONTENTS CONTENTS Pendahuluan Pendahuluan Preface Preface Ah

Ah lli Pani Pan elel

Panel Members Panel Members .. .._……….._……….. .. ..………..……….. ii iiii 1. ALGEBRA 1. ALGEBRA AL

AL GEBRAGEBRA

1.1 1.1 1.2 1.2 1.3 1.3 1.4 1.4 1.5 1.5 1.6 1.6 1.7 1.7 1.8 1.8 1.9 1.9 1.10 1.10 1.11 1.11 1.12 1.12 Indeks Indeks Indices Indices _………._………. Logaritma Logaritma Logarithm Logarithm_………_……… Surd Surd Surd Surd _………._………. Nombor Kompleks Nombor Kompleks Complex Numbers Complex Numbers ……….………. Nilai Mutlak Nilai Mutlak Absolute Value Absolute Value_……….._……….... Matriks Matriks Matrices ... Matrices ... Punca Persamaan Kuadratik

Punca Persamaan Kuadratik

Roots of Quadratic Equation ... Roots of Quadratic Equation ... Janjang Aritmetik Janjang Aritmetik Arithmetic Progression ... Arithmetic Progression ... ... Janjang Geometri Janjang Geometri Geometric Progression ... Geometric Progression ... Kembangan Binomial Kembangan Binomial Binomial Expansion ... Binomial Expansion ... Algebra Linear Algebra Linear Linear Algebra ... Linear Algebra ... Ruang Vektor Ruang Vektor Vector Spaces ... Vector Spaces ... 1 1 1 1 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 5 5 6 6

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2. GEOMETRI GEOMETRY 2.1 2.2 2.3 2.4 2.5 2.6 2.7 Luas Area ……….. Isipadu Volume ………. Sukatan Membulat Circular Measures ..._………... Bulatan Circle _……….... Geometri Koordinat Coordinate Geometry ... Transformasi: Pembesaran Transformation: Enlargement ... Keratan Kon Conic Sections ... 7 7 7 8 8 9 9 3. TRIGONOMETRI TRIGONOMETRY 3.1 3.2 Identiti Trigonometri Trigonometric Identities………... Penyelesaian Segitiga Solution of Triangles _………. 11 12 4. KALKULUS CALCULUS 4.1 4.2 4.3 4.4 4.5 4.6

Fungsi dan Had

Functions and Limits ………...

Petua L’Hopital L’ Hopital’ s Rule ... ……… Pembezaan Differentiation_{………..………...}. Pengamiran Integration _………...

Luas di bawah Lengkung

Area under a Curve _………... Isi Padu Bungkah Perkisaran

Volume of Solid of Revolution

13 13 13 15 16 16

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5. STATISTIK STATISTICS 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 Kebarangkalian Probability……….

Ukuran Kecenderungan Memusat

Measures of Central Tendency………...

Ukuran Kedudukan Measures of Position _………... Ukuran Serakan Measures of Dispersion _……….... Pilih Atur Permutation _……….... Gabungan Combination ………... Taburan Binomial Binomial Distribution ……….

Taburan Normal Piawai

Standard Normal Distribution _……….. Taburan Pensampelan Sampling Distribution _………... Anggaran Selang Interval Estimation _……… Pengujian Hipotesis Hypothesis Testing ………... Analisis Varians

Analysis of Variance (ANOVA)_………. Regresi Linear Linear Regression_……….. 17 17 18 19 20 20 20 20 21 22 23 25 26 6. MATEMATIK KEWANGAN FINANCIAL MATHEMATICS 6.1 6.2 6.3 6.4 6.5 6.6

Faedah dan Diskaun

Interest and Discount ... Anuiti Annuity ... Pelunasan Pinjaman Amortisation of Loan ... ... Dana Terikat Sinking Fund ... Aliran Tunai Cash Flow ... Nombor Indeks Index Number ... 27 28 28 29 29 30

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7. SIFIR STATISTIK STATISTICAL TABLES 7.1 7.2 7.3 7.4 7.5 7.6 Pekali Binomial Binomial Coefficients………....

Pekali Pilih Atur

Permutation Coefficients .……….………

Sifir Taburan-F

F-DistributionTables _……….. Sifir Taburan Normal Piawai

Standard Normal Distribution Table…….

………...

Sifir Taburan Khi Kuasa Dua

Chi-Square Distribution Table……….

………. Sifir Taburan-t t-Distribution Table .………... 31 32 33 36 37 38 BIBLIOGRAFI BIBLIOGRAPHY ……….. 39

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1. ALGEBRA

ALGEBRA

1.1 Indeks Indices

(i) a  , a 

(ii) m n m n a a a _ _  (iii) m n m n a a a _ _  (iv)

 

a m n  amn

(v) n _n a a   (vi) n n a b b a               (vii) n

 

n m m a a _ 1.2 Logaritma Logarithm

Bagi sebarang a, b, m, n  R dan a > 0, b > 0, m > 0, n > 0, For any a, b, m, n  R and a > 0, b > 0, m > 0, n > 0,

(i) log_a mnlog_a m log_a n

(ii) log m log n

n m

log_a  _a  _a

(iii) log_a m n  nlog_a m

(iv) log_a a 

(v) log_a  (vi) , b a log m log m log b b a (vii) a log b log_a



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1.3 Surd Surd

(i) ab  a b

(ii) b a b a  (iii) a a a



(iv) a _ b



a _ b



_ a _b

(v) b a b a b a







1.4 Nombor Komp leks Complex Numbers

(i) a bi c di a c ( b d )i

(ii) a bi c di a c ( b d )i

(iii) a bi c di ac bd ( ad bc )i

(iv) i d c ad bc d c bd ac di c bi a























(v) Bentuk kutub nombor kompleks Polar form of complex numbers



kos i sin



r iy

x  

(vi) Pendaraban nombor kompleks dalam bentuk kutub Multiplication of complex numbers in polar form























  



    sin i kos r r ) sin i kos ( r sin i kos r

(vii) Pembahagian nombor kompleks dalam bentuk kutub Division of complex numbers in polar form

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(viii) Teorem De Moivre De Moivre's theorem



_



_{ }





 



_

 

_

1.5 Nilai Mutlak Abso lu te Val ue (i) (ii) 1.6 Matriks Matrices

(i) , A ad bc

d b c a A _         (ii)

_

















 A , a b c d A A

1.7 Punca Persamaan Kuadr atik Roots of Quadratic Equation

(i) Jika



dan



_{ialah punca-punca suatu persamaan kuadratik,}

maka persamaan kuadratik tersebut ialah

If



and



are roots of a quadratic equation, then the quadratic equation is





   

(ii) Jika ax2 _bx _ c _0_{, maka}

If ax2 _ bx_ c _0_{, then} a ac b b x _           x x x x x x x x jika jika if if

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1.8 Janjang Arit metik

Ar it hm eti c Pr og res si on (i) T a



n



d n   (ii) d T n_ T n (iii) S_n n a n d l a n (iv) n  n  n S S T 1.9 Janjang Geometri Geometric Progression (i)   n n ar T (ii) n n T T r _  (iii)





   r r a S n n , r  (iv)





r r a S n n    , r  (v) r a S    , r 

1.10 Kembangan Bin omial Binomial Expansion (i)



a  b



n n r r n r n n n n n n b b a C b a C b a C a _ _ _ _ _ _   

...



...

n r r n n n n b b a r n b a n b a n a

_













































  

...



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(ii) Untuk

|| < 1

dan

 ∈ 

, For

|| < 1

and

 ∈ 

,





n r n x ... x r ... r n ... n n ... x n n nx x _ _               ( ) ( ) ( )

1.11 Alg ebra Linear Linear Algebra

(i) Jika A ialah matriks tak singular, maka

If A is a non-singular (invertible) matrix, then

( ) =

(ii) Jika A dan B ialah matriks-matriks tak singular n x n, maka If A and B are n x n non-singular (invertible) matrices, then ( ) = 

(iii) Jika A ialah matriks tak singular, maka

If A is a non-singular (invertible) matrix, then (

(iv) Petua Cramer

Cramer’s Rule

Jika A ialah matriks tak singular n x n, untuk sebarang b dalam





, penyelesaian unik bagi A x = b ialah

If A is a non-singular (invertible) n x n matrix, then for any b in





, the unique solution of A x = b is ,

(v) Rumus songsangan Inverse formula

Jika A ialah matriks tak singular n x n, maka

If A is a non-singular (invertible) n x n matrix, then

=

(vi) Persamaan ciri

Characteristic equation

) = nilai eigen/eigen value

  AB   ) =( ) x = 1 2  adj A de (A = 0,

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1.12 Ruang Vektor Vector Spaces

(i)   kos  x x  y

(ii) ˆ  y x y x    i j

(iii)   kos  x x  y y  z z

(iv) ˆ  = z y xi  j  k y z y x  

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2. GEOMETRI

GEOMETRY

2.1 Luas Ar ea

(i) Luas segi empat selari = b

×

h

Area of parallelogram (ii) Luas trapezium

Area of trapezium =



a  b



h (iii) Luas permukaan melengkung silinder

Curved surface area of cylinder

= r h

(iv) Luas permukaan melengkung kon = r r h rl

Curved surface area of cone (v) Luas permukaan sfera

Surface area of sphere =

r

2.2 Isi Padu Volume

(i) Isi padu silinder = r h

Volume of cylinder (ii) Isi padu kon

Volume of cone = r h (iii) Isi padu piramid

Volume of pyramid =  A h (iv) Isi padu sfera

Volume of sphere = r

2.3 Sukatan Membulat Circul ar Measures

(i) Panjang lengkok, s  r  dalam radian

Arc length,  _{in radians}

A  dalam radian

 _{in radians}

(ii) Luas sektor, Area of sector,

 

r

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2.4 Bulatan Circle

(i) Panjang lengkok,

Arc length, s _ r

(ii) Luas sektor,

Area of sector, A   r

2.5 Geometri Koor dinat Coordin ate Geometry (i) Titik tengah,

Mid-point, M _        x x , y y

(ii) Jarak antara dua titik,

Distance between two points, d 



x  x

 

 y  y



(iii) Jarak serenjang dari titik



x₁ , y₁



ke garis lurus axbyc0

Perpendicular distance from a point



x₁ , y₁



to a straight line

0   by c ax b a c by ax    

(iv) Koordinat-koordinat bagi titik P



x, y



yang membahagi dalam

tembereng garis yang menghubungkan A



x₁ , y₁



dan B



x₂ , y₂



mengikut nisbah m : n

Coordinates of a point P



x, y



that divides internally a line segment between A



x₁ , y₁



and B



x₂ , y₂



in the ratio of

m : n

















n m my ny , n m mx nx (v) Luas segitiga Area of triangle =









 y x y x y x y x y x y x     

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2.6 Transform asi: Pembesaran Transformation: Enlargement

(i) Luas imej = k 2_× _{luas objek}

Area of image = k 2× area of object (ii) Isi padu imej = k 3_× _{isi padu objek}

Volume of image = k 3_× _{volume of object}

2.7 Keratan Kon Conic Sections 2.7.1 Bulatan

Circle

Bentuk piawai bagi persamaan bulatan dengan pusat (h, k) dan jejari r

Standard form of the equation of a circle with centre (h, k) and radius r

   

x h  y k r

2.7.2 Parabola Parabola

Bentuk piawai bagi persamaan parabola dengan bucu (h, k)

Standard form of the equation of a parabola with vertex (h, k)

(i)



y k



 p( xh)

(ii)



x h



 p( yk)

2.7.3 Elips Ellipse

Bentuk piawai bagi persamaan elips dengan pusat (h, k)

Standard form of the equation of an ellipse with centre (h, k)

b a c , b a , b h x a k y b k y a h x 2 

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2.7.4 Hiperbola Hyperbola

Bentuk piawai persamaan bagi hiperbola dengan pusat (h, k)

Standard form of the equation of a hyperbola with centre (h, k)

(i)

(ii)

(iii) Persamaan_–persamaan asimptot Equations of asymptotes







x h



k b a y k h x a b y         b a c , a h x b k y b k y a h 2 x

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3. TRIGONOMETRI

TRIGONOMETRY

3.1 Identiti Trigon ometri Trigonometric Identities

(i) sin  

(ii) sek _ _ tan

(iii) kosek _ _ kot

(iv) sin



A_ B



_ sin A kos B_ kos A sinB

(v) kos



A_ B



_ kos A kos B  sin A sinB

(vi)





B tan A tan B tan A tan B A tan    

(vii) sin A sin A kos A

(viii) kos A_ kos A_ sin A

A sin     kos A (ix) A tan A tan A tan





(x) sin A sin A kos A

(xi) kos A kos A  sin A

A sin     A kos (xii) tan kos A tan A tan A  

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(xiii)               

 sin B sin A B kos A B

A sin (xiv)











 











 





sin B kos A B sin A B A sin (xv)











 











 





kos B kos A B kos A B A kos (xvi)











 











 







kos B sin A B sin A B A kos 3.2 Penyelesaian Segiti ga Solution of Triangles 3.2.1 Petua Sinus Sine Rule C sin c B sin b A sin a  

3.2.2 Petua Kos inu s Cosine Rule A kos bc c b a _ _ B kos ac c a b _ _ C kos ab b a c _ _ 3.2.3 Luas Segiti ga Ar ea of Tri ang le

A sin bc  B sin ac  C sin ab    

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4. KALKULUS

CALCULUS

4.1 Fungs i dan Had

Functions and Limits

(i) (ii) (iii) (iv) 4.2 Petua L L Hopital s Rule ) x ( ' g ) x ( ' f lim ) x ( g ) x ( f lim c x c x   4.3 Pembezaan Differentiation (i)   n n nx x dx d

(ii)

   

ax  b n  na ax b dx d (iii)

   

dx du v dx dv u dx dy   (iv) v u y v dx dv u dx du v dx dy            x x kos lim x x sin lim sin lim x kos lim x x x x x Hopital  n

(22)

(v) y f

_{ }

u , u g

_{ }

x dx du du dy dx dy   (vi)



→

dx dy x y  (vii) x x ln dx d  (viii)





b ax a b ax ln dx d    (ix) x x e e dx d  (x)  ax  b  ax  b e a e dx d   

(xi) sin x kos x dx

d



(xii) kos x sin x dx

d

 

(xiii) tan x sek x dx

d



(xiv) kosek x kosek x kot x dx

d

 

(xv) sek x sek x tan x dx

d



(xvi) kot x kosek x dx

d _

(xvii) sin



ax



b a kos



ax



b dx

d _ _ _

(xviii) kos



ax



b a sin



ax



b dx

d _ __ _

(23)

(xx) sin



ax



b a n sin



ax



b kos



ax



b dx

d n _ _ n _ _

(xxi) kos



ax



b a n kos



ax



b sin



ax



b dx d _n _n      

(xxii) tan



ax



b a n tan



ax



b sek



ax



b dx d n _ _ n _ _ 4.4 Pengamiran Integration (i)















c , n n x dx x n n (ii) c , n ax dx ax n n

_









n (iii)



























c n n a b ax dx b ax n n (iv) dx ln x c x  



(v) ln



ax b



c a dx b ax









(vi) _dx _f _x _c x f x f ln ' (vii)



e x dx



ex



c (viii) c k e dx e kx kx

_

_



(ix) c k kx sin dx kx kos ; c x sin dx x kos    



(x) c k kx kos dx kx sin ; c x kos dx x sin     



(xi) c k kx tan dx kx sek ; c x tan dx x sek  



 



,

(24)

(xii) c k kx kot dx kx kosek ; c x kot dx x kosek (xiii) e f x f ' x dx e f x c (xiv)

∫

   

∫

 

(xv)



tan x dx



ln sek x



c

(xvi)



kot x dx



ln sin x



c

(xvii)



kosek x dx





ln kosek x



kot x



c

(xviii)



sek x dx



ln sek x



tan x



c

4.5 Luas di bawah Lengkung Ar ea un der a Cur ve

(i) b a x ydx A (ii) b a y xdy A

4.6 Isi padu Bung kah Perki saran Volume of Solid of Revoluti on (i) V y dx b a x (ii) dy x V b a y 



(25)

5. STATISTIK

STATISTICS

5.1 Kebarangkalian Probability (i)

 

S n A n A P  (ii)  P

 

A (iii)



  ) ( n i i E P

(iv) P A' 1 P A

(v) Jika A dan B tak bersandar, maka If A and B are independent, then

(vi) Jika A dan B peristiwa saling tak eksklusif, maka If A and B are not mutually exclusive events, then

(vii) Jika A dan B peristiwa saling eksklusif, maka If A and B are mutually exclusive events, then

(viii)









 

A P B A P A | B P





5.2 Ukuran Kecenderungan Memus at Measures of Central Tendency (i) Min Aritmetik

Arithmetic Mean (a) n x x 



(b)



 f fx x



A B



P( A) P(B) P   



A B

     

P A P B P A B



P     



A B



P B P A P B P   A

(26)

(c)



  f fd x x _a (d)

_





















C f fd x x _a (ii) Median Median = f C F N L m (iii) _Mod Mode = d d C d L 5.3 Ukuran Keduduk an Measures of Position (i) Kuartil Quartile , , k , C f F kn L Q k Q k (ii) Desil Decile f F kn L D k D k (iii) Persentil Percentile ..., , , , k , C f F kn L P k P k ..., , , , k , C 99

(27)

5.4 Ukuran Serakan Measures of Dispersio n (i) Populasi Population (a) N x = N x ─



(b) N x = 2 N x (c)







  f x f = f x f (d) (ii) Sampel Sample (a) n x x s (b) n x x s f x f f d f f d f

(28)

5.5 Pilih Atur Permutation





! ! r n n P_r n   = n

   

n n



nr 



5.6 Gabungan Combination





! ! ! r n r n r C n r n

















5.7 Taburan Bino mial Binomial Distribution (i) E X np (ii) Var

 

X  npq (iii)  npq (iv) P



X  x



 nC _x p x q n x , x , , ,



,n 

5.8 Taburan Normal Piawai

Standard Normal Distributi on (i) (ii) n , s x x t x z

(29)

5.9 Taburan Pensampelan Sampling Distribu tion

5.9.1 Bagi populasi tak terhingga atau dengan pengembalian For infinite population or with replacement;

(i) Min Mean



x

(ii) Varians bagi min Variance for mean

n

x 

(iii) Ralat piawai min (Sisihan piawai bagi min)

Standard error of the mean (SEM) (Standard deviation for mean)

n

x 

5.9.2 Bagi populasi terhingga atau tanpa pengembalian For finite population or without replacement

(i) Min Mean



x

(ii) Varians bagi min Variance for mean

















(iii) Ralat piawai min (sisihan piawai bagi min)

Standard error of the mean (SEM)(standard deviation for mean)    N n N n x N n N n x

(30)

5.10 Anggaran Selang Interval Estimation

(i) Selang keyakinan

_



_



1 100% bagi  jika 2 diketahui Confidence interval

_



_



1 100% for  if 2 is known

















n z x , n z x

(ii) Selang keyakinan







1 100% bagi  jika 2 tak diketahui dan

30



n

Confidence interval

_



_



1 100% for  if 2is not known and

30  n

















n s z x , n s z x

(iii) Selang keyakinan

_



_



1 100% bagi  jika 2 tak diketahui dan sampel kecil, n 30

Confidence interval

_



_



1 100% for  if 2is not known and small sample, n 30

















  _n s t x , n s t x n , n ,

(iv) Selang keyakinan

_



_

 1 100% bagi kadaran p Confidence interval





 1 100% for proportion p

















n qˆ pˆ z pˆ , n qˆ pˆ z pˆ (v) Selang keyakinan







1 100% bagi populasi varians

Confidence interval



1_



100% for population variance

L R s ) n ( s ) n (  _ _ 

(31)

5.11 Penguj ian Hipotesis Hypothesis Testing

5.11.1 Ujian Satu Sampel One-sample Test (i) Ujian min

Mean test Apabila 2  diketahui When 2  is known n x z   Apabila 2  tak diketahui when 2is not known

n s x z  

Apabila _2 _{tak diketahui dan} ₃₀  n when 2is not known and n_30

n s x t  

(ii) Ujian perkadaran Proportion test n pq p p z   ˆ

(iii) Ujian varians Variance test s ) n (  

(32)

5.11.2 Ujian Dua Sampel Two-sample Test

(i) Ujian perbezaan antara dua min populasi

Test of the difference between two population means

n n x x z      ( ) ( )

(ii) Ujian-t tak bersandar Independent t-test (a) Varians yang sama

Equal variances assumed

n n S ) ( ) x x ( t p      dengan n n s ) n ( s ) n ( S_p dan

df



n



n



(b) Varians yang tidak sama Equal variances not assumed

n s n s ) ( ) x x ( t      _;















































n s n n s n n s n s df

(iii) Ujian-t bersandar Dependent t-test

(33)

5.11.3 Ujian Hipot esis Khi Kuasa Dua Chi-square Hypot hesis Test

(i) Apabila jangkaan frekuensi adalah sama When expected frequencies are equal

k n E _

(ii) Apabila jangkaan frekuensi adalah tak sama When expected frequencies are not all equal

np

E



(iii) Ujian statistik Khi kuasa dua Chi-square Test Statistics



   k i i i i E ) E O (

5.12 Analis is Vari ans

An aly si s o f Vari anc e (ANOVA)

(i) Ujian statistik ANOVA sehala (saiz sampel sama)

Test statistic for one-way ANOVA (equal sample sizes)

p x

s

ns

F



(ii) Ujian statistik ANOVA sehala (saiz sampel tak sama) Test statistic for one-way ANOVA (unequal sample sizes)



















































i i i i i n s n k x x n F

(34)

(iii) Hasil tambah kuasa dua ANOVA ANOVA Sum of squares

SST ₌





N x x



 SSB ₌ SSW ₌ SST _– SSB F = k N SSW k SSB



k) (N df ) (k df W B     5.13 Regresi Linear Linear Regression

(i) Persamaan ramalan Prediction equation x b b y _o







 

   x x n y x xy n b x b y b_o   (ii) Reja Residual e = y - ŷ

(iii) Pekali Korelasi

Correlation coefficient

  















   

(

y

n

)

x

(

x

n

y

x

xy

n

r

N x n T i i 

)

y

(35)

6. MATEMATIK KEWANGAN

FINANCIAL MATHEMATICS

6.1 Faedah dan Diskaun Interest and Discount

(i) Faedah mudah Simple interest

  

(ii) Diskaun mudah Simple discount

  

(iii) Faedah kompaun Compound interest

   



 



  



(iv) Nilai kini

Present value

  

_{}



(v) Kompaun berterusan Continuous compounding

  



(vi) Diskaun kompaun Compound discount

  ( )



(vii) Kadar faedah efektif Effective interest rate

(36)

6.2 Anuiti Annu it y

(i) Anuiti serta-merta Immediate annuity

  

_{ }





  

_{ }

−

(ii) Anuiti matang Annuity due

  

_

+





  

_

−−



(iii) Anuiti tertunda Deferred annuity

  

_{ }





   [

−



−+

]

(iv) Anuiti dengan kompaun berterusan Annuity with continuous compounding

  

_









  

_



 

−

6.3 Pelunasan Pinjaman Am or ti sat io n o f L oan

(37)

6.4 Dana Terikat Sinking Fund

  

_{}





6.5 Alir an Tunai Cash Flow

(i) Nilai kini bersih Net present value

  ∑ 









=

(ii) Kadar pulangan dalaman Internal Rate of Return (IRR)

  



 

_

_

_



_

_×









(iii) Harga belian bon

Purchase price of bonds

  

_{  }

−

(iv) Kadar inflasi Rate of inflation

   

+









(v) Kadar pulangan sebenar Real rate of return

(38)

6.6 Nombor Indeks Index Number (i) Harga relatif

Relative price   o n n , o h h I

(ii) Relatif rantai Chain relative     i i i , i h h I

(iii) Purata indeks relatif Average relative index

N h h I o n



















(iv) Indeks harga relatif berpemberat Weighted relative price index



   n i i n i i i w w I I

(39)

31 7. SIFIR STATISTIK

STATISTICAL TABLES

7.1 Pekali Bino mial

Binomial Coefficients Nilai-nilai  n k! ! k ! n k n n k n C _k n                  k n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 ₁ ₁ 2 ₁ ₂ ₁ 3 ₁ ₃ ₃ ₁ 4 ₁ ₄ ₆ ₄ ₁ 5 ₁ ₅ ₁₀ ₁₀ ₅ ₁ 6 ₁ ₆ ₁₅ ₂₀ ₁₅ ₆ ₁ 7 ₁ ₇ ₂₁ ₃₅ ₃₅ ₂₁ ₇ ₁ 8 ₁ ₈ ₂₈ ₅₆ ₇₀ ₅₆ ₂₈ ₈ ₁ 9 ₁ ₉ ₃₆ ₈₄ ₁₂₆ ₁₂₆ ₈₄ ₃₆ ₉ ₁ 10 ₁ ₁₀ ₄₅ ₁₂₀ ₂₁₀ ₂₅₂ ₂₁₀ ₁₂₀ ₄₅ ₁₀ ₁ 11 ₁ ₁₁ ₅₅ ₁₆₅ ₃₃₀ ₄₆₂ ₄₆₂ ₃₃₀ ₁₆₅ ₅₅ ₁₁ ₁ 12 ₁ ₁₂ ₆₆ ₂₂₀ ₄₉₅ ₇₉₂ ₉₂₄ ₇₉₂ ₄₉₅ ₂₂₀ ₆₆ ₁₂ ₁ 13 ₁ ₁₃ ₇₈ ₂₈₆ ₇₁₅ _1,287 _1,716 _1,716 _1,287 ₇₁₅ ₂₈₆ ₇₈ ₁₃ ₁ 14 ₁ ₁₄ ₉₁ ₃₆₄ _1,001 _2,002 _3,003 _3,432 _3,003 _2,002 _1,001 ₃₆₄ ₉₁ ₁₄ ₁ 15 ₁ ₁₅ ₁₀₅ ₄₅₅ _1,365 _3,003 _5,005 _6,435 _6,435 _5,005 _3,003 _1,365 ₄₅₅ ₁₀₅ ₁₅ ₁ 16 ₁ ₁₆ ₁₂₀ ₅₆₀ _1,820 _4,368 _8,008 _11,440 _12,870 _11,440 _8,008 _4,368 _1,820 ₅₆₀ ₁₂₀ ₁₆ 17 ₁ ₁₇ ₁₃₆ ₆₈₀ _2,380 _6,188 _12,376 _19,448 _24,310 _{24,3 10} _19,448 _12,376 _6,188 _2,380 ₆₈₀ ₁₃₆ 18 ₁ ₁₈ ₁₅₃ ₈₁₆ _3,060 _8,568 _18,564 _31,824 _43,758 _48,620 _43,758 _31,824 _18,564 _8,568 _3,060 ₈₁₆ 19 ₁ ₁₉ ₁₇₁ ₉₆₉ _3,876 _11,628 _27,132 _50,388 _75,582 _92,378 _92,378 _75,582 _50,388 _27,132 _11,628 _3,876 20 ₁ ₂₀ ₁₉₀ _1,140 _4,845 _15,504 _38,760 _77,520 _125,970 _167,960 _184,756 _167,960 _125,970 _77,520 _38,760 _15,504 21 ₁ ₂₁ ₂₁₀ _1,330 _5,985 _20,349 _54,264 _116,280 _203,490 _293,930 _352,716 _352,716 _293,930 _203,490 _116,280 _54,264 22 ₁ ₂₂ ₂₃₁ _1,540 _7,315 _26,334 _74,613 _170,544 _319,770 _497,420 _646,646 _705,432 _646,646 _497,420 _319,770 _170,544 23 ₁ ₂₃ ₂₅₃ _1,771 _8,855 _33,649 _100,947 _245,157 _490,314 _817,190 _1,144,066 _1,352,078 _1,352,078 _1,144,066 _817,190 _490,314 24 ₁ ₂₄ ₂₇₆ _2,024 _10,626 _42,504 _134,596 _346,104 _735,471 _1,307,504 _1,961,256 _2,496,144 _2,704,156 _2,496,144 _1,961,256 _1,307,504 25 ₁ ₂₅ ₃₀₀ _2,300 _12,650 _53,130 _177,100 _480,700 _1,081,575 _2,042,975 _3,268,760 _4,457,400 _5,200,300 _5,200,300 _4,457,400 _3,268,760 26 ₁ ₂₆ ₃₂₅ _2,600 _14,950 _65,780 _230,230 _657,800 _1,562,275 _3,124,550 _5,311,735 _7,726,160 _9,657,700 _10,400,600 _9,657,700 _7,726,160 27 ₁ ₂₇ ₃₅₁ _2,925 _17,550 _80,730 _296,010 _888,030 _2,220,075 _4,686,825 _8,436,285 _13,037,895 _17,383,860 _20,058,300 _20,058,300 _17,383,860 28 ₁ ₂₈ ₃₇₈ _3,276 _20,475 _98,280 _376,740 _1,184,040 _3,108,105 _6,906,900 _13,123,110 _21,474,180 _30,421,755 _37,442,160 _40,116,600 _37,442,160 29 ₁ ₂₉ ₄₀₆ _3,654 _23,751 _118,755 _475,020 _1,560,780 _4,292,145 _10,015,005 _20,030,010 _34,597,290 _51,895,935 _67,863,915 _77,558,760 _77,558,760 30 ₁ ₃₀ ₄₃₅ _4,060 _27,405 _142,506 _593,775 _2,035,800 _5,852,925 _14,307,150 _30,045,015 _54,627,300 _86,493,225 _119,759,850 _145,422,675 _155,117,520

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7.2 Pekali Pilih Atur

Permutation Coefficients Nilai-nilai  n r! ! n P r n   r n 1 2 3 4 5 6 7 8 9 10 11 12 1 ₁ 2 ₂ ₂ 3 ₃ ₆ ₆ 4 ₄ ₁₂ ₂₄ ₂₄ 5 ₅ ₂₀ ₆₀ ₁₂₀ ₁₂₀ 6 ₆ ₃₀ ₁₂₀ ₃₆₀ ₇₂₀ ₇₂₀ 7 ₇ ₄₂ ₂₁₀ ₈₄₀ _2,520 _5,040 _5,040 8 ₈ ₅₆ ₃₃₆ _1,680 _6,720 _20,160 _40,320 _40,320 9 ₉ ₇₂ ₅₀₄ _3,024 _15,120 _60,480 _181,440 _362,880 _362,880 10 ₁₀ ₉₀ ₇₂₀ _5,040 _30,240 _151,200 _604,800 _1,814,400 _3,628,800 _3,628,800 11 ₁₁ ₁₁₀ ₉₉₀ _7,920 _55,440 _332,640 _1,663,200 _6,652,800 _19,958,400 _39,916,800 _39,916,800 12 ₁₂ ₁₃₂ _1,320 _11,880 _95,040 _665,280 _3,991,680 _19,958,400 _79,833,600 _239,500,800 _479,001,600 _479,001,600 13 ₁₃ ₁₅₆ _1,716 _17,160 _154,440 _1,235,520 _8,648,640 _51,891,840 _259,459,200 _{1,037,836,800} _{3,113,510,400} _{6,227,020,800} 14 ₁₄ ₁₈₂ _2,184 _24,024 _240,240 _2,162,160 _17,297,280 _121,080,960 _726,485,760 _{3,632,428,800} _{14,529,715,200} _{43,589,145,600} 15 ₁₅ ₂₁₀ _2,730 _32,760 _360,360 _3,603,600 _32,432,400 _259,459,200 _{1,816,214,400} _{10,897,286,400} _{54,486,432,000} _{217,945,728,000} 16 ₁₆ ₂₄₀ _3,360 _43,680 _524,160 _5,765,760 _57,657,600 _518,918,400 _{4,151,347,200} _{29,059,430,400} _{174,356,582,400} _{871,782,912,000} 17 ₁₇ ₂₇₂ _4,080 _57,120 _742,560 _8,910,720 _98,017,920 _980,179,200 _{8,821,612,800} _{70,572,902,400} _{494,010,316,800} _{2,964,061,900,800} 18 ₁₈ ₃₀₆ _4,896 _73,440 _1,028,160 _13,366,080 _160,392,960 _{1,764,322,560} _{17,643,225,600} _{158,789,030,400} _{1,270,312,243,200} _{8,892,185,702,400} 19 ₁₉ ₃₄₂ _5,814 _93,024 _1,395,360 _19,535,040 _253,955,520 _{3,047,466,240} _{33,522,128,640} _{335,221,286,400} _{3,016,991,577,600} _{24,135,932,620,800} 20 ₂₀ ₃₈₀ _6,840 _116,280 _1,860,480 _27,907,200 _390,700,800 _{5,079,110,400} _{60,949,324,800} _{670,442,572,800} _{6,704,425,728,000} _{60,339,831,552,000}

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33 7.3 Sifi r Taburan-F F-Distribution Tables d 1 d 2 1 2 3 4 5 6 7 8 9 10 12 15 20 24 30 40 60 120 inf 1 _39.86 _49.50 _53.59 _55.83 _57.24 _58.20 _58.91 _59.44 _59.86 _60.19 _60.71 _61.22 _61.74 _62.00 _62.26 _62.53 _62.79 _63.06 _63.33 2 _8.53 _9.00 _9.16 _9.24 _9.29 _9.33 _9.35 _9.37 _9.38 _9.39 _9.41 _9.42 _9.44 _9.45 _9.46 _9.47 _9.47 _9.48 _9.49 3 _5.54 _5.46 _5.39 _5.34 _5.31 _5.28 _5.27 _5.25 _5.24 _5.23 _5.22 _5.20 _5.18 _5.18 _5.17 _5.16 _5.15 _5.14 _5.13 4 _4.54 _4.32 _4.19 _4.11 _4.05 _4.01 _3.98 _3.95 _3.94 _3.92 _3.90 _3.87 _3.84 _3.83 _3.82 _3.80 _3.79 _3.78 _3.76 5 _4.06 _3.78 _3.62 _3.52 _3.45 _3.40 _3.37 _3.34 _3.32 _3.30 _3.27 _3.24 _3.21 _3.19 _3.17 _3.16 _3.14 _3.12 _3.10 6 _3.78 _3.46 _3.29 _3.18 _3.11 _3.05 _3.01 _2.98 _2.96 _2.94 _2.90 _2.87 _2.84 _2.82 _2.80 _2.78 _2.76 _2.74 _2.72 7 _3.59 _3.26 _3.07 _2.96 _2.88 _2.83 _2.78 _2.75 _2.72 _2.70 _2.67 _2.63 _2.59 _2.58 _2.56 _2.54 _2.51 _2.49 _2.47 8 _3.46 _3.11 _2.92 _2.81 _2.73 _2.67 _2.62 _2.59 _2.56 _2.54 _2.50 _2.46 _2.42 _2.40 _2.38 _2.36 _2.34 _2.32 _2.29 9 _3.36 _3.01 _2.81 _2.69 _2.61 _2.55 _2.51 _2.47 _2.44 _2.42 _2.38 _2.34 _2.30 _2.28 _2.25 _2.23 _2.21 _2.18 _2.16 10 _3.29 _2.92 _2.73 _2.61 _2.52 _2.46 _2.41 _2.38 _2.35 _2.32 _2.28 _2.24 _2.20 _2.18 _2.16 _2.13 _2.11 _2.08 _2.06 11 _3.23 _2.86 _2.66 _2.54 _2.45 _2.39 _2.34 _2.30 _2.27 _2.25 _2.21 _2.17 _2.12 _2.10 _2.08 _2.05 _2.03 _2.00 _1.97 12 _3.18 _2.81 _2.61 _2.48 _2.39 _2.33 _2.28 _2.24 _2.21 _2.19 _2.15 _2.10 _2.06 _2.04 _2.01 _1.99 _1.96 _1.93 _1.90 13 _3.14 _2.76 _2.56 _2.43 _2.35 _2.28 _2.23 _2.20 _2.16 _2.14 _2.10 _2.05 _2.01 _1.98 _1.96 _1.93 _1.90 _1.88 _1.85 14 _3.10 _2.73 _2.52 _2.39 _2.31 _2.24 _2.19 _2.15 _2.12 _2.10 _2.05 _2.01 _1.96 _1.94 _1.91 _1.89 _1.86 _1.83 _1.80 15 _3.07 _2.70 _2.49 _2.36 _2.27 _2.21 _2.16 _2.12 _2.09 _2.06 _2.02 _1.97 _1.92 _1.90 _1.87 _1.85 _1.82 _1.79 _1.76 16 _3.05 _2.67 _2.46 _2.33 _2.24 _2.18 _2.13 _2.09 _2.06 _2.03 _1.99 _1.94 _1.89 _1.87 _1.84 _1.81 _1.78 _1.75 _1.72 17 _3.03 _2.64 _2.44 _2.31 _2.22 _2.15 _2.10 _2.06 _2.03 _2.00 _1.96 _1.91 _1.86 _1.84 _1.81 _1.78 _1.75 _1.72 _1.69 18 _3.01 _2.62 _2.42 _2.29 _2.20 _2.13 _2.08 _2.04 _2.00 _1.98 _1.93 _1.89 _1.84 _1.81 _1.78 _1.75 _1.72 _1.69 _1.66 19 _2.99 _2.61 _2.40 _2.27 _2.18 _2.11 _2.06 _2.02 _1.98 _1.96 _1.91 _1.86 _1.81 _1.79 _1.76 _1.73 _1.70 _1.67 _1.63 20 _2.97 _2.59 _2.38 _2.25 _2.16 _2.09 _2.04 _2.00 _1.96 _1.94 _1.89 _1.84 _1.79 _1.77 _1.74 _1.71 _1.68 _1.64 _1.61 21 _2.96 _2.57 _2.36 _2.23 _2.14 _2.08 _2.02 _1.98 _1.95 _1.92 _1.87 _1.83 _1.78 _1.75 _1.72 _1.69 _1.66 _1.62 _1.59 22 _2.95 _2.56 _2.35 _2.22 _2.13 _2.06 _2.01 _1.97 _1.93 _1.90 _1.86 _1.81 _1.76 _1.73 _1.70 _1.67 _1.64 _1.60 _1.57 23 _2.94 _2.55 _2.34 _2.21 _2.11 _2.05 _1.99 _1.95 _1.92 _1.89 _1.84 _1.80 _1.74 _1.72 _1.69 _1.66 _1.62 _1.59 _1.55 24 _2.93 _2.54 _2.33 _2.19 _2.10 _2.04 _1.98 _1.94 _1.91 _1.88 _1.83 _1.78 _1.73 _1.70 _1.67 _1.64 _1.61 _1.57 _1.53 25 _2.92 _2.53 _2.32 _2.18 _2.09 _2.02 _1.97 _1.93 _1.89 _1.87 _1.82 _1.77 _1.72 _1.69 _1.66 _1.63 _1.59 _1.56 _1.52 26 _2.91 _2.52 _2.31 _2.17 _2.08 _2.01 _1.96 _1.92 _1.88 _1.86 _1.81 _1.76 _1.71 _1.68 _1.65 _1.61 _1.58 _1.54 _1.50 27 _2.90 _2.51 _2.30 _2.17 _2.07 _2.00 _1.95 _1.91 _1.87 _1.85 _1.80 _1.75 _1.70 _1.67 _1.64 _1.60 _1.57 _1.53 _1.49 28 _2.89 _2.50 _2.29 _2.16 _2.06 _2.00 _1.94 _1.90 _1.87 _1.84 _1.79 _1.74 _1.69 _1.66 _1.63 _1.59 _1.56 _1.52 _1.48 29 _2.89 _2.50 _2.28 _2.15 _2.06 _1.99 _1.93 _1.89 _1.86 _1.83 _1.78 _1.73 _1.68 _1.65 _1.62 _1.58 _1.55 _1.51 _1.47 30 _2.88 _2.49 _2.28 _2.14 _2.05 _1.98 _1.93 _1.88 _1.85 _1.82 _1.77 _1.72 _1.67 _1.64 _1.61 _1.57 _1.54 _1.50 _1.46 40 _2.84 _2.44 _2.23 _2.09 _2.00 _1.93 _1.87 _1.83 _1.79 _1.76 _1.71 _1.66 _1.61 _1.57 _1.54 _1.51 _1.47 _1.42 _1.38 60 _2.79 _2.39 _2.18 _2.04 _1.95 _1.87 _1.82 _1.77 _1.74 _1.71 _1.66 _1.60 _1.54 _1.51 _1.48 _1.44 _1.40 _1.35 _1.29 120 _2.75 _2.35 _2.13 _1.99 _1.90 _1.82 _1.77 _1.72 _1.68 _1.65 _1.60 _1.55 _1.48 _1.45 _1.41 _1.37 _1.32 _1.26 _1.19 inf _2.71 _2.30 _2.08 _1.94 _1.85 _1.77 _1.72 _1.67 _1.63 _1.60 _1.55 _1.49 _1.42 _1.38 _1.34 _1.30 _1.24 _1.17 _1.00

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34 d 1 d 2 1 2 3 4 5 6 7 8 9 10 12 15 20 24 30 40 60 120 inf 1 _161.45 _199.50 _215.71 _224.58 _230.16 _233.99 _236.77 _238.88 _240.54 _241.88 _243.91 _245.95 _248.01 _249.05 _250.10 _251.14 _252.20 _253.25 _254.31 2 _18.51 _19.00 _19.16 _19.25 _19.30 _19.33 _19.35 _19.37 _19.38 _19.40 _19.41 _19.43 _19.45 _19.45 _19.46 _19.47 _19.48 _19.49 _19.50 3 _10.13 _9.55 _9.28 _9.12 _9.01 _8.94 _8.89 _8.85 _8.81 _8.79 _8.74 _8.70 _8.66 _8.64 _8.62 _8.59 _8.57 _8.55 _8.53 4 _7.71 _6.94 _6.59 _6.39 _6.26 _6.16 _6.09 _6.04 _6.00 _5.96 _5.91 _5.86 _5.80 _5.77 _5.75 _5.72 _5.69 _5.66 _5.63 5 _6.61 _5.79 _5.41 _5.19 _5.05 _4.95 _4.88 _4.82 _4.77 _4.74 _4.68 _4.62 _4.56 _4.53 _4.50 _4.46 _4.43 _4.40 _4.36 6 _5.99 _5.14 _4.76 _4.53 _4.39 _4.28 _4.21 _4.15 _4.10 _4.06 _4.00 _3.94 _3.87 _3.84 _3.81 _3.77 _3.74 _3.70 _3.67 7 _5.59 _4.74 _4.35 _4.12 _3.97 _3.87 _3.79 _3.73 _3.68 _3.64 _3.57 _3.51 _3.44 _3.41 _3.38 _3.34 _3.30 _3.27 _3.23 8 _5.32 _4.46 _4.07 _3.84 _3.69 _3.58 _3.50 _3.44 _3.39 _3.35 _3.28 _3.22 _3.15 _3.12 _3.08 _3.04 _3.01 _2.97 _2.93 9 _5.12 _4.26 _3.86 _3.63 _3.48 _3.37 _3.29 _3.23 _3.18 _3.14 _3.07 _3.01 _2.94 _2.90 _2.86 _2.83 _2.79 _2.75 _2.71 10 _4.96 _4.10 _3.71 _3.48 _3.33 _3.22 _3.14 _3.07 _3.02 _2.98 _2.91 _2.85 _2.77 _2.74 _2.70 _2.66 _2.62 _2.58 _2.54 11 _4.84 _3.98 _3.59 _3.36 _3.20 _3.09 _3.01 _2.95 _2.90 _2.85 _2.79 _2.72 _2.65 _2.61 _2.57 _2.53 _2.49 _2.45 _2.40 12 _4.75 _3.89 _3.49 _3.26 _3.11 _3.00 _2.91 _2.85 _2.80 _2.75 _2.69 _2.62 _2.54 _2.51 _2.47 _2.43 _2.38 _2.34 _2.30 13 _4.67 _3.81 _3.41 _3.18 _3.03 _2.92 _2.83 _2.77 _2.71 _2.67 _2.60 _2.53 _2.46 _2.42 _2.38 _2.34 _2.30 _2.25 _2.21 14 _4.60 _3.74 _3.34 _3.11 _2.96 _2.85 _2.76 _2.70 _2.65 _2.60 _2.53 _2.46 _2.39 _2.35 _2.31 _2.27 _2.22 _2.18 _2.13 15 _4.54 _3.68 _3.29 _3.06 _2.90 _2.79 _2.71 _2.64 _2.59 _2.54 _2.48 _2.40 _2.33 _2.29 _2.25 _2.20 _2.16 _2.11 _2.07 16 _4.49 _3.63 _3.24 _3.01 _2.85 _2.74 _2.66 _2.59 _2.54 _2.49 _2.42 _2.35 _2.28 _2.24 _2.19 _2.15 _2.11 _2.06 _2.01 17 _4.45 _3.59 _3.20 _2.96 _2.81 _2.70 _2.61 _2.55 _2.49 _2.45 _2.38 _2.31 _2.23 _2.19 _2.15 _2.10 _2.06 _2.01 _1.96 18 _4.41 _3.55 _3.16 _2.93 _2.77 _2.66 _2.58 _2.51 _2.46 _2.41 _2.34 _2.27 _2.19 _2.15 _2.11 _2.06 _2.02 _1.97 _1.92 19 _4.38 _3.52 _3.13 _2.90 _2.74 _2.63 _2.54 _2.48 _2.42 _2.38 _2.31 _2.23 _2.16 _2.11 _2.07 _2.03 _1.98 _1.93 _1.88 20 _4.35 _3.49 _3.10 _2.87 _2.71 _2.60 _2.51 _2.45 _2.39 _2.35 _2.28 _2.20 _2.12 _2.08 _2.04 _1.99 _1.95 _1.90 _1.84 21 _4.32 _3.47 _3.07 _2.84 _2.68 _2.57 _2.49 _2.42 _2.37 _2.32 _2.25 _2.18 _2.10 _2.05 _2.01 _1.96 _1.92 _1.87 _1.81 22 _4.30 _3.44 _3.05 _2.82 _2.66 _2.55 _2.46 _2.40 _2.34 _2.30 _2.23 _2.15 _2.07 _2.03 _1.98 _1.94 _1.89 _1.84 _1.78 23 _4.28 _3.42 _3.03 _2.80 _2.64 _2.53 _2.44 _2.37 _2.32 _2.27 _2.20 _2.13 _2.05 _2.01 _1.96 _1.91 _1.86 _1.81 _1.76 24 _4.26 _3.40 _3.01 _2.78 _2.62 _2.51 _2.42 _2.36 _2.30 _2.25 _2.18 _2.11 _2.03 _1.98 _1.94 _1.89 _1.84 _1.79 _1.73 25 _4.24 _3.39 _2.99 _2.76 _2.60 _2.49 _2.40 _2.34 _2.28 _2.24 _2.16 _2.09 _2.01 _1.96 _1.92 _1.87 _1.82 _1.77 _1.71 26 _4.23 _3.37 _2.98 _2.74 _2.59 _2.47 _2.39 _2.32 _2.27 _2.22 _2.15 _2.07 _1.99 _1.95 _1.90 _1.85 _1.80 _1.75 _1.69 27 _4.21 _3.35 _2.96 _2.73 _2.57 _2.46 _2.37 _2.31 _2.25 _2.20 _2.13 _2.06 _1.97 _1.93 _1.88 _1.84 _1.79 _1.73 _1.67 28 _4.20 _3.34 _2.95 _2.71 _2.56 _2.45 _2.36 _2.29 _2.24 _2.19 _2.12 _2.04 _1.96 _1.91 _1.87 _1.82 _1.77 _1.71 _1.65 29 _4.18 _3.33 _2.93 _2.70 _2.55 _2.43 _2.35 _2.28 _2.22 _2.18 _2.10 _2.03 _1.94 _1.90 _1.85 _1.81 _1.75 _1.70 _1.64 30 _4.17 _3.32 _2.92 _2.69 _2.53 _2.42 _2.33 _2.27 _2.21 _2.16 _2.09 _2.01 _1.93 _1.89 _1.84 _1.79 _1.74 _1.68 _1.62 40 _4.08 _3.23 _2.84 _2.61 _2.45 _2.34 _2.25 _2.18 _2.12 _2.08 _2.00 _1.92 _1.84 _1.79 _1.74 _1.69 _1.64 _1.58 _1.51 60 _4.00 _3.15 _2.76 _2.53 _2.37 _2.25 _2.17 _2.10 _2.04 _1.99 _1.92 _1.84 _1.75 _1.70 _1.65 _1.59 _1.53 _1.47 _1.39 120 _3.92 _3.07 _2.68 _2.45 _2.29 _2.18 _2.09 _2.02 _1.96 _1.91 _1.83 _1.75 _1.66 _1.61 _1.55 _1.50 _1.43 _1.35 _1.25 inf _3.84 _3.00 _2.60 _2.37 _2.21 _2.10 _2.01 _1.94 _1.88 _1.83 _1.75 _1.67 _1.57 _1.52 _1.46 _1.39 _1.32 _1.22 _1.00