Buku Rumus Sifir Matematik 2019
May 14, 2019 | Author: ஆனந்த ராஜ் முனுசாமி | Category: N/A
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PENDAHULUAN
Falsafah Pendidikan Kebangsaan (FPK) dan Falsafah Pendidikan Guru (FPG) menekankan kepada perkembangan potensi individu guru secara seimbang dan bersepadu dari segi jasmani, emosi, rohani dan intelek. Hasrat ini dapat dicapai menerusi pelaksanaan kurikulum pendidikan guru yang menyepadukan aspek-aspek pengetahuan, kemahiran ikhtisas dan amalan nilai keguruan.
Berasaskan kepada FPK dan FPG, matematik telah menjadi bidang ilmu yang dapat melatih minda supaya berfikir secara mantik dan bersistem dalam menyelesaikan masalah dan membuat keputusan. Sejajar dengan itu, buku Rumus Matematik dan Sifir Statistik ini telah diolah dan disusun semula dengan mengambil kira kesinambungan yang berterusan mengikut kursuskursus yang ditawarkan oleh Institut Pendidikan Guru Malaysia (IPGM). Buku ini diharapkan dapat menjadi bahan rujukan dan membantu siswa pendidik dalam menyelesaikan masalah matematik menerusi rumus-rumus penting dan sifir statistik yang disediakan.
Buku ini dibahagikan kepada enam tajuk utama iaitu Algebra, Geometri, Trigonometri, Kalkulus, Statistik dan Matematik Kewangan. Setiap tajuk pula dibahagikan kepada topik-topik kecil supaya proses pembelajaran dapat dilakukan dengan lebih mudah, teratur dan bermakna. Semoga buku ini dapat memberi manfaat kepada siswa pendidik di institut pendidikan guru.
i
AHL I PANEL PENYEMAK AN SEMUL A
1.
Dr. Jong Cherng Meei IPG Kampus Pulau Pinang Pulau Pinang
2.
Dr. Teong Mee Mee IPG Kampus Pulau Pinang Pulau Pinang
3.
Dr. Lam Kah Kei IPG Kampus Tengku Ampuan Afzan Kuala Lipis, Pahang
4.
Dr. Ng Kok Fu IPG Kampus Sultan Abdul Halim Sungai Petani, Kedah
5.
Datin Hjh. Zaitun binti Othman IPG Kampus Pendidikan Islam Bangi, Selangor
6.
Pn. Manisah Mohd Shah IPG Kampus Pendidkan Teknik Nilai, Negeri Sembilan
7.
Pn. Lee Yen Ting Institut Pendidikan Guru Malaysia Kementerian Pendidikan Malaysia
ii
KANDUNGAN CONTENTS
Pendahuluan Preface
..………………………………………………………..
i
..………………………………………………………..
ii
Ah l i Pan el Panel Members
1.
ALGEBRA AL GEBRA
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12
Indeks Indices ……………………………………………………. Logaritma Logarithm ………………………………………………… Surd Surd ………………………………………………………. Nombor Kompleks Complex Numbers ………………………………………. Nilai Mutlak Absolute Value …………………………………………... Matriks Matrices ....................................................................... Punca Persamaan Kuadratik Roots of Quadratic Equation ........................................ Janjang Aritmetik Arithmetic Progression ....................................... .......... Janjang Geometri Geometric Progression ................................................ Kembangan Binomial Binomial Expansion ..................................................... Algebra Linear Linear Algebra ............................................................. Ruang Vektor Vector Spaces .............................................................
iii
1 1 2 2 3 3 3 4 4 4 5 6
2.
GEOMETRI GEOMETRY
2.1 2.2 2.3 2.4 2.5 2.6 2.7
3.
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
TRIGONOMETRI TRIGONOMETRY
3.1 3.2
4.
Identiti Trigonometri Trigonometric Identities………………………………..... Penyelesaian Segitiga Solution of Triangles …………………………………….
11 12
KALKULUS CALCULUS
4.1 4.2
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 …………………………. ’
’
4.3 4.4 4.5 4.6
13
’
iv
13 13 15 16 16
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
6.
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
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 ..............................................................
v
27 28 28 29 29 30
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
…………………………………………………..
vi
39
1.
ALGEBRA ALGEBRA 1.1
Indeks Indices (i)
a
(ii)
a
(iii)
a
(iv)
a
(v)
a
,
m
m
n
a
a
m n
1.2
a
m n
m n
a mn
n
a b
n
m
(vii)
a
n
a
(vi)
a
n
n
b a
m
a
n
n
a
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 a
(iii)
log a m n
(iv)
log a a
(v)
log a
(vi)
log a m
(vii)
log a b
m n
log a m
log a n
n log a m
log b m log b a
,
b
log b a
1
1.3
Surd Surd (i)
ab
a
(ii)
b
a
(v)
b
a b
a
(iv)
a
(iii)
1.4
a a b
a b
a
b
a
b
a b a b
Nombor Komp leks Complex Numbers (i)
a
bi
c
di
a
c
( b d )i
(ii)
a
bi
c
di
a
c
( b
(iii)
a
bi c
(iv) (v)
di
bd
( ad bc )i
bi ac bd bc ad i c di c d c d
a
Bentuk kutub nombor kompleks Polar form of complex numbers x iy r kos
(vi)
ac
i sin
Pendaraban nombor kompleks dalam bentuk kutub Multiplication of complex numbers in polar form
r kos i sin r ( kos r r kos i sin (vii)
d )i
i sin
)
Pembahagian nombor kompleks dalam bentuk kutub Division of complex numbers in polar form r kos
i sin
r ( kos
i sin
r )
r
2
kos
i sin
(viii)
Teorem De Moivre De Moivre's theorem
1.5
Nilai Mutlak Abso lu te Val ue (i)
x
jika x
x
if
(ii) x
jika x
x
if
1.6
1.7
x
x
Matriks Matrices (i)
a b , A c d
(ii)
A
A ad bc
d b , A c a
A
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 ax If ax
x
2
2
bx c
bx c
b
b
0 , maka
0 , then
ac
a
3
1.8
Janjang Arit metik Ar it hm eti c Pr og res si on (i)
T n
(ii)
d T n
(iii)
n
S n
1.9
a
n
a
S n
d
l
S n
Janjang Geometri Geometric Progression (i)
T n
(ii)
r
(iii)
(iv)
(v)
1.10
T n
T n
n
(iv)
n d
a
S n
S n
S
ar
n
T n T n
a r
r
a
r
r
r
a
,
n
r
n
,
r
,
r
Kembangan Bin omial Binomial Expansion (i)
a b n a
n
n
C a
n
b
n
C a
n
b
...
n
C r a
n r
r
b
... b n
n n n a n a n b a n b ... a n r b r ... b n r
4
(ii)
|| < 1 ∈ || < 1 ∈
Untuk For
1.11
dan and
,
(
n
x
,
nx
n n
)
(
x
...
) ...( n r )
n n
x
r
... x
n
... r
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
,
x
(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
,
5
1.12
Ruang Vektor Vector Spaces (i)
(ii)
ˆ
kos
xi x
(iii)
(iv)
ˆ
xi
x
y y
y
y j
x x
y j
kos
=
y
x x
k
z
z
6
y y
z z
2.
GEOMETRI GEOMETRY 2.1
2.2
2.3
Luas Ar ea = b
×
(i)
Luas segi empat selari Area of parallelogram
(ii)
Luas trapezium Area of trapezium
(iii)
Luas permukaan melengkung silinder Curved surface area of cylinder
=
(iv)
Luas permukaan melengkung kon Curved surface area of cone
=
r r
(v)
Luas permukaan sfera Surface area of sphere
=
r
=
h
a b h r h
Isi Padu Volume (i)
Isi padu silinder Volume of cylinder
=
(ii)
Isi padu kon Volume of cone
=
(iii)
Isi padu piramid Volume of pyramid
=
(iv)
Isi padu sfera Volume of sphere
=
r h
r h
A h
r
Sukatan Membulat Circul ar Measures (i)
(ii)
Panjang lengkok, Arc length, Luas sektor, Area of sector,
s r
dalam
A r
7
radian in radians
dalam
radian in radians
h
r l
2.4
Bulatan Circle (i)
(ii)
2.5
Panjang lengkok, Arc length, Luas sektor, Area of sector,
s
A
r
r
Geometri Koor dinat Coordin ate Geometry (i)
x x
Titik tengah, Mid-point,
M
y
y ,
(ii)
Jarak antara dua titik, Distance between two points,
(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
d
x
x
y
y
ax by c 0
ax
a
(iv)
by
c
b
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
nx mx m n (v)
,
my m n
ny
Luas segitiga Area of triangle
x
=
8
y
x
y
x
y
x
y
x
y
x
y
2.6
2.7
Transform asi: Pembesaran Transformation: Enlargement (i)
Luas imej Area of image
= k 2× luas objek = k 2× area of object
(ii)
Isi padu imej Volume of image
= k 3× isi padu objek = k 3× volume of object
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
(ii)
x
k
h
p( 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) x h
y k
a
b
y k
x h
a
b
,
a
9
b , c 2
a
b
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)
x h
y k
a
b
y k
x h
b
a
,
c 2
a
b
Persamaan –persamaan asimptot Equations of asymptotes y
y
b a a b
x h k x h k
10
3.
TRIGONOMETRI TRIGONOMETRY 3.1
Identiti Trigon ometri Trigonometric Identities (i)
sin
kos
(ii)
sek
(iii)
kosek
(iv)
sin A B sin A kos B kos A sin B
(v)
kos A B kos A kos B sin A sin B
(vi)
tan A B
(vii)
sin A
(viii)
kos A kos A sin A
tan
kot
kos
sin A
(xi)
kos A kos
tan A
tan A A
sin
A
kos
kos
A
A
A
A
A
sin
sin
tan
(xii)
A
tan A
(x)
sin A
tan A
tan A tan B
sin A kos A
(ix)
tan A tan B
tan
A
11
3.2
A B A B kos
(xiii)
sin A sin B
sin
(xiv)
sin A sin B
kos
(xv)
kos A kos B
(xvi)
kos A kos B sin
A B A B sin A B A B kos
kos
A B A B sin
Penyelesaian Segiti ga Solution of Triangles 3.2.1 Petua Sinus Sine Rule a sin A
b
c
sin B
sin C
3.2.2 Petua Kos inu s Cosine Rule a
b
c
bc kos A
b
a
c
ac kos B
c
a
b
ab kos C
3.2.3 Luas Segiti ga Ar ea of Tri ang le
bc sin A
ac sin B
ab sin C
s s a s b s c , 12
s
a b c
4.
KALKULUS CALCULUS 4.1
Fungs i dan Had Functions and Limits
(i)
(ii)
(iii)
lim kos x
lim sin x x
lim
sin x
x
x
(iv)
kos
lim
Petua L Hopital L Hopital s Rule lim x c
4.3
x
x
x
4.2
x
f ( x )
lim
g ( x )
x c
f ' ( x ) g' ( x )
Pembezaan Differentiation (i)
(ii) (iii)
d
x
dx
d dx
nx
n
ax b n na ax bn
dy dx
(iv)
n
y
u
dv dx
du
v
u
dx
u
v
v
dy dx
du dx
dv dx
v
13
(v)
y
dy
dx
(vi)
(vii)
(viii)
(ix)
(x)
(xi)
(xii)
(xiii)
(xiv)
(xv)
(xvi)
(xvii)
(xviii)
(xix)
f u ,
dy du
→ d
dy
d
e
x
dx d
e
dx
x
ln ax
dx
e
b
ax b
sin x
dx d
d dx d
kos x
dx
a
dx
d dx d dx d dx d dx
ax b
e
ax b
kos x
sin x
tan x sek x
kosek x
dx
d
a
x
dx d
g x
dx
x
ln x
du
y
dx
d
u
sek x
kosek x kot x
sek x tan x
kot x kosek x
sin ax b a kos ax b
kos ax b a sin ax b
tan ax b a sek ax b
14
(xx)
(xxi)
(xxii)
4.4
d
sin n ax b a n sin n ax b kos ax b
dx d
kos n ax
dx d dx
b a n kos n
ax b sin ax b
tan n ax b a n tan n ax b sek ax b
Pengamiran Integration x
n
(i)
x
(ii)
ax
(iii)
ax b n ax b dx a n c ,
(iv)
x dx ln x c
(v)
ax b dx a ln ax b c
dx
n
n
c , n
n
dx
ax
n
c ,
n
n
n
'
f x
(vi)
f x
dx
l n f
x
c
(vii)
e
(viii)
e
(ix)
(x)
sin x dx kos x c
(xi)
x
dx
kx
n
e x c
dx
e kx k
c
kos x dx sin x c ;
kos kx dx
;
sek x dx tan x c ;
15
sin kx
sin kx dx
k
c
kos kx
sek kx dx
k tan kx k
c
c
4.5
(xii)
kosek
(xiii)
e f x f ' x dx
x dx
kot x
e f
x
;
kosek
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
Luas di bawah Lengkung Ar ea un der a Cur ve (i)
b
A x
y dx a
(ii)
b
A y
x dy a
4.6
c
Isi padu Bung kah Perki saran Volume of Solid of Revoluti on (i)
b
V x
y dx a
(ii)
b
V y
x
dy
a
16
kx dx
kot kx k
c
5.
STATISTIK STATISTICS 5.1
Kebarangkalian Probability (i)
P A
n S
n A
(ii)
P A n
(iii)
P ( E
i
)
i
(iv)
P A
(v)
Jika A dan B tak bersandar, maka If A and B are independent, then
'
1
P A B
(vi)
P A
P ( A) P (B)
Jika A dan B peristiwa saling tak eksklusif, maka If A and B are not mutually exclusive events, then
P A B
(vii)
P A P B P A B
Jika A dan B peristiwa saling eksklusif, maka If A and B are mutually exclusive events, then
P A B
(viii)
5.2
P A
B
P B | A
P A
P A B
P B
P A
Ukuran Kecenderungan Memus at Measures of Central Tendency (i)
Min Aritmetik Arithmetic Mean (a)
x
x
n
(b)
x
fx f 17
x x a
(c)
fd f
fd (d) x x a f C (ii)
(iii)
5.3
N F
Median Median
= L
Mod Mode
=
f m
d
L d
C
C d
Ukuran Keduduk an Measures of Position (i)
Kuartil Quartile kn Q k
(ii)
L
f Q k
C ,
k
, ,
C ,
k
, , , ...,
Desil Decile kn D k
(iii)
F
L
F
f D k
Persentil Percentile kn P k
L
f P k
F C ,
18
k
, , , ...,
5.4
Ukuran Serakan Measures of Dispersio n (i)
Populasi Population x
(a)
N
=
N
=
=
(d)
(ii)
N
─
x
(b)
(c)
x
x
2
N
f x f f x
f x
f
f
f d
f d
f
f
Sampel Sample (a)
(b)
s
x
x
n
x
x
s n
19
5.5
Pilih Atur Permutation n!
n
P r
=
5.6
n
r !
n n
n
n r
Gabungan Combination
n n n! C r r ! n r ! r
5.7
Taburan Bino mial Binomial Distribution (i)
E X
(ii)
Var X
(iii) (iv)
5.8
np
npq
npq
P X x
n
C x p x q n
x
,
Taburan Normal Piawai Standard Normal Distributi on (i)
(ii)
z
t
x
x
x s
,
n
20
x
, , , , n
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
x
n
(iii)
Ralat piawai min (Sisihan piawai bagi min) Standard error of the mean (SEM) (Standard deviation for mean) x
n
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
x
(iii)
N n n N
Ralat piawai min (sisihan piawai bagi min) Standard error of the mean (SEM)(standard deviation for mean)
x
N
N
n
n
21
5.10
Anggaran Selang Interval Estimation (i)
Selang keyakinan 1 100% bagi Confidence interval 1 100% for
x z (ii)
, x
z
n
s
, x
z
is known
jika
2
tak diketahui dan
if
2
is not known and
n
n
Selang keyakinan 1 100% bagi sampel kecil, n 30
x t
s , n
, x
t
n
jika
if
2
2
tak diketahui dan
is not known and
n
s , n
Selang keyakinan 1 100% bagi kadaran p Confidence interval 1 100% for proportion p
p z
pq ˆ ˆ
ˆ
(v)
2
s
Confidence interval 1 100% for small sample, n 30
(iv)
diketahui
30
x z (iii)
if
2
30
Confidence interval 1 100% for n
n
Selang keyakinan 1 100% bagi n
jika
, p z ˆ
n
pq
n ˆ ˆ
Selang keyakinan 1 100% bagi populasi varians Confidence interval 1 100% for population variance
( n ) s R
( n ) s L
22
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 x z
n
Apabila 2 tak diketahui when 2 is not known x z
s n
Apabila when
t
x
2
2
tak diketahui dan
is not known and
s n
(ii)
Ujian perkadaran Proportion test p ˆ
z
p
pq n
(iii)
Ujian varians Variance test ( n
) s
23
n
n
30
30
5.11.2 Ujian Dua Sampel Two-sample Test (i)
Ujian perbezaan antara dua min populasi Test of the difference between two population means z
( x
)(
x
)
n
(ii)
n
Ujian-t tak bersandar Independent t-test (a) Varians yang sama Equal variances assumed
t
( x
x ) (
S p
n
n
( n
dengan S p dan df n
)
) s n
n
( n
) s
n
(b) Varians yang tidak sama Equal variances not assumed
t
( x
x ) ( s n
s
;
n
s s n n df s n n n (iii)
)
s n
Ujian-t bersandar Dependent t-test
t
D
N D ( D ) N 24
df
N
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 E
(ii)
n
k
Apabila jangkaan frekuensi adalah tak sama When expected frequencies are not all equal
E (iii)
np
Ujian statistik Khi kuasa dua Chi-square Test Statistics k
i
( Oi E i ) E i
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)
F (ii)
ns x s p
Ujian statistik ANOVA sehala (saiz sampel tak sama) Test statistic for one-way ANOVA (unequal sample sizes)
F
ni x i x k ni s i ni
25
(iii)
Hasil tambah kuasa dua ANOVA ANOVA Sum of squares
=
SST
x
x N
T i
SSB
=
SSW
=
x
n i
N
SST – SSB
SSB
df B
k SSW
F =
df W
(k
)
(N k)
N k
5.13
Regresi Linear Linear Regression (i)
Persamaan ramalan Prediction equation
y b o b x b
b o (ii)
n xy x y
n x x
y b x
Reja Residual
e = y - ŷ (iii)
Pekali Korelasi Correlation coefficient
r
n xy x y
n x
( x )
26
n y
( y )
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
27
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
− 28
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
29
6.6
Nombor Indeks Index Number (i)
Harga relatif Relative price I o , n
(ii)
h o
Relatif rantai Chain relative I i
(iii)
h n
,i
hi hi
Purata indeks relatif Average relative index
h n ho I
N
(iv)
Indeks harga relatif berpemberat Weighted relative price index n
I w i
I
i
i n
w
i
i
30
7. SIFIR STATISTIK
STATISTICAL TABLES 7.1
Pekali Bino mial
Binomial Coefficients Nilai-nilai
k
n
C k
n n n ! k ! k n k k ! n
0
1
2
3
4
5
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 171 190 210 231 253 276 300 325 351 378 406 435
1 4 10 20 35 56 84 120 165 220 286 364 455 560 680 816 969 1,140 1,330 1,540 1,771 2,024 2,300 2,600 2,925 3,276 3,654 4,060
1 5 15 35 70 126 210 330 495 715 1,001 1,365 1,820 2,380 3,060 3,876 4,845 5,985 7,315 8,855 10,626 12,650 14,950 17,550 20,475 23,751 27,405
1 6 21 56 126 252 462 792 1,287 2,002 3,003 4,368 6,188 8,568 11,628 15,504 20,349 26,334 33,649 42,504 53,130 65,780 80,730 98,280 118,755 142,506
6
7
8
9
10
11
12
13
14
15
n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
1 7 28 84 210 462 924 1,716 3,003 5,005 8,008 12,376 18,564 27,132 38,760 54,264 74,613 100,947 134,596 177,100 230,230 296,010 376,740 475,020 593,775
1 8 36 120 330 792 1,716 3,432 6,435 11,440 19,448 31,824 50,388 77,520 116,280 170,544 245,157 346,104 480,700 657,800 888,030 1,184,040 1,560,780 2,035,800
1 9 45 165 495 1,287 3,003 6,435 12,870 24,310 43,758 75,582 125,970 203,490 319,770 490,314 735,471 1,081,575 1,562,275 2,220,075 3,108,105 4,292,145 5,852,925
1 10 55 220 715 2,002 5,005 11,440 24,3 10 48,620 92,378 167,960 293,930 497,420 817,190 1,307,504 2,042,975 3,124,550 4,686,825 6,906,900 10,015,005 14,307,150
31
1 11 66 286 1,001 3,003 8,008 19,448 43,758 92,378 184,756 352,716 646,646 1,144,066 1,961,256 3,268,760 5,311,735 8,436,285 13,123,110 20,030,010 30,045,015
1 12 78 364 1,365 4,368 12,376 31,824 75,582 167,960 352,716 705,432 1,352,078 2,496,144 4,457,400 7,726,160 13,037,895 21,474,180 34,597,290 54,627,300
1 13 91 455 1,820 6,188 18,564 50,388 125,970 293,930 646,646 1,352,078 2,704,156 5,200,300 9,657,700 17,383,860 30,421,755 51,895,935 86,493,225
1 14 105 560 2,380 8,568 27,132 77,520 203,490 497,420 1,144,066 2,496,144 5,200,300 10,400,600 20,058,300 37,442,160 67,863,915 119,759,850
1 15 120 680 3,060 11,628 38,760 116,280 319,770 817,190 1,961,256 4,457,400 9,657,700 20,058,300 40,116,600 77,558,760 145,422,675
1 16 136 816 3,876 15,504 54,264 170,544 490,314 1,307,504 3,268,760 7,726,160 17,383,860 37,442,160 77,558,760 155,117,520
7.2
Pekali Pilih Atur
Permutation Coefficients
2
n !
n
Nilai-nilai
P r
3
n
4
r !
r
1
5
6
7
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
8
9
10
11
12
n
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
32
7.3
Sifi r Taburan-F
F-Distribution Tables
d 1
1
2
3
4
5
6
7
8
1
39.86 8.53 5.54 4.54 4.06 3.78 3.59 3.46 3.36 3.29 3.23 3.18 3.14 3.10 3.07 3.05 3.03 3.01 2.99 2.97 2.96 2.95 2.94 2.93 2.92 2.91 2.90 2.89 2.89 2.88 2.84 2.79 2.75 2.71
49.50 9.00 5.46 4.32 3.78 3.46 3.26 3.11 3.01 2.92 2.86 2.81 2.76 2.73 2.70 2.67 2.64 2.62 2.61 2.59 2.57 2.56 2.55 2.54 2.53 2.52 2.51 2.50 2.50 2.49 2.44 2.39 2.35 2.30
53.59 9.16 5.39 4.19 3.62 3.29 3.07 2.92 2.81 2.73 2.66 2.61 2.56 2.52 2.49 2.46 2.44 2.42 2.40 2.38 2.36 2.35 2.34 2.33 2.32 2.31 2.30 2.29 2.28 2.28 2.23 2.18 2.13 2.08
55.83 9.24 5.34 4.11 3.52 3.18 2.96 2.81 2.69 2.61 2.54 2.48 2.43 2.39 2.36 2.33 2.31 2.29 2.27 2.25 2.23 2.22 2.21 2.19 2.18 2.17 2.17 2.16 2.15 2.14 2.09 2.04 1.99 1.94
57.24 9.29 5.31 4.05 3.45 3.11 2.88 2.73 2.61 2.52 2.45 2.39 2.35 2.31 2.27 2.24 2.22 2.20 2.18 2.16 2.14 2.13 2.11 2.10 2.09 2.08 2.07 2.06 2.06 2.05 2.00 1.95 1.90 1.85
58.20 9.33 5.28 4.01 3.40 3.05 2.83 2.67 2.55 2.46 2.39 2.33 2.28 2.24 2.21 2.18 2.15 2.13 2.11 2.09 2.08 2.06 2.05 2.04 2.02 2.01 2.00 2.00 1.99 1.98 1.93 1.87 1.82 1.77
58.91 9.35 5.27 3.98 3.37 3.01 2.78 2.62 2.51 2.41 2.34 2.28 2.23 2.19 2.16 2.13 2.10 2.08 2.06 2.04 2.02 2.01 1.99 1.98 1.97 1.96 1.95 1.94 1.93 1.93 1.87 1.82 1.77 1.72
59.44 9.37 5.25 3.95 3.34 2.98 2.75 2.59 2.47 2.38 2.30 2.24 2.20 2.15 2.12 2.09 2.06 2.04 2.02 2.00 1.98 1.97 1.95 1.94 1.93 1.92 1.91 1.90 1.89 1.88 1.83 1.77 1.72 1.67
9
10
12
15
20
24
30
40
60
120
inf
62.26 9.46 5.17 3.82 3.17 2.80 2.56 2.38 2.25 2.16 2.08 2.01 1.96 1.91 1.87 1.84 1.81 1.78 1.76 1.74 1.72 1.70 1.69 1.67 1.66 1.65 1.64 1.63 1.62 1.61 1.54 1.48 1.41 1.34
62.53 9.47 5.16 3.80 3.16 2.78 2.54 2.36 2.23 2.13 2.05 1.99 1.93 1.89 1.85 1.81 1.78 1.75 1.73 1.71 1.69 1.67 1.66 1.64 1.63 1.61 1.60 1.59 1.58 1.57 1.51 1.44 1.37 1.30
62.79 9.47 5.15 3.79 3.14 2.76 2.51 2.34 2.21 2.11 2.03 1.96 1.90 1.86 1.82 1.78 1.75 1.72 1.70 1.68 1.66 1.64 1.62 1.61 1.59 1.58 1.57 1.56 1.55 1.54 1.47 1.40 1.32 1.24
63.06 9.48 5.14 3.78 3.12 2.74 2.49 2.32 2.18 2.08 2.00 1.93 1.88 1.83 1.79 1.75 1.72 1.69 1.67 1.64 1.62 1.60 1.59 1.57 1.56 1.54 1.53 1.52 1.51 1.50 1.42 1.35 1.26 1.17
63.33 9.49 5.13 3.76 3.10 2.72 2.47 2.29 2.16 2.06 1.97 1.90 1.85 1.80 1.76 1.72 1.69 1.66 1.63 1.61 1.59 1.57 1.55 1.53 1.52 1.50 1.49 1.48 1.47 1.46 1.38 1.29 1.19 1.00
d 2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 inf
59.86 9.38 5.24 3.94 3.32 2.96 2.72 2.56 2.44 2.35 2.27 2.21 2.16 2.12 2.09 2.06 2.03 2.00 1.98 1.96 1.95 1.93 1.92 1.91 1.89 1.88 1.87 1.87 1.86 1.85 1.79 1.74 1.68 1.63
33
60.19 9.39 5.23 3.92 3.30 2.94 2.70 2.54 2.42 2.32 2.25 2.19 2.14 2.10 2.06 2.03 2.00 1.98 1.96 1.94 1.92 1.90 1.89 1.88 1.87 1.86 1.85 1.84 1.83 1.82 1.76 1.71 1.65 1.60
60.71 9.41 5.22 3.90 3.27 2.90 2.67 2.50 2.38 2.28 2.21 2.15 2.10 2.05 2.02 1.99 1.96 1.93 1.91 1.89 1.87 1.86 1.84 1.83 1.82 1.81 1.80 1.79 1.78 1.77 1.71 1.66 1.60 1.55
61.22 9.42 5.20 3.87 3.24 2.87 2.63 2.46 2.34 2.24 2.17 2.10 2.05 2.01 1.97 1.94 1.91 1.89 1.86 1.84 1.83 1.81 1.80 1.78 1.77 1.76 1.75 1.74 1.73 1.72 1.66 1.60 1.55 1.49
61.74 9.44 5.18 3.84 3.21 2.84 2.59 2.42 2.30 2.20 2.12 2.06 2.01 1.96 1.92 1.89 1.86 1.84 1.81 1.79 1.78 1.76 1.74 1.73 1.72 1.71 1.70 1.69 1.68 1.67 1.61 1.54 1.48 1.42
62.00 9.45 5.18 3.83 3.19 2.82 2.58 2.40 2.28 2.18 2.10 2.04 1.98 1.94 1.90 1.87 1.84 1.81 1.79 1.77 1.75 1.73 1.72 1.70 1.69 1.68 1.67 1.66 1.65 1.64 1.57 1.51 1.45 1.38
d 1
1
2
3
4
5
6
7
8
1
161.45 18.51 10.13 7.71 6.61 5.99 5.59 5.32 5.12 4.96 4.84 4.75 4.67 4.60 4.54 4.49 4.45 4.41 4.38 4.35 4.32 4.30 4.28 4.26 4.24 4.23 4.21 4.20 4.18 4.17 4.08 4.00 3.92 3.84
199.50 19.00 9.55 6.94 5.79 5.14 4.74 4.46 4.26 4.10 3.98 3.89 3.81 3.74 3.68 3.63 3.59 3.55 3.52 3.49 3.47 3.44 3.42 3.40 3.39 3.37 3.35 3.34 3.33 3.32 3.23 3.15 3.07 3.00
215.71 19.16 9.28 6.59 5.41 4.76 4.35 4.07 3.86 3.71 3.59 3.49 3.41 3.34 3.29 3.24 3.20 3.16 3.13 3.10 3.07 3.05 3.03 3.01 2.99 2.98 2.96 2.95 2.93 2.92 2.84 2.76 2.68 2.60
224.58 19.25 9.12 6.39 5.19 4.53 4.12 3.84 3.63 3.48 3.36 3.26 3.18 3.11 3.06 3.01 2.96 2.93 2.90 2.87 2.84 2.82 2.80 2.78 2.76 2.74 2.73 2.71 2.70 2.69 2.61 2.53 2.45 2.37
230.16 19.30 9.01 6.26 5.05 4.39 3.97 3.69 3.48 3.33 3.20 3.11 3.03 2.96 2.90 2.85 2.81 2.77 2.74 2.71 2.68 2.66 2.64 2.62 2.60 2.59 2.57 2.56 2.55 2.53 2.45 2.37 2.29 2.21
233.99 19.33 8.94 6.16 4.95 4.28 3.87 3.58 3.37 3.22 3.09 3.00 2.92 2.85 2.79 2.74 2.70 2.66 2.63 2.60 2.57 2.55 2.53 2.51 2.49 2.47 2.46 2.45 2.43 2.42 2.34 2.25 2.18 2.10
236.77 19.35 8.89 6.09 4.88 4.21 3.79 3.50 3.29 3.14 3.01 2.91 2.83 2.76 2.71 2.66 2.61 2.58 2.54 2.51 2.49 2.46 2.44 2.42 2.40 2.39 2.37 2.36 2.35 2.33 2.25 2.17 2.09 2.01
238.88 19.37 8.85 6.04 4.82 4.15 3.73 3.44 3.23 3.07 2.95 2.85 2.77 2.70 2.64 2.59 2.55 2.51 2.48 2.45 2.42 2.40 2.37 2.36 2.34 2.32 2.31 2.29 2.28 2.27 2.18 2.10 2.02 1.94
9
10
12
15
20
24
30
40
60
120
inf
d 2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 inf
240.54 19.38 8.81 6.00 4.77 4.10 3.68 3.39 3.18 3.02 2.90 2.80 2.71 2.65 2.59 2.54 2.49 2.46 2.42 2.39 2.37 2.34 2.32 2.30 2.28 2.27 2.25 2.24 2.22 2.21 2.12 2.04 1.96 1.88
34
241.88 19.40 8.79 5.96 4.74 4.06 3.64 3.35 3.14 2.98 2.85 2.75 2.67 2.60 2.54 2.49 2.45 2.41 2.38 2.35 2.32 2.30 2.27 2.25 2.24 2.22 2.20 2.19 2.18 2.16 2.08 1.99 1.91 1.83
243.91 19.41 8.74 5.91 4.68 4.00 3.57 3.28 3.07 2.91 2.79 2.69 2.60 2.53 2.48 2.42 2.38 2.34 2.31 2.28 2.25 2.23 2.20 2.18 2.16 2.15 2.13 2.12 2.10 2.09 2.00 1.92 1.83 1.75
245.95 19.43 8.70 5.86 4.62 3.94 3.51 3.22 3.01 2.85 2.72 2.62 2.53 2.46 2.40 2.35 2.31 2.27 2.23 2.20 2.18 2.15 2.13 2.11 2.09 2.07 2.06 2.04 2.03 2.01 1.92 1.84 1.75 1.67
248.01 19.45 8.66 5.80 4.56 3.87 3.44 3.15 2.94 2.77 2.65 2.54 2.46 2.39 2.33 2.28 2.23 2.19 2.16 2.12 2.10 2.07 2.05 2.03 2.01 1.99 1.97 1.96 1.94 1.93 1.84 1.75 1.66 1.57
249.05 19.45 8.64 5.77 4.53 3.84 3.41 3.12 2.90 2.74 2.61 2.51 2.42 2.35 2.29 2.24 2.19 2.15 2.11 2.08 2.05 2.03 2.01 1.98 1.96 1.95 1.93 1.91 1.90 1.89 1.79 1.70 1.61 1.52
250.10 19.46 8.62 5.75 4.50 3.81 3.38 3.08 2.86 2.70 2.57 2.47 2.38 2.31 2.25 2.19 2.15 2.11 2.07 2.04 2.01 1.98 1.96 1.94 1.92 1.90 1.88 1.87 1.85 1.84 1.74 1.65 1.55 1.46
251.14 19.47 8.59 5.72 4.46 3.77 3.34 3.04 2.83 2.66 2.53 2.43 2.34 2.27 2.20 2.15 2.10 2.06 2.03 1.99 1.96 1.94 1.91 1.89 1.87 1.85 1.84 1.82 1.81 1.79 1.69 1.59 1.50 1.39
252.20 19.48 8.57 5.69 4.43 3.74 3.30 3.01 2.79 2.62 2.49 2.38 2.30 2.22 2.16 2.11 2.06 2.02 1.98 1.95 1.92 1.89 1.86 1.84 1.82 1.80 1.79 1.77 1.75 1.74 1.64 1.53 1.43 1.32
253.25 19.49 8.55 5.66 4.40 3.70 3.27 2.97 2.75 2.58 2.45 2.34 2.25 2.18 2.11 2.06 2.01 1.97 1.93 1.90 1.87 1.84 1.81 1.79 1.77 1.75 1.73 1.71 1.70 1.68 1.58 1.47 1.35 1.22
254.31 19.50 8.53 5.63 4.36 3.67 3.23 2.93 2.71 2.54 2.40 2.30 2.21 2.13 2.07 2.01 1.96 1.92 1.88 1.84 1.81 1.78 1.76 1.73 1.71 1.69 1.67 1.65 1.64 1.62 1.51 1.39 1.25 1.00
d 1
1
2
3
4
5
6
7
1
4052.18 98.50 34.12 21.20 16.26 13.75 12.25 11.26 10.56 10.04 9.65 9.33 9.07 8.86 8.68 8.53 8.40 8.29 8.18 8.10 8.02 7.95 7.88 7.82 7.77 7.72 7.68 7.64 7.60 7.56 7.31 7.08 6.85 6.63
4999.50 99.00 30.82 18.00 13.27 10.92 9.55 8.65 8.02 7.56 7.21 6.93 6.70 6.51 6.36 6.23 6.11 6.01 5.93 5.85 5.78 5.72 5.66 5.61 5.57 5.53 5.49 5.45 5.42 5.39 5.18 4.98 4.79 4.61
5403.35 99.17 29.46 16.69 12.06 9.78 8.45 7.59 6.99 6.55 6.22 5.95 5.74 5.56 5.42 5.29 5.18 5.09 5.01 4.94 4.87 4.82 4.76 4.72 4.68 4.64 4.60 4.57 4.54 4.51 4.31 4.13 3.95 3.78
5624.58 99.25 28.71 15.98 11.39 9.15 7.85 7.01 6.42 5.99 5.67 5.41 5.21 5.04 4.89 4.77 4.67 4.58 4.50 4.43 4.37 4.31 4.26 4.22 4.18 4.14 4.11 4.07 4.04 4.02 3.83 3.65 3.48 3.32
5763.65 99.30 28.24 15.52 10.97 8.75 7.46 6.63 6.06 5.64 5.32 5.06 4.86 4.69 4.56 4.44 4.34 4.25 4.17 4.10 4.04 3.99 3.94 3.90 3.85 3.82 3.78 3.75 3.73 3.70 3.51 3.34 3.17 3.02
5858.99 99.33 27.91 15.21 10.67 8.47 7.19 6.37 5.80 5.39 5.07 4.82 4.62 4.46 4.32 4.20 4.10 4.01 3.94 3.87 3.81 3.76 3.71 3.67 3.63 3.59 3.56 3.53 3.50 3.47 3.29 3.12 2.96 2.80
5928.36 99.36 27.67 14.98 10.46 8.26 6.99 6.18 5.61 5.20 4.89 4.64 4.44 4.28 4.14 4.03 3.93 3.84 3.77 3.70 3.64 3.59 3.54 3.50 3.46 3.42 3.39 3.36 3.33 3.30 3.12 2.95 2.79 2.64
8
9
10
12
15
20
24
30
40
60
120
inf
d 2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 inf
5981.07 99.37 27.49 14.80 10.29 8.10 6.84 6.03 5.47 5.06 4.74 4.50 4.30 4.14 4.00 3.89 3.79 3.71 3.63 3.56 3.51 3.45 3.41 3.36 3.32 3.29 3.26 3.23 3.20 3.17 2.99 2.82 2.66 2.51
6022.47 99.39 27.35 14.66 10.16 7.98 6.72 5.91 5.35 4.94 4.63 4.39 4.19 4.03 3.89 3.78 3.68 3.60 3.52 3.46 3.40 3.35 3.30 3.26 3.22 3.18 3.15 3.12 3.09 3.07 2.89 2.72 2.56 2.41
35
6055.85 99.40 27.23 14.55 10.05 7.87 6.62 5.81 5.26 4.85 4.54 4.30 4.10 3.94 3.80 3.69 3.59 3.51 3.43 3.37 3.31 3.26 3.21 3.17 3.13 3.09 3.06 3.03 3.00 2.98 2.80 2.63 2.47 2.32
6106.32 99.42 27.05 14.37 9.89 7.72 6.47 5.67 5.11 4.71 4.40 4.16 3.96 3.80 3.67 3.55 3.46 3.37 3.30 3.23 3.17 3.12 3.07 3.03 2.99 2.96 2.93 2.90 2.87 2.84 2.66 2.50 2.34 2.18
6157.28 99.43 26.87 14.20 9.72 7.56 6.31 5.52 4.96 4.56 4.25 4.01 3.82 3.66 3.52 3.41 3.31 3.23 3.15 3.09 3.03 2.98 2.93 2.89 2.85 2.81 2.78 2.75 2.73 2.70 2.52 2.35 2.19 2.04
6208.73 99.45 26.69 14.02 9.55 7.40 6.16 5.36 4.81 4.41 4.10 3.86 3.66 3.51 3.37 3.26 3.16 3.08 3.00 2.94 2.88 2.83 2.78 2.74 2.70 2.66 2.63 2.60 2.57 2.55 2.37 2.20 2.03 1.88
6234.63 99.46 26.60 13.93 9.47 7.31 6.07 5.28 4.73 4.33 4.02 3.78 3.59 3.43 3.29 3.18 3.08 3.00 2.92 2.86 2.80 2.75 2.70 2.66 2.62 2.58 2.55 2.52 2.49 2.47 2.29 2.12 1.95 1.79
6260.65 99.47 26.50 13.84 9.38 7.23 5.99 5.20 4.65 4.25 3.94 3.70 3.51 3.35 3.21 3.10 3.00 2.92 2.84 2.78 2.72 2.67 2.62 2.58 2.54 2.50 2.47 2.44 2.41 2.39 2.20 2.03 1.86 1.70
6286.78 99.47 26.41 13.75 9.29 7.14 5.91 5.12 4.57 4.17 3.86 3.62 3.43 3.27 3.13 3.02 2.92 2.84 2.76 2.69 2.64 2.58 2.54 2.49 2.45 2.42 2.38 2.35 2.33 2.30 2.11 1.94 1.76 1.59
6313.03 99.48 26.32 13.65 9.20 7.06 5.82 5.03 4.48 4.08 3.78 3.54 3.34 3.18 3.05 2.93 2.83 2.75 2.67 2.61 2.55 2.50 2.45 2.40 2.36 2.33 2.29 2.26 2.23 2.21 2.02 1.84 1.66 1.47
6339.39 99.49 26.22 13.56 9.11 6.97 5.74 4.95 4.40 4.00 3.69 3.45 3.25 3.09 2.96 2.84 2.75 2.66 2.58 2.52 2.46 2.40 2.35 2.31 2.27 2.23 2.20 2.17 2.14 2.11 1.92 1.73 1.53 1.32
6365.86 99.50 26.13 13.46 9.02 6.88 5.65 4.86 4.31 3.91 3.60 3.36 3.17 3.00 2.87 2.75 2.65 2.57 2.49 2.42 2.36 2.31 2.26 2.21 2.17 2.13 2.10 2.06 2.03 2.01 1.80 1.60 1.38 1.00
7.4
Sifir Taburan Normal Piawai
Standard Normal Distribution Table
() 0.0250
Contoh bacaan P(z > 1.96) = 0.0250
0
1 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 3.1
3.2
0
1
2
3
4
5
6
7
8
9
.5000 .4602 .4207 .3821 .3446 .3085 .2743 .2420 .2119 .1841 .1587 .1357 .1151 .0968 .0808 .0668 .0548 .0446 .0359 .0287 .0228 .0179 .0139 .0107
.4960 .4562 .4168 .3783 .3409 .3050 .2709 .2389 .2090 .1814 .1562 .1335 .1131 .0951 .0793 .0655 .0537 .0436 .0351 .0281 .0222 .0174 .0136 .0104
.4920 .4522 .4129 .3745 .3372 .3015 .2676 .2358 .2061 .1788 .1539 .1314 .1112 .0934 .0778 .0643 .0526 .0427 .0344 .0274 .0217 .0170 .0132 .0102
.4880 .4483 .4090 .3707 .3336 .2981 .2643 .2327 .2033 .1762 .1515 .1292 .1093 .0918 .0764 .0630 .0516 .0418 .0336 .0268 .0212 .0166 .0129
.4840 .4443 .4052 .3669 .3300 .2946 .2611 .2296 .2005 .1736 .1492 .1271 .1075 .0901 .0749 .0618 .0505 .0409 .0329 .0262 .0207 .0162 .0125
.4801 .4404 .4013 .3632 .3264 .2912 .2578 .2266 .1977 .1711 .1469 .1251 .1056 .0885 .0735 .0606 .0495 .0401 .0322 .0256 .0202 .0158 .0122
.4761 .4364 .3974 .3594 .3228 .2877 .2546 .2236 .1949 .1685 .1446 .1230 .1038 .0869 .0721 .0594 .0485 .0392 .0314 .0250 .0197 .0154 .0119
.4721 .4325 .3936 .3557 .3192 .2843 .2514 .2206 .1922 .1660 .1423 .1210 .1020 .0853 .0708 .0582 .0475 .0384 .0307 .0244 .0192 .0150 .0116
.4681 .4286 .3897 .3520 .3156 .2810 .2483 .2177 .1894 .1635 .1401 .1190 .1003 .0838 .0694 .0571 .0465 .0375 .0301 .0239 .0188 .0146 .0113
.4641 .4247 .3859 .3483 .3121 .2776 .2451 .2148 .1867 .1611 .1379 .1170 .0985 .0823 .0681 .0559 .0455 .0367 .0294 .0233 .0183 .0143 .0110
.00990
.00964
.00939
.00914 .00889
.00866
.00842
.00676 .00508 .00379 .00280 .00205 .00149 .00107
.00657 .00494 .00368 .00272 .00199 .00144 .00104
.00639 .00480 .00357 .00264 .00193 .00139 .00100
.000762
.000736
.000711
.00820 .00621 .00466 .00347 .00256 .00187 .00135 .000968
.000687
.00798 .00604 .00453 .00336 .00248 .00181 .00131 .000935
.000664
.00776 .00587 .00440 .00326 .00240 .00175 .00126 .000904
.000641
.00755
.00734
.00570 .00427 .00317 .00233 .00169 .00122
.00554 .00415 .00307 .00226 .00164 .00118
.00714 .00539 .00402 .00298 .00219 .00159 .00114
.000874
.000845
.000816
.000619
.000483
.000466
.000450
.000434
.000419
3.4
.000337 .000233 .000159 .000108 .000072 .000048
.000325 .000224 .000153 .000104 .000069 .000046
.000313 .000216 .000147 .000100 .000067 .000044
.000302 .000208 .000142 .000096 .000064 .000042
.000291 .000200 .000136 .000092 .000062 .000041
3.6 3.7 3.8 3.9
.000789
.000598
3.3
3.5
.00695 .00523 .00391 .00289 .00212 .00154 .00111
.000577
.000557
.000538
.000519
.000501
.000404 .000280 .000193 .000131 .000088 .000059 .000039
.000390 .000270 .000185 .000126 .000085 .000057 .000037
.000376 .000260 .000178 .000121 .000082 .000054 .000036
.000362 .000251 .000172 .000117 .000078 .000052 .000034
.000349 .000242 .000165 .000112 .000075 .000050 .000033
36
2
1.96
3
4
5
6
7
8
9
24 24 23 22 22 20 19 18 16 15 14 12 11 10 8 7 6 5 4 4 3 2 2 2 15 14 13 11 9 7 6 4 3 2 19 17 15 13 11 10 8 6 4 3
28 28 27 26 25 24 23 21 19 18 16 14 13 11 10 8 7 6 5 4 3 3 2 2 18 16 15 13 11 8 7 5 4 3 22 20 17 15 13 11 9 7 5 3
32 32 31 30 29 27 26 24 22 20 19 16 15 13 11 10 8 7 6 5 4 3 3 2 20 18 17 15 12 9 8 6 4 3 25 22 20 18 15 13 10 8 6 4
36 36 35 34 32 31 29 27 25 23 21 18 17 14 13 11 9 8 6 5 4 4 3 2 23 21 19 17 14 10 9 6 4 4 28 25 22 20 17 14 12 9 7 5
TOLAK
4 4 4 4 4 3 3 3 3 3 2 2 2 2 1 1 1 1 1 1 0 0 0 0 3 2 2 2 2 1 1 1 0 0 3 3 2 2 2 2 1 1 1 0
8 8 8 7 7 7 7 6 5 5 5 4 4 3 3 2 2 2 1 1 1 1 1 1 5 5 4 4 3 2 2 1 1 1 6 6 5 4 4 3 3 2 1 1
12 12 12 11 11 10 10 9 8 8 7 6 6 5 4 4 3 3 2 2 1 1 1 1 8 7 6 6 5 3 3 2 1 1 9 8 7 7 6 5 4 3 2 1
16 16 15 15 14 14 13 12 11 10 9 8 7 6 6 5 4 4 3 2 2 2 1 1 10 9 8 7 6 5 4 3 2 2 13 11 10 9 8 6 5 4 3 2
20 20 19 19 18 17 16 15 14 13 12 10 9 8 7 6 5 4 4 3 2 2 2 1 13 12 11 9 8 6 5 4 2 2 16 14 12 11 9 8 7 5 4 2
7.5
Sifir Khi Kuasa Dua
Chi-Square ( ) Table
0
v
0.995
0.990
0.975
0.950
0.900
0.100
0.050
0.025
0.010
0.005
1
0.000 0.010 0.072 0.207 0.412 0.676 0.989 1.344 1.735 2.156 2.603 3.074 3.565 4.075 4.601 5.142 5.697 6.265 6.844 7.434 8.034 8.643 9.260 9.886 10.520 11.160 11.808 12.461 13.121 13.787 20.707 27.991 35.534 43.275 51.172 59.196 67.328
0.000 0.020 0.115 0.297 0.554 0.872 1.239 1.646 2.088 2.558 3.053 3.571 4.107 4.660 5.229 5.812 6.408 7.015 7.633 8.260 8.897 9.542 10.196 10.856 11.524 12.198 12.879 13.565 14.256 14.953 22.164 29.707 37.485 45.442 53.540 61.754 70.065
0.001 0.051 0.216 0.484 0.831 1.237 1.690 2.180 2.700 3.247 3.816 4.404 5.009 5.629 6.262 6.908 7.564 8.231 8.907 9.591 10.283 10.982 11.689 12.401 13.120 13.844 14.573 15.308 16.047 16.791 24.433 32.357 40.482 48.758 57.153 65.647 74.222
0.004 0.103 0.352 0.711 1.145 1.635 2.167 2.733 3.325 3.940 4.575 5.226 5.892 6.571 7.261 7.962 8.672 9.390 10.117 10.851 11.591 12.338 13.091 13.848 14.611 15.379 16.151 16.928 17.708 18.493 26.509 34.764 43.188 51.739 60.391 69.126 77.929
0.016 0.211 0.584 1.064 1.610 2.204 2.833 3.490 4.168 4.865 5.578 6.304 7.042 7.790 8.547 9.312 10.085 10.865 11.651 12.443 13.240 14.041 14.848 15.659 16.473 17.292 18.114 18.939 19.768 20.599 29.051 37.689 46.459 55.329 64.278 73.291 82.358
2.706 4.605 6.251 7.779 9.236 10.645 12.017 13.362 14.684 15.987 17.275 18.549 19.812 21.064 22.307 23.542 24.769 25.989 27.204 28.412 29.615 30.813 32.007 33.196 34.382 35.563 36.741 37.916 39.087 40.256 51.805 63.167 74.397 85.527 96.578 107.565 118.498
3.841 5.991 7.815 9.488 11.070 12.592 14.067 15.507 16.919 18.307 19.675 21.026 22.362 23.685 24.996 26.296 27.587 28.869 30.144 31.410 32.671 33.924 35.172 36.415 37.652 38.885 40.113 41.337 42.557 43.773 55.758 67.505 79.082 90.531 101.879 113.145 124.342
5.024 7.378 9.348 11.143 12.833 14.449 16.013 17.535 19.023 20.483 21.920 23.337 24.736 26.119 27.488 28.845 30.191 31.526 32.852 34.170 35.479 36.781 38.076 39.364 40.646 41.923 43.195 44.461 45.722 46.979 59.342 71.420 83.298 95.023 106.629 118.136 129.561
6.635 9.210 11.345 13.277 15.086 16.812 18.475 20.090 21.666 23.209 24.725 26.217 27.688 29.141 30.578 32.000 33.409 34.805 36.191 37.566 38.932 40.289 41.638 42.980 44.314 45.642 46.963 48.278 49.588 50.892 63.691 76.154 88.379 100.425 112.329 124.116 135.807
7.879 10.597 12.838 14.860 16.750 18.548 20.278 21.955 23.589 25.188 26.757 28.300 29.819 31.319 32.801 34.267 35.718 37.156 38.582 39.997 41.401 42.796 44.181 45.559 46.928 48.290 49.645 50.993 52.336 53.672 66.766 79.490 91.952 104.215 116.321 128.299 140.169
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 50 60 70 80 90 100
37
7.6
Sifi r Taburan-t
t-Distri bution Table
α
0.1
0.05
0.025
0.01
0.005
0.001
0.0005
3.078 1.886 1.638 1.533 1.476 1.440 1.415 1.397 1.383 1.372 1.363 1.356 1.350 1.345 1.341 1.337 1.333 1.330 1.328 1.325 1.323 1.321 1.319 1.318 1.316 1.315 1.314 1.313 1.311 1.310 1.296 1.289 1.282
6.314 2.920 2.353 2.132 2.015 1.943 1.895 1.860 1.833 1.812 1.796 1.782 1.771 1.761 1.753 1.746 1.740 1.734 1.729 1.725 1.721 1.717 1.714 1.711 1.708 1.706 1.703 1.701 1.699 1.697 1.671 1.658 1.645
12.706 4.303 3.182 2.776 2.571 2.447 2.365 2.306 2.262 2.228 2.201 2.179 2.160 2.145 2.131 2.120 2.110 2.101 2.093 2.086 2.080 2.074 2.069 2.064 2.060 2.056 2.052 2.048 2.045 2.042 2.000 1.980 1.960
31.821 6.965 4.541 3.747 3.365 3.143 2.998 2.896 2.821 2.764 2.718 2.681 2.650 2.624 2.602 2.583 2.567 2.552 2.539 2.528 2.518 2.508 2.500 2.492 2.485 2.479 2.473 2.467 2.462 2.457 2.390 2.358 2.326
63.657 9.925 5.841 4.604 4.032 3.707 3.499 3.355 3.250 3.169 3.106 3.055 3.012 2.977 2.947 2.921 2.898 2.878 2.861 2.845 2.831 2.819 2.807 2.797 2.787 2.779 2.771 2.763 2.756 2.750 2.660 2.617 2.576
318.309 22.327 10.215 7.173 5.893 5.208 4.785 4.501 4.297 4.144 4.025 3.930 3.852 3.787 3.733 3.686 3.646 3.610 3.579 3.552 3.527 3.505 3.485 3.467 3.450 3.435 3.421 3.408 3.396 3.385 3.232 3.160 3.090
636.619 31.599 12.924 8.610 6.869 5.959 5.408 5.041 4.781 4.587 4.437 4.318 4.221 4.140 4.073 4.015 3.965 3.922 3.883 3.850 3.819 3.792 3.768 3.745 3.725 3.707 3.690 3.674 3.659 3.646 3.460 3.373 3.291
df 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 60 120 ∞
38
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