list of formulae exam for statistics
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STAtisticsTRANSCRIPT
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LIST OF FORMULAE
INSTITUT MATEMATIK KEJURUTERAAN
Sample Mean
xx
n
Sample Mean for Grouped Data
1
1
n
i i
i
n
i
i
f xfx
x orf
f
Binomial Probability
Distribution
~ ( , )
( ) 1n xn x
x
X B n p
P X x C p p
Sample Variance
2
2
2
2
1
=1
i
i
i
x xs
n
xx
n
n
Sample Variance for Grouped Data
2
2
2
2( )
1 1
fxfx
f x x fs
f f
Poisson Probability Distribution
~ ( )
( )!
o
x
X P
eP X x
x
Sample Standard Deviation
2s s
Lower and Upper Fences
Upper Fence = Q3 + 1.5(IQR)
Lower Fence = Q1 - 1.5(IQR)
Interquartile Range
IQR= Q3-Q1
Mode for Grouped Data
1
1 2
x L c
Median for Grouped Data
12
f
j
j
Fx L c
f
Normal Distribution 2~ ( , )X N
X
Z
(population)
X x
Zs
(sample)
Sampling Distribution for:
Sample Mean 2
~ ,x Nn
Sample Proportion
(1 ) ~ ,
p pp N p
n
Difference between Two Sample Means 2 2
1 21 2 1 2
1 2
~ ,x x Nn n
Difference between Two Sample Proportions
1 1 2 21 2 1 2
1 2
(1 ) (1 ) ~ ,
p p p pp p N p p
n n
Confidence Interval for Single Population Mean:
2
x zn
, is known
, 12n
sx t
n
, is unknown, 30n
2
sx z
n
, is unknown, 30n
Confidence Interval for Single Population
Proportion:
2
(1 )
p pp z
n
where x
pn
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INSTITUT MATEMATIK KEJURUTERAAN
Confidence Interval for Two Population Means
i. 2 21 2, known
2 2
1 21 2 2
1 2
x x zn n
ii. 2 21 2, unknown, 1 230, 30n n
2 2
1 21 2 2
1 2
s sx x z
n n
iii. 2 2 2 21 2 1 2 1 2, unknown, , 30, 30n n
2 2
1 21 2 , 2
1 2
v
s sx x t
n n , 1 2with min( 1, 1) v n n
iv. 2 21 2, unknown, assume 2 21 2 , 1 230, 30n n
1 2 21 2
1 1px x z S
n n
2 21 1 2 21 2
1 1, with
2p
n s n sS
n n
v. 2 21 2, unknown, assume 2 21 2 , 1 230, 30n n
1 2 , 21 2
1 1v px x t S
n n
2 21 1 2 21 2
1 2
1 1,
2with min( 1, 1) and p
n s n sS
n nv n n
Confidence Interval for Two Population Proportions
1 1 2 2
1 22
1 2
1 1
p p p pp p z
n n
Sample Size:
2
/2znE
, E : margin of error
2
2z sn
E
unknown
Sample Size:
2
/2 (1 )z
n p pE
, where x
pn
Hypothesis Testing for :
Test Statistic:
i. 0 0,1 , knownx
Z N
n
Hypothesis Testing for p:
Test Statistic:
0
0 0
0,1 , where
1
p p xZ N p
np p
n
ii. 0 0,1 , unknown, 30x
Z N ns
n
iii. 0 with 1, unknown < 30 x
t t v n nvs
n
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INSTITUT MATEMATIK KEJURUTERAAN
Hypothesis Testing for 1 - 2
Test Statistic:
i. 1 2 1 2 1 22 2
1 2
1 2
0,1 , ( , ) knownx x
Z N
n n
ii. 1 2 1 2 1 2 1 22 2
1 2
1 2
0,1 , ( , ) unknown, 30, 30x x
Z N n ns s
n n
iii. 2 21 2 1 2 1 1 2 2
1 2 1 2 1 2
1 2
1 2
1 10,1 , ( , ) unknown,assume , 30, 30,with
21 1p
p
x x n s n sZ N n n S
n nS
n n
iv. 2 21 2 1 2 1 1 2 2
1 2 1 2 1 2
1 2
1 2
( ) 1 1, ( , ) unknown,assume , 30, 30,with
21 1v p
p
x x n s n st t n n S
n nS
n n
1 2with min( 1, 1) v n n
v.
1 2 1 2 1 2 1 2 1 2 1 22 2
1 2
1 2
( ), ( , ) unknown,assume , 30, 30, min 1, 1v
x xt t n n v n n
s s
n n
Hypothesis Testing for p1 - p2
Test Statistic: 1 2 1 2
1 2
( )
1 1
p p p pZ
pqn n
1 2
1 2
with , = 1x x
p q pn n
Simple Linear Regression
Linear Regression Model: 0 1 y x
2
12
1
n
ini
xx i
i
x
S xn
2
12
1
n
ini
yy i
i
y
S yn
1 1
1
n n
i ini i
xy i i
i
x y
S x yn
1 1 111 2
22
11 1
n n nn
i i i ii ixyi i ii
nn n
xxi
i ii
i i
n x y x yx x y yS
Sx x n x x
1
1 10 1
n n
i i
i i
y x
y xn
2
2 xy
xx yy
S SSRr
S S SST
1
xySSR S yySST S
2
0 1
1 1 1
2 2
n n n
i i i i
i i i
est
y y x ySSE
s MSEn n
Test Statistic:
11 2 1
1
1 , 2
yy xy
n
xx
S St t Var
n SVar
Standard error of 1 :
1
22
2
11 1
1
est est
nn n xx
ii i
ii i
s s MSEs
Sx x x x
n
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INSTITUT MATEMATIK KEJURUTERAAN
Non-Parametric Test
Wilcoxon Signed-Rank Statistics:
( ) , ( ) , min( , )T R d T R d T T T
Kruskal-Wallis Statistic:
2
1
1 2
123 1
1
where sum of ranks of -th group , ... , number of groups
ki
ii
i k
RH N
N N n
R i N n n n k
Mann-Whitney Statistic:
1 1 1 where T R R = ranks from group with smaller sample size
*1 1 1 2 1 1 1 21 where 1 , : Critical value from Mann-Whitney Table U L LT n n n T T n n n T T
Chi-Square Test
2
2
1
ki i
calc
i i
o e
e
; : observed frequency for -th cell , : expected frequency for -th celli io i e i
row total column total*expected frequency =
table total
One-Way ANOVA
2
22
1 1 1 1 1
1i in nk k k
ij ij i
i j i j i
SST y Y y YN
222
1 1 1
1( )
k k ki
i i i
i i ii
YSSTR SSB n y Y Y
n N
2
2
1 1 1
( ) 1ink k
ij i i i
i j i
SSE SSW y y n s
Two-Way ANOVA without replication
2
1 1
k n
ij
i j
SST y Y
2
1
k
i
i
SSTR n y Y
2
22 1
1
n
jnj
j
j
YY
SSBL k y Yk kn
2
1 1
k n
ij i j
i j
SSE y y y Y
number of treatments, number of blocksk n
Two-Way ANOVA with replication
2
22
1 1 1 1 1 1
a b r a b r
ijk ijk
i j k i j k
YSST y Y y
abr
2
22
1
1
a
iai
i
i
YY
SSA br y Ybr abr
2
22 1
1
b
jbj
j
j
YY
SSB ar y Yar abr
2
22
1
ija
ij i j
i
YY
SSAB r y y y Y SSA SSBr abr
2
1 1 1
a b r
ijk ij
i j k
SSE y y