Download Highest value - Lowest value Number of classes x x n = 1

ENGINEERING STATISTICS EQT 271 SEM 2 2012/2013
SUMMARY OF CHAPTER 1
1.1 DESCRIPTIVE STATISTICS
1.1.1 Frequency Table
1.
Finding the number of classes
a)
Sturges Rule, 1  3.3logc
b) n
2. Class Width
Highest value - Lowest value
Number of classes
1.1.2 Measures of Central Tendency
Ungrouped Data:Mean
x
x
n
Median
n 1
- for n is odd
2
n
- for n is even
2
Mode
Highest frequency in data set
Grouped data:Mean
SYAFAWATI AB. SAAD
ENGINEERING STATISTICS EQT 271 SEM 2 2012/2013
x
 fx
f
Median
 f

 Fi 1 

2
x  Lc

fj




L  Lower class boundary of median class
c  the size of median class interval
Fj 1  the sum of frequencies of all classes lower than
the median class
f j  the frequency of median class
Mode
 1 
xˆ  L  c 

 1   2 
L  lower class boundary of mode class
c  the size of mode class interval
1  difference between the modal class frequency and before it
 2  difference between the modal class frequency and after it
1.1.3 Measures of Dispersion
Range = Largest value – Smallest value
Ungrouped data
s2 
 
2
 (x  x )
Grouped data
2
n 1
 ( x  x )2
n
(for sample)
(for population)
s2 
2 
 fx
2
 fx

n 1
 fx
2
2
n
 fx

n
n
(for sample)
2
(for population)
1.2 Probability Distributions
SYAFAWATI AB. SAAD
ENGINEERING STATISTICS EQT 271 SEM 2 2012/2013
1.2.1 Discrete Probability Distribution
1.2.1.1 Binomial Distribution
P ( X  x )  nC x p x q n  x
Mean  E ( X )  np
Variance  Var ( X )  npq
n relatively large, Table Binomial used (P( X  k )
follow the guidelines below:
i)P( X  x)  P( X  x)  P( X  x  1)
ii)P( X  x)  P( X  x  1)
iii)P( X  x)  1  P( X  x)
iv)P( X  x)  1  P( X  x  1)
v)P( x1  X  x2 )  P( X  x2  1)  P( X  x1 )
1.2.1.2 Poisson Distribution
x
ee 
x
P
P(( X
X
 xx)) 

for x  0,1, 2,3,...
xx!
E( X )  
Var ( X )  
1.2.1.3 Poisson Approximation of Binomial Probabilities
Rule : n  30, np  5, nq  5
X ~ B(n, p)  X ~ Po (np)
1.3.1 Continuous Probability Distribution
1.3.1.1 Normal Distribution
SYAFAWATI AB. SAAD
ENGINEERING STATISTICS EQT 271 SEM 2 2012/2013
X ~ N ( , 2 )
E( X )  
Var ( X )   2
Z
X 
1.3.1.2

Normal Approximation
c .c
a) P( X  x) 
 P( x  0.5  X  x  0.5)
c .c
b) P( X  x) 
 P( X  x  0.5)
c .c
c) P( X  x) 
 P( X  x  0.5)
c .c
d) P( X  x) 
 P( X  x  0.5)
c .c
e) P( X  x) 
 P( X  x  0.5)
c.c  continuous correction factor
1.3.1.3
Normal Approximation of Binomial Distribution
Rule : n  30, np  5, nq  5
X ~ B(n, p)  X ~ N (np, npq)
1.3.1.4 Normal Approximation of Poisson Distribution
Rule :   10
X ~ Po ( )  X ~ N ( ,  )
1.4 Sampling Distribution
1.4.1 Sampling Distribution of One Sample Mean
SYAFAWATI AB. SAAD
ENGINEERING STATISTICS EQT 271 SEM 2 2012/2013
x  

x 
Z
n
X 

n
If n  30, X ~ N (  ,
2
n
)
If n  30, variance is known X ~ N (  ,
2
)
n
If n  30, variance is unknown, t distribution
with n-1 degree
T
X 
s2
n
~ tn 1
1.4.2 Sampling Distribution for Two Sample Mean
X 1  X 2 ~ N ( 1  2 ,
 12
n1

 22
n2
)
1.4.3 Sampling Distribution of Sample Proportion
X
x
and pˆ 
N
n
N  total number of elements in population
X  number of elements in population
n  total number of elements in sample
p
x  number of elements in the sample
pˆ ~ N ( p,
pq
)
n
 pˆ  p,  pˆ 
pq
n
1.4.4 Sampling Distribution of Two Sample Proportion
SYAFAWATI AB. SAAD
ENGINEERING STATISTICS EQT 271 SEM 2 2012/2013
pˆ1 ~ N ( p1 ,
p1 (1  p1 )
)
n1
pˆ 2 ~ N ( p2 ,
p2 (1  p2 )
)
n2
pˆ1  pˆ 2 ~ N ( p1  p2 ,
p1 (1  p1 ) p2 (1  p2 )

)
n1
n2
SYAFAWATI AB. SAAD