fundamentals of engineering supplied-reference handbook - Ventech!

LINEAR REGRESSION
Least Squares
bx ˆ y = â + , where
y- intercept : a ˆ = y −bx
ˆ ,
and slope : b ˆ = SS /SS ,
n
⎛ ⎞
( ) ⎜∑i⎟ ⎝ i= 1 ⎠
n
⎛ ⎞
( ) ⎜∑i⎟ ⎝ i= 1 ⎠
⎛ ⎞⎛ ⎞
( ) ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠
S = x y − 1n / x y ,
i= 1 i= 1 i= 1
⎛ ⎞
( ) ⎜ ⎟
⎝ ⎠
n n
2
S = x − 1n / x ,
xx i i
i= 1 i= 1
n = sample size,
y = 1n / y ,and
x = 1n / x .
xy xx
n n n
∑ ∑ ∑
xy i i i i
∑ ∑
Standard Error of Estimate
S S − S
2 xx yy xy
S = = MSE
e
S
yy
xx
2
( − 2)
S n
n
= ∑ y
i=
1
2
i
−
2
⎛ n ⎞
1 ⎜ ∑ yi
⎟
⎝ i=
1 ⎠
( n)
Confidence Interval for a
x
â t , n
MSE
n S ⎟
xx
⎟
⎛ 2
1 ⎞
± ⎜
α 2 −2
⎜
+
⎝ ⎠
Confidence Interval for b
b t ˆ ±
α 2, n−2
MSE
S
Sample Correlation Coefficient
S xy
r =
S S
xx
yy
xx
,
2
where
2 n FACTORIAL EXPERIMENTS
Factors: X1, X2, …, Xn
Levels of each factor: 1, 2 (sometimes these levels are
represented by the symbols – and +, respectively)
r = number of observations for each experimental
condition (treatment),
Ei = estimate of the effect of factor Xi, i = 1, 2, …, n,
Eij = estimate of the effect of the interaction between
factors Xi and Xj,
Yik
= average response value for all r2 n–1 observations
having Xi set at level k, k = 1, 2, and
192
INDUSTRIAL ENGINEERING (continued)
km
Y ij = average response value for all r2 n–2 observations
having Xi set at level k, k = 1, 2, and Xj set at level
m, m = 1, 2.
Ei
= Yi2
− Yi1
22 21 12 11
( Yij
− Yij
) − ( Yij
− Yij
)
Eij
=
2
ONE-WAY ANALYSIS OF VARIANCE (ANOVA)
Given independent random samples of size ni from k
populations, then:
k ni
2
∑∑(
xij − x)
i= 1 j=
1
k ni
2 k
2
= ∑∑( xij − x) + ∑ni(
xi−x) or
i= 1 j= 1 i=
1
SSTotal = SSError + SSTreatments
Let T be the grand total of all N=Σini observations and Ti be
the total of the ni observations of the ith sample. See the
One-Way ANOVA table on page 196.
C = T 2 /N
SS
SS
k ni
2
Total = ∑∑xij
i=
1j= 1
k
Treatments = ∑ i
i=
1
− C
2 ( T n )
i
− C
SSError = SSTotal – SSTreatments
RANDOMIZED BLOCK DESIGN
The experimental material is divided into n randomized
blocks. One observation is taken at random for every
treatment within the same block. The total number of
observations is N=nk. The total value of these observations
is equal to T. The total value of observations for treatment i
is Ti. The total value of observations in block j is Bj.
C = T 2 /N
k n
2
SSTotal
= ∑ ∑ x
ij
− C
i=
1j= 1
n
= ∑ ⎜
⎛ 2
SS
⎟
⎞
Blocks B −
=
⎝ j
k C
j
⎠
1
k
SS = ∑ ⎜
⎛ 2
T n⎟
⎞
Treatments
− C
i=
⎝ i ⎠
1
SSError = SSTotal – SSBlocks – SSTreatments
See Two-Way ANOVA table on page 196.