8/12/2019 Dec Sci 2 Formulas

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DECSCI2 - De La Salle University Final ExaminationLecturer: Mrs. Marian T. Reyes List of Formulas

Chapter 8 - Single Parameter Estimation

x z2

n

x t2

sn

p z2

p(1p)

n

n=(z

2)22

e2

n=(z

2)2p(1p)

e2

Chapter 9 - Single Parameter Hypothesis

Testing

z= X o

n

t= X o

sn

z= p po

po(1po)n

Rejection Regions for ztests:upper-tailed test Z > zlower-tailed test Z < ztwo-tailed test|Z| > z/2Rejection Regions for ttests:upper-tailed test T > tlower-tailed test T < ttwo-tailed test|T| > t/2

pvalue for ztests:upper-tailed test P(Z >test stat)

lower-tailed test P(Z |test stat|)

Chapter 10a - Double Parameter Estimation

(x1 x2) z2

21

n1+

22

n2

(x1 x2) z2

s21

n1+

s22

n2

(x1 x2) t2

s2p(

1

n1+

1

n2) where

s2p=(n1 1)s21+ (n2 1)s22

n1+ n2 2

(x1 x2) t2

s21

n1+

s22

n2where

df =(

s21

n1+

s22

n2)2

1

n1 1 (s21

n1 )2

+ 1

n2 1 (s22

n2 )2

(p1 p2) z2

p1(1 p1)

n1+

p2(1p2)

n2

Chapter 10b - Double Parameter Hypothesis

Testing

z=(x1 x2) (1 2)

21

n1+

22

n2

z=(x1 x2) (1 2)

s2

1

n1+

s22

n2

t = (x1 x2) (1 2)

s2p(1

n1+

1

n2)

with n1 + n2 2 d.f.

s2p=(n1 1)s21+ (n2 1)s22

n1+ n2 2F =

s21

s22

Reject H0 if F F/2(n1 1, n2 1) or F1

F/2(n2 1, n1 1)t=

(x1 x2) (1 2)s21n1

+ s22n2

where

df =(

s21

n1+

s22

n2)2

1

n1 1 (s21

n1)2 +

1

n2 1 (s22

n2)2

t= d D

sdn

withn 1 d.f.

z= (p1 p2) (p1 p2)

p(1 p)(1

n1+

1

n2)

where p = Y1+ Y2n1+ n2

Chapter 12 - Chi Square Tests

2 =i

(oi ei)2ei

with (r 1) (c 1) d.f. fortest of independence,

ei=(row total) (column total)

grand totalFor goodness of fit tests,d.f.=k 1 no. of parameters estimated

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Chapter 13 - ANOVA CRD

SST =

nij

ki

(Yij Y..)2 =nij

ki

Y2ij T2..

N

where N=ki

ni and T..=

nij

ki

Yij

SSTr =

ki n

i(Yi. Y..)2

=

ki

T2i.

ni T2..

N where

Ti.=

nij

Yij

SSE = SST - SSTr

MSTr = SSTr

k 1 and MSE = SSE

N korMSTr = ns2

Y where s2

Y is the variance of the k

treatment meansMSE = mean of the k treatment variances

F =MSTr

MSE with k 1 numerator d.f. and N k

denominator d.f.

For pairwise comparison tests Ho : r = sabsolute difference is Q = |Yr. Ys.| and the

HSD is

MSE

2 (

1

nr+

1

ns) Q,k,Nk and the deci-

sion rule is to reject H0 if the absolute difference >HSD

Chapter 13 - ANOVA CRBD

SST =bj

ki

(Yij Y..)2 =bj

ki

Y2ij T2..

N

where N=k

ini and T..=

b

jk

iYij

SSTr =ki

T2i.b T

2

..

N where Ti.=

bj

Yij

SSB =

ki

T2.j

k T

2

..

N where T.j=

ki

Yij

SSE = SST - SSTr - SSB

MSTr = SSTr

k 1 and MSB = SSB

b 1 MSE =SSE

(b 1)(k 1)F =

MSTr

MSE with k 1 numerator d.f. and

(b

1)(k

1) denominator d.f.

F = MSBMSE

with b 1 numerator d.f. and(b 1)(k 1) denominator d.f.

Chapter 14 - Simple Linear Regression

Yi = b0+ b1xi

b1 = Sxy

Sxx=

ni=1

xiYi (

ni=1

xi)(ni=1

Yi)

n

ni=1

x2i(

n

i=1

xi)2

n

and

b0 =

ni=1

Yi

n b1

ni=1

xi

n

r= b1

Sxx

Syyor r =

SxySxx

Syywhere

Syy =n

i=1

Y2i (

ni=1

Yi)2

n

r2 = b2

1SxxSyy

For test ofHo : 1 = 0 using a t test:t=

b1 s2eSxx

where s2e = Syy b21Sxx

n 2

For test ofHo : 1 = 0 using an F test:F =

MSReg

MSE where MSReg =b21Sxx and

MSE =s2eFor test ofHo : = 0 using a t test:t=

r

n 21 r2

Prediction interval for a single response ofY :

Y t/2se

1 +1

n+

(x0 x)2Sxx

Prediction interval for the mean response Y :

Y t/2se

1

n+

(x0 x)2Sxx

Confidence interval for 1:

b1 t/2seSxx

Confidence interval for 0:

b0 t/2se n

i=1x2

i

n

Sxx