SUMMARY FOR EQT271 Semester 1 2014/2015 Maz Jamilah Masnan, Inst. of Engineering Mathematics, Univ. Malaysia Perlis

SUMMARY FOR EQT271Semester 1 2014/2015

Maz Jamilah Masnan, Inst. of Engineering Mathematics, Univ. Malaysia Perlis

2

EQT 271ENGINEERING

STATISTICS

1. Basic Statistics 2. Statistics

Inference

4. Simple Linear

Regression

3. ANOVA

5. Nonparametric

Statistics

maz jamilah masnan/sem 1 2014/2015

3

Chapter 1. Basic Statistics

Statistics in Engineering Collecting Engineering Data (data type, group vs

ungroup) Data Summary and Presentation – 1. graphically (table, charts, graph etc) and 2. numerically (MCT – mean, mode, median,

MOD – range, variance, std. dev., MOP – quartile, z-score, percentile, outlier, boxplot ≈ 5-number-summary)


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1. Basic Statistics

Probability Distributions - Discrete Probability Distribution (Binomial & Poisson,

Poisson Approximation of Binomial Probabilities –

[ ]) - Continuous Probability Distribution (Normal & Normal

approximation of Binomial – [ +

continuous correction factor] & Poisson – [ 10 +

continuous correction factor])

λ


5

Population 1 Population 2

S2

S3

S4

Sn

S2

S3

S4

Sn

S1

.

.

.

.

X1

X2

X3

X4

Xn

X

S1

.

.

.

.

X1

X2

X3

X4

Xn

X X n

X n

S2S3

S4

Sn

S2S3

S4

Sn

Concept of Sampling Distribution of the Sample Mean


6

X1

X2

X3

X4

Xn

.

.

.

.

X 799.6

X n

12.16

X


7

1. Basic Statistics

Uses Central of Limit Theorem, std. dev. = std. error i.e.

If a population is normal with mean μ and standard deviation σ, the sampling distribution of is also normally distributed with and .

, ,

Z-value for the sampling distribution of is

X X n

( )XZ

n

XX x

X

Sampling Distribution of the Sample Mean


8

Properties and Shape of the Sampling Distribution

of the Sample Mean. If n ≥30, is normally distributed, where

Note: If the unknown then it is estimated by .

If n<30 and variance is known. is normally distributed

If n<30 and variance is unknown. t distribution

with n-1 degree of freedom is used

2

~ ,X Nn

X

2s2

X

X2

~ ,X Nn

12~ n

XT t

sn


9

Population 1 Population 2

S1

.

.

.

.

S1

.

.

.

.

S2S3

S4

Sn

S2S3

S4

Sn

1

2

3

4

n

p

p

p

p

p

1

2

3

4

n

p

p

p

p

p

Pp

Pp

P

pq

n

P

pq

n


10

Sampling Distribution of the Sample Proportion

The population and sample proportion are denoted by p and , respectively, are calculated as,

and For the large values of n (n ≥ 30), the sampling distribution is very closely normally distributed.

Mean and Standard Deviation of Sample Proportion

pX

pN

ˆx

pn

ˆ ,pq

p N pn

~

Pp

P

pq

n


11

Statistical inference is a process of drawing an inference about the data statistically. It concerned in making conclusion about the characteristics of a population based on information contained in a sample. Since populations are characterized by numerical descriptive measures called parameters, therefore, statistical inference is concerned in making inferences about population parameters.

Chapter 2. Statistical Inference


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1.Estimation (point & interval estimate [a < < b])

- Confidence interval estimation for mean (μ ) and proportion (p) - Determining sample size

2. Hypothesis Testing - Test for one and two means - Test for one and two proportions

x

Estimation & Hypothesis Testing


13

X

X

μ

-1.96 X

x +1.96 X xObserved X

+1.96 x-1.96 x μ μ

95% of the s lie in this interval

X

F( )

μ

X1

X2

X3

X4

Xn

X

n interval estimates computed by using

±1.96 X x


14

Confidence Interval (Mean)


15

Confidence Interval Estimates for the differences between two population mean,

i) Variance and are known

ii) If the population variances, and are unknown, then the following tables shows the different formulas that may be used depending on the sample sizes and the assumption on the population variances.

Estimation (Confidence Interval – Difference in Means)

1 2

21

22

2 2

1 21 2

1 22

X X Zn n

21

22


16

Equality of variances, when are unknown

Sample size

1 230 30n , n 2 2

1 2, 1 230 30n , n

2 21 2

2 2

1 21 2

1 22

s sX X Z

n n

2 21 2

1 21 22

22 21 2

1 22 22 2

1 2

1 2

1 2

1 1

,v

s sX X t

n n

s sn n

vs sn n

n n

2 21 2

1 21 22

2 21 1 2 22

1 2

1 1

1 1

2

p

p

X X Z Sn n

n s n sS

n n

1 21 22

2 21 1 2 22

1 2

1 2

1 1

1 1

2

2

p,v

p

X X t Sn n

n s n sS

n n

v n n


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Confidence Interval (Proportion)


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Hypothesis Testing1. Test for one and two population

means2. Test for one and two population

proportions

Hypothesis Testing

Require understanding of:-Definition of hypothesis test, null and alternative hypothesis, tests statistics,

critical region (rejection region), critical value, p-value.


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Procedure for hypothesis testing

1. Define the question to be tested and formulate a hypothesis for a stating the problem.

2. Choose the appropriate test statistic and calculate the sample statistic value. The choice of test statistics is dependent upon the probability distribution of the random variable involved in the hypothesis.

3. Establish the test criterion by determining the critical value and critical region.

4. Draw conclusions, whether to accept or to reject the null hypothesis.

1

: a or a or a

: a or a or > aoH

H


20maz jamilah masnan/sem 1

2014/2015


2014/2015


2014/2015

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Hypothesis testing for the differences between two population mean,

Test hypothesis

Test statisticsi) Variance and are known, and both and are samples of

any sizes.

ii) If the population variances, and are unknown, then the following tables shows the different formulas that may be used depending on the

sample sizes and the assumption on the population variances.

Hypothesis testing for the differences between two population mean,

Test hypothesis

Test statisticsi) Variance and are known, and both and are samples of

any sizes.

ii) If the population variances, and are unknown, then the following tables shows the different formulas that may be used depending on the

sample sizes and the assumption on the population variances.

1 2

21

22

1 2 0

2 21 2

1 2

test

X XZ

n n

1n 2n

21 2

2

0 1 2

1 1 2 0

1 1 2 0

1 1 2 0 2 2

: 0

: 0 Reject when

: 0 Reject when

: 0 Reject when or Z

test

test

H

H H Z Z

H H Z Z

H H Z z z


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Equality of variances, when are unknown

Sample sizeEquality of variances, when are unknown

Sample size

1 230 30n , n

2 21 2,

1 230 30n , n

2 21 2

1 2 0

2 21 2

1 2

test

X XZ

s sn n

1 2 0

2 21 2

1 2

22 21 2

1 22 22 2

1 2

1 2

1 2

1 1

test

X Xt

s sn n

s sn n

vs sn n

n n

2 21 2

1 2

1 2

2 21 1 2 2

1 2

1 1

1 1

2

test

g

g

X XZ

Sn n

n s n sS

n n

1 2 0

1 2

2 21 1 2 2

1 2

1 2

1 1

1 1

2

2

test

g

g

X Xt

Sn n

n s n sS

n n

v n n

maz jamilah masnan/sem 1

2014/2015


2014/2015


2014/2015

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For the single mean & proportion Confidence Interval vs Hypothesis Testing

0000 :@: ppHμμH

At the same level in confidence interval and hypothesis testing, when the null hypothesis is rejected, the confidence interval for the mean and proportion will not contain the hypothesized mean/proportion.

Likewise, when we fail to reject null hypothesis the confidence interval will contain the hypothesized mean/ proportion.

•** Applies only for two-tailed test. •Allan Bluman, pg. 458

α


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For the difference of means & proportions Confidence Interval vs Hypothesis Testing

[-8.5 , 8.5]

Contains zero

=If the CI contains zero, we fail to reject H0

(Means that the there is NO DIFFERENCE in population means or proportions)

[5.45 , 12.45]No zero

= If the CI does not contain zero, we reject H0

(Mean/proportion for population 1 is GREATER than the mean/proportion for population 2)

[-7.3 , -3.3]

No zero

= If the CI does not contain zero, we reject H0

(Mean/proportion for population 1 is LESS than the mean/proportion for population 2)

0:@0: 210210 ppHμμH

0:@0: 210210 ppHμμH

0:@0: 210210 ppHμμH



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Outcomes for hypothesis result

Ho (Claim)

Reject Ho----------------------

---There is sufficient

evidenceto reject the

claim.

Fail to Reject Ho

-------------------------

There is insufficient evidence

to reject the claim.

H1 (Claim)Reject H1

-------------------------

There is sufficient evidence

to support the claim.

Fail to Reject H1

-------------------------

There is insufficient evidence

to support the claim.

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Chapter 3. ANALYSIS OF VARIANCE (ANOVA)

1. 1-way-ANOVA[Completely Randomized Design]

2. 2-way-ANOVA (without replication)[Randomized Completely Block Design]

3. 2-way-ANOVA (with replication)[Factorial Design]

* Testing 3 or more population means


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1. 1-way-ANOVA [Completely Randomized Design]

• Hypothesis:H0: µ1 = µ2 = ... = µt *H1: µi µj for at least one pair (i,j)(At least one of the treatment group means differs from the rest. OR At least two of the population means are not equal)

The populations from which the samples were obtained must be normally or approximately normal distributed

The variance of the response variable, denoted 2, is the same for all of the populations.

The observations (samples) must be independent of each other

Assumptions for Analysis of Variance


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Source ofVariation

Sum ofSquares

DF MeanSquar

e

F p-Value

Treatments(between group var.)

k-1

Error (within group var.)

N-k

Total N-1


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Source ofVariation

Sum ofSquares

Degrees ofFreedom

MeanSquare

F p-Value

Treatments

SSTR t-1

Error SSE N-t

Total SST N-1

-1

SSTRMSTR

t

SSEMSE

N t

MSTR

MSE



34

CONCLUSION

Fail to Reject H0

No difference in mean

Between- group variance estimate

approximately equal to the within-

group variance

F test value approximately equal to

1

Reject H0

Difference in mean

Between- group variance estimate will be larger than

within-group variance

F test value = greater than 1

* All treatments are equal

* Treatments are not equal

1. 1-way-ANOVA [Completely Randomized Design]


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Sampling Distribution of MSTR/MSE

MSTR/MSEMSTR/MSE

Sampling Distributionof MSTR/MSE

aDo Not Reject H0Do Not Reject H0

Reject H0Reject H0

Critical ValueCritical ValueFF

Comparing the Variance Estimates: The F Test

If F_ratio < F_(critical value) , FAIL to REJECT Ho

If F_ratio > F_(critical value), REJECT Ho


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2-way-ANOVA (without replication) [Randomized Completely Block Design]

12345

2730292831

3328313030

2928303231

Sample MeanSample Variance

ObservationWaxType 1

WaxType 2

WaxType 3

2.5 3.3 2.529.0 30.4 30.0

12345

2730292831

3328313030

2928303231

Sample MeanSample Variance

BatchWaxType 1

WaxType 2

WaxType 3

2.5 3.3 2.529.0 30.4 30.0

Treatment

Treatment

Block

*Treatment can be in column or row

(in 1-way-ANOVA)

*Treatment and block can either be in column or row

(in 2-way-ANOVA)


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• First Hypothesis:Treatment Effect:

H0: 1 = 2 = ... = t =0H1: j 0 at least one j

OR


• Second Hypothesis:Block Effect:

H0: i = 0 for each value of i through nH1: i ≠ 0 at least one i

OR

H0: µ1 = µ2 = ... = µt *H1: µi µj for at least one pair (i,j)

H0: µ1 = µ2 = ... = µt *H1: µi µj for at least one pair (i,j)


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Source ofVariation

Sum ofSquares

DF MeanSquare

F p-Value

Treatments

k-1

Blocksn-1

Error(k-1) *(n-1)

Total kn-1



39

Source ofVariation

Sum ofSquares

Degrees ofFreedom

MeanSquare

F p-Value

Treatments

SSTR t-1

Blocks SSBL n-1

Error SSE (t-1)(n-1)

Total SST tn-1

Source ofVariation

Sum ofSquares

Degrees ofFreedom

MeanSquare

F p-Value

Treatments

SSTR t-1

Blocks SSBL n-1

Error SSE (t-1)(n-1)

Total SST tn-1

1

SSTRMSTR

t

1

SSBLMSBL

n

1 1

SSEMSE

t n

MSTR

MSE

MSBL

MSE



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Do Not Reject H0Do Not Reject H0

Reject H0Reject H0

(Critical Value)(Critical Value)FF




DECISION TO MAKE


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Three Sets of Hypothesis:i. Factor A Effect: H0: 1 = 2 = ... = a =0

H1: at least one i 0

ii. Factor B Effect: H0: 1 = 2 = ... = b =0

H1: at least one j ≠ 0

iii. Interaction Effect: H0: ( )ij = 0 for all i,j

H1: at least one ( )ij 0

H0: µ1 = µ2 = ... = µa *

H1: µi µk for at least one pair (i,k)

H0: µ1 = µ2 = ... = µb *

H1: µi µk for at least one pair (i,k)

H0: µAB1 = µAB2 = ... = µABb *

H1: µABi µABk for at least one pair (i,k)

REMEMBER THIS

2-way-ANOVA (with replication) [Factorial Design]


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H0: There is no difference in means of factor AH1: There is a difference in means of factor AH0: There is no difference in means of factor BH1: There is a difference in means of factor BH0: There is no interaction effect between factor A and B for/on ………H1: There is an interaction effect between factor A and B for/on ………

Three Sets of Hypothesis:i. Factor A Effect:

ii. Factor B Effect:

iii. Interaction Effect:

OR USE THESE



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FIRST : Run test to check INTERACTION [Plot the interaction and test the

hypothesis]

If there is NO INTERACTION, then run a test to know which factor

effect is significance

If there EXIST INTERACTION, no need to run tests

for each factor.

44

Source ofVariation

Sum ofSquares

DF MeanSquare

F p-Value

Factor A a-1

Factor B b-1

Interaction

(a-1)(b-1)

Error ab(r-1)

Total abr-1



45


Source ofVariation

Sum ofSquares

Degrees ofFreedom

MeanSquare

F p-Value

Factor A SSA a-1

Factor B SSB b-1

Interaction

SSAB (a-1)(b-1)

Error SSE ab(r-1)

Total SST abr-1

-1

SSAMSA

a

-1

SSBMSBb

( 1)( 1)SSABMSAB

a b

( 1)

SSEMSEab r

MSAMSE

MSBMSE

MSABMSE


46

Do Not Reject H0Do Not Reject H0

Reject H0Reject H0

(Critical Value)(Critical Value)FF




DECISION TO MAKE



2014/2015

Two-wheel-drive (mean) Four-wheel-drive (mean)0

5

10

15

20

25

30

35

Graph of the means of the Factors

Series1 Series2

Gasoline C

onsum

pti

on

REG-ULAR

OCTANE

Disordinal InteractionThere is a SIGNIFICANCE interaction between

……

Ordinal InteractionThere is an interaction but not significant. The main effect can be interpreted independently


5

10

15

20

25

30

35


Series1 Series2

Gasoline C

onsum

pti

on

REG-ULAR

OCTANE


5

10

15

20

25

30

35


Series1 Series2

Gasoline C

onsum

pti

on

REG-ULAR

OCTANE

No Interaction (parallel)There is no significant interaction . The main effect can be interpreted

independently

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4. Simple Linear Regression

0 1ˆ ˆY X

0 1Y X

Estimate the model using LEAST SQUARE

METHOD

0 1ˆ ˆy x

1 1

1

2

12

1

2

12

1

n n

i ini i

xy i ii

n

ini

yy ii

n

ini

xx ii

x y

S x yn

y

S yn

x

S xn

1xy

xx

S

S


49

ADEQUACY OF THE MODEL COEFFICIENT OF DETERMINATION

( OR )2R 2r

PEARSON PRODUCT MOMENT CORRELATION COEFFICIENT

(r)

.xy

xx yy

Sr

S S

2

2 xy

xx yy

SSSRr

SST S S

measure of the variation (%) of the dependent variable (Y) that is explained

by the regression line and the independent

variable (X)

measures the strength of a linear

relationship between the two variables X

and Y.

r-1 +1Wea

k Strong

Strong



50

TEST FOR LINEARITY OF REGRESSION

1. Determine the hypotheses.

2. Compute Critical Value/ level of significance.

3. Compute the test statistic.

( no linear r/ship)(exist linear r/ship)

valueportn

2,

2

0:0:

11

10

HH

xx

xyyy

Sn

SSVar

Vart

1

2

ˆ)ˆ(

)ˆ(

ˆ

11

1

1

( no linear r/ship)

2,2

2,2

or

nn

tttt

4. Determine the Rejection Rule.

Reject H0 if : orp-value < a

5.Conclusion.

t -Test F -Test1. Determine the hypotheses.

3. Compute the test statistic.

F = MSR/MSE * this value can get from ANOVA table

4. Determine the Rejection Rule. Reject H0 if :

p-value < aF test >

( no linear r/ship)(exist linear r/ship)

0:0:

11

10

HH

2. Specify the level of significance.

2,1, nF

2,1, nF valuepor

There is a significant relationship between variable X and Y.

5.Conclusion.There is a significant relationship between variable X and Y.

51

0.01,1,8 11.26F


ANOVA APPROACH FOR TESTING LINEARITY OF REGRESSION

1) Hypothesis:

2) F-distribution table:

3) Test Statistic:

F = MSR/MSE = 17.303

or using p-value approach:

0 1

1 1

: 0

: 0

H

H

0.01,1,8 11.26F

4) Rejection region:

If F statistic > F table, we reject H0 or

if p-value < alpha, we reject H0

5) Thus, there is a linear relationship between the variables X and Y.


2014/2015

5. Nonparametric Statistics

Sign Test (ST) Wilcoxon Signed Rank Test (WSRT)

Man Whitney Test (MWT)

(i.e. Wilcoxon Rank Sum Test)

Kruskal Wallis Test (KWT)

Test for 1 sample (use median) Can be performed for paired sample

but not covered in EQT271

Test for 1 sample (use median) Can be performed for paired sample

but not covered in EQT271

The parametric version is t-test

Test for 2 samples (use

medians) The parametric

version is Z test and t-test

Test for 3 or more samples (use medians)

The parametric version is ANOVA


2014/2015

* Please double check the summary with the notes. Some

of the complete descriptions and formulae might not be available in the summary. *

* Please do more exercises for the final exam preparation *

Documents

SUMMARY FOR EQT271 Semester 1 2014/2015 Maz Jamilah Masnan, Inst. of Engineering Mathematics, Univ. Malaysia Perlis