ITK 226 2 Statistics

8/3/2019 ITK 226 2 Statistics

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Dicky Dermawanwww.dickydermawan.net78.net

[email protected]


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I

n statistics we are concerned with method for designing andevaluating experiments to obtain information about practical

problems.

In most cases the inspection of each item ofpopulation would

be too expensive, time-consuming, or even impossible. Hencea few ofsample are drawn at random and from this inspection

conclusion about the population are inferred.


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POPULATION SAMPLE

n

y

y

n

1i

i!!!Q

1n

yy

S

2n

1i

i22

!!

!

W

!

!N

1j

jj )x(fxQMean Average

VarianceVariance )x(fx jj

2

j

2 ! QW

Size large number N Size small number n

Probability function/density f(x) Relative frequency function

Distribution function F(x) Cumulative frequency function


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Sample of 100 Values of the Splitting Tensile Strength (lb/in2)

320 380 340 410 380 340 360 350 320 370

350 340 350 360 370 350 380 370 300 420

370 390 390 440 330 390 330 360 400 370

320 350 360 340 340 350 350 390 380 340

400 360 350 390 400 350 360 340 370 420

420 400 350 370 330 320 390 380 400 370

390 330 360 380 350 330 360 300 360 360

360 390 350 370 370 350 390 370 370 340

370 400 360 350 380 380 360 340 330 370

340 360 390 400 370 410 360 400 340 360


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0

2

4

6

8

10

12

14

16

18

300 310 320 330 340 350 360 370 380 390 400 410 420 430 440

AbsoluteFrequency

Tensile Strength


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0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

300 310 320 330 340 350 360 370 380 390 400 410 420 430 440

Relati

veFrequency

Tensile Strength


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0

20

40

60

80

100

120

300 310 320 330 340 350 360 370 380 390 400 410 420 430 440

CumulativeAbsoluteFrequency

Tensile Strength


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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

300 310 320 330 340 350 360 370 380 390 400 410 420 430440CumulativeRelativeFrequency

Tensile Strength


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Min

Lower Quartile

Middle Quartile = Median

Upper Quartile

Interquartile range

Max


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DOX 6E Montgomery 10


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Experimental errorHypothesis testing: null hypothesis, alternative hypothesisType I error : rejecting a true hypothesisType II error : accepting a false hypohesis

One-tail test vs Two-tail testConfidence level = Significance LevelP-valueConfidence interval


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I

f Y1, .Yn are independent normal random variables each ofwhich has mean Q and variance 2, then the normal randomvariable:

Is normal with the mean Q and variance 2/n and the random

variable

Is normal with the mean 0 and variance 1

The confidence interval for Q is

)Y....YYY(n

1Y n321 !

n/YZW

Q!

So far we have regarded the value y1, y2, .of a sample as n observedvalue of a single random variable Y. We may equally well regard these nvalues as single observations of n random variables Y1, Y2,.that have thesame distribution and are independent

ee

n

cy

n

cyCONF

WQ

W


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A vendor submits lots of fabric to a textile manufacturer.

The manufacturer wants to know if the lot average breakingstrength exceeds 200 psi. If so, she wants to accept the lot.

Past experience indicates that a reasonable value for thevariance of breaking strength is 100 (psi)2.

Four speciments are randomly selected, and the average

breaking strength observed is psi.214y !


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The hypothesis to be tested are:

This is a one-sided alternative hypothesis

The value ofthe test statistic is:

If the confidence level of 95% is chosen, i.e. type I error = 0.05, wefind Z = 1.645

Thus the difference is significant: H0 is rejected and we conclude that

the lot average breaking strength exceeds 200 psi.Thus, we accept the lot.

The confidence interval for Q at 95% confidence level is 205.8 Q 222.2. Clearly, 200 is outside the interval.

The P-value is 0.0026.

200:H

200:H

1

0

"

!

Q

Q

80.2

n/

yZ 00 !

!

W

Q


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n/S

Yt 00 Q!

ee

n

Sty

n

StyCONF Q

The test statistic is

The confidence interval is

At (n-1) degree of freedom

The same as previous, but we use..t distribution instead of normal distribution

Sample standard deviationS instead of tNormal

S

W


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If Variance Known

2

2

2

1

2

1

210

nn

yyZWW

!The test statistic is

The confidence interval is

ee1

2

1

1

2

12/2121

1

2

1

1

2

12/21

nnZyyyy

nnZyyCONF

WWWWEE

Normal

nnn

y

2

2

2

1

2

1

2

WWW

Q


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If Variance Unknown, but 12 = 2

2

1 20

1 2


1 1

p

y yt

Sn n

!

)1n()1n(

S)1n(S)1n(S

21

222

2112

P !

Choose confidence level, usually 95%, then find critical t value at associated degree offreedom, i.e. tE/2,RIf |t0|> t E/2,R, we have enough reason to reject null hypothesis and conclude that thetwo method differ significantlyAlternatively, calculate P value, i.e. the risk ofwrongly rejecting the null hypothesisOr set confidence interval and reject null hypothesis if 0 is not included in the interval

2nn 21 !R

2nn,tNormal

n

1

n

1S

n

y

21

21

P

2

!

R

W

Q


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If Variance Unknown, 12 2

2

1n1n

2

n

S

n

S

2

2

2

1

2

1

2

2

2

2n

22S

1

2

1n

21S

2

22

1

21

tNormal

n

S

n

S

n

y

!

R

W

Q

2

2

2

1

2

1

210

n

S

n

S

yy

t

!



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Tension bond strength of portland cementmortar is an important characteristics of the

product. An engineer is interested incomparing the strength of a modified

formulation in which polymer latex

emulsions have been added during mixing tothe strength of the unmodified mortar. Hecollected 10 observations (Table 2.1)

Plot the dot diagram.

Plot the Box & Whisker plot

Are the two formulations really different?

Or perhaps the observed difference is theresults of sampling fluctuation and the two

formulations are really identical?


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Bloking is a design technique used to improve the precision with which the

comparisons among the factors of interest are made. Often blocking is used to reduceor eliminate the variability transmitted from nuisance factors, i.e.factors that may influence the experimental response but in which we are notinterested.

The term block refers to a relatively homogeneous experimental unit, and the block

represents a restriction on complete randomization because the treatmentcombinations are only randomized within the block. Blocking is carried out by makingcomparisons within matched pairs of experimental material.

The confidence interval based on paired analysis usually much narrower than thatfrom the independent analysis. This illustrates the noise reduction property of

blocking.


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S

tatistical model 4 complete randomization:

with (2ni -1) degree of freedom

Statistical model with blocking:

with only (ni pair -1) degree of freedom

The test statistic:

The confidence interval for 2-sided test:

n/S

dt

d

0 !

!

!!

i

ijiij

n1,2,...,j

1,2iy IQ

!

!!

i

ijjiijn1,2,...,j

1,2iy IFQ

j2j1j yyd !

n/Stddn,2/

sE


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Consider a hardness testing machine that presses a rod with a pointed tip into a

metal specimen with a known force. Two different tips are available for this machine,and it is suspected that one tip produces different hardness readings than the other.

The test could be performed as follows: a number of metal specimens couldrandomly be selected. Half are tested by tip 1 and the other half by tip 2.

The metal specimens might be cut from different bar stock that were not exactlydifferent in their hardness. To protect against this possibility, an alternativeexperimental design should be considered: divide each specimen into two part andrandomly assign each tip to of each specimen


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- Use the paired data to determine a 95% confidence interval for the difference- What if we use pooled or independent analysis?

Speciment Tip 1 Tip 2

1 7 6

2 3 3

3 3 5

4 4 3

5 8 8

6 3 2

7 2 4

8 9 9

9 5 4

10 4 5


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In some experiments it is the comparison of variability in the data that is important.

For example, in chemical laboratories, we may wish to compare the variability of twoanalytical methods.Unlike the tests on means, the procedures for tests on variances are rather sensitiveto the normality assumption.

Suppose we wish to test the hypothesis weather or not the variance of a normalpopulation equals a constant, viz. 0

2 . The test statistic is:

The appropriate distribution for G02 is chi-square distribution with (n-1) degree of

freedom. The confidence interval for 02 is

2

0

2

2

0

2

0

S)1n(SS

WWG !!

2

1n,1

22

2

1n,

2

22

S)1n(S)1n(

ee

EE GW

G


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Suppose we wish to test equality of the variances of two normal populations.If independent random samples of size n1 and n2 are taken from populations 1 & 2,respectively, the test statistic for:

Is the ratio of the sample variances:

The appropriate distribution for F0 is the F distribution with (n1-1) numerator degree

of freedom and (n2-1) denominator degree of freedom. The null hypothesis would berejected if F0 > F/2,n1-1,n2-1

The confidence interval for 12 / 2

2 is

2

2

2

10

S

SF !

1n,1n,22

2

1

22

2

1

1n,1n,122

2

1

122122

FS

SF

S

S

ee EEW

W

2

0

2

1

2

0

2

0

:H

:H

WW

WW

{

!

1n,1n,1n,1n,1

122

212 F

1F:Note

!E

E


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Probability plotting is a graphical technique for determining whether sample data

conform to a hypothesized distribution based on a subjective visual examination ofthe data.

To construct a probability plot, the observation in the sample are first rank fromsmallest to largest. That is, the sample y1,y2,,yn is arranged as y(1) ,y(2) ,.,y(n) wherey(1) is the smallest observation, with y(n) the largest.The ordered observations y(j) are then plotted against their observed cumulativefrequency (j-0.5)/n.The cumulative frequency scale has been arranged so that if the hypothesizeddistribution adequately describes the data, the plotted points will fall approximatelyalong a straight line. Usually, this is subjective.


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An experiment is a test or a series of tests Experiments are used widely in the engineering

world

Process characterization & optimization Evaluation of material properties

Product design & development

Component & system tolerance determination

All experiments are designed experiments,some are poorly designed, some are well-

designed


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Randomization

Running the trials in an experiment in random order

Notion of balancing out effects oflurking variables

Replication

Sample size (improving precision of effect estimation,estimation of error or background noise)

Replication versus repeat measurements? (see page 13)

Blocking

Dealing with nuisance factors


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Best-guess experiments Used a lot

More successful than you might suspect, but there are

disadvantages One-factor-at-a-time (OFAT) experiments

Sometimes associated with the scientific orengineering method

Devastated by interaction, also very inefficient Statistically designed experiments

Based on Fishers factorial concept

ITK 226 2 Statistics

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