Statistisk Försöksplanering, Design of Experiments

7.5 ECTS

Ladok code: 41T09B The exam is given to: KININ12 Exam Code: Date of exam: 15 January 2016 Time: 9:00 – 13:00 Means of assistance: Calculator, Dictionary: English- Native Language

Total amount of point on exam: 50 Points Requirements for grading: A =44, B = 38, C=32, D=26, E=20, F<20

Additional information:

Next re-exam date: The marking period is, for the most part, 15 working days, otherwise it’s the following date: Important! Do not forget to write the ExamCode on each paper you hand in. Good Luck! Examiner: Sara Lorén Phone number: 033-4354622 076 13 64 871

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Question1(10points)Explain the following words (terms) in connection with design of experiment

a) Level

b) Pesudoreplicate

c) Blocking

d) Alias

e) Main effect

f) Sparsity of effects principle

g) Noise variable

h) Resolution

i) Screening experiment

j) Steepest ascent

Question2(6points)Consider a 26-2 fraction factorial design

a) How many factors does this design have? (1p)

b) How many runs are involved in this design? (1p)

c) How many levels have each factor? (1p)

d) How many generators does this design have? (1p)

e) Write a design matrix (test matrix) for this design. Where no main effects are alias with other main effects. (2p)

Question3(3points)Answer the following questions a) What is experimental design? Explain. (2p)

b) Why should not more than 25% of the available resources be invested in the first experiment? (1p)

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Question4(11points) A person is interested in the effects of speed (A), geometry (B) and angle (C) on the life of a machine tool. Two levels of each factor are chosen and two replicates of a 23 factorial design are run. The result is in Table 1: The sum of all observations are 652 ∑ 652 . The sum of all observations in square is 28362 ∑ 28362 Table 1: Data for question 4

Coded factor   

Run  A  B  C Replicate 

1 Replicate 

2  Total 

(1)  ‐   ‐   ‐  22  25  47 

a  +   ‐   ‐  32  29  61 

b  ‐   +   ‐  35  50  85 

ab  +   +   ‐  55  46  101 

c  ‐   ‐   +  44  38  82 

ac  +   ‐   +  40  36  76 

bc  ‐   +   +  60  54  114 

abc  +   +   +  39  47  86 

a) Estimate the factor effects. Which effects appear to be large? (2p)

b) Use the analysis of variance to conform your conclusions for part a. Use significance level 0.05. (5p)

c) Write down a regression model for predicting tool life based on the results of this experiment.(1p)

d) Calculate the residuals for run bc. (1p)

e) Draw an interaction plot for the largest interaction in this experiment. (2p)

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Question5(9points)An experiment was run in a semiconductor fabrication plant in an effort to increase yield. Five factors, each at two levels, were studied. The factors (and levels) were A = aperture setting (small, large), C= exposure time (20% below nominal, 20% above nominal), C = development time (30 and 45 s), D = mask dimension (small, large), and E etch time (14.5 and 15.5). An unreplicated 25 design with four center point was down. The results for the four center point runs were 68, 74, 76 and 70. The result from the 25 is in Table 3 and the sum of all observations in Table 3 is 977. In Table 4 we have an incomplete ANOVA table for the observations in Table3. Table 3: Data for question 5

Run  data  Run  data  Run  data  Run  data 

(1)  7  d  8  e  8  de  6 

a  9  ad  10  ae  12  ade  10 

b  34  bd  32  be  35  bde  30 

ab  55  abd  50  abe  52  abde  53 

c  16  cd  18  ce  15  cde  15 

ac  20  acd  21  ace  22  acde  20 

bc  40  bcd  44  bce  45  bcde  41 

abc  60  abcd  61  abce  65  abcde  63 

Table 4: Incomplete ANOVA table for data in Table 3. Source  SS  df  MS 

A  1116.3  1  1116.3 

B  9214.0  1  9214.0 

C  750.8  1  750.8 

D  5.3  1  5.3 

E  1.5  1  1.5 

AB  504.0  1  504.0 

Curvature  ?  ?    

Error  ?  ?   

Lack of Fit    ?  ?   

Pure Error    ?  ?   

Total  17818.3     

a) Calculate the four missing sum of square SSCurvature, SSError, SSLack of fit and SSPure Error. (3p)

b) Calculate the four missing degree of freedoms dfCurvature, dfError, dfLack of fit and dfPure Error. (1p)

c) Test for curvature in this experiment use significance level =0.01. Explain the result.(2p)

d) Do the F-test for lack of fit use significance level =0.01. Explain the result of this test, which conclusions can you make from the test. (2p)

e) Using the information in c) and d) and propose an experimental design for the next experiment. (1p)

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Question6(5points)An experiment was done and analyzed. The following second order model was fitted

570 8 34 19 30 110

The three factors are temperature , Agitation rate and Pressure . The response variable y is the viscosity. Assume also that the pressure is a noise variable ( 1). The residual mean square from fitting the response model is MSerror=100. The different levels for the factors are in Table 5 both for coded and uncoded units.

Table 5: Levels and settings for the factors. In coded units and uncoded units. Level Temp Agiation

rate Pressure

High 200 10.3 25 +1 +1 +1

Middle 175 7.5 20 0 0 0

Low 150 5.0 15 -1 -1 -1

a) Write down the mean and the variance model. (2p)

b) Fit the response model for the viscosity as close as possible to 600 and that minimizes the variability transmitted from the noise variable pressure. (Use the models in a)) (2p)

c) From the result in b) what are the optimal setting for and in uncoded units. (1p)

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Question7(6points)An experiment was done for investigate the effect of material type (A) and phosphor type (B) on the response y. The number of levels for A is two and the number of levels for B is three. An analysis of variance was done and in Figure 1 four different residuals plots from the analysis is presented.

a) From the four figures in Figure 1 explain what you see and make a conclusion one from each figure. (3p)

b) In Figure 2 the main effect plot for factor A is shown. Estimate the main effect from the figure. (2p)

c) From the interaction plot for AB in Figure 3 what can you say about the interaction between A and B? (1p)

Figure 1: Residuals plots for question 7.

100-10

99

90

50

10

1

Residual

Perc

ent

300280260240220

10

0

-10

Fitted Value

Resi

dual

151050-5-10

4

3

2

1

0

Residual

Freq

uenc

y

18161412108642

10

0

-10

Observation Order

Resi

dual

Normal Probability Plot Versus Fits

Histogram Versus Order

Residual Plots for Y

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Figure 2: Main effect plot for factor A. 1 and 2 corresponds two the two different levels of factor A.

Figure 3: Interaction plot between A and B

21

290

280

270

260

250

240

230

A

Mea

nMain Effect Plot

321

310

300

290

280

270

260

250

240

230

220

B

Mea

n

12

A

Interaction Plot for YData Means

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FormulaDoE2015Distributions

Normal √

, ∞ ∞

Exponential , 0

Binomial 1 , 0,1, … ,

1

Poisson !, 0,1, ….

Sample variance

2∑ 2

1

1

∑ 2 ∑ 12

1

1

Pooled sample variance

2 1 1 12

2 1 22

1 2 2

Confidence interval X random variable with unknown mean μ and known variance σ2. A two sided confidence interval 100(1-α)% for μ

/2√

/2√

X normal distributed random variable with unknown mean μ and unknown σ2. A two sided confidence interval 100(1-α)% for μ

2, 1 √ 2

, 1 √

X normal distributed random variable with unknown mean μ and unknown σ2. A two sided confidence interval 100(1-α)% for σ2

1 2

2, 12

21 2

12, 1

2

Hypothesis testing X random variable with unknown mean μ and known variance σ2

Test the mean 00

√

X normal random variable with unknown mean μ and unknown variance σ2

Test the mean 00

√

X normal random variable with unknown mean μ and unknown variance σ2

Test the variance 02 1 2

02

Test μ1-μ2 variance known, normal population 01 2 Δ0

12

1

22

2

Test μ1-μ2 variance known, normal population 01 2 Δ0

1

1

1

2

Test variance two independent normal distributions

0: 12

22 , 1: 1

222 0

22

12 reject if 0 , 2 1, 1 1

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Linear contrast

∑ ∑ 0

∑ ∑ Variance estimation ∑ 2

Oneway ANOVA

Source SS Df MS F0

Factor SSFactor a-1 MSFactor

Error SSError a(n-1) MSError

Total SSTotal an-1  

∑ ∑ ..

∙ ∑ . ..

∙

Oneway ANOVA with block

Source SS Df MS F0

Factor SSFactor a-1 MSFactor

Block SSB b-1 MSBlock

Error SSError (a-1)(b-1) MSError  

Total SSTotal ab-1    

∑ ∑ ..

∙ ∑ . ..

∙ ∑ . ..

∙

Twoway ANOVA Source SS Df MS F0

A SSA a-1 1

B SSB b-1 1

Interaction SSAB (a-1)(b-1) 1 1

Error SSE ab(n-1) 1

Total SST abn-1

∑ ∑ ∑ 2111

…2

∑ ..2

1…2

∑ . .2

1…2

∑ ∑ .2

11…2

2

.2

10

Random factor

1, … , , 1, … ,2,

~ 0, 2 ~ 0, 2 assume and are independent

.. .. ..

and Simple Linear regression Model

Estimations 1 1∑ 1

∑ 21

∑∑ ∑ 11

1

∑ 2 ∑2

1 1

and ̅

Matrix ′ 1 ′

∑2

1 ∑ 2 ∑2

1 1 ∑2

1

12 ∑ 2

1

2

1

2

1

2

1

2

∑ 21

Confidence interval | ∗ ∶ 0 1 ∗ 1 2, 22 1 ∗ 2

∑ 21

Prediction interval | ∗:

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Percentage Points of the F distribution F0.05, v1, v2

12

Percentage Points of the F distribution, F0.01, v1, v2

13

CumulativeStandardNormaldistribution

14

Cumulative Standard Normal distribution

15

Percentage Points of the t Distribution