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Experimental design STAT 430 / 830

Assignment #2

October 2018

Purpose of this assignment: let you explore the material of chapters 5, 6, 7 and 8 (factorial experiments),

and thus help you to prepare for the midterm exam.

 Deadline for submitting your solutions: Friday, October 19, 10.30am. Solutions submitted after this

deadline will be graded as 0.

 How to submit? You will get an invitation to submit your solutions on Crowdmark.

 Rules of the game: It is allowed to work on this assignment together with a fellow student, but you must

write up your solutions individually and each must submit their own solutions.

 The assignment will be discussed in the tutorial on Friday, October 19.

 Establishing a rule for the stopping distance of cars

Suppose that the Ministry of Transportation is planning an information campaign for drivers. The purpose

of the campaign is to make drivers aware that when they hit the brakes, it may take much longer for the car

to come to a standstill than many drivers suspect. For the campaign, the Ministry wants to create a table

showing the stopping distance for various driving speeds, but they believe that maybe also the weight of a

car should be taken into account. Let鈥檚 suppose they hire you to do a study on the basis of which they can

make the table.

Design a suitable experiment that establishes the relationship between driving speed and weight of a car

on the one hand, and stopping distance on the other. Present the following:

a) What is the 饾憣饾憣 variable? Describe what a single run of the experiment looks like and how the 饾憣饾憣

b) What are the 饾憢饾憢 variables? Choose appropriate high and low levels for each of them.

c) Mention nuisance variables that should be taken into account, and describe how your experimental
variable will be measured.
design will deal with them.

d) What experimental design do you choose? Briefly motivate your choice.

e) Create the chosen design in Minitab, and present the worksheet (list of runs and the settings for
the factors)

Some issues that you need to consider.

o There鈥檚 not a single correct answer, so you can choose what you think will work best. However, if you

make challenging choices and as a result the experiment is not a good experiment, then grades will

be deducted.

o Stopping distance is the sum of reaction distance (the distance traveled in the time that it takes a

driver to perceive an obstacle and react) and the braking distance (the distance traveled from the

point the brakes are fully applied to when the car comes to a complete stop). You need to decide

whether you want the 饾憣饾憣 variable to be the total stopping distance, or only the braking distance.

o Different drivers will have different reaction speeds, and varying road conditions or variability in tires

o There is a budget to conduct 15 runs at most.

The 鈥渄on鈥檛 try this at home鈥?bit: don鈥檛 actually conduct the experiment! That should be done by

professionals, taking effective safety measures, and on a closed circuit.

may cause variability in braking distance.

 Battery life

Context: batteries for photo cameras. The battery life is the number of pictures that you can take with a full

battery until it is empty. The specification is that battery life should be at least 300 pictures. The

engineering team who designs a new model has conducted an experiment in which they investigated the

 Component X (3鈥?%): content of a certain component in the chemical composition of the battery.

effects of two 饾憢饾憢 variables onto battery life:

 Ambient temperature (5鈥?5掳C): it is believed that the temperature under which the battery is used

affects the battery鈥檚 performance.

The results of the 12 runs are summarized in this table.
degrees of freedom (DF) does it have?

a) What experimental design did they use?

b) Complete the missing column in the table, with the contrast for the 饾惔饾惔饾惔饾惔 interaction effect.

c) Estimate the effect of factor 饾惔饾惔 (Component X) by computing the effect contrast.

d) Also determine the associated sum of squares 饾憜饾憜饾憜饾憜饾惔饾惔 from the effect contrast.

e) (For STAT 430): Calculate a 饾憖饾憖饾憜饾憜饾惛饾惛 based on the 4 observations in the center point. How many
(For STAT 830): Calculate a 饾憜饾憜饾憜饾憜饾惛饾惛 based on the 2 replicates in each design point (it should have 4 DF).Also calculate a 饾憜饾憜饾憜饾憜饾惛饾惛 based on the 4 measurements in the center point. Now calculate a 饾憖饾憖饾憜饾憜饾惛饾惛 bypooling these to 饾憜饾憜饾憜饾憜饾惛饾惛鈥檚, which should have 7 DF total.
 f) Calculate the 饾憞饾憞 value for the curvature test. How many degrees of freedom does the 饾憽饾憽 distribution
饾懄饾懄锟?286+56饾懃饾懃1+74饾懃饾懃2鈭?饾懃饾懃1饾懃饾懃2
 g) Suppose that the fitted model in coded units is:
What is the fitted model in uncoded units?
 h) The camera鈥檚 user manual specifies that the camera can be used at temperatures of 5掳C or higher.
have?
Suppose that the fitted model above is correct:
o What value for component X do you recommend?
o With that value, what will the mean battery life be (in # pictures) when the camera is used at 5掳Cand at 15掳C? (a point estimate is sufficient, you don鈥檛 need to give a C.I.)

 Lasercoding experiment

Context: lasercoding process. Engineers optimize the settings of the laserprinting machine and the coding

process.

 饾憣饾憣 variable: readability.

 Five 饾憢饾憢 variables:

o Speed (750鈥?50ms)

o Frequency (17.5鈥?2.5kHz)

o Focal distance (179鈥?81mm)

o Current (25鈥?7Amp)

o Surface reflectivity (0.4鈥?.8)

a 2饾憠饾憠5鈭? experiment.

The data are given below this question, and they can be downloaded from LEARN. The engineers conducted
It is not useful to include 3, 4 and 5factor interactions in the analysis. Why not?

a)

b) Analyze the experiment.
o For the first analysis, include main effects and 2factor interactions.
o Note: in the workflow presented in Ch. 6, you don鈥檛 start creating a factorial design. Instead, youneed to apply the procedure explained on slides 21鈥?2 of Ch. 6 (鈥淎nalyzing a givenexperiment鈥?.
 c)
After removing the nonsignificant effects, what is the final model?
Is there significant evidence for curvature? If so, then what do we need to do? If not, then whatdoes this mean for the experiment?
 d) In the final model, how many degrees of freedom are there for the 饾憖饾憖饾憜饾憜饾惛饾惛? Explain this number by
listing what contrasts are included in the 饾憖饾憖饾憜饾憜饾惛饾惛.
 e) What setting for Frequency maximizes Readability? Hint: make factorial plots (main effects plots
and interaction plots)
 f) Suppose the engineers have a budget for only 8 runs. Create a fractional factorial experiment in
Minitab for 5 factors with 8 runs. What is the resolution of this design? What practicalconsequences does that have?
 g) Minitab will give you the alias structure of this design with 8 runs. Show how to determine the
aliases of the A, AB and ABC effects.
 StdOrder RunOrder CenterPt Blocks
Speed
Freq
Focal
Current
Refl
22.5
22.5
22.5
17.5
22.5
22.5
22.5
17.5
17.5
17.5
17.5
22.5
17.5
17.5
17.5
22.5
Readability3.02.75.25.72.91.33.13.81.64.00.84.01.44.60.73.62.42.92.31.30.6
0.8
0.8
0.8
0.6
0.4
0.6
0.4
0.4
0.4
0.8
0.4
0.6
0.4
0.8
0.8
0.4
0.8
0.8
0.4

 All effects nonsignificant?

(for STAT 430 only)

An experimenter conducted a 23 experiment with 2 replicates (16 runs total). After the first analysis, she

finds that all effects are nonsignificant (see the table). Indicate for each of the explanations below whether

it could explain why all the pvalues are high, and briefly motivate (5 lines per item max)

a) The three factors were studied in too small a range (that is, the low and high level for each factor
were chosen too close to each other)

b) Curvature has baffled the analysis

c) All three factors have no or only very minor effects on 饾憣饾憣

d) The 饾憣饾憣 measurements were done with a very imprecise measurement system, which resulted in a
very large error variance.
Coded Coefficients
Constant
A
B
C
A*B
A*C
B*C
ABC
0.201
0.269
0.024
0.461
0.089
0.361
0.071
23.038
0.101
0.134
0.012
0.231
0.044
0.181
0.036
0.247
0.247
0.247
0.247
0.247
0.247
0.247
0.247
93.10
0.41
0.54
0.05
0.93
0.18
0.73
0.14
0.000
0.695 1.00
0.602 1.00
0.963 1.00
0.379 1.00
0.862 1.00
0.486 1.00
0.889 1.00

 Computer experiments

(for STAT 830 only)

Especially in product development, experiments are often not done with a physical product but with a

simulation model in a computer. Simulation models such as a finite elements models are based on physics

laws, and they can be used to predict relevant characteristics of the final product, such as tensions in its

material. Many of those simulation models are deterministic models. This means that they will give exactly

the same 饾懄饾懄 values each time for a given set of 饾懃饾懃 values 鈥?that is, the error variance is zero.
For studying the effects of the 饾懃饾懃 variables onto a 饾懄饾懄 variable, would a 23 factorial with 3 replicates be a good design in this case? Given that you can do 24 runs, roughly describe what a better design would be,
 and motivate why that would be a better or more useful experiment.