NAME:STUDENTID:

UNIVERSITY OF TORONTOFaculty of Arts and Science

APRIL / MAY EXAMINATIONS 1999

STA 332S / 1004S

Duration - 3 hours

Examination Aids: One (1) 8" x 11" Aid SheetOne (1) Non-Programmable Calculator

1. (7 marks)What is the difference between a 2-factor design with one observation per celland a randomized complete block design? What do these designs have incommon?

\f\aue <£.

. I obs£n/

locus

one f's a. riu-isounce •fio&nsY' useJ -fe Coirtfrcs/ your i&bffi'

offs

Page 1 of 14

NAME:STUDENTID:

2. An experiment was conducted to study the effect of soil type, level of nitrogenfertilizer, and level of potassium fertilizer on the yield of dry herbage. Two levelsof each of the fertilizers were chosen because they were of specific interest tothe experimenter; 3 soil types were randomly selected from those available. It isknown from past experience that there are no interactions involving bothnitrogen and potassium. Four observations were made for each of theexperimental conditions.

(a) (1 mark)What design was used in planning this experiment?

(b) (3 marks)State the model that you would use to analyse these data, including allassumptions that are required. •£ = n/'frcx?i£^ % •* so< (

iid M<To,c Z ( : r ^^ - o

(c) (4 marks)Give the degrees of freedom and the expected mean square for eachterm in the model that you stated in (b) above.

"Potass/urn I—i I x- .ea-2.5. 16 d^c 4-

H >< S o°* 8 (T2 J-

Error 33 £Tbfal

Page 2 of 14

NAME:STUDENTID:

(d) (4 marks)After the data were collected and the sums of squares were computed, itwas found that SSsoil, SSsoilxnitrogen, and SSE were each equal to 31.2. Whatevidence do these data provide that soil type has an impact on the yieldof dry herbage?

Ho- 0£-0^a-O >O

= 19.5

P<TFai33 > (3.5^) * < O. 0(

:. daJia. 'provide sS r̂o»oq e\/td&Y~icJZ +fa-<x* ^^ll

of

(e) (4 marks)How would your conclusions in (d) change if the levels of nitrogen hadbeen randomly selected?

Meed ~fo rec^ciuJiaJe. £/ViS

Source _d£

/ 3^O 4- S +

Error

H-a : crj > oP - MSso,u / MS so,,, v^t -P'V^LtUL = PCPj/A > 0 > O.

Page 3 of 14.'. axxu^i CouveLu^U -fi/uajt Soil

NAME:STUDENTID:

3. A study is to be designed to compare k new treatments with a controltreatment. A completely randomized design will be used to plan the study.

One of the decisions that must be made by the researchers is whether an equalnumber of experimental units should be allocated to each of the k+1 groups.Specifically, if nc experimental units will be randomly allocated to the controlgroup, and n experimental units will be allocated to each of the k newtreatments, then how large should nc be in relation to n?

To help answer this question, the researchers have decided to choose nc tominimize the variance of the difference between the control effect and anytreatment effect. In other words, the variance of fc - ft must be minimized (/ =

/, 2, ..., k). In this notation, fc is the effect of the control treatment, and f. is theeffect of the i-th new treatment.

(a) (£ marks)Suppose that the total number of experimental units available is N, andthat this number is fixed. How should n and nc be chosen to minimizethe variance of fr - f ?

(b) (0 marks)Write the expression for the 95% confidence interval for TC- ft

± -fc.025,ar 1/MSE IJ -̂̂ S"-^ d-P-* df fTrr Bmrr

Page 4 of 14

NAME:STUDENTID:

4. (7 marks)An experiment is carried out to investigate the deterioration in a product afterstorage for different lengths of time at different temperatures. The experimentaldesign is a 2-factor study with factor A being time at a levels and factor B beingstorage temperature at b levels. However, one of the levels of A is zero, andzero storage time is the same for all temperatures. Thus there are b(a-l)+lexperimental conditions and n observations for each condition. How would youanalyse the data?

Treat -tt/v^s a-s a. otne, -fa-cfrn' s-fu^Uj aJ^rt/i b(A -f)-H

CoiA_trvas.-hs

-for-

Page 5 of 14

NAME:STUDENTID:

5. A researcher studying stairway safety is interested in screening several factors todetermine which ones influence stairway safety. The factors of interest arelighting (bright versus normal), handrail height (high versus low), stairway pitch(steep versus normal), and flooring material (high friction versus moderatefriction).

There is a shortage of volunteers, and the researcher would like to test as manyexperimental conditions as possible in each subject. Each subject is availablefor only one day. Each set of experimental conditions takes one hour to test andvolunteers usually rest for 30 minutes between tests. The researcher feels itwould be unreasonable to require subjects to complete more than 5 tests inone day.

(a) (10 marks)The researcher has conducted many studies to examine each of thefactors individually but has never had the opportunity to examine thefactors in combination. Design a study with one complete replicate thatwould provide the researcher with information that would complementexisting data. Explain how you would choose your design.

&*- 5

C&uu&L /L^PJ^LCA^LL « -X

4

3

* t'.-c A ,Bt AB

Q, A*

^-3-

Page 7 of 15A ., AGC-D

NAME:STUDENTID:

(b) (3 marks)Does the study that you described in part (a) completely meet theresearcher's needs? Explain.

(c) (2 marks)How would you go about testing hypotheses in the design that youspecified in part (a) above?

i-vo CL-f.

!

Page 8 of 15

NAME:STUDENTID:

6. Four gasoline additives are to be studied to determine which would be mosteffective is reducing automobile emissions of oxides of nitrogen. To study theeffectiveness of the additives, cars will be driven through a test course andemissions will be measured during the test.

Although each driver may do their best to drive in a manner required by the test,systematic differences between drivers may exist which could affectperformance. For this reason, drivers will be included in the design of the study.Similarly, although the cars to be used in the study are all of the same model, itis possible that there are systematic differences between cars which wouldaffect performance. Therefore, cars will also be included as a factor in thisstudy.

Four drivers will each test each of 4 cars. Also, each additive will be used witheach car exactly once, and each driver will test each additive exactly once.

(a) (1 mark)Which experimental design would you use to plan this study?

(b) (3 marks)How would you randomize the study based on the design that youspecified in part (a)?

orve Uxh 1/1QU possible 4x'<4

(c) (2 marks)What biases does this study design guard against?

cars

Page 9 of 15

NAME:STUDENTID:

(d) (5 marks)The researchers randomized the study as indicated in the table below.The additives used for each test are shown in brackets and are denotedA,, AZ AS, and A4; the figures presented in the table are the levels of theoxides of nitrogen for each test. Use the data in the table along with thefollowing summaries to construct an ANOVA table for this study.

Driver

I

II

III

rvAverage

Car

1

21 (A,)

23 (A4)

15 (A2)

17 (A3)

19

2

26 (A2)

26 (A3)

13 (A4)

15 (A,)

20

3

20 (A4)

20 (A,)

16 (A3)

20 (A2)

19

4

25 (A3)

27 (A2)

16 (A,)

20 (A4)

22

Average

23

24

15

18

20

Additive Averages: A, 18 A222 A321 A419

= 6,696

Source ofVariation

AddUnVes

"Drivers

Tof-al

ANOVA Table

d.f.

33a6

Sum ofSquares

564.

MeanSquare F-ratio p-value

Page 10 of 15

NAME:STUDENTID:

(e) (4 marks)This design proved to be quite difficult for the company to execute, dueto complexities in scheduling drivers to test the different cars. If possible,they would like to be able to avoid including cars as a factor in any futurestudies. Do the data from this study indicate that including cars as afactor provides any advantage over not including them as a factor?

••

<?•?*

3MS,e -

to '

I ( J

7. A quality assurance test consists of taking porosity readings on condenser paperfor each of 3 lots produced on a given day. Four rolls of paper are randomlyselected from each lot, and 3 measurements are taken for each roll.

(a) (2 marks)What experimental design was used in planning this experiment?

I COO

Page 11 of 15

NAME:STUDENTID:

(b) (5 marks)Complete the ANOVA table given below for this study.

ANOVA Table

Source ofVariation

Lots

Rolls

Error

Total

d.f.

Si

33J4-35

Sum ofSquares

<o. I5&4WI

36.36

21.80

& &> 31 3 S

MeanSquare

>.3.c?qa±W}

4.04

o, 9o83 33

F-ratio

3.39

4-. 44 7 -7

p-value

(.05, ,10

<0.0j

(c) (3 marks)What conclusions would you draw about the interaction between lotsand rolls of paper? Explain.

vj e i i/vfo

Page 12 of 15

NAME:STUDENTID:

8. (a) (3 marks)What are the advantages and disadvantages of using a balancedincomplete block design compared to a randomized complete blockdesign?

of e-u "s /'i/i.

(b) (2 marks)Which aspects of a balanced incomplete block design are balanced?

of

(c) (7 marks)For a balanced incomplete block design, why is it incorrect to estimatethe difference in the effects of treatments / andy as Yin - YJn ? What is

the correct least squares estimate?

,' b

Page 13 of 15

u

NAME:STUDENTID:

9. The following experiment was described in an article which appeared in theJournal of Quality Technology in 1985. A replicated fractional factorial study wasused to investigate the effect of 5 factors on the free height of leaf springs usedin an automotive application. The 5 factors are: furnace temperature (A),heating time (B), transfer time (C), hold down time (D), and quench oiltemperature (E). The data that resulted from this study are given in the tablebelow.

FactorA-

+-+-

+-

+-

+-

+-

+-+

B-

-++--++--

++--

++

C-

---

++++----

++++

D-

++-+--+-++-

+--+

E-

-------++++++++

Free HeightMeasurements

7.78

8.15

7.50

7.59

7.54

7.69

7.56

7.56

7.50

7.88

7.50

7.63

7.32

7.56

7.18

7.81

7.78

8.18

7.56

7.56

8.00

8.09

7.52

7.81

7.25

7.88

7.56

7.75

7.44

7.69

7.18

7.5&

7.81

7.88

7.50

7.75

7.88

8.06

7.44

7.69

7.12

7.44

7.50

7.56

7.44

7.62

7.25

7.59

(a) (3 marks)Write out the alias structure for this design. What is ffte resofcitfeft of thisdesign? :c=- /4BcD 1?esoiu-rTc^-\ IVA 64 BCD"B (A&C.D) *=*

D ABC

46 CD

Page 14 of 15

A 6

NAME:STUDENTID:

(b) (4 marks)Is this the best possible design for 5 factors in 16 runs? Can you find afractional factorial design for 5 factors in 16 runs with a higher resolutionthan this onef State the design and its resolution.

A- 'rKSotufn'qi/A V

bu L U X c t A L T -

(c) (4 marks)Construct the ANOVA table for the design used in. thes^ement of ttie problem, giving source of variati€J!iaiaB^€}egrees offreedom. You do not need to calculate SS, MSsFr

4 I ; MB I

e

k r

AOAD

Dt1 1

r

• &1 47-

Page 15 af 15