NAME: STUDENTID: UNIVERSITY OF TORONTO Faculty of Arts and Science APRIL / MAY EXAMINATIONS 1999 STA 332S / 1004S. Duration - 3 hours

Transcription

1 UNIVERSITY OF TORONTO Faculty of Arts and Science APRIL / MAY EXAMINATIONS 1999 STA 332S / 1004S Duration - 3 hours Examination Aids: One (1) 8" x 11" Aid Sheet One (1) Non-Programmable Calculator 1. (7 marks) What is the difference between a 2-factor design with one observation per cell and a randomized complete block design? What do these designs have in common? \f\aue <.. I obs n/ locus one f's a. riu-isounce fio&nsy' usej -fe Coirtfrcs/ your i&bffi' f s of Page 1 of 14

2 2. An experiment was conducted to study the effect of soil type, level of nitrogen fertilizer, and level of potassium fertilizer on the yield of dry herbage. Two levels of each of the fertilizers were chosen because they were of specific interest to the experimenter; 3 soil types were randomly selected from those available. It is known from past experience that there are no interactions involving both nitrogen and potassium. Four observations were made for each of the experimental 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 all assumptions 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 each term in the model that you stated in (b) above. "Potass/urn I i I x-.ea d^c 4- H >< S o * 8 (T 2 Error 33 Tbfal J- Page 2 of 14

3 (d) (4 marks) After the data were collected and the sums of squares were computed, it was found that SS soil, SS soilxnitrogen, and SS E were each equal to What evidence do these data provide that soil type has an impact on the yield of dry herbage? Ho- 0-0 ^a-o >O = 19.5 P<TFa i33 > (3.5^) * < O. 0( :. dajia. 'provide ss^ro»oq e\/td&y~icjz fa-<x* ^^ll of (e) (4 marks) How would your conclusions in (d) change if the levels of nitrogen had been randomly selected? Meed ~fo rec^ciujiaje. /ViS Source _d / 3^O 4- S Error H-a : crj > o P - MSso,u / MS so,,, v^t - P'V^LtUL = PCPj /A > 0 > O. Page 3 of 14.'. axxu^i CouveLu^U -fi/uajt Soil

4 3. A study is to be designed to compare k new treatments with a control treatment. 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 equal number of experimental units should be allocated to each of the k1 groups. Specifically, if n c experimental units will be randomly allocated to the control group, and n experimental units will be allocated to each of the k new treatments, then how large should n c be in relation to n? To help answer this question, the researchers have decided to choose n c to minimize the variance of the difference between the control effect and any treatment effect. In other words, the variance of f c - f t must be minimized (/ = /, 2,..., k). In this notation, f c is the effect of the control treatment, and f. is the effect of the i-th new treatment. (a) ( marks) Suppose that the total number of experimental units available is N, and that this number is fixed. How should n and n c be chosen to minimize the variance of f r - f? (b) (0 marks) Write the expression for the 95% confidence interval for T C - f t ± -fc.025,ar 1/MSE IJ^-^S" -^ d-p-* df ftrr Bmrr Page 4 of 14

5 4. (7 marks) An experiment is carried out to investigate the deterioration in a product after storage for different lengths of time at different temperatures. The experimental design is a 2-factor study with factor A being time at a levels and factor B being storage temperature at b levels. However, one of the levels of A is zero, and zero storage time is the same for all temperatures. Thus there are b(a-l)l experimental conditions and n observations for each condition. How would you analyse 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

6 5. A researcher studying stairway safety is interested in screening several factors to determine which ones influence stairway safety. The factors of interest are lighting (bright versus normal), handrail height (high versus low), stairway pitch (steep versus normal), and flooring material (high friction versus moderate friction). There is a shortage of volunteers, and the researcher would like to test as many experimental conditions as possible in each subject. Each subject is available for only one day. Each set of experimental conditions takes one hour to test and volunteers usually rest for 30 minutes between tests. The researcher feels it would be unreasonable to require subjects to complete more than 5 tests in one day. (a) (10 marks) The researcher has conducted many studies to examine each of the factors individually but has never had the opportunity to examine the factors in combination. Design a study with one complete replicate that would provide the researcher with information that would complement existing 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 15 A., AGC-D

7 (b) (3 marks) Does the study that you described in part (a) completely meet the researcher's needs? Explain. (c) (2 marks) How would you go about testing hypotheses in the design that you specified in part (a) above? i-vo CL-f.! Page 8 of 15

8 6. Four gasoline additives are to be studied to determine which would be most effective is reducing automobile emissions of oxides of nitrogen. To study the effectiveness of the additives, cars will be driven through a test course and emissions 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 affect performance. 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, it is possible that there are systematic differences between cars which would affect performance. Therefore, cars will also be included as a factor in this study. Four drivers will each test each of 4 cars. Also, each additive will be used with each 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 you specified in part (a)? QU possible 4x'<4 orve Uxh 1/1 (c) (2 marks) What biases does this study design guard against? cars Page 9 of 15

9 (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 denoted A,, AZ AS, and A 4 ; the figures presented in the table are the levels of the oxides of nitrogen for each test. Use the data in the table along with the following summaries to construct an ANOVA table for this study. Car Driver Average I II III rv Average 21 (A,) 23 (A 4 ) 15 (A 2 ) 17 (A 3 ) (A 2 ) 26 (A 3 ) 13 (A 4 ) 15 (A,) (A 4 ) 20 (A,) 16 (A 3 ) 20 (A 2 ) (A 3 ) 27 (A 2 ) 16 (A,) 20 (A 4 ) Additive Averages: A, 18 A 2 22 A 3 21 A 4 19 = 6,696 ANOVA Table Source of Variation d.f. Sum of Squares Mean Square F-ratio p-value AddUnVes "Drivers 3 3 a Tof-al Page 10 of 15

10 (e) (4 marks) This design proved to be quite difficult for the company to execute, due to 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 future studies. Do the data from this study indicate that including cars as a factor 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 paper for each of 3 lots produced on a given day. Four rolls of paper are randomly selected 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

11 (b) (5 marks) Complete the ANOVA table given below for this study. ANOVA Table Source of Variation d.f. Sum of Squares Mean Square F-ratio p-value Lots Si <o. I5&4WI >.3.c?qa±W} 3.39 (.05,,10 Rolls Error Total 3 3J & &> 31 3 S 4.04 o, 9o <0.0j (c) (3 marks) What conclusions would you draw about the interaction between lots and rolls of paper? Explain. vj e i i/vfo Page 12 of 15

12 8. (a) (3 marks) What are the advantages and disadvantages of using a balanced incomplete block design compared to a randomized complete block design? 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 estimate the difference in the effects of treatments / andy as Y in - Y Jn? What is the correct least squares estimate?,' b Page 13 of 15

13 NAME: u 9. The following experiment was described in an article which appeared in the Journal of Quality Technology in A replicated fractional factorial study was used to investigate the effect of 5 factors on the free height of leaf springs used in an automotive application. The 5 factors are: furnace temperature (A), heating time (B), transfer time (C), hold down time (D), and quench oil temperature (E). The data that resulted from this study are given in the table below. A (a) B Factor C D E Free Height Measurements & (3 marks) Write out the alias structure for this design. What is ffte resofcitfeft of this design? :c=- /4BcD 1?esoiu-rTc^-\ IV A 64 BCD" B (A&C.D) *=* D 46 ABC CD Page 14 of 15 A6

14 (b) (4 marks) Is this the best possible design for 5 factors in 16 runs? Can you find a fractional factorial design for 5 factors in 16 runs with a higher resolution than this onef State the design and its resolution. A- 'rksotufn'qi/a V bu LUXctAL T - (c) (4 marks) Construct the ANOVA table for the design used in. the s^ement of ttie problem, giving source of variati J!ia i ab^ }egrees of freedom. You do not need to calculate SS, MSsFr 4 I ; M B I e k r AO AD Dt 1 1 r &1 47- Page 15 af 15