MATH 581: Experimental Design and Analysis

Sample MIDTERM

                         

                                                                                                                     Name ____________________________

Note: To receive full credits you need to show all your work. You may use two pages of notes, tables and a calculator but no other reference materials. Talking during the exam will be considered as cheating. Use separate papers if necessary. The exam is exactly 1 and a half hour (no exception).

 

  1. A manufacturer has conducted an experiment to investigate the effect of the level of supply of raw material (Supply) and the ratio of its assignment (Ratio) to the two product manufacturing lines on the profit per unit of raw materials. The ultimate goal was to be able to choose the best ratio to match each day¡¯s supply of raw materials. The levels of supply of the raw material chosen for the experiment were 15, 18, and 21 tons. The levels of the ratio of allocation to the two product lines were ¨ö, 1, and 2. The response was the profit (in cents) per unit of raw material supply obtained from a single day¡¯s production. Three replications of each combination were conducted in a random sequence. The data for the 27 days and the SAS (Proc GLM) output of ANOVA are shown in the followings.

 

 

Ratio of Raw Material Allocation

Raw Material Supply

15

18

21

¨ö

22, 20, 21

21, 19, 20

19, 18, 20

1

21, 20, 19

23, 24, 22

20, 19, 21

2

17, 18, 16

21, 11, 20

20, 22, 24

 

SAS Output #1

 

   Dependent Variable: profit  

                                       Sum of

 Source                     DF        Squares    Mean Square   F Value   Pr > F

 

 Model                       8     93.1851852     11.6481481      2.54   0.0482

 

 Error                      18     82.6666667      4.5925926

 

 Corrected Total            26    175.8518519

 

 

              R-Square     Coeff Var      Root MSE    profit Mean

 

              0.529907      10.75500      2.143034       19.92593

 

 

 Source                     DF      Type I SS    Mean Square   F Value   Pr > F

 

 ratio                       2    22.29629630    11.14814815      2.43   0.1166

 supply                      2     4.96296296     2.48148148      0.54   0.5917

 ratio*supply                4    65.92592593    16.48148148      3.59   0.0255

 

 

 

 

         Level of     Level of           ------------profit-----------

         ratio        supply       N             Mean          Std Dev

 

         0.5          15           3       21.0000000       1.00000000

         0.5          18           3       20.0000000       1.00000000

         0.5          21           3       19.0000000       1.00000000

         1            15           3       20.0000000       1.00000000

         1            18           3       23.0000000       1.00000000

         1            21           3       20.0000000       1.00000000

         2            15           3       17.0000000       1.00000000

         2            18           3       17.3333333       5.50757055

         2            21           3       22.0000000       2.00000000

 

(a)     Draw conclusions based on the analysis of variance shown above. Use 0.05 level of the significance

 

(b)     Identify the two best combinations of Supply and Ratio. Are theses two combinations significantly different? Use the Tukey procedure that limits the error rate of all pairwise comparisons of combinations to be 0.05.

 

(c)     Mr. Flippantly, the experimenter, made a mistake by fitting an ANOVA model without interactions and drew a wrong conclusion. Fill in the ANOVA table that Mr. Flippantly would have. What would be his conclusion?

 

 

                              Sum of

 Source             DF        Squares    Mean Square   F Value   Pr > F

 


 Model            

 Error           

 Corrected Total 

 

  1. Consider the data in problem 1. Since the Supply factor do not seem to be significant, the experimenter decided to consider the Ratio as only factor in the study (one-factor ANOVA). At this time, suppose that the three levels of Supply factor were chosen randomly from many different levels.

 

(a)     Perform a hypothesis testing for the factor effect. Write the hypotheses carefully.

 

(b)     Let sm2 and s2 be population variance for the factor effect and the error, respectively. Construct 95% confidence intervals for sm2/s2 and for s2.

 

(c)     Using the lower and upper bounds in (b). Construct an approximate 95% confidence interval for sm2.

 

 

  1. The marketing research group of a corporation examined the public response to the introduction of a new TV game module by comparing weekly sales volumes (in thousand dollars) for three different store chains.

 

Week

Chain

1

2

3

1-Week

35 42 35

17 30 35 43

7 22 15

2-Week

30 48 38 26

22 28 40

12 19 20 23

 

As we have discussed in class, an appropriate regression model can be written as

 

where

 

The SAS output of the regression analysis is given below 

 

 

 

SAS Output #2

 

                               The GLM Procedure

 

Dependent Variable: sales  

 

                                       Sum of

 Source                     DF        Squares    Mean Square   F Value   Pr > F

 

 Model                       5    1434.869048     286.973810      4.23   0.0134

 

 Error                      15    1018.083333      67.872222

 

 Corrected Total            20    2452.952381

 

 

               R-Square     Coeff Var      Root MSE    sales Mean

 

               0.584956      29.47320      8.238460      27.95238

 

 

 Source                     DF      Type I SS    Mean Square   F Value   Pr > F

 

 X1                          1       0.416017       0.416017      0.01   0.9386

 X2                          1    1321.142857    1321.142857     19.47   0.0005

 X3                          1      80.000649      80.000649      1.18   0.2948

 X1X2                        1      27.523810      27.523810      0.41   0.5338

 X1X3                        1       5.785714       5.785714      0.09   0.7743

 

 

                                         Standard

       Parameter         Estimate           Error    t Value    Pr > |t|

 

       Intercept      27.87500000      1.81640968      15.35      <.0001

       X1             -0.12500000      1.81640968      -0.07      0.9460

       X2              8.54166667      2.56879121       3.33      0.0046

       X3              2.75000000      2.56879121       1.07      0.3013

       X1X2            1.04166667      2.56879121       0.41      0.6908

       X1X3            0.75000000      2.56879121       0.29      0.7743

 

(a)     Write the fitted regression line and calculate the estimated sales for 1-week and chain 2 under the regression model. Also calculate that under the cell means model. What did you find? Discuss.

 

 

 

(b)     From the output above SAS Output #2, which factor seems to be the most significant? Write a reduced model with the factor of your choice only and perform a general linear F-test to see if the reduced model is acceptable.

 

GOOD LUCK.