Solutions to 402 Exam 1 from Spring 2001

1 a) F=4.11, p=0.0243, Yes, there are significant
differences.  Not all the means are the same.

1 b) 4

1 c) 1.45

1 d) (0.926, 7.074)

1 e) A and C are significantly different from the
control based on Dunnett's method.

1 f) A, B, C (according to Hsu's Comparison with Best)
2 a) df=5(model), 6(error), 11(c. total)
   
  b) SS=676.75
  
  c) MS_model=135.35   MSE=5.196666666
 
  d) Quadratic fit seems to be appropriate.  The 
     information provided allows you to construct
     the following table

Source    DF      SS   F_o  P-Value
x          1  189.78  32.1  <0.05
x*x        1  483.48  81.7  <0.05
LOF        3    3.49  0.59  >0.05

The LOF line stands for lack of fit.  It is obtained
by adding the x*x*x, x*x*x*x, and x*x*x*x*x terms.
The p-values can be roughly determined by comparing
to an F-table.  The first 2 F-statistics would be
compared to an F with 1 and 6 df.  The appropriate
table value is 5.99.  Because both F stats are way
bigger than 5.99, the p-values are much less than
0.05.  The LOF F-stat would be compared to the F
distribution with 3 and 6 df.  The appropriate table
value is 4.76.  Because the F-stat is way less than
4.76  (0.59<<<4.76), we can conclude that there is
not significant lack of fit for the quadratic model.

3 a) 

Trials  Random Digits   Speed
------  -------------   -----
1           84            55
2           71            45
3           32            40
4           81            50
5           78            50
6           24            35
7           64            45
8           08            30
9           34            40
10          28            35
11          90            55
12          05            30
  b) gas used
  c) driving speeds of 30, 35, 40, 45, 50, and 55 mph.
  d) trials (there are 12 altogether)
  e) trials (the observational units=experimental units
     in this case)
  f) 2
  g) Using one car makes the experimental units (the 12
     trials) more homogenous.  This makes it easier to 
     compare treatments.  The bad thing is that we only
     get information about how driving speed affects
     gas used for the one car used in the experiment.
     Would the result hold for cars in general?  We
     can't be sure.
  h) Use car A for the first 6 trials.  Use car B for the
     second 6 trials.  Assign the 6 speeds randomly to the
     first 6 trials.  Assign the 6 speeds randomly to the
     second 6 trials.  This is and RCBD with the first
     6 trials using car A as one block and the second 6
     trials using car B as a second block.