Generous Motors wants to study the effect of seven variables
they suspect may affect their new car performance. In the past they
have used the traditional "Change one thing at a time" approach.
This time they will try a designed experiment. The seven variables
are called INPUT FACTORS. GM chooses two values (called LEVELS)
for each factor, and codes them 0 or 1. The car performance will
be based on two OUTPUTS:

1. Does the engine "ping"?

2. Miles per gallon at 55 mph

INPUT FACTORS                       CODED LEVELS
                                     0        1
A  Tune up                         Before   After
B  Gas type                        Regular  Ethyl
C  Tire pressure                   25 PSI   35 PSI
D  Air conditioner                  Off      On
E  Lights                           Off      On
F  Altitude                        0-2000'  5000-8000'
G  Driver                          Joan     Mike

This plan requires only 8 tests to evaluate all 7 factors.
Here are the test plan and the results of the tests.

             INPUT FACTORS               OUTPUTS
             A B C D E F G            PING     MPG
Test #1      0 0 0 0 0 0 0             Yes     20
Test #2      0 0 0 1 1 1 1             Yes     20
Test #3      0 1 1 0 0 1 1             No      25
Test #4      0 1 1 1 1 0 0             No      19
Test #5      1 0 1 0 1 0 1             Yes     27
Test #6      1 0 1 1 0 1 0             Yes     27
Test #7      1 1 0 0 1 1 0             No      28
Test #8      1 1 0 1 0 0 1             No      22


Assignment:

(a) How many times does each level of each factor occur in the experiment?
    How many times does each combination of two factors occur in the experiment?
    Check for all combinations involving factor A; the results are
    similar for other factors.

    The conclusion from (a) is that the effect of each factor is just the 
    difference of averages computed separately for the observations where 
    the factor is set to 1, and the observations where the factor is set to 0. 
    That is, the effects of all other factors cancel each other due to the
    symmetry of the design.

(b) What are the factors that affect pinging of the car?

(c) Is there any difference in driving habits of the two drivers
    indicated by different gas consumption?

(d) Does turning the A/C on decrease mileage per gallon? If yes, by how much?

(e) How many times does each combination of three factors occur in the 
    experiment? Check the combinations involving A and B. You'll find that 
    not every combination is present. The fact that the combination AB=00 
    occurs only together with C=0 means that you won't be able to distinguish 
    the effect of changing from AB=00 to another combination of levels of A 
    and B, from the effect of changing C=0 to C=1. Thus, the interaction AB 
    is confounded with factor C. 
    
    As was mentioned in class, the trade-off for a smaller design is that one
    cannot estimate (some of) the interactions, such as "what is the combined 
    effect of tune-up and a different gas type?" (AB). 

    What are the meanings of interactions DE and CF? What are those 
    interactions confounded with?

(Example due to M. Johnson)