STA 6166 UNIT 5 Section 2 Answers
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# Engineering

Randomized Block Design with Replication

An experiment was designed to examine the effectiveness of three types of gasoline additives on boosting the miles per gallon on a specific type of car. Three randomly selected cars (Block factor at levels Car_1, Car_2 and Car_3) of the same model/year were purchased direct from the factory. Each additive (Treatment factor at levels A, B and C) was tested four times on each of the three cars. The assignment of additive by replicates were randomly assigned within each car. The same driver performed all tests on the same track under controlled conditions of temperature and humidity. The data follow. Determine whether there is a significant difference in the average gas milage for the three additives.

```
CAR_1	A	1	18.24
CAR_1	A	2	18.73
CAR_1	A	3	19.17
CAR_1	A	4	19.25
CAR_1	B	1	17.95
CAR_1	B	2	18.07
CAR_1	B	3	18.07
CAR_1	B	4	17.64
CAR_1	C	1	16.92
CAR_1	C	2	16.91
CAR_1	C	3	16.39
CAR_1	C	4	16.25
CAR_2	A	1	21.97
CAR_2	A	2	21.78
CAR_2	A	3	21.93
CAR_2	A	4	21.80
CAR_2	B	1	21.35
CAR_2	B	2	21.28
CAR_2	B	3	20.72
CAR_2	B	4	20.99
CAR_2	C	1	19.75
CAR_2	C	2	19.43
CAR_2	C	3	19.18
CAR_2	C	4	20.00
CAR_3	A	1	21.32
CAR_3	A	2	21.96
CAR_3	A	3	22.14
CAR_3	A	4	22.02
CAR_3	B	1	20.54
CAR_3	B	2	20.64
CAR_3	B	3	21.57
CAR_3	B	4	21.15
CAR_3	C	1	19.38
CAR_3	C	2	19.20
CAR_3	C	3	19.81
CAR_3	C	4	19.14
```

The analysis is performed in SAS. The program can be found here (SAS program). Note that the main effects for the blocking factor, Car, as well as the treatment factor, Additive, are significant. The significant blocking main effect tells us that we were correct to block on car, obviously each car has its own overall average MPG. Clearly the additive produce significant and sustainable differences between MPG across cars. A multiple comparison procedure on the additive means is also attached to show which additives are better than which other ones. Clearly Additive A is better than the other two.

```
Dependent Variable: MPG

Sum of
Source                      DF         Squares     Mean Square    F Value    Pr > F
Model                        8     105.2067889      13.1508486     112.10    <.0001
Error                       27       3.1675000       0.1173148
Corrected Total             35     108.3742889

R-Square     Coeff Var      Root MSE      MPG Mean
0.970773      1.730250      0.342513      19.79556

Source                      DF       Type I SS     Mean Square    F Value    Pr > F
Car                          2     71.81173889     35.90586944     306.06    <.0001
Additive                     2     33.28417222     16.64208611     141.86    <.0001
Car*Additive                 4      0.11087778      0.02771944       0.24    0.9154

Source                      DF     Type III SS     Mean Square    F Value    Pr > F
Car                          2     71.81173889     35.90586944     306.06    <.0001
Additive                     2     33.28417222     16.64208611     141.86    <.0001
Car*Additive                 4      0.11087778      0.02771944       0.24    0.9154

Tukey's Studentized Range (HSD) Test for MPG

Alpha                                   0.05
Error Degrees of Freedom                  27
Error Mean Square                   0.117315
Critical Value of Studentized Range  3.50643
Minimum Significant Difference        0.3467
Means with the same letter are not significantly different.