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6166 UNIT 5 Section 2 Answers

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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 Additive Rep MPG 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 RSquare 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. Tukey Grouping Mean N Additive A 20.8592 12 A B 19.9975 12 B C 18.5300 12 C