Analysis of Variance
The Case of the Different Gasoline Types
A young, cost-conscious college student was concerned that he wasn’t getting the best value for his gasoline dollar. After all, didn’t the gasoline companies advertise that the higher grades of gasoline would lead to higher gas mileage? The student knew that the higher grades cost more but wondered if the higher cost would be offset by the higher number of miles per gallon. Always willing to save a buck, the student decided to run an experiment.
The student found nine friends, all of whom owned cars, that were willing to be a part of the experiment. The student explained that the ten of them (including himself) would keep track of the next three times they filled up their gas tanks. On the first fill-up, they would all use Regular gasoline, on the second they would use Super gasoline and on the third they would use Ultra gasoline. At each fill-up the student conducting the research instructed his friends to compute the miles per gallon they had gotten from each of the brands of gas.
At the end of the study, the student researcher collected the miles per gallon information from each student and plotted it into a table like the one seen below. Now, all he had to do was figure out how to appropriately analyze the data!
Car 1 2 3 4 5 6 7 8 9 10
Regular 22 15 14 25 12 15 15 9 15 12
Super 22 15 14 25 12 15 15 9 15 15
Ultra 24 17 12 22 14 11 16 11 14 9
1. What is the hypothesis that the student is investigating?
2. What is the independent variable? What are the levels of the independent variable?
3. What is the dependent variable?
4. Which statistical test would he use to test his hypothesis?
5. For each of the sets of output below, what can you tell about the dependent variable? What decision would the student make?
Case A:
Gas Type N Mean Standard Standard Deviation Error of
the Mean
Regular 10 15.4 4.7422 1.4996
Super 10 15.7 4.5959 1.4533
Ultra 10 15.00 4.8762 1.5420
All Brands 30 15.37 4.5825 .8366
Sum of Degrees of Mean F value p value Squares Freedom Square
MPG Between 2.467 2 1.233 .055 .947 Groups Within 606.500 27 22.463 Groups
Total 608.967 29
Comparison Mean Difference Standard Error p value
Regular to Super .30 2.120 .990
Regular to Ultra .40 2.120 .982
Super to Ultra .70 2.120 .947
Case B:
Gas Type N Mean Standard Standard Deviation Error of
the Mean
Regular 10 20.40 4.4771 1.4158
Super 10 15.70 4.5959 1.4533
Ultra 10 15.00 4.8762 1.5420
All Brands 30 17.03 5.1090 .9328
Sum of Degrees of Mean F value p value Squares Freedom Square
MPG Between 172.467 2 86.233 3.983 .030 Groups Within 584.500 27 21.648 Groups
Total 756.967 29
Comparison Mean Difference Standard Error p value
Regular to Super 4.7 2.081 .097
Regular to Ultra 5.4 2.081 .049
Super to Ultra .70 2.081 .945
Case C:
Gas Type N Mean Standard Standard Deviation Error of
the Mean
Regular 10 20.4 4.4771 1.4158
Super 10 15.7 4.5959 1.4533
Ultra 10 45.0 6.2716 1.9833
All Brands 30 27.03 13.9913 2.5545
Sum of Degrees of Mean F value p value Squares Freedom Square
MPG Between 4952.47 2 2476.2 92.282 .000 Groups Within 724.500 27 26.833 Groups
Total 5676.97 29
Comparison Mean Difference Standard Error p value
Regular to Super 4.7 2.317 .147
Regular to Ultra 24.6 2.317 .000
Super to Ultra 29.3 2.317 .000