# Analysis of Variance

Review attached case study and answer all questions.

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