# ANOVA

Case Study - The Higher the Better?

Max knew that gas companies advertise the higher octane of gas result in higher gas mileage? He wanted to know if the higher price would be offset by the higher amount of miles per gallon. So, he found some friends who would help him with an experiment.

He asked 9 of his buddies (total of 10 for the experiment) who drove cars if they would help him find out. Max told everyone to keep track of their mileage for the next three times they filled their tanks with gas. The first time they filled their tank they must use Regular Unleaded 87 octane gas, on the second time, Mid-grade Unleaded 89 octane gas, and on the last fill-up they would use the expensive Super-Unleaded 92 octane gas. At each fill-up Max instructed his buddies to calculate the miles per gallon they had received from each of the different octane of gas.

When the experiment was over, Max collected the miles per gallon information from each of his nine buddies and recorded the data in a table like the one below.

1. What is the hypothesis that Max 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?

Â© BrainMass Inc. brainmass.com December 24, 2021, 10:14 pm ad1c9bdddfhttps://brainmass.com/statistics/analysis-of-variance/experimenting-different-octane-gases-454528

## SOLUTION This solution is **FREE** courtesy of BrainMass!

Please see the answers

Case Study - The Higher the Better?

Max knew that gas companies advertise the higher octane of gas result in higher gas mileage? He wanted to know if the higher price would be offset by the higher amount of miles per gallon. So, he found some friends who would help him with an experiment.

He asked 9 of his buddies (total of 10 for the experiment) who drove cars if they would help him find out. Max told everyone to keep track of their mileage for the next three times they filled their tanks with gas. The first time they filled their tank they must use Regular Unleaded 87 octane gas, on the second time, Mid-grade Unleaded 89 octane gas, and on the last fill-up they would use the expensive Super-Unleaded 92 octane gas. At each fill-up Max instructed his buddies to calculate the miles per gallon they had received from each of the different octane of gas.

When the experiment was over, Max collected the miles per gallon information from each of his nine buddies and recorded the data in a table like the one below.

Car 1 2 3 4 5 6 7 8 9 10

Regular 22 15 14 25 12 15 15 9 15 12

Mid-grade 22 15 14 25 12 15 15 9 15 15

Super 24 17 12 22 14 11 16 11 14 9

1. What is the hypothesis that Max is investigating?

H0: There is no significant difference in the mean mileage when using different octane of gas.

H1: There is significant difference in the mean mileage when using different octane of gas.

2. What is the independent variable?

Type of gas.

What are the levels of the independent variable?

Regular, Mid-grade, and Super

3. What is the dependent variable?

The mean gas mileage of the group.

4. Which statistical test would he use to test his hypothesis?

F test (ANOVA)

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 (SD=Standard Deviation, SE=Standard Error of the Mean)

Gas Type N MEAN SD SE Comparison Mean Difference Standard Error p value

Regular 10 15.4 4.7422 1.4996 Regular to Mid .30 2.120 .990

Mid-grade 10 15.7 4.5959 1.4533 Regular to Super .40 2.120 .982

Super 10 15.0 4.8762 1.5420 Mid to Super .70 2.120 .947

All Brands 30 15.37 4.5825 .8366

Miles per Gallon Sum of Squares Degrees of Freedom Mean Square F value p value

Between Groups 2.467 2 1.233 .055 .947

Within Groups 606.50 27 22.463

Total 608.967 29

Conclusion: Fails to reject the null hypothesis. The sample does not provide enough evidence to support the claim that there is significant difference in the mean mileage when using different octane of gas.

CASE B (SD=Standard Deviation, SE=Standard Error of the Mean)

Gas Type N MEAN SD SE Comparison Mean Difference Standard Error p value

Regular 10 20.40 4.4771 1.4158 Regular to Mid 4.7 2.081 .097

Mid-grade 10 15.7 4.5959 1.4533 Regular to Super 5.4 2.081 .049

Super 10 15.0 4.8762 1.5420 Mid to Super .70 2.081 .945

All Brands 30 17.03 5.1090 .9328

Miles per Gallon Sum of Squares Degrees of Freedom Mean Square F value p value

Between Groups 172.467 2 86.233 3.983 .030

Within Groups 584.50 27 21.648

Total 756.967 29

Conclusion: Reject the null hypothesis. The sample provides enough evidence to support the claim that there is significant difference in the mean mileage when using different octane of gas. The pairwise comparison of different type of gas suggests that there is significant difference in the mean values of regular and super gas type. Clearly the mean mileage of Regular type gas is significantly higher than that of Super type gas.

CASE C (SD=Standard Deviation, SE=Standard Error of the Mean)

Gas Type N MEAN SD SE Comparison Mean Difference Standard Error p value

Regular 10 20.40 4.4771 1.4158 Regular to Mid 4.7 2.317 .147

Mid-grade 10 15.7 4.5959 1.4533 Regular to Super 24.6 2.317 .000

Super 10 45.0 6.2716 1.9833 Mid to Super 29.3 2.317 .000

All Brands 30 27.03 13.9913 2.5545

Miles per Gallon Sum of Squares Degrees of Freedom Mean Square F value p value

Between Groups 4952.47 2 2476.2 92.282 .000

Within Groups 724.50 27 26.833

Total 5676.97 29

Conclusion: Reject the null hypothesis. The sample provides enough evidence to support the claim that there is significant difference in the mean mileage when using different octane of gas. The pairwise comparison of different type of gas suggests that there is significant difference in the mean values of (Regular -Super) and (Mid -Super). Clearly the mean mileage of Super type gas is higher than that of Mid-grade and Regular type.

Â© BrainMass Inc. brainmass.com December 24, 2021, 10:14 pm ad1c9bdddf>https://brainmass.com/statistics/analysis-of-variance/experimenting-different-octane-gases-454528