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Does Octane Affect Miles per Gallon? A researcher wants to know if the octane level of gasoline affects the gas mileage of a car. She randomly selects 10 cars and puts 5 gallons of 87 -octane gasoline in the tank. On a closed track, each car is driven at 50 miles per hour until it runs out of gas. The experiment is repeated, with each car getting 5 gallons of 92 -octane gasoline. The miles per gallon for each car are then computed. The results are shown. Would you recommend purchasing 92 octane? Why or why not? $$ \begin{array}{cc|cc} \text { 87 Octane } & \text { 92 Octane } & \text { 87 Octane } & \text { 92 Octane } \\ \hline 18.0 & 18.5 & 23.4 & 22.8 \\ \hline 23.2 & 23.1 & 23.1 & 23.5 \\ \hline 31.5 & 31.9 & 19.0 & 19.5 \\ \hline 24.9 & 26.7 & 26.8 & 26.2 \\ \hline 24.1 & 25.1 & 31.8 & 30.7 \\ \hline \end{array} $$

Step by step solution

01

Understand the Data

The researcher tested two types of gasoline (87-octane and 92-octane) on 10 cars. Each car was driven with both types of gasoline, and the miles per gallon (MPG) achieved was recorded.

02

Organize the Data

List all the MPG values for both octane levels: 87-octane: [18.0, 23.2, 31.5, 24.9, 24.1, 23.4, 23.1, 19.0, 26.8, 31.8] 92-octane: [18.5, 23.1, 31.9, 26.7, 25.1, 22.8, 23.5, 19.5, 26.2, 30.7]

03

Calculate the Mean MPG for Each Octane Level

Find the mean (average) MPG for each gasoline type: Mean for 87-octane: \(\frac{18.0 + 23.2 + 31.5 + 24.9 + 24.1 + 23.4 + 23.1 + 19.0 + 26.8 + 31.8}{10} = 24.78\) MPG Mean for 92-octane: \(\frac{18.5 + 23.1 + 31.9 + 26.7 + 25.1 + 22.8 + 23.5 + 19.5 + 26.2 + 30.7}{10} = 24.8\) MPG

04

Compare the Means

Compare the mean MPG for both gasoline types. Mean for 87-octane: 24.78 MPG Mean for 92-octane: 24.8 MPG

05

Make a Recommendation

The mean MPG for 92-octane gasoline is slightly higher than that for 87-octane, but the difference is very small (0.02 MPG). Given this minimal difference, and considering that 92-octane gasoline is typically more expensive, it would not be cost-effective to recommend purchasing 92-octane gasoline based on this data alone.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Miles per Gallon Analysis

The experiment focuses on one core aspect: Miles per Gallon (MPG) for different octane levels.
MPG is a crucial measure, showing how efficiently a car uses fuel.
Higher MPG means a car can travel further on the same amount of gasoline.
In the study, each car was driven until it ran out of gas to measure MPG.
This allows for a direct comparison of fuel efficiency between 87-octane and 92-octane gasoline.
It is important to note that the results might show variability depending on car models, driving conditions, and maintenance.

Comparative Statistics

Comparative statistics are used here to understand the differences in fuel efficiency.
The researcher collected MPG data for each car using both 87-octane and 92-octane gasoline.
This data collection is crucial for drawing reliable conclusions.
Comparing each car's performance with both gasoline types shows if one type consistently outperforms the other.
In the given data, the researcher ensures that all 10 cars undergo similar conditions, reducing the influence of external variables.
Looking at the raw data points can reveal patterns, trends, and outliers, aiding in accurate analysis.

Mean Calculation

Calculating the mean MPG shows the average performance of each gasoline type.
The mean is obtained by summing all MPG values and dividing by the number of cars.
For 87-octane: \(Mean = \frac{18.0 + 23.2 + 31.5 + 24.9 + 24.1 + 23.4 + 23.1 + 19.0 + 26.8 + 31.8}{10} = 24.78 MPG\).
For 92-octane: \(Mean = \frac{18.5 + 23.1 + 31.9 + 26.7 + 25.1 + 22.8 + 23.5 + 19.5 + 26.2 + 30.7}{10} = 24.8 MPG\).
The slight difference of 0.02 MPG between 92- and 87-octane suggests similar performance.
It's critical to account for any minor variations and focus on the overall trend.

Cost-effectiveness of Fuel Types

Despite a small increase in MPG with 92-octane, the cost-effectiveness must be considered.
92-octane is typically more expensive than 87-octane.
Consumers should evaluate whether the slight MPG increase justifies the higher cost.
In this case, the difference of 0.02 MPG does not offer significant savings on fuel usage.
Considering the price gap, continuing with 87-octane is often more economical.
Drivers should also consider the specific requirements of their car's engine, as some engines may perform better with higher octane due to design.