Do cars with larger engines tend to use more fuel?

• Engine size can be measured using displacement.
• Fuel efficiency is measured in miles per gallon (MPG).
• We will examine the relationship using statistics and graphs.

We will use the built-in R dataset mtcars.

• It includes 32 cars.
• disp is engine displacement in cubic inches.
• mpg is miles driven per gallon of fuel.
• Higher MPG means better fuel efficiency.

##                                 car disp  mpg
## Mazda RX4                 Mazda RX4  160 21.0
## Mazda RX4 Wag         Mazda RX4 Wag  160 21.0
## Datsun 710               Datsun 710  108 22.8
## Hornet 4 Drive       Hornet 4 Drive  258 21.4
## Hornet Sportabout Hornet Sportabout  360 18.7

Correlation measures how strongly two numerical variables move together.

• Values near 1 show a strong positive relationship.
• Values near -1 show a strong negative relationship.
• Values near 0 show little linear relationship.

Here we expect a negative correlation: larger engines often have lower MPG.

Pearson’s correlation coefficient is

\[ r = \frac{\sum_{i=1}^{n}(x_i-\bar{x})(y_i-\bar{y})} {\sqrt{\sum_{i=1}^{n}(x_i-\bar{x})^2}\sqrt{\sum_{i=1}^{n}(y_i-\bar{y})^2}} \]

For our data, \(x\) is engine displacement and \(y\) is MPG.

The calculated correlation is -0.848.

A simple linear regression line is written as

\[ \hat{y} = b_0 + b_1x \]

Here, \(\hat{y}\) is predicted MPG and \(x\) is displacement.

Our estimated equation is

\[ \widehat{MPG} = 29.6 + (-0.0412) \times displacement \]

The slope is negative, meaning predicted MPG falls as displacement increases.

This is the main R code used for the analysis.

cars <- mtcars
cor(cars$disp, cars$mpg)
fit <- lm(mpg ~ disp, data = cars)
summary(fit)

The built-in dataset makes this analysis easy to reproduce.

The downward trend suggests larger engines generally have lower MPG.

Cars with more cylinders tend to have lower fuel efficiency in this dataset.

Hover over any point to see which car it represents.

• The correlation between displacement and MPG is -0.848.
• This shows a strong negative linear relationship in these 32 cars.
• The regression slope is -0.0412 MPG per cubic inch.
• Correlation does not prove that changing engine size alone causes the difference in MPG.

• The data show that larger engine displacement is associated with lower MPG.
• The scatterplot and correlation tell a consistent story.
• The boxplot gives another view of fuel efficiency across cylinder counts.
• Other factors, such as weight and vehicle design, can also influence MPG.

Overall, engine size is a useful predictor of fuel efficiency in the mtcars dataset.