Introduction

This Presentation explores how car weight affects fuel efficiency. Simple linear regression is important to understand as it helps us understand the relationship between one predictor variable and one response variable

The model is written as: \[ mpg_i = \beta_0 + \beta_1 wt_i + \varepsilon_i \]

Where: - \(mpg_i\) = miles per gallon for car \(i\). - \(wt_i\) = weight of car \(i\). - \(\beta_0\) = intercept (expected MPG when weight is 0). - \(\beta_1\) = slop (change in MPG for a 1 unit increase in weight). - \(\varepsilon_i\) = random error term.

Motivation

I wanted to look at if heavier cars tend to have lower fuel efficiency. So I decided to best display this using a regression model to estimate miles per gallon (MPG) from a car’s weight (wt). I used the dataset mtcars and pulled variables mpg and wt.

Response Variable: mpg Predictor Variable: wt

Data and fitting the model

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## Attaching package: 'plotly'

## The following object is masked from 'package:ggplot2':
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## The following object is masked from 'package:stats':
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##                    mpg cyl disp  hp drat    wt  qsec vs am gear carb
## Mazda RX4         21.0   6  160 110 3.90 2.620 16.46  0  1    4    4
## Mazda RX4 Wag     21.0   6  160 110 3.90 2.875 17.02  0  1    4    4
## Datsun 710        22.8   4  108  93 3.85 2.320 18.61  1  1    4    1
## Hornet 4 Drive    21.4   6  258 110 3.08 3.215 19.44  1  0    3    1
## Hornet Sportabout 18.7   8  360 175 3.15 3.440 17.02  0  0    3    2
## Valiant           18.1   6  225 105 2.76 3.460 20.22  1  0    3    1

## 
## Call:
## lm(formula = mpg ~ wt, data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.5432 -2.3647 -0.1252  1.4096  6.8727 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  37.2851     1.8776  19.858  < 2e-16 ***
## wt           -5.3445     0.5591  -9.559 1.29e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.046 on 30 degrees of freedom
## Multiple R-squared:  0.7528, Adjusted R-squared:  0.7446 
## F-statistic: 91.38 on 1 and 30 DF,  p-value: 1.294e-10

Estimated Regression Equation

We expect the regression line: \[ \hat{mpg} = b_0 +b_1 wt \]

Where: - \(\hat{mpg}\) is the predicted miles per gallon. - \(b_0\) is the estimated intercept. - \(b_1\) is the estimated slope.

If: \[ b_1 < 0 \] then heavier cars tend to have lower fuel efficiency, which is what we expect to observe in the mtcars dataset.

ggplot scatterplot of MPG vs Weight

## `geom_smooth()` using formula = 'y ~ x'

Residual Plot of mtcars Fitted MPG and Residuals

Interactive plotly plot of MPG vs Weight Plot

Example Code

## Example Code

mod = lm(mpg~wt, data = mtcars)

ggplot(mtcars, aes(x = wt, y = mpg)) +
  geom_point(color = "blue", size = 2) +
  geom_smooth(method = "lm", se = T, color = "red")+
  labs(
    title = "MPG vs Weight",
    x = "Weight (1000 lbs)",
    y = "Miles Per Gallon"
  )+
  theme_minimal()

## `geom_smooth()` using formula = 'y ~ x'

Conclusion

This presentation used simple linear regression to study the relationship between vehicle weight and fuel efficiency. The results show a negative relationship in that as a vehicle weight increases, MPG tends to decrease.