Simple linear regression models the relation between some variable \(X\) and it’s dependent variable \(Y\).

We are exploring the relationship between:
- wt: weight of the car (predictor variable)
- mpg: miles per gallon (dependent variable)

The model:

\[ Y = \beta_0 + \beta_1 X + \varepsilon \]

Assumptions

1. Linearity: The relationship between X and Y is linear.
   
2. Independence: Observations are independent of each other.
   
3. Homoscedasticity: Variance of residuals is constant.
   
4. Normality: Residuals are normally distributed.

The mtcars dataset was extracted from the 1974 Motor Trend US magazine, and comprises fuel consumption and 10 aspects of automobile design and performance for 32 automobiles (1973-74 models).

dim(mtcars)

## [1] 32 11

We’ll use:
- mpg: miles per gallon
- wt: weight (1000 lbs)
- hp: horsepower

head(mtcars)

##                    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

ggplot(mtcars, aes(x = wt, y = mpg)) +
  geom_point() +
  geom_smooth(method = "lm", se = TRUE, color = "blue") +
  labs(title = "MPG vs Weight", x = "Weight (1000 lbs)", y = "Miles per Gallon")

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

model <- lm(mpg ~ wt, data = mtcars)
summary(model)

## 
## 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

plot_ly(mtcars, x = ~wt, y = ~hp, z = ~mpg,
        type = "scatter3d", mode = "markers",
        marker = list(size = 5)) %>%
  layout(title = "MPG vs Weight + HP")

\[ \beta_1 = \frac{\sum(x_i - \overline{x})(y_i - \overline{y})}{\sum(x_i - \overline{x})^2}\\ \beta_0 = \overline{y} - \beta_1\overline{x} \]

x <- mtcars$wt
y <- mtcars$mpg

x_bar <- mean(x)
y_bar <- mean(y)

beta_1 <- sum((x - x_bar) * (y - y_bar)) / sum((x - x_bar)^2)
beta_0 <- y_bar - beta_1 * x_bar

cat("Estimated slope (β1):", round(beta_1, 4), "\n")

## Estimated slope (β1): -5.3445

cat("Estimated intercept (β0):", round(beta_0, 4), "\n")

## Estimated intercept (β0): 37.2851

ggplot(data = model, aes(x = .fitted, y = .resid)) +
  geom_point() +
  geom_hline(yintercept = 0, linetype = "dashed", color = "red") +
  labs(title = "Residuals vs Fitted", x = "Fitted Values", y = "Residuals")

• Intercept (β₀ = 37.2851): Estimated mpg when weight = 0.
  This isn’t realistic (no car weighs 0), but it’s needed to anchor the regression line.
• Slope (β₁ = -5.3445): For every additional 1000 lbs of weight, the car’s fuel efficiency is expected to decrease by about 5.34 mpg.
  This indicates a strong negative relationship between weight and fuel efficiency.

Residual Interpretations

• Residual = Actual mpg − Predicted mpg
• A positive residual means the model underestimated the mpg.
• A negative residual means the model overestimated the mpg.
  
• In a good model, residuals should be randomly scattered around 0 (as we saw in the residual plot).

• Strong negative relationship means there is an inverse correlation between mpg and wt.
• Model assumptions are fairly satisfied.
• This provides realisitc insight into how fuel efficiency changes with vehicle weight.

Questions? Feedback?
Feel free to reach out:
arao75 at asu dot edu.