week 8

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

Call:
lm(formula = mpg ~ wt, data = mtcars)

Coefficients:
(Intercept)           wt  
     37.285       -5.344  
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
# Scatter plot with regression line
plot(mtcars$wt, mtcars$mpg, pch=19, col="blue", 
     xlab="Weight (1000 lbs)", ylab="Miles per Gallon (mpg)", 
     main="Effect of Vehicle Weight on Fuel Efficiency")
plot(mtcars$wt, mtcars$mpg, 
     pch=19, 
     col="blue", 
     xlab="Weight (1000 lbs)", 
     ylab="Miles per Gallon (mpg)", 
     main="Effect of Vehicle Weight on Fuel Efficiency")

abline(model, col="red", lwd=2)

dev.off()
null device 
          1 
# Step 1: Fit the linear model
model <- lm(mpg ~ wt, data=mtcars)

# Step 2: Create a scatter plot
plot(mtcars$wt, mtcars$mpg, 
     pch=19, 
     col="blue", 
     xlab="Weight (1000 lbs)", 
     ylab="Miles per Gallon (mpg)", 
     main="Effect of Vehicle Weight on Fuel Efficiency")

# Step 3: Add the regression line from the model
abline(model, col="red", lwd=2)

For every additional 1,000 lbs of vehicle weight, fuel efficiency decreases by approximately 5.34 mpg. This effect is statistically significant with a p-value of 1.29e-10, and the model explains about 75% of the variability in fuel efficiency.

Calculate and interpret the coefficients

# Fit the linear model
model <- lm(mpg ~ wt, data=mtcars)
# View summary of the model
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

The slope tells us that for every 1 unit increase in weight (1000 lbs), the fuel efficiency (mpg) decreases by 5.3445 mpg.

  • This is a negative relationship, meaning heavier cars are less fuel efficient.

  • The p-value (1.29e-10) is much smaller than 0.05, so this coefficient is statistically significant.

The t-value of -9.559 indicates the strength of this relationship.

# View ANOVA table for the model
anova(model)
Analysis of Variance Table

Response: mpg
          Df Sum Sq Mean Sq F value    Pr(>F)    
wt         1 847.73  847.73  91.375 1.294e-10 ***
Residuals 30 278.32    9.28                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Conclusion

  • The model shows a strong negative relationship between the weight of a car and its fuel efficiency.

  • For every 1000 lb increase in car weight, the fuel efficiency decreases by 5.34 mpg.

  • The relationship is statistically significant (p < 0.05).

  • The R-squared value of 75.28% indicates that the model explains most of the variability in mpg.

  • The ANOVA table confirms the significance of the predictor, as shown by the low p-value for the F-test.