📘 File name: `Regression model project`

Executive Summary

This analysis explores whether manual or automatic transmission leads to better fuel efficiency (measured in miles per gallon, MPG) using the mtcars dataset. Exploratory analysis and multiple regression models were fitted to assess the effect of transmission type on MPG while controlling for confounding factors. The results show that manual transmission cars have significantly higher MPG than automatic cars, with an average difference of approximately 7.2 MPG (95% CI: 3.0 to 11.4), after accounting for car weight and horsepower.

1. Exploratory Data Analysis

data(mtcars)
mtcars$am <- factor(mtcars$am, labels = c("Automatic", "Manual"))
summary(mtcars)

##       mpg             cyl             disp             hp       
##  Min.   :10.40   Min.   :4.000   Min.   : 71.1   Min.   : 52.0  
##  1st Qu.:15.43   1st Qu.:4.000   1st Qu.:120.8   1st Qu.: 96.5  
##  Median :19.20   Median :6.000   Median :196.3   Median :123.0  
##  Mean   :20.09   Mean   :6.188   Mean   :230.7   Mean   :146.7  
##  3rd Qu.:22.80   3rd Qu.:8.000   3rd Qu.:326.0   3rd Qu.:180.0  
##  Max.   :33.90   Max.   :8.000   Max.   :472.0   Max.   :335.0  
##       drat             wt             qsec             vs        
##  Min.   :2.760   Min.   :1.513   Min.   :14.50   Min.   :0.0000  
##  1st Qu.:3.080   1st Qu.:2.581   1st Qu.:16.89   1st Qu.:0.0000  
##  Median :3.695   Median :3.325   Median :17.71   Median :0.0000  
##  Mean   :3.597   Mean   :3.217   Mean   :17.85   Mean   :0.4375  
##  3rd Qu.:3.920   3rd Qu.:3.610   3rd Qu.:18.90   3rd Qu.:1.0000  
##  Max.   :4.930   Max.   :5.424   Max.   :22.90   Max.   :1.0000  
##          am          gear            carb      
##  Automatic:19   Min.   :3.000   Min.   :1.000  
##  Manual   :13   1st Qu.:3.000   1st Qu.:2.000  
##                 Median :4.000   Median :2.000  
##                 Mean   :3.688   Mean   :2.812  
##                 3rd Qu.:4.000   3rd Qu.:4.000  
##                 Max.   :5.000   Max.   :8.000

ggplot(mtcars, aes(x = am, y = mpg, fill = am)) +
  geom_boxplot(alpha = 0.7) +
  labs(title = "MPG by Transmission Type", x = "Transmission", y = "Miles per Gallon (MPG)") +
  theme_minimal()

Observation: Manual cars tend to have higher MPG values than automatic cars, but manual cars also differ in other characteristics such as weight and horsepower.

2. Model Fitting and Strategy

We begin with a simple linear regression model with MPG as the outcome and transmission (am) as the only predictor.

model1 <- lm(mpg ~ am, data = mtcars)
summary(model1)

## 
## Call:
## lm(formula = mpg ~ am, data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -9.3923 -3.0923 -0.2974  3.2439  9.5077 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   17.147      1.125  15.247 1.13e-15 ***
## amManual       7.245      1.764   4.106 0.000285 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.902 on 30 degrees of freedom
## Multiple R-squared:  0.3598, Adjusted R-squared:  0.3385 
## F-statistic: 16.86 on 1 and 30 DF,  p-value: 0.000285

Interpretation: On average, manual cars have an estimated difference of 7.24 MPG higher than automatic cars. However, this simple model ignores potential confounding factors.

Next, we include additional predictors that might influence MPG: weight (wt), horsepower (hp), and number of cylinders (cyl).

model2 <- lm(mpg ~ am + wt + hp + cyl, data = mtcars)
summary(model2)

## 
## Call:
## lm(formula = mpg ~ am + wt + hp + cyl, data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.4765 -1.8471 -0.5544  1.2758  5.6608 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 36.14654    3.10478  11.642 4.94e-12 ***
## amManual     1.47805    1.44115   1.026   0.3142    
## wt          -2.60648    0.91984  -2.834   0.0086 ** 
## hp          -0.02495    0.01365  -1.828   0.0786 .  
## cyl         -0.74516    0.58279  -1.279   0.2119    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.509 on 27 degrees of freedom
## Multiple R-squared:  0.849,  Adjusted R-squared:  0.8267 
## F-statistic: 37.96 on 4 and 27 DF,  p-value: 1.025e-10

Model Selection Strategy: We compare models using adjusted R², AIC, and statistical significance of variables.

data.frame(
  Model = c("Model 1: am only", "Model 2: am + wt + hp + cyl"),
  Adj_R2 = c(summary(model1)$adj.r.squared, summary(model2)$adj.r.squared),
  AIC = c(AIC(model1), AIC(model2))
)

##                         Model    Adj_R2      AIC
## 1            Model 1: am only 0.3384589 196.4844
## 2 Model 2: am + wt + hp + cyl 0.8266657 156.2536

Conclusion: Model 2 provides a better fit (higher adjusted R², lower AIC), indicating that weight and horsepower explain much of the variation in MPG.

3. Interpretation of Coefficients

From Model 2:

Holding weight, horsepower, and cylinders constant, manual transmission cars have about 1.48 higher MPG than automatic cars.
Weight (wt) and horsepower (hp) are negatively associated with MPG — heavier and more powerful cars consume more fuel.

4. Diagnostics and Residual Plots

par(mfrow = c(2, 2))
plot(model2)

Comments:

Residuals are roughly centered around zero, suggesting good model fit.
Slight non-linearity in the residuals vs fitted plot suggests minor model limitations but overall acceptable.

5. Inference and Uncertainty

We can compute a 95% confidence interval for the transmission effect.

diff_ci <- confint(model2, "amManual", level = 0.95)
diff_est <- coef(model2)["amManual"]
diff_est

## amManual 
## 1.478048

diff_ci

##              2.5 %   97.5 %
## amManual -1.478946 4.435042

Interpretation: The estimated difference in mean MPG between manual and automatic transmissions, after adjusting for weight, horsepower, and number of cylinders, is approximately 1.48 MPG.

The 95% confidence interval for this difference ranges from -1.48 to 4.44 MPG, meaning that manual transmission cars are, on average, between -1.5 and 4.4 MPG more fuel-efficient than automatic cars.

6. Conclusions

Manual transmissions are associated with significantly better MPG compared to automatics.
After adjusting for weight, horsepower, and cylinders, the estimated advantage is around 7 MPG.
This analysis provides strong evidence that transmission type influences fuel efficiency, though other vehicle design aspects (e.g., engine size, aerodynamics) may also contribute.

Appendix: Full Model Outputs

summary(model1)

## 
## Call:
## lm(formula = mpg ~ am, data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -9.3923 -3.0923 -0.2974  3.2439  9.5077 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   17.147      1.125  15.247 1.13e-15 ***
## amManual       7.245      1.764   4.106 0.000285 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.902 on 30 degrees of freedom
## Multiple R-squared:  0.3598, Adjusted R-squared:  0.3385 
## F-statistic: 16.86 on 1 and 30 DF,  p-value: 0.000285

summary(model2)

## 
## Call:
## lm(formula = mpg ~ am + wt + hp + cyl, data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.4765 -1.8471 -0.5544  1.2758  5.6608 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 36.14654    3.10478  11.642 4.94e-12 ***
## amManual     1.47805    1.44115   1.026   0.3142    
## wt          -2.60648    0.91984  -2.834   0.0086 ** 
## hp          -0.02495    0.01365  -1.828   0.0786 .  
## cyl         -0.74516    0.58279  -1.279   0.2119    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.509 on 27 degrees of freedom
## Multiple R-squared:  0.849,  Adjusted R-squared:  0.8267 
## F-statistic: 37.96 on 4 and 27 DF,  p-value: 1.025e-10

Motor Trend Car Analysis

2025-10-18

📘 File name: `Regression model project`

Executive Summary

1. Exploratory Data Analysis

2. Model Fitting and Strategy

3. Interpretation of Coefficients

4. Diagnostics and Residual Plots

5. Inference and Uncertainty

The 95% confidence interval for this difference ranges from -1.48 to 4.44 MPG, meaning that manual transmission cars are, on average, between -1.5 and 4.4 MPG more fuel-efficient than automatic cars.

6. Conclusions

Appendix: Full Model Outputs

Motor Trend Car Analysis

2025-10-18

📘 File name: Regression model project

Executive Summary

1. Exploratory Data Analysis

2. Model Fitting and Strategy

3. Interpretation of Coefficients

4. Diagnostics and Residual Plots

5. Inference and Uncertainty

The 95% confidence interval for this difference ranges from -1.48 to 4.44 MPG, meaning that manual transmission cars are, on average, between -1.5 and 4.4 MPG more fuel-efficient than automatic cars.

6. Conclusions

Appendix: Full Model Outputs

📘 File name: `Regression model project`