Introduction

In this analysis, we explore the relationship between a used Honda Accord’s mileage and its price using the AccordPrice dataset from the Stat2Data package.

Our main research question is: Can you price a used Honda Accord just by reading its odometer? Additionally, we want to know how far off our regression line’s predictions are and how much we should trust them.

Step 1: Data Setup and Initial Model

We begin by loading the required packages, preparing our dataset, exploring the scatterplot, and calculating the correlation coefficient (\(r\)).

library(Stat2Data)
library(dplyr)
library(ggplot2)

data("AccordPrice")
my_data <- AccordPrice

# Scatterplot of Price vs. Mileage
ggplot(my_data, aes(x = Mileage, y = Price)) +
  geom_point() +
  labs(
    title = "Used Honda Accord Price vs. Mileage",
    x = "Mileage (thousands of miles)",
    y = "Price (thousands of dollars)"
  )

# Calculate correlation coefficient r
cor(my_data$Mileage, my_data$Price)
## [1] -0.8489441

Next, we fit the linear regression model with Price as the response variable (\(y\)) and Mileage as the explanatory variable (\(x\)).

model <- lm(Price ~ Mileage, data = my_data)
coef(model)
## (Intercept)     Mileage 
##  20.8096129  -0.1198119

Interpretation of Model Parameters

  • Correlation (\(r\)): \(-0.849\)
  • Intercept: \(20.8096129\) (thousands of dollars)
  • Slope: \(-0.1198119\) (thousands of dollars per thousand miles)

The negative correlation (\(r = -0.849\)) and negative slope indicate a strong inverse relationship: as a car’s mileage increases, its price tends to decrease.

Step 2: Scatterplot with Fitted Line and Hand Residual Calculation

Using the fitted slope and intercept, we superimpose the regression line onto our scatterplot.

ggplot(my_data, aes(x = Mileage, y = Price)) +
  geom_point() +
  geom_abline(intercept = 20.8096129, slope = -0.1198119, color = "steelblue") +
  labs(
    title = "Used Honda Accord Price vs. Mileage with Fitted Line",
    x = "Mileage (thousands of miles)",
    y = "Price (thousands of dollars)"
  )

Hand Calculation of the First Residual

We inspect the first row of our dataset:

my_data %>%
  select(Mileage, Price) %>%
  head(1)
##   Mileage Price
## 1    74.9    12

For the first car in the dataset: * Actual \(x\) (Mileage): \(74.9\) thousand miles * Actual \(y\) (Price): \(12.0\) thousand dollars

Using our regression equation \(\hat{y} = -0.1198119 \cdot x + 20.8096129\): \[\text{Predicted } y = -0.1198119(74.9) + 20.8096129 = 11.835694 \text{ thousand dollars}\]

\[\text{Residual} = \text{Actual } y - \text{Predicted } y = 12 - 11.835694 = 0.164306 \text{ thousand dollars}\]

Interpretation: This data point sits above the regression line by \(0.1643\) thousand dollars (about \(\$164.30\)). In context, for a Honda Accord with \(74.9\) thousand miles, the actual price is \(12\) thousand dollars, which is slightly higher than our model’s prediction.

Step 3: Residual Analysis and Residual Plot

We add fitted values and residuals to our dataset, then generate a residual plot to check model assumptions.

my_data <- my_data %>%
  mutate(
    predicted = fitted(model),
    residual = residuals(model)
  )

my_data %>%
  select(Mileage, Price, predicted, residual) %>%
  head()
##   Mileage Price predicted   residual
## 1    74.9  12.0  11.83570  0.1643021
## 2    53.0  17.9  14.45958  3.4404204
## 3    79.1  15.7  11.33249  4.3675123
## 4    50.1  12.5  14.80703 -2.3070342
## 5    62.0   9.5  13.38127 -3.8812720
## 6     4.8  21.5  20.23452  1.2654845
ggplot(my_data, aes(x = Mileage, y = residual)) +
  geom_point(alpha = 0.6) +
  geom_hline(yintercept = 0, color = "steelblue") +
  labs(
    title = "Residual Plot for Used Honda Accord Model",
    x = "Mileage (thousands of miles)",
    y = "Residual (thousands of dollars)"
  )

Residual Plot Evaluation: The residual plot exhibits a fairly random, “boring” cloud of points around zero without any distinct curved pattern. This indicates that a straight line is an appropriate model shape for describing the relationship between mileage and price.

Step 4: Model Assessment (\(R^2\) and Residual Standard Error)

Finally, we evaluate how well the linear model fits the data using the summary statistics.

summary(model)
## 
## Call:
## lm(formula = Price ~ Mileage, data = my_data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.5984 -1.8169 -0.4148  1.4502  6.5655 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  20.8096     0.9529   21.84  < 2e-16 ***
## Mileage      -0.1198     0.0141   -8.50 3.06e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.085 on 28 degrees of freedom
## Multiple R-squared:  0.7207, Adjusted R-squared:  0.7107 
## F-statistic: 72.25 on 1 and 28 DF,  p-value: 3.055e-09

Key Summary Metrics

  • Multiple R-squared (\(R^2\)): \(0.7207\)
  • Residual Standard Error (\(s\)): \(3.085\) thousand dollars

Summary Interpretations

  1. \(R^2\) (\(72.07\%\)): Approximately \(72.07\%\) of the variability in used Honda Accord prices is explained by the linear relationship with mileage.
  2. Residual Standard Error (\(s = 3.085\)): Predictions made by this linear model are typically off by about \(3.085\) thousand dollars (\(\$3,085\)) from the actual car prices.