Honda Accord Price Analysis

Introduction and Setup

In this analysis, we explore whether we can price a used Honda Accord just by reading its odometer. Mileage serves as our explanatory variable (\(x\)), and price serves as our response variable (\(y\)).

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(alpha = 0.6) +
  labs(
    title = "Honda Accord Price vs. Mileage",
    x = "Mileage (thousands of miles)",
    y = "Price (thousands of dollars)"
  )

# Compute correlation
cor_value <- cor(my_data$Mileage, my_data$Price)
print(cor_value)

## [1] -0.8489441

The correlation coefficient is \(r = -0.84\).

Next, we fit a linear model to our data to find the slope and intercept.

model <- lm(Price ~ Mileage, data = my_data)
coef(model)

## (Intercept)     Mileage 
##  20.8096129  -0.1198119

The values obtained are: - \(r = -0.84\) - Slope = \(-0.119\) - Intercept = \(20\)

The negative sign of the slope matches the negative sign of \(r\), as expected.

Examining a Single Residual by Hand

To understand how far off our line is for a specific car, we add the fitted line to our scatterplot and inspect the first row of our dataset.

# Scatterplot with fitted line
ggplot(my_data, aes(x = Mileage, y = Price)) +
  geom_point(alpha = 0.6) +
  geom_abline(intercept = 20, slope = -0.119, color = "steelblue", linewidth = 1) +
  labs(
    title = "Honda Accord Price vs. Mileage with Fitted Line",
    x = "Mileage (thousands of miles)",
    y = "Price (thousands of dollars)"
  )

# Inspect first row of data
my_data %>% select(Mileage, Price) %>% head(1)

##   Mileage Price
## 1    74.9    12

For the first car in the dataset, the actual price is $12,000. Evaluating the residual manually shows that the point is above the line by about \(0.2\) thousand dollars ($200).

Residuals and Residual Plot

To check whether a straight line is appropriate for these data, we compute the predicted values and residuals for all data points and construct a residual plot.

# Add predicted values and residuals
my_data <- my_data %>%
  mutate(predicted = fitted(model), residual = residuals(model))

# View columns
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

# Residual plot
ggplot(my_data, aes(x = Mileage, y = residual)) +
  geom_point(alpha = 0.6) +
  geom_hline(yintercept = 0, color = "steelblue", linewidth = 1) +
  labs(
    title = "Residual Plot for Honda Accord Price",
    x = "Mileage (thousands of miles)",
    y = "Residual (thousands of dollars)"
  )

The residual plot appears boring (a random cloud of points centered around zero with no distinct pattern), which indicates that using a straight line model is appropriate for these data.

Model Summary: R-squared and Residual Standard Error

Finally, we examine the summary output of our linear model to evaluate overall fit and error.

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

Based on the model summary: - R-squared: Approximately 72% of the variation in price is explained by mileage. - Residual standard error: Measures the typical distance that data points fall from the regression line, in the units of thousands of dollars.