Simple Linear Regression is a basic yet powerful statistical technique used to model the relationship between a single explanatory variable and a dependent variable.

The simple linear regression model is represented by the equation:

\[ y = \beta_0 + \beta_1 x + \varepsilon \]

Where:
- \(y\): Response variable
- \(x\): Predictor variable
- \(\beta_0\): Intercept
- \(\beta_1\): Slope
- \(\varepsilon\): Random error

1. Linearity
   
2. Independence of errors
   
3. Homoscedasticity (equal variance)
   
4. Normality of residuals

We will explore the relationship between car weight (wt) and miles per gallon (mpg) using the built-in mtcars dataset in R.

library(ggplot2)
ggplot(mtcars, aes(x = wt, y = mpg)) +
  geom_point(color = "darkblue") +
  geom_smooth(method = "lm", se = TRUE, color = "red") +
  labs(title = "Miles per Gallon vs Weight of Car",
       x = "Weight (1000 lbs)",
       y = "Miles per Gallon")

model <- lm(mpg ~ wt, data = mtcars)
res <- residuals(model)
ggplot(data.frame(wt = mtcars$wt, res = res), aes(x = wt, y = res)) +
  geom_point(color = "purple") +
  geom_hline(yintercept = 0, linetype = "dashed", color = "black") +
  labs(title = "Residuals vs Weight",
       x = "Weight (1000 lbs)",
       y = "Residuals")

library(plotly)

## 
## Attaching package: 'plotly'

## The following object is masked from 'package:ggplot2':
## 
##     last_plot

## The following object is masked from 'package:stats':
## 
##     filter

## The following object is masked from 'package:graphics':
## 
##     layout

plot_ly(data = mtcars, x = ~wt, y = ~mpg,
        type = 'scatter', mode = 'markers',
        marker = list(color = 'green', size = 10)) %>%
  layout(title = "Interactive MPG vs Weight",
         xaxis = list(title = "Weight"),
         yaxis = list(title = "MPG"))

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

\[ \text{mpg} = 37.2851 - 5.3445 \cdot \text{wt} \]

• The slope is negative, indicating that as weight increases, mpg decreases.
  
• \(R^2 = 0.7528\): About 75% of the variation in mpg is explained by weight.
  
• The p-value for wt is very small (\(< 0.001\)), so it’s a statistically significant predictor.

• Simple linear regression is effective for modeling relationships between two continuous variables.
• Visualization using ggplot2 and plotly helps us understand patterns and validate model assumptions.
• This is widely used in fields such as engineering, economics, and data science.

• R Documentation
  
• “An Introduction to Statistical Learning” by James, Witten, Hastie, Tibshirani
  
• mtcars dataset