• Explore how one variable (predictor) affects another (response).
• Using R to model and visualize the relationship.
• Key takeaway: Regression simplifies relationships into a linear equation.

• Definition: A method to model the relationship between two variables:
  • Independent Variable (x): Predictor or explanatory variable.
  • Dependent Variable (y): Response or outcome variable.
• Goal: To find the best-fit line that predicts \(y\) from \(x\).
• Mathematical Formula: \[ y = \beta_0 + \beta_1x + \epsilon \]
  • \(\beta_0\): Intercept (value of \(y\) when \(x = 0\)).
  • \(\beta_1\): Slope (change in \(y\) for one unit increase in \(x\)).
  • \(\epsilon\): Residual error (difference between observed and predicted \(y\)).

• Before applying regression, it’s essential to visualize the relationship between \(x\) and \(y\).
• Example: Weight vs. Miles per Gallon (mpg) using the mtcars dataset.

• Using the lm() function in R to fit a linear regression model.
• Example: Predicting mpg (miles per gallon) based on wt (weight).

# Fit the regression model
model <- lm(mpg ~ wt, data = mtcars)

## 
## 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

• Add the regression line to the scatter plot to visualize the model fit.

• Coefficients:
  • Intercept (\(\beta_0\)): 37.29 – Predicted mpg when wt is 0.
  • Slope (\(\beta_1\)): -5.34 – For every 1-unit increase in wt, mpg decreases by 5.34.
• Model Fit:
  • R-squared: 0.75 – 75% of the variability in mpg is explained by wt.
  • p-value for wt: < 0.001 – Indicates the relationship is statistically significant.
• Takeaway:
  • Heavier cars are less fuel-efficient.

• Explore the relationship between car weight (wt) and fuel efficiency (mpg) interactively.

• Key Insights from the Analysis:
  • Scatter Plot: Showed a clear negative trend between car weight (wt) and fuel efficiency (mpg).
  • Regression Line: Quantified the relationship with:
    • Slope (\(\beta_1 = -5.34\)): Heavier cars have lower mpg.
    • Intercept (\(\beta_0 = 37.29\)): Predicted mpg when weight is 0.
  • Model Fit:
    • \(R^2 = 0.75\): 75% of mpg variability explained by wt.
    • Strong statistical significance (\(p < 0.001\)).

• Real-World Application:
  • Predicting fuel efficiency.
  • Other use cases: sales forecasting, price prediction, and more.

Linear regression is a powerful starting point for understanding relationships between variables and making predictions. Mastering it builds a foundation for more advanced statistical methods!