2026-02-08
Simple Linear Regression is a statistical tool used to: - Summarize and study relationships between two quantitative variables.
Predict an outcome based on a single input .
Determine how much the dependent variable changes when the independent variable changes by one unit.
A line drawn on scatter plot to help summarize the trend between two quantitative variable
The equation to find the best fitting line is \[\hat{y}_i = b_0 + b_1 x_i\]
-\(y_i\): denotes the observed response for experimental unit i
-\(x_i\): denotes the predictor value for experimental unit i
-\(\hat{y}_i\): is the predicted response (or fitted value) for experimental unit
Using \[\hat{y}_i = b_0 + b_1 x_i\] to predict the actual response \(y_i\), we make prediction error of size:
\[\epsilon_i = y_i - \hat{y}_i\]
-To illustrate the application of a simple linear regression, a built-in data “cars” in RStudio will be used.
summary(cars)
## speed dist ## Min. : 4.0 Min. : 2.00 ## 1st Qu.:12.0 1st Qu.: 26.00 ## Median :15.0 Median : 36.00 ## Mean :15.4 Mean : 42.98 ## 3rd Qu.:19.0 3rd Qu.: 56.00 ## Max. :25.0 Max. :120.00
-Plotting the data to show the relationship between two quantitative variable
If a function is a non-linera regression function, the residuals depart from 0 in some systematic manner.
Such as being positive for small x values
Negative for medium x values
Positive again for large x values
Any systemic pattern is sufficient to suggest that teh regression is not linear
-using lm() R-studios
