Least-Squares Regression

Load the data from Table 1 in Section 4.1 into R.

Table1 <- read.csv("https://sullystats.github.io/Statistics6e/Data/Chapter4/Table1.csv")
head(Table1,n=4)

##   Speed Distance
## 1   100      257
## 2   102      264
## 3   103      274
## 4   101      266

Rather than reading the data from Github, we could manually enter the data into R.

Table1a=data.frame("Speed"=c(100, 102, 103, 101, 105, 100, 99, 105), "Distance"=c(257, 264, 274, 266, 277, 263, 258, 275))

Base R

Find the Least-Squares Regression Line

To find the Least-Squares regression line between two variables, use the following command:

lm(y-variable ~ x-variable, data = df_name)

lm(Distance ~ Speed,data=Table1)

## 
## Call:
## lm(formula = Distance ~ Speed, data = Table1)
## 
## Coefficients:
## (Intercept)        Speed  
##     -55.797        3.166

Letting x represent the speed and y represent the distance, the least-squares regression model is

\(\hat{y}=3.166x - 55.797\)

Graph Least-Squares Regression Line on Scatter Diagram

To graph the least-squares regression line on the scatter diagram, first store the regression line in a variable called golf_model. Then, draw the scatter diagram using the plot command. Finally, use the abline command to draw the regression line on the scatter diagram.

golf_model <- lm(Distance ~ Speed, data=Table1)
plot(Table1$Speed, Table1$Distance,xlab="Club-Head Speed (mph)",ylab="Distance (yards)",main="Driving Distance versus Club-Head Speed")
abline(golf_model)   # Draws regression line on the scatter diagram.

Make Predictions Using a Least-Squares Regression Model

predict(golf_model,data.frame(Speed=103))

##        1 
## 270.3119

We predict the mean distance the golf ball will travel when hit at 103 mph is 270.3 yards.

Obtaining the Sum of Squared Residuals of a Least-Squares Regression Model

resids <- residuals(golf_model)
sum(resids^2)

## [1] 49.85763

The sum of squared residuals is 49.86.

Mosaic

install.packages("Mosaic")

To find the least-squares regression model, use the lm command.

lm(y-variable ~ x-variable, data = df_name)

library(mosaic)
lm(Distance ~ Speed,data=Table1)

## 
## Call:
## lm(formula = Distance ~ Speed, data = Table1)
## 
## Coefficients:
## (Intercept)        Speed  
##     -55.797        3.166