head(cars)
##   speed dist
## 1     4    2
## 2     4   10
## 3     7    4
## 4     7   22
## 5     8   16
## 6     9   10

Plot Data In this linear model, speed is the independent variable and stopping distance is the dependent variable.

The plot shows that stopping distance increases as speed increases.

plot(cars, xlab = "Speed", ylab = "Stopping distance")

Linear Model This linear model is based on a single factor regression. speed is the independent variable (input) and stopping distance is the dependent variable (output).

The intercept is -17.5791. The slope is 3.9324.

The one factor linear model is:

stopping distance = -17.5791 + 3.9324 * speed

cars.lm <- lm(dist ~ speed, data = cars)
summary(cars.lm)
## 
## Call:
## lm(formula = dist ~ speed, data = cars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -29.069  -9.525  -2.272   9.215  43.201 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -17.5791     6.7584  -2.601   0.0123 *  
## speed         3.9324     0.4155   9.464 1.49e-12 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 15.38 on 48 degrees of freedom
## Multiple R-squared:  0.6511, Adjusted R-squared:  0.6438 
## F-statistic: 89.57 on 1 and 48 DF,  p-value: 1.49e-12

Linear Model Summary Upon analyzing the linear model, the residuals distribution suggests a normal distribution, indicating a good fit.

The standard error for the speed coefficient is approximately 9.4 times the coefficient value, indicating a relatively low variability. This aligns with the expectation of a good model, where the standard error should be significantly smaller than the coefficient.

The p-value of 1.49e-12 suggests that the speed coefficient is highly significant and relevant in predicting the stopping distance. This means that the speed variable plays a crucial role in the model.

The intercept also holds significance in the model, as indicated by its p-value of 0.0123.

The multiple R-squared value of 0.6511 implies that the model explains approximately 65.11% of the variation in the data. This demonstrates a moderate to strong level of explanatory power.

#Plot of Linear Model
plot(cars, xlab = "Speed", ylab = "Stopping distance")
abline(cars.lm)

Plot the Risiduals

The plot below shows that the residuals look uniformly distributed around zero. The residuals appear to be uniformly scattered above and below zero.

plot(fitted(cars.lm), resid(cars.lm))

Normal Q-Q Plot The plot below suggests that there’s some skew to the right.

qqnorm(resid(cars.lm))
qqline(resid(cars.lm))