Introduction

The dataset is from R preloaded libraries and it consists of speed / distance values.

Load datasets

library(ggplot2)
df<-get("cars")
df

Correlation Coefficient

Correlation coefficient provides us the relationship between the variables. If it is greater then a strong relationship exists and if it is low then the relationship is weak or none.

cor_df<-cor(df$speed, df$dist)
cor_df  
## [1] 0.8068949

The Correlation coefficient is ~ 0.8 and this shows speed has strong impact over stopping distance.

Linear Model

lm_speed <- lm(dist ~ speed, data = df)

lm_speed
## 
## Call:
## lm(formula = dist ~ speed, data = df)
## 
## Coefficients:
## (Intercept)        speed  
##     -17.579        3.932
summary(lm_speed)
## 
## Call:
## lm(formula = dist ~ speed, data = df)
## 
## 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
ggplot(data = df, aes(x = speed, y = dist)) + 
  geom_point(color='blue') +
  geom_smooth(method = "lm", se = FALSE)

Plot Residuals

It is necessary to plot residuals and see if there are any variances.

hist(lm_speed$residuals)

qqnorm(lm_speed$residuals)
qqline(lm_speed$residuals)  

plot(lm_speed$residuals ~ df$dist)

par(mfrow=c(2,2))
plot(lm_speed)

Conclusion

We can conclude by saying that the data satisfies the conditions of linear model. Below are explanations.

Linearity - The data is linear as the residuals are following theoretical line.

Homoscedasticity - From Scale location plot we can see that the line is horizontal and not angled. This explains the homoscedasticity is satisfied.

Independence - Speed and Stopping distance are both independent variables

Normality - By looking at the QQ Plot we can see that the data is distributed normally