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102 CHAPTER 7. SIMPLE LINEAR REGRESSION
7.2.2 Residuals
If we think of our model as “Response = Prediction + Error,” we can then write
it as
= ̂ + .
We then define a residual to be the observed value minus the predicted value.
= − ̂
Let’s calculate the residual for the prediction we made for a car traveling 8 miles
per hour. First, we need to obtain the observed value of for this value.
which(cars$speed == 8)
## [1] 5
cars[5, ]
## speed dist
## 5 8 16
cars[which(cars$speed == 8), ]
## speed dist
## 5 8 16
We can then calculate the residual.
= 16 − 13.88 = 2.12
16 - (beta_0_hat + beta_1_hat * 8)
## [1] 2.119825
The positive residual value indicates that the observed stopping distance is
actually 2.12 feet more than what was predicted.
