Page 142 - Applied Statistics with R

142 CHAPTER 8. INFERENCE FOR SIMPLE LINEAR REGRESSION

8.6.2 Significance of Regression, t-Test

We pause to discuss the significance of regression test. First, note that
based on the above distributional results, we could test and against any
1
0
particular value, and perform both one and two-sided tests.
However, one very specific test,

∶ = 0 vs ∶ ≠ 0
1
0
1
1
is used most often. Let’s think about this test in terms of the simple linear
regression model,

= + + .
0

1
If we assume the null hypothesis is true, then = 0 and we have the model,
1
= + .
0

In this model, the response does not depend on the predictor. So then we could
think of this test in the following way,

• Under there is not a significant linear relationship between and .
0
• Under there is a significant linear relationship between and .
1
For the cars example,

• Under there is not a significant linear relationship between speed and
0
stopping distance.
• Under there is a significant linear relationship between speed and
1
stopping distance.

Again, that test is seen in the output from summary(),

p-value = 1.4898365 × 10 −12 .

With this extremely low p-value, we would reject the null hypothesis at any rea-
sonable level, say for example = 0.01. So we say there is a significant linear
relationship between speed and stopping distance. Notice that we emphasize
linear.