Page 138 - Applied Statistics with R

138 CHAPTER 8. INFERENCE FOR SIMPLE LINEAR REGRESSION

and

̂
̂
̂
± /2, −2 ⋅ SE[ ] ± /2, −2 ⋅
1
1
1
√
where /2, −2 is the critical value such that ( −2 > /2, −2 ) = /2.
8.5 Hypothesis Tests

“We may speak of this hypothesis as the ‘null hypothesis’, and it
should be noted that the null hypothesis is never proved or estab-
lished, but is possibly disproved, in the course of experimentation.”
— Ronald Aylmer Fisher

Recall that a test statistic (TS) for testing means often take the form:

EST − HYP
TS =
SE
where EST is an estimate for the parameter of interest, HYP is a hypothesized
value of the parameter, and SE is the standard error of the estimate.
So, to test

∶ = 00 vs ∶ ≠ 00
0
0
0
1
we use the test statistic
̂
̂
− 00 − 00
0
0
= =
̂
SE[ ] √ +
1
2
̄
0

which, under the null hypothesis, follows a distribution with − 2 degrees of
freedom. We use 00 to denote the hypothesized value of .
0
Similarly, to test
∶ = 10 vs ∶ ≠ 10
0
1
1
1
we use the test statistic
̂
̂
− −
= 1 10 = 1 10
̂
SE[ ] /√

1
which again, under the null hypothesis, follows a distribution with −2 degrees
of freedom. We now use 10 to denote the hypothesized value of .
1