Page 165 - Applied Statistics with R

9.2. SAMPLING DISTRIBUTION 165

9.2.2 Confidence Intervals

̂
Since is our estimate for and we have

̂
E[ ] =

as well as the standard error,

̂
SE[ ] = √

̂
and the sampling distribution of is Normal, then we can easily construct

̂
confidence intervals for each of the .

̂
± /2, − ⋅ √

We can find these in R using the same method as before. Now there will simply
be additional rows for the additional .

confint(mpg_model, level = 0.99)

## 0.5 % 99.5 %
## (Intercept) -25.052563681 -4.222720208
## wt -0.007191036 -0.006078716
## year 0.632680051 0.890123859

9.2.3 Confidence Intervals for Mean Response

As we saw in SLR, we can create confidence intervals for the mean response,
that is, an interval estimate for E[ ∣ = ]. In SLR, the mean of was only
dependent on a single value . Now, in multiple regression, E[ ∣ = ] is
dependent on the value of each of the predictors, so we define the vector to
0
be,

1
⎡ ⎤
⎢ 01 ⎥
= ⎢ 02 ⎥ .
0
⎢ ⋮ ⎥
⎣ 0( −1)⎦

Then our estimate of E[ ∣ = ] for a set of values is given by
0
0