Page 145 - Applied Statistics with R

8.7. CONFIDENCE INTERVAL FOR MEAN RESPONSE 145

8.7 Confidence Interval for Mean Response

In addition to confidence intervals for and , there are two other common
1
0
interval estimates used with regression. The first is called a confidence inter-
val for the mean response. Often, we would like an interval estimate for the
mean, [ ∣ = ] for a particular value of .

In this situation we use ̂( ) as our estimate of [ ∣ = ]. We modify our
notation slightly to make it clear that the predicted value is a function of the
value.

̂
̂
̂ ( ) = +
0
1
Recall that,
E[ ∣ = ] = + .
0
1

Thus, ̂( ) is a good estimate since it is unbiased:

E[ ̂( )] = + .
0
1
We could then derive,
1 ( − ̄ ) 2
2
Var[ ̂( )] = ( + ) .

Like the other estimates we have seen, ̂( ) also follows a normal distribution.

̂
̂

Since and are linear combinations of normal random variables, ̂( ) is as
1
0
well.
1 ( − ̄ ) 2
2
̂ ( ) ∼ ( + , ( + ))
1
0

And lastly, since we need to estimate this variance, we arrive at the standard
error of our estimate,
1 ( − ̄ ) 2
SE[ ̂( )] = √ + .

We can then use this to find the confidence interval for the mean response,

1 ( − ̄ ) 2
̂ ( ) ± /2, −2 ⋅ √ +