Applied Statistics Using SPSS, STATISTICA, MATLAB and R

7.1 Simple Linear Regression 283
This sampling distribution allows us to compute confidence intervals for the
predicted values. Figure 7.4 shows with dotted lines the 95% confidence interval
for the cork-stopper Example 7.1. Notice how the confidence interval widens as we
move away from ( x , y)
.
Example 7.6
Q: The observed value of ART for PRT = 1612 is 882. Determine the 95%
confidence interval of the predicted ART value using the ART(PRT) linear
regression model derived in Example 7.1.
A: Using the MSE and sPRT values as described in Example 7.2, and taking into
account that PRT = 710.4, we compute:
2
xk − = (1612–710.4) 2 = 812882.6; ∑ xi −
( x)
⎛
2 ⎞
ˆ ⎜ 1 ( xk
− x)
s[
Y = +
⎟
k ] MSE
= 73.94.
⎜
2 ⎟
⎝
n ∑ ( xi
− x)
⎠
( x
2
)
= 19439351;
Since t148,0.975 = 1.976 we obtain yˆ k ∈ [882 – 17, 882 + 17] with 95%
confidence level. This corresponds to the 95% confidence interval depicted in
Figure 7.4.
7.1.3.4 Prediction of New Observations
Imagine that we want to predict a new observation, that is an observation for new
predictor values independent of the original n cases. The new observation on y is
viewed as the result of a new trial. To stress this point we call it:
Y .
k(
new)
If the regression parameters were perfectly known, one would easily find the
confidence interval for the prediction of a new value. Since the parameters are
usually unknown, we have to take into account two sources of variation:
– The location of E[ Y k(
new)
] , i.e., where one would locate, on average, the
new observation. This was discussed in the previous section.
– The distribution of Y k(
new)
, i.e., how to assess the expected deviation of the
new observation from its average value. For the normal regression model,
the variance of the prediction error for the new prediction can be obtained
as follows, assuming that the new observation is independent of the original
n cases:
V
ˆ 2
= V[ Y −Y
] = σ + V[ Yˆ
] .
pred k(
new)
k
k