CHAPTER 13 Simple Linear Regression

548 CHAPTER THIRTEEN Simple Linear Regression
PREDICTION INTERVAL FOR AN INDIVIDUAL RESPONSE, Y
Yˆ
± t S 1 + h
i n−2
YX i
(13.21)
where h i
, Yˆ
i , S YX
, n, and X i
are defined as in Equation (13.20) on page 546 and Y X = Xi
is a
future value of Y when X = X i
.
To construct a 95% prediction interval of the annual sales for an individual store that contains
4,000 square feet (X = 4), you first compute Yˆ
i . Using the prediction line:
= 7.
6439 (millions of dollars)
Also, given the following:
X = 2. 9214 S = 0.
9664
From Table E.3, t 12
= 2.1788. Thus,
where
Yˆ
− t S 1+ h ≤ Y ≤ Yˆ
+ t S 1+
h
i n− 2 YX i X= Xi
i n−2
YX i
Yˆ = 0. 9645 + 1.
6699X
i
= 0. 9645 + 1. 6699( 4)
n
∑
YX
SSX = ( X − X ) = 37.
9236
i=
1
i
Yˆ
± t S + h
i n−2 YX 1 i
2
i
h
i
1
= +
n
( X − X)
n
∑
i=
1
i
( X − X)
i
2
2
so that
ˆ 1 ( Xi
− X)
Yi ± tn−2SYX
1 + +
n SSX
1 ( 4 − 2. 9214)
= 7. 6439 ± ( 2. 1788)( 0. 9664)
1 + +
14 37.
9236
= 7. 6439 ± 2.
2104
2
2
so
5.4335 ≤ Y X=4
≤ 9.8543
Therefore, with 95% confidence, you predict that the annual sales for an individual store with
4,000 square feet is between $5,433,500 and $9,854,300.