102 CHAPTER 7 Correlation, Regression, and Logistic Regression(^) Yˆˆ=+a bX or a Y bX= −ˆˆ.
To illustrate solving a simple linear regression problem, we use
weight and height data obtained for 10 individuals. The data and the
calculations are illustrated in Table 7.2.
We fi rst calculate b and a.
(^) ∑(XXYY−−=ˆˆ)( )^201 and
(^) ∑()XX−=ˆ^2 9108 90..
(^) So b==201 9108 90 0 0221/. ..
Then (^) aYbX=− = −ˆˆ 63 (.0 0221 154 10 59 59). =..
Now let us see how we would get confi dence intervals for new
values of Y when X = x. First let us defi ne a few terms.
- Sum of squares error: SSE = Σ ( Y i − Y^ )^2
2. Standard error of estimate: Syx. =−[/()]SSE n 2
Table 7.2
Calculations for Inference about the Predicted Y - Value and the
Slope of the Regression Line
Subject
IDX = Weight
(lbs)X − X
^
( X − X^
)^2 Y^ = Height
(in)Predicted
height Y pY − Y
^
( Y − Y^
)^201 148 − 6.1 37.21 64 62.87 1.13 1.29
02 172 17.9 320.41 63 63.40 − 0.40 0.16
03 203 48.9 2391.21 67 64.08 2.92 8.52
04 109 − 45.1 2034.01 60 62.00 − 2.00 4.01
05 110 − 44.1 1944.81 63 62.02 0.97 0.95
06 134 − 20.1 404.01 62 62.56 − 0.56 0.31
07 195 40.9 1672.81 59 63.90 − 4.90 24.05
08 147 − 7.1 50.41 62 62.84 − 0.84 0.71
09 153 − 1.1 1.21 66 62.98 3.02 0.15
10 170 15.9 252.81 64 63.35 0.65 0.42
Total 1541 9108.9 49.56