Geoscience Reference
In-Depth Information
8. As a check, add up the original normal equations and substitute the solutions for b 1 , b 2 , and
b 3 .
13,724.3216 b 1 + 6,617.1787 b 2 + 7,854.6 073 b 3 = 12,388.9287
12,388.92869 ≈ 12,388.9287
Given the values of b 1 , b 2 , and b 3 we can now compute
aY bX
=− −
bX
bX
= −
11 7320
.
11
22
33
Thus, after rounding of the coefficients, the regression equation is
ˆ
Y
=−
11 732
.
+
0 539
.
X
+
0 474
.
X
+
0 236
.
X
1
2
3
It should be noted that in solving the normal equations more digits have been carried than would be
justified by the rules for number of significant digits. Unless this is done, the rounding errors may
make it difficult to check the computations.
7.17.2.1 Tests of Significance
Tests of significance refer to the methods of inference used to support or reject claims based on sam-
ple data. To test the significance of the fitted regression, the outline for the analysis of variance is
Source of Variation
Degrees of Freedom
Reduction due to regression on X 1 , X 2 , and X 3
3
Residuals
24
Total
27
The degrees of freedom for the total are equal to the number of observations minus 1. The total
sum of squares is
2
Total
SS
=
y
=
5974.7143
The degrees of freedom for the reduction are equal to the number of independent variables fitted, in
this case 3. The reduction sum of squares for any least squares regression is
Reduction
SS =
(Estimatedcoefficients) (righ
ttsideoftheir normalequations)
In this example there are three coefficients estimated by the normal equations, and so
= (
) +
(
) + (
)
∑∑∑
Reduction SS
bx yb
xy
b
x y
1
1
2
2
3
3
3
df
=
((.
0 53926
)(
6428 7858
.
)
+
(.
0 47368
)(
2632 2143
.
)
+ ((.
0 23598
)(
3327 9286
.
)
=
5498 9389
.
The residual df and sum of squares are obtained by subtraction. Thus the analysis becomes
Source of Variation
Degrees of Freedom
Sums of Squares
Mean Squares
Reduction due to X 1 , X 2 , and X 3
3
5498.9389
1832.9796
Residuals
24
475.7754
19.8240
Total
27
5974.7143
 
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