Page 251 - Statistics for Environmental Engineers

L1592_frame_C28.fm Page 254 Tuesday, December 18, 2001 2:48 PM

For this experiment, n = 8 gives:

1
s y = ----------- –( ( 2.6) + 0.3 + … + 1.1 ) = 1.332
2
2
2
2
48()
and
s y = 1.15
The main and interaction effects are estimated using the model matrix given in Table 28.4. The average is:
8
y = 1 ∑ y i
---
8
i=1
and the estimate of each effect is:
8
Effect j() = 1 --- ∑ X ij y i
4
i=1
where X ij is the ith element of the vector in column j.
y
The variance of the average is:
2
2
Var y() = ( 1/8) 8σ y = σ y
2
------
8
and the variance of the main and interaction effects is:
2
(
Var Effect) = ( 1/4) 8σ y = σ y
2
2
------
2
2 2
s y
Substituting for σ y gives the standard errors of the average and the estimated effects:
2
------------- =
SE y() = ---- = 1.332 0.41
s y
8 8
and
2
------------- =
SE Effect j ) = ---- = 1.332 0.82
(
s y
2 2
The estimated effects and their standard errors are given in Table 28.5.
The 95% confidence interval for the true value of the effects is bounded by:
Effect j ± t ν=8,α /2 =0.025 SE (Effect j )
Effect j ± 2.306(0.82) = Effect j ± 1.9
from which we can state, with 95% confidence, that effects larger than 1.9, or smaller than −1.9, represent
real effects. All four main effects and the two-factor interaction SL + DF are significant.
Alternately, the estimated effects can be viewed in relation to their relevant reference distribution
shown in Figure 28.2. This distribution was constructed by scaling a t distribution with ν = 8 degrees
of freedom according to the estimated standard error of the effects, which means, in this case, using a
scaling factor of 0.83. The calculations are shown in Table 28.6.
The main effects of all four variables are far out on the tails of the reference distribution, indicating
that they are statistically significant. The bounds of the confidence interval (±1.9) could be plotted on
this reference distribution, but this is not necessary because the results are clear. Stirring (S), on average,
© 2002 By CRC Press LLC