Statistics for Environmental Engineers

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The average of the 24 observations is y = 0.06673 mg/L. Table 25.2 also gives the averages for the duplicate measurements on each specimen, ybs, and the average of the eight measurements made on each batch of waste (yb).


Analysis of Variance Table for the Foundry Waste Example


Source of Variation


Sum of Squares


df


MS


F


Average


0.10693


1


Batches


0.00367


2


0.001835


367


Specimens


0.00624


9


0.000693


139


Tests


0.00006


12


0.000005


Total


0.11690


24


The sum of squares for each variance component is:


nb ns nt


Total SSt=    = 0.082 + 0.0842+•••+ 0.0442= 0.11690


b=1 s=1 t=1


Average SSave= nbnsn,y2 = 3 (4)(2)(0.06675)2= 0.106934


nb


Batch SSb = ^n^(yb — y)2


b=1


= (4)(2)[(0.0840 — 0.06675)2+ (0.0606 — 0.06675)2+ (0.0556 — 0.06675)2]


= 0.003671


nb ns


Specimen SSs = nt^^(ybs — Уь)2


b=1 s= 1


= 2[(0.083 — 0.0840)2+ (0.1085 — 0.0840)2+ ••• + (0.047 — 0.0556)2]


= 0.000624


nb ns nt


Test SSt = XXX(ybst — ybs)2


b=1 s=1 t=1


= (0.082 — 0.083)2+ (0.084 — 0.083)2+ •••+( 0.044 — 0.0470)2= 0.000057


Table 25.4 gives the full analysis of variance table, with sums of squares, degrees of freedom, and mean square values. The mean squares (the sums of squares divided by the respective degrees of freedom) are MSb= 0.001835, MSs= 0.000693, and MSt= 0.000005. The mean squares are used to estimate the variances. Table 25.3 shows that MStestimates a,2, MSsestimates nta2s + a2, and MSbestimates nsntal + n,a2s + a2. Using these relations with the computed mean square values gives the following estimates:

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