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Water treatment

H 0 : H a : Treatment

H 0 : H a : Treatment levels working in parallel in the MSF pilot system do not present statistically significant differences in the mean removal efficiencies. This means that τ 1 =τ 2 =τ 3 , in the case of filtration rates in DyGF stage, and τ 1 =τ 2 =τ 3 =τ 4 =τ 5 , in the case of the CGF lines or the SSF units working in series with the CGF lines. τi ≠ τk, for any i ≠ k The process to accept or reject the null hypothesis (H 0 ), with an established level of significance, involves the F-Test, after the British statistician, Sir Ronald A. Fisher, who developed the method called analysis of variance (ANOVA) on which the F-test rests (Fisher, 1947, quoted by Mesa, 1999; Rowntree, 1981). ANOVA allows knowing if the observed values divided in three or more groups might all belong to the same population, regardless of group, or whether the observations in at least one of the groups seem to come from a different population. The answer is obtained by comparing the variability of the values within groups with the variability of the values between groups. If the null hypothesis is false (and there is a real difference in population means), the between groups estimate of variance should be larger than the within groups estimate. If the variance ratio or F-ratio – dividing the between groups estimate by the within group estimate – is greater than 1 it is necessary to check if the difference is large enough to be confident that it is attributable not simply to random, sampling variation. In other words, it is necessary to check if the difference between the groups is a reliable one that could be repeated in similar tests. To do this the calculated F-ratio is checked against the F-distribution, which is a family of curves varying according to the size and the number of groups being compared. In general, the smaller or the fewer the groups from the experimental work, the bigger must the calculated F-ratio be in order to obtain significance (Rowntree, 1981). To apply ANOVA technique with the statistical model represented by equation 3.1 the total variance in the experiment is distributed among the variance due to treatment levels (between), blocks (between), and experimental error (within). This is represented by equation 3.2. Table 3.5 is prepared with the results obtained after applying this equation. Annex 4 includes an example of F-test application. Sum of squares (SS) total = SS Treatment + SS Blocks + SS Error b 2 2 2 ∑ ∑( yij − y) = b∑( yi − y) + t∑ ( y j − y) + ∑ ∑ j= 1 t t i= 1 i= 1 j= 1 j= 1 in which y = b ∑ j= 1 t ∑ i= 1 yij ; tb yi = b ∑ j= 1 b y b ij ; yj = t ∑ i= 1 t y ij b t i= 1 ( yij − yi − y j + y) 2 (3.2) The F-ratio calculated with the experimental data is identified as F c in table 3.5. H 0 is rejected when F c > F (α, t-1, (t-1)(b-1)) , where F (α, t-1, (t-1)(b-1)) is the F distribution with t-1 degrees of freedom for the numerator, (t-1)(b-1) degrees of freedom for the denominator and α is the significance level of the test. 87

Table 3.5 Analysis of Variance (ANOVA) to the randomised block design Source of Variation Degrees of freedom Sum of Squares (SS) Mean Square (MS) F c (1) Treatment (between) t-1 SSTreat. SSTreat/(t-1) = MSTreat MSTreat/MSE Blocks (between) b-1 SSBlocks SSBlocks/(b-1) = MSBlocks MSBlocks/MSE Error (within) (t-1)(b-1) SSE SSE/((t-1)(b-1)) = MSE Total tb-1 SST If H 0 is rejected other techniques can be used to compare sample means of all possible pairs of treatment levels included in each main treatment stage of the MSF pilot system. Tukey test will be used if all treatment levels have the same number of observed values and Bonferroni methodology if the number of observed values are different (Mendenhall, 1997). Tukey test is employed to make all possible comparisons of means based on the Minimum Significant Difference (MSD) calculated as follows (Reyes, 1980) MSD q , t, n1) = ( α ∗ ( MSE b ) In which q (α, t, n1) is a critical value of Studentized Range; α is the significance level; t is the number of means being compared; n1 = (t – 1)(b – 1), the degrees of freedom of MSE (Mean Square of error); and b is the data number for each treatment level. The difference between each pair of means (D) is calculated as D = ⎺Y i – ⎺Y k for i ≠ k. The decision rules are as follows: If D > MSD, the two means are statistically different (⎺Y i ≠⎺Y k ), otherwise they are not statistically different (⎺Y i = ⎺Y k ). Bonferroni test is based on calculating a B ij value as follows (Mendenhall, 1997) (3.3) B ( MSE) ∗ ( 1 1 ) i k ij = ( α1 / 2) ∗ + t b b (3.4) In which, t (α1/2) is a critical value of t –Student distribution; α 1 = α/(p(p-1)/2) being α the significance level; p is the number of means to be compared; MSE mean square of within; b i and b k the number of data used to calculate the means i and k respectively. The difference between each pair of means (D) is calculated as D = ⎺Y i – ⎺Y k for i ≠ k. The decision rules are as follows: If D > B ij , the two means are statistically different (⎺Y i ≠ ⎺Y k ), otherwise they are not statistically different (⎺Y i = ⎺Y k ). Statistical analyses were accomplished using the commercial software SPSS (Statistical Package for the Social Sciences) version. 8.0 for Windows 95 or 98. 88

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