to produce platykurtic or flattened distribution curves.
5. Analysis of variances
The analyses have been carried out on the average
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j-
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particle size parameters (namely the mean, the standard deviation, the
skewness and the kurtosis) in relation to distance and depth. The
experimental design shown in tables 83 ~ 91 is a two-way classification,
analysis of variance experiment. It has been designed to compare two
procedures of sampling pediment sediments. The samples have been collected
so that they will represent a number of sites on a pediment traverse at
fixed distance and depth intervals. Thus for the experimental layout there
are two factors A and B, where A has r levels A^, ..., A^ and B has
p levels Bj,, • ••, B • To study the effect of distance (factor A) and
depth (factor B) on the grain size parameters (variable X) a sample of
n
« rp values x^ (i = 1, ...r, K = 1,
p) have been collected and
analyzed. The results of the size analysis are those random variables
(which are obtained as the moment statistics for the frequency distribution
of/
'•
(1-118)
of particle size) on which the effect of factors A and B will be
performed. In this tvzo factorial design the value X^, is an observation on X when level A. of the factor A and level. B. of the factor B are
i k
present. Thus X^ would be regarded as an observed value of a random
variable X..with the following hypotheses; .
Hop: states that all those mean values are equal, that is to say, those random variables X^^ have exactly the same normal distribution. This would mean that factors A and B have no effect on X (distance and depth have no effect on the particle size parameters), and the sample values differ only because of random variation and experimental errors.
HogS states that the mean values are not equal, that is to say, that those random variables X., have different distribution and that factor A
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has effect on X (meaning that the distance has effects on the grain size distribution) and the sample values differ only by factor A while factor B has no effect.
Ho^: states that the mean values are not equal because of factor B, that is to say, that the values of the variables X^ are different because the depth factor has an effect on the grain size distribution and that distance has no effect on the variables.
Ho,: states that the mean values are not equal because of both Factor A and B, that is to say, that the value of the variables X., are different
Xiv
because both, the depth and distance, factors have effect on the grain size
distribution. .
To test these hypotheses the computation for the two-way analysis of variance/
“ : ’ " (1-119)
variance without replication was carried out according- to the standard
formulas (see Griffiths 1967 and Kreyszig 197O)«. The results for the analysis of variance are given in tables 83 - 91* The computation
includes the row means, the column means, the grand mean, the correction factor for the sums of squares, the column sum of squares, the error sum of squares, the row mean square, the column mean square and the error mean square. The P (mean square) ratios are then given. If the P ratios are significant, the standard errors of the difference between the row and the column means are given.
Prom the results in tables 83 *■* 86 we have four depth variables (B^) for pediment A and five depth variables for pediment B and five distance localities (A^) for pediment A and six distance localities for pediment B. The degrees of freedom for rows (B factor) is n ~ 1 ~ 4 ~ 1 ~ 3 and columns
(A factor) isn~l=5~l~4 with (n - l) (m ~ l) - 3 x 4 « 12 degrees of freedom. The P ratio (Pp) for the phi means of samples from pediment (A) is « Pp = 3*9452, and for the standard deviations of samples from the same pediment the P ratio (Pr) is - Pr - 3*1462, the skewnesses P ratio is equal Pr « 1.3225, and kurtosises P ratio is Pp ~ I.7848. Prom the
critical values of the P distribution tables we have the value of Cq at a -
0.10 with (3,12) degrees of freedom Cq = 2.6055? and the value of C2 at a ~
0.10 with (4,12) degrees of freedom C2 ~ 2.48OI. The P ratio for the phi means of - samples from pediment A is ~ 3*9452 that means that in comparing Pr value with either Cq or C2 we have a significant variation due to factors A and B at a » 0.10 level. Considering C-j and C9 values at another level a/
—X-.3 ?-ZV4'Y'
■ ■. (1-120)
a - 0.05 we have 0^ 3*49035 and C? = 3*2592* Again we have here a significant variation due to the combined factors A and B, The F ratio of the standard deviations of samples from pediment A F?., 3*1462 has a significant variation at a = 0,10 level only due to factors A and B. But neither skewness nor kurtosis has any significant variation due to either factors. Therefore, for the phi means and the standard deviations of
. samples from pediment A (Khashm Qraydan) we reject the hypotheses Hoj, Ho?, Ho^ and say that all those mean values are not equal due to effects of
factors A (distance) and B (depth). Thus we accept the hypothesis (H04) which states that the difference is due to the fact that the size
distribution has been changed significantly by the depth and distance
factors. (For summary of the results and test of significance see table 9l) Unfortunately there is no way at present to isolate either factor to see which factor is more significant than the other because there is no replication for either level (a^) and/or (B, ). This would be carried out in a future work when one c.an add a third factor to this (estimate of the operators errors). At the moment we can only say that the distance and depth in pediment A have significant effects on the grain size parameters
(means and standard deviations). But we accept the first hypothesis for the skewness and kurtosis. That is to say that these random variables have exactly the same normal distribution which means that distance a,nd depth do not effect their behaviour. The results of the analysis of variance for pediment A samples agree with the t-test discussed earlier.
For pediment B (Wadi A1 Quway’iyah) the results of the analysis of variance/