# Turbid water is muddy or cloudy water. Sunlight is necessary for most life forms; thus turbid water

Turbid water is muddy or cloudy water. Sunlight is necessary for most life forms; thus turbid water is considered a threat to wetland ecosystems. Passive filtration systems are commonly used to reduce turbidity in wetlands. Suspended solids are measured in mg/l. Is there a relation between input and output turbidity for a passive filtration system and, if so, is it statistically significant? At a wetlands environment in Illinois, the inlet and outlet turbidity of a passive filtration system have been measured. A random sample of measurements are shown below. (Reference: EPA Wetland Case Studies.)
Reading 1 2 3 4 5 6 7 8 9 10 11 12
Inlet (mg/l) 59.1 25.7 70.5 71.0 37.6 43.5 13.1 24.2 16.7 49.1 67.6 31.7
Outlet (mg/l) 18.2 14.3 15.3 17.5 13.1 8.0 4.1 4.4 4.3 5.8 16.3 7.1
Use a 1% level of significance to test the claim that there is a monotone relationship (either way) between the ranks of the inlet readings and outlet readings.
(a) Rank-order the inlet readings using 1 as the largest data value. Also rank-order the outlet readings using 1 as the largest data value. Then construct a table of ranks to be used for a Spearman rank correlation test.
Rank x Oulet
Rank y d = x - y d2
1
2
3
4
5
6
7
8
9
10
11
12Σd2 =
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Kristina Petty
${r}_{s}=1-\frac{6\sum {d}^{2}}{n\left({n}^{2}-1}$
Spearman's correlation coefficient is given by the formula given above.
Here n= 12
Reading 1 2 3 4 5 6 7 8 9 10 11 12
Inlet (mg/l) 59.1 25.7 70.5 71 37.6 43.5 13.1 24.2 16.7 49.1 67.6 31.7
Outlet (mg/l) 18.2 14.3 15.3 17.5 13.1 8 4.1 4.4 4.3 5.8 16.3 7.1
Inlet Rank X 9 4 11 12 6 7 1 3 2 8 10 5
Outlet Rank Y 12 8 9 11 7 6 1 3 2 4 10 5
d= x-y -3 -4 3 1 -1 1 0 0 0 4 0 0
d^2 9 16 9 1 1 1 0 0 0 16 0 0
( Summation d^2) = (9+16+9+1+1+1+16)
= 53