A 9.6 -acre low-density residential site in Anchorage, Alaska, was monitored by the USGS during a study of the hydrology and water quality characteristics of the area (Brabets, 1987). Data for rainfall depth, runoff depth, and suspended solids, SS (in units of lb and lb/ac-in.) are given in the table. These data are for discrete rainfall events. Note that pounds per acre-inch has units of mass/ volume and is a concentration, whereas pounds of SS are a load (mass) and calculated by multiplying concentration x volume of runoff. (Appropriate conversion factors are included, of course.)
Rain (in.) | Runoff (in.) | SS (lb) | SS (lb/ac-in.) |
0.08 | 0.024 | 12.4 | 53.8 |
0.08 | 0.010 | 6.4 | 66.7 |
0.31 | 0.060 | 6.0 | 10.4 |
0.50 | 0.133 | 9.6 | 7.5 |
0.19 | 0.030 | 3.9 | 13.5 |
0.23 | 0.110 | 5.1 | 4.8 |
0.30 | 0.090 | 8.3 | 9.6 |
0.22 | 0.060 | 5.7 | 9.9 |
0.11 | 0.025 | 21.1 | 87.9 |
0.13 | 0.029 | 19.0 | 68.2 |
0.16 | 0.026 | 9.7 | 38.9 |
0.18 | 0.046 | 1 7.8 | 40.3 |
0.17 | 0.041 | 4.3 | 10.9 |
0.48 | 0.129 | 57.4 | 46.4 |
0.10 | 0.039 | 22.5 | 60.1 |
0.06 | 0.010 | 1.2 | 12.5 |
0.14 | 0.022 | 5.0 | 23.7 |
0.54 | 0.133 | 55.4 | 43.4 |
0.08 | 0.037 | 9.5 | 26.7 |
0.55 | 0.090 | 1 7.1 | 19.8 |
0.23 | 0.059 | 22.9 | 40.4 |
0.23 | 0.042 | 16.9 | 41.9 |
0.16 | 0.040 | 6.5 | 16.9 |
0.35 | 0.110 | 25.6 | 24.2 |
Check Sums: 5.58 | 1.395 | 369.3 | 778.4 |
(a) Demonstrate the units conversion computation for the first row in the table (i.e., that 53.8 lb/ac-in. with a runoff of 0.024 in. results in 12.4 lb of SS).
(b) Rainfall is to be considered as an explanatory variable (independent variable) for the prediction of runoff, SS (lb), and SS (lb/ac-in.). Perform the three indicated linear regressions. Test the significance of the regressions at the 95% level (alpha = 5%). Plot the data points for runoff vs. rainfall on one graph and for both SS values vs. rainfall on one or two other graphs. If the regression is significant, include the predicted straight line.
Although your software may test the significance automatically, list the “table” T-value that must be exceeded for the regression to be significant. Obtain this value from a statistics book.
(c) What other causative factors (that would vary with each storm) might be included in a multiple linear regression of runoff vs. rainfall (depths)?
Note: This problem illustrates an example of “spurious correlation” for the SS-vs.-rainfall data. Load is the product of a constant × runoff depth × concentration. Since runoff is correlated with rainfall, the dependent variable (load) “includes” rainfall as part of its value. Hence, load will always correlate better with rainfall than will concentration.
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