3.12. A simulation model for peak water flow from watersheds was tested by compa

Question

3.12. A simulation model for peak water flow from watersheds was tested by comparing measured peak flow (cfs) from 10 storms with predictions of peak flow obtained from the simulation model. Qo and Qp are the observed and predicted peak lows, respectively. Four independent variables were recorded: X1 = area of watershed (ma), = average slope of watershed (in percent), X,-surface absorbency index (0 = complete absorbency, 100 X4-peak intensity of rainfall (in/hr) computed on half-hour time intervals

Explanation / Answer

Solution:

we will use minitab software to do this problem

1) in all case we assume significance level = 0.95

Regression Analysis: y versus x1, x2, x3, x4

The regression equation is
y = 0.087 - 0.0354 x1 + 0.00473 x2 - 0.00191 x3 - 0.120 x4

Predictor Coef SE Coef T P
Constant 0.0873 0.3119 0.28 0.791
x1 -0.035384 0.009336 -3.79 0.013
x2 0.004726 0.004765 0.99 0.367
x3 -0.001913 0.004189 -0.46 0.667
x4 -0.12008 0.02381 -5.04 0.004

S = 0.0411912 R-Sq = 92.9% R-Sq(adj) = 87.2%

Analysis of Variance

Source DF SS MS F P
Regression 4 0.110577 0.027644 16.29 0.005
Residual Error 5 0.008484 0.001697
Total 9 0.119060

here from the above ANOVA table's p-value = 0.005<0.05 that implies the regression is worth while of y on x1 x2 x3 x4.

Also we can comment that the significance of covariates using the p-values corresponding to the co variates, we can see that only x4 and x1 has p-value <0.05.so except x4 and x1 all covariate is insignificant.

b) Ignoring the intercept

Call:
lm(formula = y ~ 0 + x1 + x2 + x3 + x4)

Coefficients:
Estimate Std. Error t value Pr(>|t|)
x1 -0.0330400 0.0037987 -8.698 0.000128 ***
x2 0.0057518 0.0028025 2.052 0.085956 .
x3 -0.0007525 0.0005484 -1.372 0.219124
x4 -0.1215292 0.0213807 -5.684 0.001279 **

here also we can see from p-value x1 and x4 <0.05 so they are significant.

c)   Estimate Std. Error t value Pr(>|t|)
x1 -0.033182 0.003892 -8.525 2.76e-05 ***
x4 -0.124274 0.012236 -10.156 7.56e-06 ***

the estimate gets change as the estimation space previous is with 4 variable now it is 2 variable so that this change has occure

d) in compare to (a) and (b) in all cases the SE has higher in (a) than (b). also that is conclusion for (c) and (d), is for X1 the SE decreses in part c than b and for X4 it is increses in part b than c

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