A Study Reports the Accompanying Data on Discharge Amount $q , \text { in } m ^ { 3 } / \mathrm { sec }$

question 12

Essay

A study reports the accompanying data on discharge amount ( $q , \text { in } m ^ { 3 } / \mathrm { sec }$ ), flow area ( $a , \text { in } m ^ { 2 }$ ), and slope of the water surface (b, in m/m) obtained at a number of floodplain stations. The study proposed a multiplicative power model $Q = \alpha a ^ { \beta } b ^ { y} \in$ . $$\begin{array} { c c c c c c } \hline q & 17.6 & 23.8 & 5.7 & 3.0 & 7.5 \\\hline a & 8.4 & 31.6 & 5.7 & 1.0 & 3.3 \\\hline b & .0048 & .0073 & .0037 & .0412 & .0413 \\\hline & & & & & \\\hline q & 89.2 & 60.9 & 27.5 & 13.2 & 12.2 \\\hline a & 41.1 & 26.2 & 16.4 & 6.7 & 9.7 \\\hline b & .0063 & .0061 & .0036 & .0039 & .0025 \\\hline\end{array}$$
a. Use an appropriate transformation to make the model linear and then estimate the regression parameters for the transformed model. Finally, estimate $\alpha , \beta \text {, and } \gamma$
(the parameters of the original model). What would be your prediction of discharge amount when flow area is 10 and slope is .01?
b. Without actually doing any analysis, how would you fit a multiplicative exponential model
?
c. After the transformation to linearity in part (a), a 95% CI for the value of the transformed regression function when a = 3.3 and b = .0046 was obtained from computer output as (.217, 1.755). Obtain a 95% CI for $\alpha a ^ { \mathcal { \beta } } b ^ { y }$
when a = 3.3 and b = .0046.