91Ó°ÊÓ

(Calculus needed.) Derive the weighted least squares normal equations For letting a simple linear regression function when \(\sigma_{i}^{2}=k X_{i},\) where \(k\) is a proportionality constant.

Crop yield. An agronomist studied the effects of moisture \(\left(X_{1}, \text { in inches }\right)\) and temperature \(\left(X_{2}, \text { in } C\right)\) on the yield of a new hybrid tomato \((Y)\). The experimental data follow. The agronomist expects that second-order polynomial regression model (8.7) with independent normal error terms is appropriate here. a Fit a second-order polynomial regression model omitting the interaction term and the quadratic effect term for temperature. b. Construct a contour plot of the fitted surface obtained in pert ( \(a\) ). c. Use the lowest method to obtain a non parametric estimate of the yield response surface as a function of moisture and temperature. Employ weight function \((11.53), q=9 / 25\) and a Euclidean distance measure with unsealed variables. Obtain fitted values \(\hat{Y}_{h}\) for the \(9 \times 9\) rectangular grid of \(\left(X_{11}, X_{n 2}\right)\) values where \(X_{n 1}=6.7 \ldots \ldots 13.14\) and \(X_{n 2}=\) \(20,20.5, \ldots .23 .5 .24 .\) using a local first-order model. d. Construct a contour plot of the resulting lowest surface. Are the lowest contours convenient with the contours in part (b) for the polynomial model? Discuss.