PROBABILITY
In many polynomial regression problems, rather than fitting a “centered” regression function using $x^{\prime}=x-\bar{x}$, computational accuracy can be improved by using a function of the standardized independent variable $x^{\prime}=(x-\bar{x}) / s_{x}$, where $s_{x}$ is the standard deviation of the $x_{i}$s. Consider fitting the cubic regression function $y=\beta_{0}^{*}+\beta^{*} x^{\prime}+\beta_{2}^{*}\left(x^{\prime}\right)^{2}+\beta_{3}^{*}\left(x^{\prime}\right)^{3}$ to the following data resulting from a study of the relation between thrust efficiency y of supersonic propelling rockets and the half-divergence angle x of the rocket nozzle (“More on Correlating Data,” CHEMTECH, 1976: 266–270): $$ \begin{array}{c|ccccccc} x & 5 & 10 & 15 & 20 & 25 & 30 & 35 \\ \hline y & .985 & .996 & .988 & .962 & .940 & .915 & .878 \end{array} $$ $$ \begin{array}{crr} \text { Parameter } & \text { Estimate } & \text { Estimated SD } \\ \hline \beta_{0}^{*} & .9671 & .0026 \\ \beta_{1}^{*} & -.0502 & .0051 \\ \beta_{2}^{*} & -.0176 & .0023 \\ \beta_{3}^{*} & .0062 & .0031 \end{array} $$ a. What value of y would you predict when the half-divergence angle is 20? When x=25? b. What is the estimated regression function $\hat{\beta}_{0}+\hat{\beta}_{1} x+\hat{\beta}_{2} x^{2}+\hat{\beta}_{3} x^{3}$ for the “unstandardized” model? c. Use a level .05 test to decide whether the cubic term should be deleted from the model. d. What can you say about the relationship between SSE’s and $R^{2}$s for the standardized and unstandardized models? Explain. e. SSE for the cubic model is .00006300, whereas for a quadratic model SSE is .00014367. Compute $R^{2}$ for each model. Does the difference between the two suggest that the cubic term can be deleted?