Ch 12.6 - 102E

Chapter 12, Problem 102E

(choose chapter or problem)

When fitting polynomial regression models, we often subtract \(\bar{x}\) from each \(\bar{x}\) value to produce a "centered" regressor \(x^{\prime}=x-\bar{x}\). This reduces the effects of dependencies among the model terms and often leads to more accurate estimates of the regression coefficients. Using the data from Exercise 12-84, fit the model \(Y=\beta_{0}^{*}+\beta_{1}^{*} x^{\prime}+\beta_{11}^{*}\left(x^{\prime}\right)^{2}+\epsilon\).

(a) Use the results to estimate the coefficients in the uncentered model \(Y=\beta_{0}+\beta_{1} x+\beta_{11} x^{2}+\epsilon\). Predict \(y\) when \(x=285^{\circ} \mathrm{F}\). Suppose that you use a standardized variable \(x^{\prime}=(x-\bar{x}) / s_{x}\) where \(S_{x}\) is the standard deviation of \(x\) in constructing a polynomial regression model. Fit the model \(Y=\beta_{0}^{*}+\beta_{1}^{*} x^{\prime}+\beta_{11}^{*}\left(x^{\prime}\right)^{2}+\epsilon\).

(b) What value of \(y\) do you predict when \(x=285^{\circ} \mathrm{F}\)?

(c) Estimate the regression coefficients in the unstandardized model \(Y=\beta_{0}+\beta_{1} x+\beta_{11} x^{2}+\epsilon\).

(d) What can you say about the relationship between \(S S_{E}\) and \(R^{2}\) for the standardized and unstandardized models?

(e) Suppose that \(y^{\prime}=(y-\bar{y}) / s y\) is used in the model along with \(x^{\prime}\). Fit the model and comment on the relationship between \(S S_{E}\) and \(R^{2}\) in the standardized model and the unstandardized model.

Equation Transcription:

Text Transcription:

bar x

barx

x^prime=x-bar x

Y=beta_0^*+beta_1^*x^prime+beta_11^*x^prime ^2+epsilon

Y=beta_0+beta_1 x+beta_11 x^2+epsilon

y

x=285^circ mathrm F

x^prime=(x-barx) / s_x

S_x

X

Y=beta_0^*+beta_1^* x^prime+beta_11^*tx^prime ^2+epsilon

y

x=285^circ mathrm F

Y=beta_0+beta_1 x+beta_11 x^2+epsilon

S S_E

R^2

y^prime=(y-bary)s y

x^prime

S S_E

R^2