You will need a computer system, such as MINITAB, SAS/STAT,...

You will need a computer system, such as MINITAB, SAS/STAT, or EXPLORE to do this exercise. Sam Spade of Hilbert Hackers has discovered the following table that relates $x$ to $y$. He wants to fit a cubic to the data and is concerned about the numerical stability of the calculations. Help Sam by doing the following: (a) Fit the regression function $y=\beta_0+\beta_1 x+\beta_2 x^2+\beta_3 x^3$ to the data. Record the values of $\hat{\beta}_0, \hat{\beta}_1, \hat{\beta}_2, \hat{\beta}_3$, the estimated standard deviation of $\hat{\beta}_3, R^2$, and $s_e$. Is $\beta_3$ significant at the five percent level? (b) Standardize $x$ by the formula $x^{\prime}=(x-\bar{x}) / s_x$, and fit the regression function $y=\beta_0^*+\beta_1^* x^{\prime}+\beta_2^*\left(x^{\prime}\right)^2+\beta_3^*\left(x^{\prime}\right)^3$ to the data. Then record the same values that you recorded for (a) and make the same test. $$ \begin{array}{crcccr} {\text { Data }} \\ \hline x & y & x & y & x & y \\ 10 & 7 & 10 & 8 & 10 & 6 \\ 15 & 12 & 15 & 15 & 15 & 13 \\ 20 & 10 & 20 & 11 & 20 & 7 \\ 25 & 14 & 25 & 16 & 25 & 17 \\ \hline \end{array} $$

You will need a computer system, such as MINITAB, SAS/STAT, or EXPLORE to do this exercise. Sam Spade of Hilbert Hackers has discovered the following table that relates $x$ to $y$. He wants to fit a cubic to the data and is concerned about the numerical stability of the calculations. Help Sam by doing the following:
(a) Fit the regression function $y=\beta_0+\beta_1 x+\beta_2 x^2+\beta_3 x^3$ to the data. Record the values of $\hat{\beta}_0, \hat{\beta}_1, \hat{\beta}_2, \hat{\beta}_3$, the estimated standard deviation of $\hat{\beta}_3, R^2$, and $s_e$. Is $\beta_3$ significant at the five percent level?
(b) Standardize $x$ by the formula $x^{\prime}=(x-\bar{x}) / s_x$, and fit the regression function $y=\beta_0^*+\beta_1^* x^{\prime}+\beta_2^*\left(x^{\prime}\right)^2+\beta_3^*\left(x^{\prime}\right)^3$ to the data. Then record the same values that you recorded for (a) and make the same test.
$$
\begin{array}{crcccr}
{\text { Data }} \\
\hline x & y & x & y & x & y \\
10 & 7 & 10 & 8 & 10 & 6 \\
15 & 12 & 15 & 15 & 15 & 13 \\
20 & 10 & 20 & 11 & 20 & 7 \\
25 & 14 & 25 & 16 & 25 & 17 \\
\hline
\end{array}
$$

Key Concepts

Model Fit Metrics – R-squared and Standard Error

R-squared and the standard error of the regression are key metrics used to assess the fit of a regression model. R-squared measures the proportion of variance in the dependent variable that is explained by the independent variables in the model, indicating the strength of the relationship between the model and the data. The standard error provides an estimate of the average distance that the observed values fall from the regression line, giving insight into the model's predictive accuracy and reliability.

Hypothesis Testing of Coefficients

Hypothesis testing in the context of regression involves examining whether the regression coefficients differ significantly from zero. This is typically accomplished by calculating standard errors for the estimates and using t-tests to assess their significance at a given level, such as the five percent significance level. This process helps determine if individual predictors contribute meaningfully to the predictive power of the model.

Polynomial Regression

Polynomial regression is a form of regression analysis in which the relationship between the independent variable and the dependent variable is modeled as an nth degree polynomial. In the context of a cubic regression, the model includes terms up to x³, allowing for the capture of curvature in the data. This technique is particularly useful when the relationship between variables is non-linear, offering a flexible approach to curve fitting while still using linear least squares estimation methods.

Numerical Stability in Regression

Numerical stability refers to the sensitivity of the computed solution of a regression analysis to changes in the input data, particularly when using high-degree polynomial terms. As the degree of the polynomial increases, the matrix computations for estimating coefficients can become ill-conditioned, leading to large variances in the parameter estimates. This instability can result in imprecise estimates, and hence, standardization or other scaling techniques are often employed to improve the reliability of the estimates.

Standardization of Variables

Standardization involves transforming a variable by subtracting its mean and dividing by its standard deviation. This process results in a new variable with a mean of zero and a standard deviation of one. Standardizing predictors is especially important in polynomial regression, as it mitigates issues related to numerical instability and multicollinearity, allowing for more robust and interpretable coefficient estimates during the regression analysis.

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