Question

This exercise requires the use of a computer package. The accompanying data resulted from a study of the relationship between \(y=\) brightness of finished paper and the independent variables \(x_{1}=\) hydrogen peroxide (\% by weight), \(x_{2}=\) sodium hydroxide (\% by weight), \(x_{3}=\) silicate \((\%\) by weight \()\), and \(x_{4}=\) process temperature ("Advantages of CE-HDP Bleaching for High Brightness Kraft Pulp Production," TAPPI [1964]: 107A-173A). $$ \begin{array}{ccccc} x_{1} & x_{2} & x_{3} & x_{4} & y \\ \hline .2 & .2 & 1.5 & 145 & 83.9 \\ .4 & .2 & 1.5 & 145 & 84.9 \\ .2 & .4 & 1.5 & 145 & 83.4 \\ .4 & .4 & 1.5 & 145 & 84.2 \\ .2 & .2 & 3.5 & 145 & 83.8 \\ .4 & .2 & 3.5 & 145 & 84.7 \\ .2 & .4 & 3.5 & 145 & 84.0 \\ .4 & .4 & 3.5 & 145 & 84.8 \\ .2 & .2 & 1.5 & 175 & 84.5 \\ .4 & .2 & 1.5 & 175 & 86.0 \\ .2 & .4 & 1.5 & 175 & 82.6 \\ .4 & .4 & 1.5 & 175 & 85.1 \\ .2 & .2 & 3.5 & 175 & 84.5 \\ .4 & .2 & 3.5 & 175 & 86.0 \\ .2 & .4 & 3.5 & 175 & 84.0 \\ .4 & .4 & 3.5 & 175 & 85.4 \\ .1 & .3 & 2.5 & 160 & 82.9 \\ .5 & .3 & 2.5 & 160 & 85.5\\\ .3 & .1 & 2.5 & 160 & 85.2 \\ .3 & .5 & 2.5 & 160 & 84.5 \\ .3 & .3 & 0.5 & 160 & 84.7 \\ .3 & .3 & 4.5 & 160 & 85.0 \\ .3 & .3 & 2.5 & 130 & 84.9 \\ .3 & .3 & 2.5 & 190 & 84.0 \\ .3 & .3 & 2.5 & 160 & 84.5 \\ .3 & .3 & 2.5 & 160 & 84.7 \\ .3 & .3 & 2.5 & 160 & 84.6 \\ .3 & .3 & 2.5 & 160 & 84.9 \\ .3 & .3 & 2.5 & 160 & 84.9 \\ .3 & .3 & 2.5 & 160 & 84.5 \\ .3 & .3 & 2.5 & 160 & 84.6 \end{array} $$ a. Find the estimated regression equation for the model that includes all independent variables, all quadratic terms, and all interaction terms. b. Using a \(.05\) significance level, perform the model utility test. c. Interpret the values of the following quantities: SSResid, \(R^{2}, s_{e}\)

Short answer

An answer cannot be provided without running the statistical analysis using software like R, SPSS, or Minitab. However, after running the regressions as described in the step by step solution, one can obtain the regression equation, F-statistic for model utility, and values of SSResid, \(R^{2}\), and \(s_{e}\).

Step-by-step solution

Constructing the Regression Model

Import the given data into a statistical software package and use its multiple regression function to construct a model that includes all independent variables, their squared terms, and all their interactions.

Perform the Model Utility Test

Perform the model utility test with a significance level of 0.05 using an F-test. The Null Hypothesis will assume that all regression coefficients are zero. If the p-value obtained from the F-statistic is less than the significance level, the null hypothesis is rejected and the alternative hypothesis is accepted.

Interpret the Values

SSResid is the sum of squares of residuals. It measures the variation in the data not explained by the model. \(R^{2}\) (R-Sq) is the coefficient of determination. It measures the proportion of variation in the dependent variable that can be predicted from the independent variables. \(s_{e}\) is the standard error of the residuals, a measure of the difference between the observed and predicted values of the dependent variable.

Key concepts

Statistical Software Package

Statistical software packages are crucial tools for researchers and analysts who deal with complex data sets and require advanced analytical tools. When working with multiple regression analysis, these software packages offer automated calculations that save time and reduce the likelihood of human error in statistical computation.

For example, in the given exercise, data from a study that examines the relationship between the brightness of paper and various independent variables is imported into such a software. The software efficiently handles multiple variables, polynomial terms, and interactions, which can be quite challenging to manage without computational aid.

Statistical packages typically come with user-friendly interfaces and provide outputs such as regression equations, significance tests, and diagnostic measures. These outputs enable users to interpret their model's predictive power and the statistical significance of their findings efficiently.

Model Utility Test

The model utility test is an essential step in regression analysis to assess the overall significance of the regression model. This test examines whether the independent variables, as a group, are significantly related to the dependent variable.

The F-test is commonly employed for this purpose. In a model utility test, the null hypothesis assumes that all regression coefficients are zero—indicating no linear relationship between the predictors and the outcome. On the other hand, the alternative hypothesis suggests that at least one coefficient is not zero. If the computed p-value from the F-statistic is less than the chosen significance level, typically 0.05, we reject the null hypothesis, providing evidence that our model has utility in explaining the dependent variable.

Sum of Squares of Residuals

The sum of squares of residuals (SSResid) is a measure that reflects the variation in the observed data that the regression model does not explain. It is calculated by summing the squares of the differences between the observed and the predicted values of the dependent variable.

In the process of model fitting, the SSResid provides a numerical value that helps to gauge how well the model fits the data. A smaller SSResid suggests that the model's predictions closely match the actual data, whereas a larger SSResid indicates discrepancies between observed and predicted values. Minimizing SSResid is one of the objectives in selecting the most appropriate regression model.

Coefficient of Determination

The coefficient of determination, denoted as \(R^{2}\), is a key metric in regression analysis that estimates the proportion of variance in the dependent variable that can be explained by the independent variables in the model.

Values of \(R^{2}\) range from 0 to 1. A value of 0 indicates that the model explains none of the variability of the response data around its mean, while a value of 1 indicates that the model explains all the variability. In practice, a higher \(R^{2}\) value means a better fit of the model to the data. However, it is important not to rely solely on \(R^{2}\) to judge a model's quality since it can be influenced by the number of predictors in the model and does not account for the data's underlying structure.