Question

The accompanying Minitab output results from fitting the model described in Exercise 14.14 to data. \(\begin{array}{lrrr}\text { Predictor } & \text { Coef } & \text { Stdev } & \text { t-ratio } \\ \text { Constant } & 86.85 & 85.39 & 1.02 \\ \text { X1 } & -0.12297 & 0.03276 & -3.75 \\ \text { X2 } & 5.090 & 1.969 & 2.58 \\\ \text { X3 } & -0.07092 & 0.01799 & -3.94 \\ \text { X4 } & 0.0015380 & 0.0005560 & 2.77 \\ S=4.784 & \text { R-sq }=90.8 \% & \text { R-sq(adj) }=89.4 \%\end{array}\) Analysis of Variance \(\begin{array}{lrrr} & \text { DF } & \text { SS } & \text { MS } \\ \text { Regression } & 4 & 5896.6 & 1474.2 \\ \text { Error } & 26 & 595.1 & 22.9 \\ \text { Total } & 30 & 6491.7 & \end{array}\) a. What is the estimated regression equation? b. Using a .01 significance level, perform the model utility test. c. Interpret the values of \(R^{2}\) and \(s_{e}\) given in the output.

Short answer

a. The estimated regression equation is \(Y = 86.85 -0.12297X1 + 5.09X2 - 0.07092X3 + 0.0015380X4\). b. For the model utility test, we use the F-statistic, which is computed as the ratio of the Mean Square Regression (MSR) and the Mean Square Error (MSE), if this value is greater than the critical F value at 0.01 level of significance, the model is useful. c. \(R^{2} = 90.8\%\) implies that our model explains 90.8% of the variability of the response data around its mean, and \(s_{e} = 4.784\) indicates the average distance that the observed values fall from the regression line.

Step-by-step solution

STEP 1: Formulate the Estimated Regression Equation

The coefficients given in the Minitab output are used in the regression equation. The estimated regression equation is \[Y = B_0 + B_1X_1 + B_2X_2 + B_3X_3 + B_4X_4 + e\] where \(Y\) is the dependent variable, \(B_0, B_1, B_2, B_3, B_4\) are the coefficients of the model, \(X_1, X_2, X_3, X_4\) are the independent variables, and \(e\) is the error. Substituting the given coefficients into the equation, we get: \[Y = 86.85 -0.12297X1 + 5.09X2 - 0.07092X3 + 0.0015380X4\]

STEP 2: Perform the Model Utility Test

To execute the model utility test, we use the F-statistic and compare it with the F-distribution. The F-statistic is derived from the Mean Square Regression (MSR) and the Mean Square Error (MSE), calculated as \[F = \frac{MSR}{MSE} = \frac{1474.2}{22.9}\] If this computed F value is greater than the critical F value at a 0.01 significance level, then the model is considered useful.

STEP 3: Interpret \(R^{2}\) and \(s_{e}\) Values

\(R^{2}\) or the Coefficient of Determination is a statistical measure that shows the proportion of the variance for a dependent variable that's explained by an independent variable. In this case, \(R^{2} = 90.8\%\), which implies that 90.8% of data fit the regression model. \(s_{e}\) or the Standard Error of the estimate measures the variations in the observations around the regression line. The given \(s_{e}\) is 4.784, which reveals the average distance that the observed values deviate from the regression line.

Key concepts

Understanding the Estimated Regression Equation

When we talk about the estimated regression equation, we're referring to a mathematical representation that shows the relationship between one dependent variable and one or several independent variables. The equation is typically presented as
\[Y = B_0 + B_1X_1 + B_2X_2 + B_3X_3 + B_4X_4 + e\]
In this construction,
\(Y\) represents the predicted value of the dependent variable,
\(B_0\) is the Y-intercept (the value of Y when all independent variables are zero),
\(B_1, B_2, B_3, B_4\) are the coefficients that measure how much the dependent variable changes as the independent variables change,
\(X_1, X_2, X_3, X_4\) are the independent variables, and
\(e\) represents the error term which accounts for variability in Y that cannot be explained by the model.
Being able to determine this equation from a given dataset, as with regression analysis, allows us to make predictions or understand the influence of certain factors on an outcome of interest.

Performing the Model Utility Test

The model utility test, commonly involving an F-test, is crucial to understanding whether the multiple regression model is statistically significant. In essence, it determines if the relationship that the model establishes between the dependent and independent variables actually exists in the population from which the sample is drawn.
We look at the Mean Square Regression (MSR) and the Mean Square Error (MSE) to calculate the F-statistic:
\[F = \frac{MSR}{MSE}\]
By comparing the computed F value with the critical F value from the F-distribution tables at a given significance level, we can judge the utility of the model. If the computed F is larger than the critical value, we have evidence to say that the model provides a better fit than a model without predictors, confirming the collective effect of independent variables on the dependent variable.

Deciphering the Coefficient of Determination

What Is \(R^{2}\) and Why Is It Important?

The Coefficient of Determination, denoted as \(R^{2}\), tells us about the goodness of fit of the model. It's a value between 0 and 1, where higher values indicate a better model fit. Specifically, it represents the proportion of the variance in the dependent variable that can be explained by the independent variables.
For instance, an \(R^{2}\) of 90.8% indicates that about 91% of the variation in the output can be explained by the input variables included in the model, which means the model performs quite well in explaining the changes in the dependent variable. This is a key metric for assessing how well the model captures the real data and helps us compare the performance of different models.

Analyzing the Standard Error of the Estimate

The Standard Error of the estimate, denoted as \(s_{e}\), serves as a measure of the accuracy of predictions made with a regression model. Specifically, it calculates the average distance that the observed values fall from the regression line. So, if the standard error of the estimate is low, that means the observations are clustered closely around the regression line, indicating better prediction accuracy.
The value of \(s_{e}\) provided in the output is 4.784. A smaller value of \(s_{e}\) would indicate a tighter cluster of points around the regression line, suggesting that the model has greater predictive accuracy. Conversely, a larger standard error would point to more dispersion and might suggest the need for a model that fits the data more closely or possibly having additional or different explanatory variables.