91Ó°ÊÓ

In the context of multiple regression analysis, regression coefficients represent the influence that each independent variable has on the dependent variable. For the given exercise, we are dealing with temperature (\( x_1 \)) and reaction time (\( x_2 \)) impacting the viscosity of a chemical product (\( y \)). The regression coefficients (\( \hat{\beta}_0 = 300.00 \), \( \hat{\beta}_1 = 0.85 \), \( \hat{\beta}_2 = 10.40 \)) indicate the estimated changes in viscosity with changes in temperature and time.
- \( \hat{\beta}_0 = 300.00 \) is the intercept, indicating the expected viscosity when both temperature and reaction time are zero. However, in practical terms, settings like zero temperature and zero time may be nonsensical, but this intercept helps improve the model's fit.
- \( \hat{\beta}_1 = 0.85 \) shows that with each 1-degree increase in temperature, viscosity is expected to increase by 0.85 units, assuming reaction time remains constant.
- \( \hat{\beta}_2 = 10.40 \) demonstrates the expected increase of 10.40 units in viscosity for every additional hour of reaction time, assuming temperature remains constant.
By plugging in values such as \( x_1 = 100 \) and \( x_2 = 2 \) into the regression equation, we estimate the mean viscosity, allowing predictions based on different experimental conditions.

The F-statistic is an integral part of regression analysis, as it measures the overall significance of the regression model. It evaluates whether there is a meaningful relationship between the dependent and independent variables. To arrive at the F-statistic, one can use the formula:
\[ F = \frac{(SS_T - SS_E)/p}{SS_E/(n-p-1)} \]
In this problem, given the sums of squares (\( SS_T = 1230.50 \), \( SS_E = 120.30 \)), number of predictors (\( p = 2 \)), and number of observations (\( n = 15 \)), we calculate:
- \( SS_R \), the regression sum of squares, is \( 1110.20 \).
- The F-statistic comes out to be 55.37, which strongly exceeds the critical value of 3.89 for \( \alpha = 0.05 \).
This indicates a significant relationship, meaning the combination of temperature and reaction time provides valuable predictive information for viscosity. Thus, the regression model as a whole is deemed to be statistically significant.

The coefficient of determination, denoted as \( R^2 \), is a statistical measure that provides insight into how well the independent variables explain the variability of the dependent variable. In this specific exercise, \( R^2 \) is calculated as:
\[ R^2 = \frac{SS_R}{SS_T} = \frac{1110.20}{1230.50} \approx 0.9021 \]
Here, approximately 90.21% of the variability in viscosity is explained by the model, a very high proportion which suggests a strong predictive power.
The closer \( R^2 \) is to 1, the better the data points fit the regression line. This high value indicates that temperature and reaction time substantially account for changes in viscosity, leaving very little unexplained variance. Such insight helps validate the model's robustness.

The error sum of squares (\( SS_E \)) is crucial in assessing model fit. It represents the unexplained variance after fitting the model. Lower \( SS_E \) implies a better fitting model. Initially, \( SS_E \) was 120.30 for two predictors.
Upon adding a third predictor (\( x_3 \), stirring rate), the new \( SS_E \) becomes 117.20. Despite this decrease, one must consider the \( MS_E \) or mean square error:
- With two predictors, \( MS_E = 10.025 \).
- With three, it increases slightly to \( 10.655 \).
The rise in \( MS_E \) indicates the new variable didn’t contribute meaningfully to reducing error, emphasizing that just because a variable appears in a model, it doesn't automatically ensure beneficial results. Sometimes, superfluous predictors can increase complexity without offering predictive improvement, a vital consideration during model refinement.