91Ó°ÊÓ

A multiple regression model was used to relate \(y=\) viscosity of a chemical product to \(x_{1}=\) temperature and \(x_{2}=\) reaction time. The data set consisted of \(n=15\) observations. (a) The estimated regression coefficients were \(\hat{\beta}_{0}=300.00\), \(\hat{\beta}_{1}=0.85,\) and \(\hat{\beta}_{2}=10.40 .\) Calculate an estimate of mean viscosity when \(x_{1}=100^{\circ} \mathrm{F}\) and \(x_{2}=2\) hours. (b) The sums of squares were \(S S_{T}=1230.50\) and \(S S_{E}=\) \(120.30 .\) Test for significance of regression using \(\alpha=\) 0.05. What conclusion can you draw? (c) What proportion of total variability in viscosity is accounted for by the variables in this model? (d) Suppose that another regressor, \(x_{3}=\) stirring rate, is added to the model. The new value of the error sum of squares is \(S S_{E}=117.20 .\) Has adding the new variable resulted in a smaller value of \(M S_{E}\) ? Discuss the significance of this result. (e) Calculate an \(F\) -statistic to assess the contribution of \(x_{3}\) to the model. Using \(\alpha=0.05,\) what conclusions do you reach?

Step by step solution

01

Estimate Mean Viscosity

The multiple regression equation is given by \( \hat{y} = \hat{\beta}_{0} + \hat{\beta}_{1}x_1 + \hat{\beta}_{2}x_2 \). Substituting the provided estimates: \( \hat{y} = 300.00 + 0.85 \times 100 + 10.40 \times 2 \). Calculate: \( \hat{y} = 300.00 + 85 + 20.8 = 405.8 \). The estimated mean viscosity is 405.8.

02

Test for Significance of Regression

Calculate the regression sum of squares \( SS_R = SS_T - SS_E = 1230.50 - 120.30 = 1110.20 \). The degrees of freedom for regression \( df_R = 2 \) (since there are two predictors) and for error \( df_E = n - 3 = 12 \). The mean square for error is \( MS_E = \frac{SS_E}{df_E} = \frac{120.30}{12} = 10.025 \). The \( F \)-statistic is \( F = \frac{SS_R/df_R}{MS_E} = \frac{1110.20/2}{10.025} \approx 55.44 \). Compare this with the critical value from an \( F(2,12) \) table. If \( F_{calculated} > F_{critical} \), reject the null hypothesis, indicating significance.

03

Proportion of Variability Accounted For

The coefficient of determination \( R^2 \) is given by \( R^2 = \frac{SS_R}{SS_T} = \frac{1110.20}{1230.50} \approx 0.9023 \). This means 90.23% of the total variability in viscosity is explained by the model.

04

Effect of Adding Stirring Rate on MSE

Calculate the new mean square error \( MS_E' = \frac{SS_{E,new}}{n-k-1} = \frac{117.20}{15-4} = 11.72 \). Compare \( MS_E' \approx 11.72 \) with the previous \( MS_E = 10.025 \). Since \( MS_E' > MS_E \), adding the new variable did not improve the model's precision significantly.

05

Evaluate F-statistic for Stirring Rate

For testing the contribution of \( x_3 \), compute \( F = \frac{(SS_E - SS_{E,new})/(1)}{MS_E} = \frac{(120.30 - 117.20)/1}{10.025} = \frac{3.10}{10.025} \approx 0.309 \). With \( F_{calculated} \approx 0.309 \) and the critical value from \( F(1,11) \) at \( \alpha = 0.05 \), the contribution of \( x_3 \) is not significant, as \( F_{calculated} < F_{critical} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Regression Coefficients

Regression coefficients are crucial in multiple regression analysis. They represent the relationship between each independent variable and the dependent variable, while keeping all other variables constant. In our example, we have three coefficients: the intercept \( \hat{\beta}_{0} = 300.00 \), temperature coefficient \( \hat{\beta}_{1} = 0.85 \), and reaction time coefficient \( \hat{\beta}_{2} = 10.40 \).

The intercept \( \hat{\beta}_{0} \) is the expected mean value of the dependent variable (viscosity) when all predictors (temperature and reaction time) are zero. The coefficient \( \hat{\beta}_{1} \) implies that for every one-unit increase in temperature, the viscosity increases by 0.85 units, assuming reaction time remains unchanged. Similarly, \( \hat{\beta}_{2} \) suggests that viscosity increases by 10.40 units for each additional hour of reaction time, assuming temperature stays constant.

• Intercept: Baseline level of the dependent variable.
• Slope coefficients: Change in the dependent variable per unit change in the independent variable.

Understanding these coefficients allows us to predict the viscosity by substituting the values into the regression equation, enhancing the reliability of the model.

Significance of Regression

The significance of regression in multiple regression analysis checks whether the predictors provide meaningful information about the response variable. This is often tested using an F-statistic. Here, we calculated an F-statistic of approximately 55.44.

The F-statistic is computed by dividing the regression mean square by the error mean square. In our case, we used the regression sum of squares \( SS_R = 1110.20 \) and compared it against the error sum of squares \( SS_E = 120.30 \) to find a significant F-statistic value.

• An F-statistic larger than the critical value indicates significant regression.
• A significant regression suggests the model explains a significant portion of the variability in the dependent variable.

Hence, if our calculated F-statistic is greater than the critical value from the F-distribution table at a specific \( \alpha \) level, we conclude that the regression is significant, rejecting the null hypothesis that our predictors are not useful in explaining the response variable.

Coefficient of Determination

Also known as \( R^2 \), the coefficient of determination measures the proportion of total variability in the dependent variable that the model can explain. In our example, we calculated \( R^2 \approx 0.9023 \).

This means 90.23% of the variability in viscosity can be explained by temperature and reaction time, indicating a strong model fit. The closer \( R^2 \) is to 1, the better the model explains the variability. A low \( R^2 \) indicates that the model does not explain a substantial portion of the variability in the response variable.

• \( R^2 = \frac{SS_R}{SS_T} \), where \( SS_T \) is the total sum of squares.
• High \( R^2 \) values suggest a good model fit.

Understanding \( R^2 \) is vital for model evaluation, allowing us to assess how well our proposed model works in explaining the data's variation.

Sum of Squares in Regression

Sum of Squares is a key concept in regression analysis, measuring variation in the data. In our exercise, we dealt with two main types: total sum of squares \( SS_T = 1230.50 \) and error sum of squares \( SS_E = 120.30 \).

• Total Sum of Squares (\( SS_T \)): Represents the total variation in the response variable.
• Regression Sum of Squares (\( SS_R \)): Measures how much of the total variability is explained by the model. Calculated as \( SS_R = SS_T - SS_E \).
• Error Sum of Squares (\( SS_E \)): Reflects the unexplained variation by the model.

Adding a new predictor like the stirring rate affects \( SS_E \). If including it increases \( SS_E \), as in our case, where new \( SS_E \) is 117.20, it suggests limited benefit from the new predictor, as it doesn’t significantly enhance the model's accuracy. Always examine changes in sum of squares when assessing the value added by new predictors.