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## Checking Multiple Regression Assumptions

1. We will examine in detail the process of checking multiple regression assumptions next week. This week we’ll assume the assumptions hold for this data set
Identifying Confounding Variables
1. Open the Excel data set for Week 3. Use New Capital Expenses (X1) and Cost of Materials (X2) as the explanatory variables and End of Year Inventory as your response variable (Y)
2. Generate a simple regression equation relating New Capital Expenses to End of Year Inventory. Report the regression equation and the associated Excel regression table output.
3. Interpret the regression coefficient for New Capital Expenses including significance information. Use complete sentences.
4. Assess if the Cost of Materials variable is a potential confounder variable by first generating a multiple regression equation that includes both explanatory variables in the equation. Report the multiple regression equation
5. Examine the new regression slope coefficient for the New Capital Expense variable with the Cost of Materials variable now entered into the equation. Interpret this coefficient including significance information. Use complete sentences. Write up as appropriate (check the note pages for this week)
6. Calculate the percent of change observed in the simple regression slope coefficient to the multiple regression slope coefficient, for the New Capital Expense variable. Report the percentage and show your calculations. If the percentage is larger than 10% (in either direction), then report the Cost of Materials variable as confounding.

## Relative Importance of Explanatory Variables

1. Consider the multiple regression equation generated in #3 above. Create a table which lists just the t statistics for each explanatory variable. Ensure that your table is formatted to class expectations
2. Use the magnitude of the t statistic to determine which explanatory variable is the most significant. Report that variable

## Using Multiple Regression Model for Prediction

3. Consider the multiple regression equation generated in #3 above. Predict the value of the End of Year Inventory when the Cost of Materials is $725 and New Capital Expenses are$1369. Show your calculations.
 Simple linear regression model Regression Statistics Multiple R 0.821384888 R Square 0.674673134 Adjusted R Square 0.672245321 Standard Error 1586.927704 Observations 136 ANOVA   df SS MS F Significance F Regression 1 7E+08 7E+08 277.8934 1.77E-34 Residual 134 3.37E+08 2518340 Total 135 1.04E+09         Coefficients Standard Error t Stat P-value Lower 95.0% Upper 95.0%   Intercept 798.7778605 173.5187 4.603411 9.54E-06 455.5881 1141.967622   New Capital Expenses 2.293338698 0.137572 16.67014 1.77E-34 2.021246      2.56543152   The coefficient for the new capital expenses [2.2933] implies that the value for the end of year inventory is expected to increase by 2.2933 as a result of a unit increase in new capital expenses. The P-value for this coefficient (1.77E-34 is less than significance level (5%) and hence we conclude that the coefficient is statistically significant.  Multiple linear regression model. Regression Statistics Multiple R 0.82948959 R Square 0.68805297 Adjusted R Square 0.68336204 Standard Error 1559.78299 Observations 136 ANOVA   df SS MS F Significance F Regression 2 713708763.6 3.57E+08 146.6772 2.27183E-34 Residual 133 323578756.1 2432923 Total 135 1037287520         Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 751.017158 171.7189238 4.373526 2.45E-05 411.3637782 1090.670538 Cost of Materials 0.03286509 0.013760177 2.388421 0.018326 0.005647998 0.06008219 New Capital Expenses 1.85477276 0.22803722 8.13364 2.58E-13 1.403723969 2.305821546 The coefficient for the new capital expenses [1.8548] in the multiple linear model implies that the value for the end of year inventory is expected to increase by 1.8548 as a result of a unit increase in new capital expenses when Cost of materials is controlled or held constant. The P-value for this coefficient (2.58E-13) is less than significance level (5%) and hence we conclude that the coefficient is statistically significant in the model.  Percent of change observed $\%change= 100 - \frac{model2}{model1}*100$ $=100- \frac{1.8548}{2.2933}*100$=100-80.9 =19.1%Since the percentage is larger than 10%, then Cost of Materials is a confounding variable. The larger the t-value, the larger the differences between the groups under consideration. From the table above, New Capital Expenses has a larger t score and hence it is most significant.   Coefficients Standard Error t Stat P-value Intercept 751.017158 171.7189238 4.373526 2.45E-05 Cost of Materials 0.03286509 0.013760177 2.388421 0.018326 New Capital Expenses 1.85477276 0.22803722 8.13364 2.58E-13  Prediction End of year inventory =751.017158+0.03286509*Cost of Materials+1.854772768New Capital Expenses End of year inventory =751.017158+0.03286509*725+1.854772768*1369 End of year inventory=3314.028268