# Regression Equation and Interpretation

A CEO of a large pharmaceutical company would like to determine if he should be placing more money allotted in the budget next year for television advertising of a new drug marketed for controlling asthma. He wonders whether there is a strong relationship between the amount of money spent on television advertising for this new drug called XBC and the number of orders received. The manufacturing process of this drug is very difficult and requires stability so the CEO would prefer to generate a stable number of orders. The cost of advertising is always an important consideration in the phase I roll-out of a new drug. Data that have been collected over the past 20 months indicate the amount of money spent of television advertising and the number of orders received.

The use of linear regression is a critical tool for a manager's decision-making ability. Please carefully read the example below and try to answer the questions in terms of the problem context. The results are as follows:

Month Advertising Cost (thousands of dollars) Number of Orders

1 $55.93 4,102,000

2 70.62 3,893,000

3 79.58 5,299,000

4 58.67 4,130,000

5 69.18 4,367,000

6 70.14 4,111,000

7 73.37 3,923,000

8 68.88 4,935,000

9 80.99 5,276,000

10 75.23 4,654,000

11 71.38 4,398,000

12 52.90 2,967,000

13 61.27 3,999,000

14 79.19 4,345,000

15 60.03 3,834,000

16 78.21 4,653,000

17 93.77 5,625,000

18 62.53 3,978,000

19 78.76 4,999,000

20 92.64 5,834,000

a. Set up a scatter diagram and calculate the associated correlation coefficient. Discuss how strong you think the relationship is between the amount of money spent on television advertising and the number of orders received. Please use the Correlation procedures within Excel under Tools > Data Analysis. The Scatterplot can more easily be generated using the Chart procedure.

b. Assuming there is a statistically significant relationship, use the least squares method to find the regression equation to predict the advertising costs based on the number of orders received. Please use the regression procedure within Excel under Tools > Data Analysis to construct this equation.

c. Interpret the meaning of the slope, b1, in the regression equation.

d. Predict the monthly advertising cost when the number of orders is 4,999,000. (Hint: Be very careful with assigning the dependent variable for this problem)

e. Compute the coefficient of determination, r2, and interpret its meaning.

f. Compute the standard error of estimate, and interpret its meaning.

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#### Solution Preview

Please see the attached files.

a. Set up a scatter diagram and calculate the associated correlation coefficient. Discuss how strong you think the relationship is between the amount of money spent on television advertising and the number of orders received. Please use the Correlation procedures within Excel under Tools > Data Analysis. The Scatterplot can more easily be generated using the Chart procedure.

After copying the date, paste them into the Excel file, which I have created. You then select the whole area. Go to the Chart Wizard icon, select Scatter diagram, and the end result is presented in the following diagram.

Then use the formula CORREL in Excel to calculate the correlation coefficient. I have done so at the bottom of the column. The correlation coefficient found is 0.86.

In statistics, if the correlation coefficient is higher than 0.8 than there is a relatively strong dependence of the two variables. So in this case the dependence of order on advertising spending is relatively significant. (Note if the coefficient is larger than 0.9 than the relationship is very strong)

b. Assuming there is a statistically significant relationship, use the least squares method to find the regression equation to predict the advertising costs based on the number of orders received. Please use the regression procedure within Excel under Tools > Data Analysis to construct this equation.

First select the chart, then click on Chart menu, then Add trendline then select Linear, then go to the Options tab, then select Display equation on chart and Display R2 value. ...

#### Solution Summary

The solution provides detailed calculations and explanations for the problem. It includes a 5-page Word file and an Excel document for calculations and plots.

Statistics Problems - Regression Analysis, Autocorrelation, Multicollinearity

1. Suppose an appliance manufacturer is doing a regression analysis, using quarterly time-series data, of the factors affecting its sales of appliances. A regression equation was estimated between appliance sales (in dollars) as the dependent variable and disposable personal income and new housing starts as the independent variables. The statistical tests of the model showed large t-values for both independent variables, along with a high r2 value. However, analysis of the residuals indicated that substantial autocorrelation was present.

a. What are some of the possible causes of this autocorrelation?

b. How does this autocorrelation affect the conclusions concerning the significance of the individual explanatory variables and the overall explanatory power of the regression model?

c. Given that a person uses the model for forecasting future appliance sales, how does this autocorrelation affect the accuracy of these forecasts?

d. What techniques might be used to remove this autocorrelation from the model?

2. Suppose the appliance manufacturer discussed in Exercise 1 also developed another model, again using time-series data, where appliance sales was the dependent variable and disposable personal income and retail sales of durable goods were the independent variables. Although the r2 statistic is high, the manufacturer also suspects that serious multicollinearity exists between the two independent variables.

a. In what ways does the presence of this multicollinearity affect the results of the regression analysis?

b. Under what conditions might the presence of multicollinearity cause problems in the use of this regression equation in designing a marketing plan for appliance sales?

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