Question

To study the relationship of advertising and capital investment to corporate profits, the following data, recorded in units of \(\$ 100,000\), were collected for 10 medium-sized firms in the same year. The variable \(y\) represents profit for the year, \(x\), represents capital investment, and \(x_{2}\) represents advertising expenditures. a. Using the model $$E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}$$ and an appropriate computer software package, find the least-squares prediction equation for these data. b. Use the overall \(F\) -test to determine whether the model contributes significant information for the prediction of \(y .\) Use \(\alpha=.01\) c. Does advertising expenditure \(x_{2}\) contribute significant information for the prediction of \(y\), given that \(x\), is already in the model? Use \(\alpha=.01\). d. Calculate the coefficient of determination, \(R^{2}\). What percentage of the overall variation is explained by the model?

Short answer

** Answer: We can determine if advertising expenditure contributes significant information for the prediction of y by comparing the p-value for the advertising expenditure coefficient (x2) to the chosen significance level of α = .01. If the p-value is less than α, it indicates that advertising expenditure contributes significant information to the prediction of y when capital investment is already in the model.

Step-by-step solution

Input the data into the software

Import or input the data (profits, capital investments, and advertising expenditures) of the 10 medium-sized firms into the chosen statistical software.

Run the multiple linear regression

Using the software's built-in function for linear regression, run the regression by fitting the given model $$E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}$$ to the data.

Find the prediction equation

After obtaining the coefficients \(\beta_{0}\), \(\beta_{1}\), and \(\beta_{2}\) from the regression, write down the obtained prediction equation. b. Perform the overall F-test:

Find the F-statistic and p-value

From the regression output, find the F-statistic and its corresponding p-value for the overall model.

Check the significance

Determine if the model provides significant information for predicting \(y\) by comparing the p-value to the chosen significance level of \(\alpha = .01\). If the p-value is less than \(\alpha\), the model contributes significant information for the prediction of \(y\). c. Test advertising expenditure contribution:

Find the t-statistic and p-value for \(x_{2}\)

From the regression output, find the t-statistic and its corresponding p-value for \(x_{2}\) (advertising expenditure) coefficient.

Check the significance

Determine if \(x_{2}\) contributes significant information for the prediction of \(y\) by comparing the p-value to the chosen significance level of \(\alpha = .01\). If the p-value is less than \(\alpha\), advertising expenditure contributes significant information to the prediction of \(y\) when capital investment is already in the model. d. Calculate the coefficient of determination:

Find \(R^{2}\)

From the regression output, locate the value for the coefficient of determination, \(R^{2}\).

Calculate the percentage of variation explained

Convert \(R^{2}\) to a percentage by multiplying it by 100. The resulting value represents the percentage of overall variation that is explained by the model.

Key concepts

Least-Squares Prediction Equation

One of the fundamental aspects of multiple linear regression is the least-squares prediction equation. This equation is created using a method that minimizes the sum of the squared differences (squared residuals) between observed values and the values predicted by the model.

For the exercise in question, the prediction equation takes the form of
\( E(y)=\beta_{0}+\beta_{1}x_{1}+\beta_{2}x_{2} \), where \( E(y) \) is the expected profit, \( x_{1} \) is the capital investment, and \( x_{2} \) is the advertising expenditure. The coefficients \( \beta_{0}, \beta_{1}, \) and \( \beta_{2} \) are estimated based on the given data by employing statistical software that processes the regression model. The least-squares criterion guides the software to find these optimal values.

The resulting equation can then be used to predict new or future values of profit, given specific input values for capital investment and advertising expenditure. Simply put, it's like crafting a recipe that best fits the taste based on the ingredients measured.

F-test for Model Significance

When dealing with multiple linear regression, it is crucial to determine if the model we have created is significant. Is it reliable for making predictions? This is where the F-test for model significance enters the picture.

Statistical software calculates the F-statistic, which allows us to test the null hypothesis that all regression coefficients are equal to zero versus the alternative hypothesis that at least one coefficient is different from zero. A low p-value related to the F-statistic, often lower than the significance level \( \alpha \), indicates that our model provides significant predictive power beyond what would be expected by chance alone.

In our exercise, the significance level chosen is \( \alpha=.01 \). If the p-value obtained from the regression output is less than 0.01, we reject the null hypothesis, concluding that our model significantly aids in predicting corporate profits.

Coefficient of Determination

Understanding the effectiveness of a regression model is paramount for interpreting its predictive value. The coefficient of determination, known as \( R^{2} \), is a statistic that indicates the proportion of variance in the dependent variable (profit, in our case) that is predictable from the independent variables (capital investment and advertising).

After running the multiple linear regression, the software provides us with an \( R^{2} \) value. This value, when converted to a percentage, indicates the amount of profit variation explained by the model. For instance, if our \( R^{2} \) value is 0.85, it means that 85% of the variability in profit can be explained by variations in capital investments and advertising expenditures. The closer this percentage is to 100%, the better the model is at predicting the dependent variable.

Statistical Hypothesis Testing

Statistical hypothesis testing plays a vital role in multiple linear regression, as it helps to determine the significance of individual predictors in the model. It is a structured process of making decisions about populations based on sample data.

For our specific exercise, two tests are used: the F-test (previously discussed) to judge the significance of the model, and the t-test to check the significance of individual predictors of the model. Using the t-test, we can evaluate advertising expenditures' (\( x_{2} \)) individual contribution to the model by comparing the computed p-value of the t-statistic against the significance level \( \alpha=.01 \).

If this p-value is less than the significance level, we may conclude that advertising expenditures significantly affect corporate profits when capital investment is accounted for. This step is integral in validating each predictor's value and ensuring that the model includes only meaningful variables.