A major marketing company is interested in evaluating the return to television advertising on sales. For that they want to use the following simple linear regression model:

where qi is measured in millions of dollars, and Ai is a binary variable equals to one if firm iis advertising. The sample size is 1000.

Use the regression results to answer the following questions:

1. If the sample standard deviation of sales (qi) is 1.5, what is the \(R^{2}\) of that regression?Show your calculations. Interpret your result in terms of the goodness-of-fit of themodel.

2. Using those estimates, what is the conditional expectation of sales if firms areadvertising versus not advertising?

3. Consider the following two hypothesis:

Can you reject the null hypothesis using a 5% signicance level? Show your calculations.

4. Assume now that we can measure total advertising expenditures by ai (in thousandsof dollars). Our marketing executive decides to estimate the following linear regressionmodel by OLS:

If a firm considers increasing its advertising expenditures from 0.1 to 0.25 thousandsof dollars, what is the expected change in sales?

5.Compute a 95% confidence interval for the predicted increase sales that you foundin the previous question. Show your calculations.