LINEAR REGRESSION

2. Consider the following data for a dependent variable \(y\) and two independent variables, \(x_{1}\) and \(x_{2}\)

1. Develop an estimated regression equation relating \(y\) to \(x_{1}\). Estimate \(y\) if \(x_{1}=45\)
2. Develop an estimated regression equation relating \(y\) to \(x_{2}\). Estimate \(y\) if \(x_{2}=15\).
3. Develop an estimated regression equation relating \(y\) to \(x_{1}\) and \(x_{2}\). Estimate \(y\) if \({{x}_{1}}=4\) and \(x_{2}=15\).

5. The owner of Showtime Movie Theaters, Inc., would like to estimate weekly gross revenue as a function of advertising expenditures. Historical data for a sample of eight weeks is

1. Develop an estimated regression equation with the amount of television advertisement as: the independent variable.
2. Develop an estimated regression equation with both television advertising and newspaper advertising as the independent variables.
3. Is the estimated regression equation coefficient for television advertising expender? the same in part (a) and in part (b)? Interpret the coefficient in each case.
4. What is the estimate of the weekly gross revenue for a week when $\$ 3500$ is spent on television advertising and $\$ 1800$ is spent on newspaper advertising?

12. In exercise 2,10 observations were provided for a dependent variable \(y\) and two independent variables \(x_{1}\) and \(x_{2}\); for these data SST \(=15,182.9\), and \(\mathrm{SSR}=14,052.2\).

1. Compute \(R^{2}\).
2. Compute \(R_{\mathrm{a}}^{2}\).
3. Does the estimated regression equation explain a large amount of the variability in the data? Explain.

15. In exercise 5, the owner of Showtime Movie Theaters, Inc., used multiple regression analysis to predict gross revenue \((y)\) as a function of television advertising \(\left(x_{1}\right)\) and newspaper \(x\) advertising \(\left(x_{2}\right)\). The estimated regression equation was

\(\hat{y}=83.2+2.29 x_{1}+1.30 x_{2}\)

The computer solution provided SST \(=25.5\) and \(\mathrm{SSR}=23.435\).

1. Compute and interpret \(R^{2}\) and \(R_{\mathrm{a}}^{2}\).
2. When television advertising was the only independent variable, \(R^{2}=.653\) and \(R_{a}^{2}\) = .595. Do you prefer the multiple regression results? Explain.

23. Refer to exercise 5 .

1. Use \(\alpha=.01\) to test the hypotheses
   \(\begin{aligned}
   &H_{0}: \beta_{1}=\beta_{2}=0 \\
   &H_{\mathrm{a}}: \beta_{1} \text { and/or } \beta_{2} \text { is not equal to zero }
   \end{aligned}\)
   for the model \(y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\epsilon\), where
   \(\begin{aligned}
   &x_{1}=\text { television advertising }(\\) 1000 \mathrm{~s}) \\
   &x_{2}=\text { newspaper advertising }(\$ 1000 \mathrm{~s})
   \end{aligned}$
2. Use \(\alpha=.05\) to test the significance of \(\beta_{1}\). Should \(x_{1}\) be dropped from the model
3. Use \(\alpha=.05\) to test the significance of \(\beta_{2}\). Should \(x_{2}\) be dropped from the model

34. Management proposed the following regression model to predict sales at a fast-food outlet.

\(y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3}+\epsilon\) where

\(\begin{aligned}

x_{1} &=\text { number of competitors within one mile } \\

x_{2} &=\text { population within one mile }(1000 \mathrm{~s}) \\

x_{3} &=\left\{\begin{array}{l}

1 \text { if drive-up window present } \\

0 \text { otherwise }

\end{array}\right.\\

y &=\text { sales }(\\) 1000 \mathrm{~s})

\end{aligned}$

The following estimated regression equation was developed after 20 outlets were surveyed.

\(\hat{y}=10.1-4.2 x_{1}+6.8 x_{2}+15.3 x_{3}\)

1. What is the expected amount of sales attributable to the drive-up window?
2. Predict sales for a store with two competitors, a population of 8000 within one mile, and no drive-up window.
3. Predict sales for a store with one competitor, a population of 3000 within one mile, and a drive-up window.

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Solution: The downloadable solution consists of 12 pages, 759 words and 6 charts.
Deliverable: Word Document