Chapter 14, Problem 1SR

There are many restaurants in northeastern South Carolina. They serve beach vacationers in the summer, golfers in the fall and spring, and snowbirds in the winter. Bill and Joyce Tune all manage several restaurants in the North Jersey area and are considering moving to Myrtle Beach, SC, to open a new restaurant. Before making a final decision, they wish to investigate existing restaurants and what variables seem to be related to profitability. They gather sample information where profit (reported in $000) is the dependent variable and the independent variables are:

x₁ the number of parking spaces near the restaurant.

x₂ the number of hours the restaurant is open per week.

x₃ the distance from the SkyWheel, a landmark in Myrtle Beach.

x₄ the number of servers employed.

x₅ the number of years the current owner operated the restaurant.

The following is part of the output obtained using statistical software.

1. (a) What is the amount of profit for a restaurant with 40 parking spaces that is open 72 hours per week, is 10 miles from the SkyWheel, has 20 servers, and has been operated by the current owner for 5 years?
2. (b) Interpret the values of b₂ and b₃ in the multiple regression equation.

To determine

Find the amount of profit for a restaurant with the given conditions.

Answer to Problem 1SR

The amount of profit for a restaurant with the given conditions is $389,500.

Explanation of Solution

Calculation:

Multiple linear regression model:

A multiple linear regression model is given as $\widehat{y}=a+b_1{x}_1+b_2{x}_2+b_3{x}_3+\mathrm{...}+b_k{x}_k$ where y is the response or dependent variable, and $x_1,x_2,\mathrm{...},x_k$ are the k quantitative independent variables where k is a positive integer.

Here, a is the intercept term of the regression model, that is, the value of predicted value of y when X’s are 0 and $b_i$’s are the slopes, that is, the amount of change of the predicted value of y for one unit increase in $x_i$ when all other independent variables are constant.

In the given problem the predicted dependent variable y is the amount of profit in $1,000. The number of parking space near the restaurant, the number of hours the restaurant is opened per week, the distance from the landmark in Beach M, the number of servers and the number of years the current owner operated the restaurant are defined as $x_1,x_2,x_3,x_4\text{and}{x}_5$, respectively.

Hence, using the given information the regression equation is $\widehat{y}=2.50+3x_1+4x_2-3x_3+0.20x_4+x_5$.

Thus, the amount of profit for a restaurant with 40 parking spaces which is open 72 hours per week, 10 mile from a landmark of Beach M with 20 servers and has been operated by the current owner for 5 years, is calculated as follows:

$\begin{array}{c}\widehat{y}=2.50+3\left(40\right)+4\left(72\right)-3\left(10\right)+0.20\left(20\right)+5\\ =2.50+120+288-30+4+5\\ =389.5\end{array}$

Therefore, the amount of profit for a restaurant with the given conditions is $389,500.

To determine

Explain the values of $b_2$ and $b_3$ in the multiple regression equation.

Explanation of Solution

It is clear that, the value $b_2$ is 4 which is the coefficient of $x_2$, the number of hours the restaurant is opened per week.

Hence, it can be concluded that if the restaurant is opened one more hour per week, then the amount of profit for the restaurant would be increased by $4,000, when all the other factors are constant.

It is clear that, the value $b_3$ is –3 which is the coefficient of $x_3$, the distance from the landmark in Beach M.

Hence, it can be concluded that if the restaurant is opened one more mile away from the landmark of Beach M, then the amount of profit for the restaurant would be decreased by $3,000, when all the other factors are constant.