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Chapter 13: Problem 4

Let \(y=\) sales at a fast-food outlet \((1000 s\) of \(\$), x_{1}=\) number of competing outlets within a 1 -mile radius, \(x_{2}=\) population within a 1 -mile radius ( \(1000 \mathrm{~s}\) of people), and \(x_{3}\) be an indicator variable that equals 1 if the outlet has a drive-up window and 0 otherwise. Suppose that the true regression model is $$ Y=10.00-1.2 x_{1}+6.8 x_{2}+15.3 x_{3}+\epsilon $$ a. What is the mean value of sales when the number of competing outlets is 2 , there are 8000 people within a 1-mile radius, and the outlet has a drive-up window? b. What is the mean value of sales for an outlet without a drive-up window that has three competing outlets and 5000 people within a 1 -mile radius? c. Interpret \(\beta_{3}\).

Short Answer

Expert verified
a. $77,300; b. $40,400; c. A drive-up window increases sales by $15,300.

Step by step solution

01

Substitute Values for Part a

To find the mean value of sales, substitute \(x_1 = 2\), \(x_2 = 8\) (since it's 8000 people, use 8), and \(x_3 = 1\) into the regression equation:\[Y = 10.00 - 1.2(2) + 6.8(8) + 15.3(1) + \epsilon\]
02

Calculate the Mean Sales for Part a

Perform the arithmetic:\[Y = 10.00 - 2.4 + 54.4 + 15.3\]Combine the terms to find the mean:\[Y = 77.3 \ \]This is the mean sales, in thousands of dollars, for scenario a.
03

Substitute Values for Part b

Substitute \(x_1 = 3\), \(x_2 = 5\), and \(x_3 = 0\) into the regression equation:\[Y = 10.00 - 1.2(3) + 6.8(5) + 15.3(0) + \epsilon\]
04

Calculate the Mean Sales for Part b

Perform the arithmetic:\[Y = 10.00 - 3.6 + 34 + 0 \]Combine the terms to find the mean:\[Y = 40.4 \]This is the mean sales, in thousands of dollars, for scenario b.
05

Interpret the Coefficient 尾鈧�

The coefficient \(\beta_3 = 15.3\) indicates that having a drive-up window increases the mean sales by $15,300 (in thousands of dollars) when holding the number of competing outlets \(x_1\) and the population \(x_2\) constant.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Linear Regression
Linear regression is a fundamental statistical method used to model the relationship between a dependent variable and one or more independent variables. In simple terms, it helps us understand how changes in the independent variables influence the dependent variable.
In the context of our exercise, sales at a fast-food outlet ( Y ) is the dependent variable, while the number of competing outlets ( x_1 ), the population within a certain radius ( x_2 ), and the presence of a drive-up window ( x_3 ) are independent variables.

This regression model can be expressed mathematically as: \[ Y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \epsilon \]
  • \( \beta_0 \) is the intercept, representing the expected mean value of the dependent variable when all independent variables are zero.
  • \( \beta_1, \beta_2, \) and \( \beta_3 \) are coefficients that show the change in the dependent variable for unit changes in the corresponding independent variables.
  • \( \epsilon \) is the error term that accounts for variability not explained by the model.
The regression model allows for predictions and insights into data patterns, making it a versatile tool in various fields.
Mean Value Calculation in Regression
Calculating the mean value in regression analysis involves substituting the values of the independent variables into the regression equation. This computation provides us with an expected (or mean) outcome based on the given inputs. Let's break down how this works using parts a and b from our example exercise.

Part a: For two competing outlets, 8000 people, and a drive-up window:
Plug the values \( x_1 = 2 \), \( x_2 = 8 \), and \( x_3 = 1 \) into the regression equation. The arithmetic steps are as follows: \[ Y = 10.00 - 1.2 \times 2 + 6.8 \times 8 + 15.3 \times 1 = 77.3 \]Which results in mean sales of 77.3, or \(77,300.

Part b: Without a drive-up window, with three competing outlets and 5000 people:
Enter \( x_1 = 3 \), \( x_2 = 5 \), and \( x_3 = 0 \):\[ Y = 10.00 - 1.2 \times 3 + 6.8 \times 5 + 15.3 \times 0 = 40.4 \]Thus, the mean sales here are 40.4, or \)40,400.

By applying these values to the regression equation, you gain a clear understanding of how different factors affect outcomes.
Interpreting Coefficients in Regression
Coefficients in regression analysis are key to interpreting the relationship between dependent and independent variables. Each coefficient corresponds to a predictor variable, indicating the expected change in the dependent variable when the predictor increases by one unit, keeping all other variables constant.

In our exercise, the coefficient \( \beta_3 = 15.3 \) represents the change in sales when a fast-food outlet has a drive-up window compared to when it does not, assuming all other factors remain constant. This means:

- If the outlet doesn't have a drive-up window, sales would be \( \\(0 \) more due to \( x_3 = 0 \) (drive-up absence). - When the outlet adds a drive-up window ( x_3 = 1 ), the sales increase by \( \\)15,300 \) because \( 15.3 \times 1 = 15.3 \).Therefore, the coefficient tells us that having a drive-up window significantly boosts the sales for the outlet. Interpreting coefficients gives insights into how different elements of the model impact the overall outcome, enabling businesses to make informed strategic decisions.

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Most popular questions from this chapter

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