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Chapter 13: Problem 45

A chain restaurant that specializes in selling pizza wants to analyze how \(y=\) sales for a customer (the total amount spent by a customer on food and beverage, in pounds) depends on the location of the restaurant, which is classified as inner city, suburbia, or at an interstate exit. a. Construct indicator variables \(x_{1}\) for inner city and \(x_{2}\) for suburbia so you can include location in a regression equation for predicting the sales. b. For part a, suppose \(\hat{y}=6.9+1.2 x_{1}+0.5 x_{2} .\) Find the difference between the estimated mean sales at inner-city locations and at interstate exits.

Short Answer

Expert verified
The difference is 1.2 pounds.

Step by step solution

01

Define Indicator Variables

Indicator variables are used to represent categorical data in regression models. Since there are three categories for location (inner city, suburbia, interstate exit), we need two indicator variables. Let: - \(x_1\) be 1 if the location is 'inner city' and 0 otherwise.- \(x_2\) be 1 if the location is 'suburbia' and 0 otherwise.This means for an 'interstate exit' location, both \(x_1\) and \(x_2\) would be 0.
02

Express Mean Sales for Inner City

To find the estimated mean sales for inner-city locations, substitute \(x_1 = 1\) and \(x_2 = 0\) into the regression equation:\[\hat{y}_{\text{inner city}} = 6.9 + 1.2(1) + 0.5(0) = 6.9 + 1.2 = 8.1\]Thus, the estimated mean sales for inner-city locations is 8.1 pounds.
03

Express Mean Sales for Interstate Exit

For interstate exit locations, both indicator variables \(x_1\) and \(x_2\) are zero. Substitute \(x_1 = 0\) and \(x_2 = 0\) into the regression equation:\[\hat{y}_{\text{interstate exit}} = 6.9 + 1.2(0) + 0.5(0) = 6.9\]So, the estimated mean sales for interstate exit locations is 6.9 pounds.
04

Calculate Difference Between Estimated Means

To find the difference in estimated mean sales between inner-city locations and interstate exits, subtract the mean sales of the interstate exit from the mean sales of the inner city:\[\text{Difference} = \hat{y}_{\text{inner city}} - \hat{y}_{\text{interstate exit}} = 8.1 - 6.9 = 1.2\]This indicates that the estimated mean sales in the inner city are 1.2 pounds higher than at interstate exits.

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Key Concepts

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

Understanding Indicator Variables
Indicator variables are essential tools in regression analysis for representing categorical data. They function like switches that are turned on or off depending on the category to which a subject belongs. In contexts such as our pizza restaurant's study of sales, locations can be divided into distinct categories such as 'inner city', 'suburbia', and 'interstate exit'. Unfortunately, these location types cannot be directly used in a mathematical model because they are not numerical.

To include these categories in regression analysis, we introduce indicator variables. These variables transform categorical data into binary figures (0 or 1) that a model can process. For the three location categories, two indicator variables are sufficient:
  • Let \(x_1\) represent 'inner city'; it's 1 if the location is inner city and 0 otherwise.
  • Let \(x_2\) represent 'suburbia'; it's 1 if the location is suburbia and 0 otherwise.
When a location is neither inner city nor suburbia, it is automatically classified as an 'interstate exit', making both \(x_1\) and \(x_2\) equal to 0.
The Role of Categorical Data in Regression Models
Categorical data, like restaurant locations in this scenario, represent distinct groups or types. Unlike continuous data that range over an interval, categorical data are typically separated into specific categories without any intrinsic order. For the pizza chain, understanding how different locations (a categorical variable) affect sales is crucial.

In regression models, categorical data are included using indicator variables, which we've already set up. These allow us to compare the averages across different groups, analyzing how each location type may impact sales differently. Remember:
  • Categorical variables like these often help identify trends and patterns that might be missed if only continuous data are analyzed.
  • Incorporating categorical data through indicator variables can refine model predictions and make analyses more robust and meaningful.
Using indicator variables to analyze categorical data ensures that the nuances associated with different groups, such as geographical locations, are systematically accounted for, enriching the insights derived from the data.
Exploring Mean Sales Difference
The difference in mean sales, which illustrates variations in sales depending on location, is a valuable metric for businesses. By analyzing these differences, companies can tailor marketing strategies or improve service delivery in underperforming areas.

Within the regression equation, \[hat{y} = 6.9 + 1.2x_1 + 0.5x_2,\] the coefficients (1.2 and 0.5) behind the indicator variables show how strongly sales are associated with each location type:
  • For 'inner city' locations, substituting \(x_1 = 1\) and \(x_2 = 0\) gives an estimated mean sale of 8.1 pounds.
  • For 'interstate exit', with both \(x_1 = 0\) and \(x_2 = 0\), the estimated mean sale is 6.9 pounds.
The calculated mean sales difference between 'inner city' and 'interstate exit' locations is 1.2 pounds, showing higher average sales in inner cities. This difference stems directly from the indicator variable \(x_1\), reflecting an uplift associated with this category.By understanding these differences, businesses can better allocate resources and potentially boost their sales where needed the most.

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Most popular questions from this chapter

Suppose you fit a straightline regression model to \(x=\) time and \(y=\) population. Sketch what you would expect to observe for (a) the scatterplot of \(x\) and \(y\) and (b) a plot of the residuals against the values of time.

An entrepreneur owns two filling stations - one at an inner city location and the other at an interstate exit location. He wants to compare the regressions of \(y=\) total daily revenue on \(x=\) number of customers who visit the filling station, for total revenue listed on a daily basis at the inner city location and at the interstate exit location. Explain how you can do this using regression modeling a. With a single model, having an indicator variable for location that assumes the slopes are the same for each location. b. With separate models for each location, permitting the slopes to be different.

Graduation, gender, and race The U.S. Bureau of the Census lists college graduation numbers by race and gender. The table shows the data for graduating 25 -year-olds. $$ \begin{array}{lcc} \hline \text { College graduation } & & \\ \hline \text { Group } & \text { Sample Size } & \text { Graduates } \\ \hline \text { White females } & 31,249 & 10,781 \\ \text { White males } & 39,583 & 10,727 \\ \text { Black females } & 13,194 & 2,309 \\ \text { Black males } & 17,707 & 2,054 \\ \hline \end{array} $$ a. Identify the response variable. b. Express the data in the form of a three-variable contingency table that cross-classifies whether graduated (yes, no), race, and gender. c. When we use indicator variables for race \((1=\) white, \(0=\) black \()\) and for gender \((1=\) female \(, 0=\) male \(),\) the coefficients of those predictors in the logistic regression model are 0.975 for race and 0.375 for gender. Based on these estimates, which race and gender combination has the highest estimated probability of graduation? Why?

Price, age, and horsepower In the previous exercise, \(r^{2}=0.66\) when age is the predictor and \(R^{2}=0.69\) when both age and HP are predictors. Why do you think that the predictions of price don't improve much when HP is added to the model? (The correlation between HP and price is \(r=0.56,\) and the correlation between HP and age is \(r=-0.51 .)\)

Cancer prediction A breast cancer study at a city hospital in New York used logistic regression to predict the probability that a female has breast cancer. One explanatory variable was \(x=\) radius of the tumor (in \(\mathrm{cm}\) ). The results are as follows: Term zf Constant -2.165 radius 2.585 The quartiles for the radius were \(\mathrm{Q} 1=1.00, \mathrm{Q} 2=1.35\), and \(Q 3=1.85\) a. Find the probability that a female has breast cancer at \(\mathrm{Q} 1\) and \(\mathrm{Q} 3 .\) b. Interpret the effect of radius by estimating how much the probability increases over the middle half of the sampled radii, between \(\mathrm{Q} 1\) and \(\mathrm{Q}_{3}\).
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