Consumer Research, Inc. You are an analyst for Consumer Research, Inc. who recen
ID: 3229469 • Letter: C
Question
Consumer Research, Inc.
You are an analyst for Consumer Research, Inc. who recently conducted a study examining the purchasing behavior of credit card users. Your goal is to analyze the data using multiple regression in order to develop a model that will help firms predict the amount charged by card users. Data were collected on a user's annual income, household size, and annual credit card charges for a sample of 50 consumers.
Your first step is to analyze the relationship between each independent variable (annual income, household size) and the dependent variable (annual amount charged) using simple regression. The result and graph for each analysis is below.
After checking for multicollinearity, you proceed with the multiple regression analysis using household size and annual income and the independent variables and amount charged per year as the dependent variable. The results are as follows:
1) How would you best explain these results to a client in the financial services industry? What is the predicted annual credit card charge for a three-person household with an annual income of $40,000?
A) Household size and annual income explain some of the variation in amount charged per year when analyzed separately; a much better model exists in which both variables are used simultaneously to explain the variation in amount charged per year. As the 3D scatterplot shows, amount charged on credit cards increases as both household size and annual income increase. For example, the multiple regression model predicts that a three-person household making $40,000 per year will charge approximately $3,699.11 on credit cards.
B) Household size and annual income explain a lot of the variation in amount charged per year when analyzed separately; a much worse model exists in which both variables are used simultaneously to explain the variation in amount charged per year. As the 3D scatterplot shows, amount charged on credit cards increases as both household size and annual income increase. For example, the multiple regression model predicts that a three-person household making $40,000 per year will charge approximately $1,089.30 on credit cards.
C) Household size and annual income explain a lot of the variation in amount charged per year when analyzed separately and a much worse model exists in which both variables are used simultaneously to explain the variation in amount charged per year. As the 3D scatterplot shows, amount charged on credit cards decreases as both household size and annual income increase. For example, the multiple regression model predicts that a three-person household making $40,000 per year will charge approximately $1,048.47 on credit cards.
Now that you have analyzed each independent variable separately, you think an even better model can be constructed if both household size and annual income are used together to predict the annual credit card amount charged. For this type of analysis you decide to use multiple regression. However, you remember learning at university that inputting several independent variables into a multiple regression model raises the potential problem of multicollinearity. That is, the situation whereby the independent variables themselves are correlated with each other. Multicollinearity in multiple regression can make the coefficients hard to interpret or even misleading.
Therefore, to check for multicollinearity between the two independent variables, you decide to perform simple regression analysis on the relationship between household size and annual income. The results are as follows:
2) Based on the multicollinearity results, what are your thoughts about proceeding with a multiple regression model using both household size and annual income to predict the amount charged per year?
A) The two independent variables are not highly correlated with each other. Therefore, using both of them in a multiple regression model could cause problems with interpretation or cause the results to be misleading.
B) The two independent variables are not highly correlated with each other. Therefore, using both of them in a multiple regression model should not cause problems with interpretation or cause the results to be misleading.
C) The two independent variables are highly correlated with each other. Therefore, using both of them in a multiple regression model could cause problems with interpretation or cause the results to be misleading.
3) Which set of linear equations represent each relationship and which independent variable seems to be the better predictor and why?
A) AMOUNT = 2582 + 404size; AMOUNT = 2204 + 40.5income; income size seems to be a better predictor due to its smaller R2.
B) AMOUNT = 258 - 404size; AMOUNT = 2204 - 40.5income; household size seems to be a better predictor due to its larger R2.
C) AMOUNT = 2582 + 404size; AMOUNT = 2204 + 40.5income; household size seems to be a better predictor due to its larger R2.
TABLE.1 LINEAR REGRESSION OUTPUT FOR HOUSEHOLD SIZE AND ANNUAL CREDIT CARD AMOUNT Coefficients Standard Error p-value t Statistic 2581.9410 195.2626 0.0000 Intercept 13.2229 Household Size 404.1284 50.9979 7.9244 0.0000 RE (adjusted) S 620,793 0.567 0.558 ANNUALCREDITCARD CHARGES BY HOUSEHOLD SIZE 6,000 5,500 v 5.000 4,500 4,000 3,500 3.000 2.500 2,000 Household Size TABLE.2 LINEAR REGRESSION OUTPUT FOR ANNUAL INCOME AND ANNUAL CREDIT CARD AMOUNT Coefficients Standard Error t Statistic p-value 6.6981 2203.9996 329.0489 0.0000 Intercept 40.4798 5.6348 Household Size 0.0000 7.1839 R2 (adjusted) 0.386 731.713 0.398 ANNUALCREDIT CARD CHARGES BY ANNUAL INCOME 5,000 5,500 5,000 3.000 2,500 2,000 10 70 Annual Income (1000'sExplanation / Answer
Answer:
1) How would you best explain these results to a client in the financial services industry? What is the predicted annual credit card charge for a three-person household with an annual income of $40,000?
A) Household size and annual income explain some of the variation in amount charged per year when analyzed separately; a much better model exists in which both variables are used simultaneously to explain the variation in amount charged per year. As the 3D scatterplot shows, amount charged on credit cards increases as both household size and annual income increase. For example, the multiple regression model predicts that a three-person household making $40,000 per year will charge approximately $3,699.11 on credit cards.
2) Based on the multicollinearity results, what are your thoughts about proceeding with a multiple regression model using both household size and annual income to predict the amount charged per year?
B) The two independent variables are not highly correlated with each other. Therefore, using both of them in a multiple regression model should not cause problems with interpretation or cause the results to be misleading.
3) Which set of linear equations represent each relationship and which independent variable seems to be the better predictor and why?
C) AMOUNT = 2582 + 404size; AMOUNT = 2204 + 40.5income; household size seems to be a better predictor due to its larger R2.
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