Exam 17: Regression Models With Dummy Variables

In the model y = β₀ + β₁x + β₂d + β₃xd + ε,the dummy variable and the interaction variable cause:

Exhibit 17.8.A realtor wants to predict and compare the prices of homes in three neighboring locations.She considers the following linear models: Model A: Price = β₀ + β₁Size + β₂Age + ε, Model B: Price = β₀ + β₁Size + β₂Loc1 + β₃Loc2 + ε, Model C: Price = β₀ + β₁Size + β₂Age + β₃Loc1 + β₄Loc2 + ε, where, Price = the price of a home (in $thousands), Size = the square footage (in square feet), Loc1 = a dummy variable taking on 1 for Location 1,and 0 otherwise, Loc2 = a dummy variable taking on 1 for Location 2,and 0 otherwise. After collecting data on 52 sales and applying regression,her findings were summarized in the following table.

Exhibit 17.8.A realtor wants to predict and compare the prices of homes in three neighboring locations.She considers the following linear models: Model A: Price = β₀ + β₁Size + β₂Age + ε, Model B: Price = β₀ + β₁Size + β₂Loc1 + β₃Loc2 + ε, Model C: Price = β₀ + β₁Size + β₂Age + β₃Loc1 + β₄Loc2 + ε, where, Price = the price of a home (in $thousands), Size = the square footage (in square feet), Loc1 = a dummy variable taking on 1 for Location 1,and 0 otherwise, Loc2 = a dummy variable taking on 1 for Location 2,and 0 otherwise. After collecting data on 52 sales and applying regression,her findings were summarized in the following table.

Exhibit 17.9.A bank manager is interested in assigning a rating to the holders of credit cards issued by her bank.The rating is based on the probability of defaulting on credit cards and is as follows.

Which of the following is an estimated logistic model?

Exhibit 17.9.A bank manager is interested in assigning a rating to the holders of credit cards issued by her bank.The rating is based on the probability of defaulting on credit cards and is as follows. To estimate this probability,she decided to use the logistic model: , where, y = a binary response variable with value 1 corresponding to a default,and 0 to a no default, x₁ = the ratio of the credit card balance to the credit card limit (in percent), x₂ = the ratio of the total debt to the annual income (in percent). Using Minitab on the sample data,she arrived at the following estimates: Note: The p-values of the corresponding tests are shown in parentheses below the estimated coefficients. Refer to Exhibits 17.9.If only applicants with excellent and good ratings are qualified for a loan,find a linear relation between their balance ratio and their debt ratio that must be satisfied to be qualified.

Exhibit 17.8.A realtor wants to predict and compare the prices of homes in three neighboring locations.She considers the following linear models: Model A: Price = β₀ + β₁Size + β₂Age + ε, Model B: Price = β₀ + β₁Size + β₂Loc1 + β₃Loc2 + ε, Model C: Price = β₀ + β₁Size + β₂Age + β₃Loc1 + β₄Loc2 + ε, where, Price = the price of a home (in $thousands), Size = the square footage (in square feet), Loc1 = a dummy variable taking on 1 for Location 1,and 0 otherwise, Loc2 = a dummy variable taking on 1 for Location 2,and 0 otherwise. After collecting data on 52 sales and applying regression,her findings were summarized in the following table.

Exhibit 17.9.A bank manager is interested in assigning a rating to the holders of credit cards issued by her bank.The rating is based on the probability of defaulting on credit cards and is as follows. To estimate this probability,she decided to use the logistic model: , where, y = a binary response variable with value 1 corresponding to a default,and 0 to a no default, x₁ = the ratio of the credit card balance to the credit card limit (in percent), x₂ = the ratio of the total debt to the annual income (in percent). Using Minitab on the sample data,she arrived at the following estimates: Note: The p-values of the corresponding tests are shown in parentheses below the estimated coefficients. (Using Excel)Refer to Exhibit 17.9.Suppose that only applicants with excellent and good ratings are qualified for a loan.Assume that the balance ratio,x₁,of those who apply is normally distributed with μ₁ = 18% and σ₂ = 6%,while their debt ratio,x₂,is normally distributed with μ₂ = 30% and σ₂ = 8%.Because of limited capabilities of Excel,assume also that x₁ and x₂ are independent.Using Random Number Generator in Data Analysis of Excel,simulate 1000 applications to estimate the percent of those that are qualified for a loan.

Exhibit 17.2.To examine the differences between salaries of male and female middle managers of a large bank,90 individuals were randomly selected and the following variables considered: Salary = the monthly salary (excluding fringe benefits and bonuses), Educ = the number of years of education, Exper = the number of months of experience, Train = the number of weeks of training, Gender = the gender of an individual;1 for males,and 0 for females. Also,the following Excel partial outputs corresponding to the following models are available: Model A: Salary = β₀ + β₁Educ + β₂Exper + β₃Train + β₄Gender + ε Model B: Salary = β₀ + β₁Educ + β₂Exper + β₃Gender + ε Refer to Exhibit 17.2.A group of female managers considers a discrimination lawsuit if on average their salaries can be statistically proven to be lower by more than $500 than the salaries of their male peers with the same level of education and experience.Using Model B,what is the alternative hypothesis for testing the lawsuit condition?

Exhibit 17.8.A realtor wants to predict and compare the prices of homes in three neighboring locations.She considers the following linear models: Model A: Price = β₀ + β₁Size + β₂Age + ε, Model B: Price = β₀ + β₁Size + β₂Loc1 + β₃Loc2 + ε, Model C: Price = β₀ + β₁Size + β₂Age + β₃Loc1 + β₄Loc2 + ε, where, Price = the price of a home (in $thousands), Size = the square footage (in square feet), Loc1 = a dummy variable taking on 1 for Location 1,and 0 otherwise, Loc2 = a dummy variable taking on 1 for Location 2,and 0 otherwise. After collecting data on 52 sales and applying regression,her findings were summarized in the following table.

Exhibit 17.9.A bank manager is interested in assigning a rating to the holders of credit cards issued by her bank.The rating is based on the probability of defaulting on credit cards and is as follows.

Exhibit 17.9.A bank manager is interested in assigning a rating to the holders of credit cards issued by her bank.The rating is based on the probability of defaulting on credit cards and is as follows. To estimate this probability,she decided to use the logistic model: , where, y = a binary response variable with value 1 corresponding to a default,and 0 to a no default, x₁ = the ratio of the credit card balance to the credit card limit (in percent), x₂ = the ratio of the total debt to the annual income (in percent). Using Minitab on the sample data,she arrived at the following estimates: Note: The p-values of the corresponding tests are shown in parentheses below the estimated coefficients. Refer to Exhibit 17.9.Assuming the debt ratio 30%,compute the increase in the probability of defaulting when the balance ratio goes up from 5% to 15%.

Exhibit 17.9.A bank manager is interested in assigning a rating to the holders of credit cards issued by her bank.The rating is based on the probability of defaulting on credit cards and is as follows. To estimate this probability,she decided to use the logistic model: , where, y = a binary response variable with value 1 corresponding to a default,and 0 to a no default, x₁ = the ratio of the credit card balance to the credit card limit (in percent), x₂ = the ratio of the total debt to the annual income (in percent). Using Minitab on the sample data,she arrived at the following estimates: Note: The p-values of the corresponding tests are shown in parentheses below the estimated coefficients. Refer to Exhibits 17.9.Bob has a balance ratio of 10%,an annual income of $80,000,and $15,000 in total debt.Only applicants with excellent and good ratings are qualified for a loan.Find the maximum amount of loan Bob can get if he is required to maintain his excellent or good rating after getting this amount.

For a linear regression model with a dummy variable d and an interaction variable xd,we:

Exhibit 17.8.A realtor wants to predict and compare the prices of homes in three neighboring locations.She considers the following linear models: Model A: Price = β₀ + β₁Size + β₂Age + ε, Model B: Price = β₀ + β₁Size + β₂Loc1 + β₃Loc2 + ε, Model C: Price = β₀ + β₁Size + β₂Age + β₃Loc1 + β₄Loc2 + ε, where, Price = the price of a home (in $thousands), Size = the square footage (in square feet), Loc1 = a dummy variable taking on 1 for Location 1,and 0 otherwise, Loc2 = a dummy variable taking on 1 for Location 2,and 0 otherwise. After collecting data on 52 sales and applying regression,her findings were summarized in the following table. Note: The values of relevant test statistics are shown in parentheses below the estimated coefficients. Refer to Exhibit 17.8.Using Model C,what is the conclusion for testing the joint significance of the two dummy variables at the 1% significance level?

Exhibit 17.7.To examine the differences between salaries of male and female middle managers of a large bank,90 individuals were randomly selected and the following variables considered: Salary = the monthly salary (excluding fringe benefits and bonuses), Educ = the number of years of education, Exper = the number of months of experience, Gender = the gender of an individual;1 for males,and 0 for females. The regression results for the models, Model A: Salary = β₀ + β₁Educ + β₂Exper + β₃Gender + β₄Exper × Gender + ε, Model B: Salary = β₀ + β₁Educ + β₂Exper + ε,are summarized below. Note.The values of relevant test statistics are shown in parentheses below the estimated coefficients. Refer to Exhibit 17.7.What is the value of the test statistic for testing the joint significance of Exper and Exper × Gender in Model A?

If the number of dummy variables representing a qualitative variable equals the number of categories of this variable,one deals with the problem of perfect multicollinearity.

Exhibit 17.9.A bank manager is interested in assigning a rating to the holders of credit cards issued by her bank.The rating is based on the probability of defaulting on credit cards and is as follows. To estimate this probability,she decided to use the logistic model: , where, y = a binary response variable with value 1 corresponding to a default,and 0 to a no default, x₁ = the ratio of the credit card balance to the credit card limit (in percent), x₂ = the ratio of the total debt to the annual income (in percent). Using Minitab on the sample data,she arrived at the following estimates: Note: The p-values of the corresponding tests are shown in parentheses below the estimated coefficients. Refer to Exhibit 17.9.What is the estimated logistic model?

Exhibit 17.8.A realtor wants to predict and compare the prices of homes in three neighboring locations.She considers the following linear models: Model A: Price = β₀ + β₁Size + β₂Age + ε, Model B: Price = β₀ + β₁Size + β₂Loc1 + β₃Loc2 + ε, Model C: Price = β₀ + β₁Size + β₂Age + β₃Loc1 + β₄Loc2 + ε, where, Price = the price of a home (in $thousands), Size = the square footage (in square feet), Loc1 = a dummy variable taking on 1 for Location 1,and 0 otherwise, Loc2 = a dummy variable taking on 1 for Location 2,and 0 otherwise. After collecting data on 52 sales and applying regression,her findings were summarized in the following table.

Exhibit 17.5.An over-the-counter drug manufacturer wants to examine the effectiveness of a new drug in curing an illness most commonly in older patients.Thirteen patients are given the new drug and 13 patients are given the old drug.To avoid bias in the experiment,they are not told which drug is given to them.To check how the effectiveness depends on the age of patients,the following data has been collected. Assuming the variables: Effectiveness = the response variable measured on a scale from 0 to 100, Age = the age of a patient (in years), Drug = a binary variable with 1 for the new drug,and 0 for the old drug,the regression model, Effectiveness = β₀ + β₁Age + β₂Drug + β₃Age × Drug,is considered,and the following Excel results are available: Refer to Exhibit 17.5.Which of the following is true?