Math Problem Statement

Suppose the following table was generated from a sample of 20 CEOs relating annual salary to years of experience and the industry in which they are employed. Indicator (dummy) variables were created for each industry (1 if the CEO works in that industry, 0 otherwise). The Health Care industry is considered the base-level industry.

Coefficients Standard Error t Stat P-Value Intercept 142712.946698 4039.569273 35.328753 0.000000 Experience 2885.800829 190.013764 15.187325 0.000000 Service 12526.980087 3792.667105 3.302947 0.004830 Financial −19574.362623 3849.361835 −5.085093 0.000134 Manufacturing 20351.863484 3776.640067 5.388881 0.000075

According to the results, if a CEO in the Health Care industry and a CEO in the Financial industry both have 6 years of experience, what is the estimated difference in annual salary between the two CEOs? Enter your answer below, rounded to two decimal places.

Solution

To find the estimated difference in annual salary between a CEO in the Health Care industry and a CEO in the Financial industry, we will use the coefficients from the regression model.

Steps:

  1. The base industry is Health Care, so the salary for a CEO in this industry is based only on the intercept and the experience coefficient.

    • Salary for Health Care CEO with 6 years of experience: SalaryHealth Care=Intercept+(Experience Coefficient×Years of Experience)\text{Salary}_{\text{Health Care}} = \text{Intercept} + (\text{Experience Coefficient} \times \text{Years of Experience}) SalaryHealth Care=142712.95+(2885.80×6)\text{Salary}_{\text{Health Care}} = 142712.95 + (2885.80 \times 6)
  2. For a CEO in the Financial industry, we also include the Financial industry coefficient in the calculation:

    • Salary for Financial CEO with 6 years of experience: SalaryFinancial=Intercept+(Experience Coefficient×Years of Experience)+Financial Coefficient\text{Salary}_{\text{Financial}} = \text{Intercept} + (\text{Experience Coefficient} \times \text{Years of Experience}) + \text{Financial Coefficient} SalaryFinancial=142712.95+(2885.80×6)19574.36\text{Salary}_{\text{Financial}} = 142712.95 + (2885.80 \times 6) - 19574.36
  3. The estimated difference in salary between the two CEOs is the difference between the two equations: Difference=SalaryHealth CareSalaryFinancial\text{Difference} = \text{Salary}_{\text{Health Care}} - \text{Salary}_{\text{Financial}}

Let's calculate this step by step.The estimated difference in annual salary between a CEO in the Health Care industry and a CEO in the Financial industry, both with 6 years of experience, is $19,574.36.

Do you have any questions or want further details on this calculation?

Here are some related questions you might find useful:

  1. How do dummy variables affect the interpretation of a regression model?
  2. What is the meaning of the intercept in a regression model?
  3. How can you interpret the t-statistics and p-values in the regression output?
  4. How would the estimated salary difference change if the CEO had 10 years of experience?
  5. How do you interpret the negative coefficient for the Financial industry?

Tip: The intercept in a regression model typically represents the estimated outcome when all predictor variables are equal to zero.

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Math Problem Analysis

Mathematical Concepts

Regression Analysis
Dummy Variables
Linear Algebra

Formulas

Salary = Intercept + (Experience Coefficient × Years of Experience) + Industry Coefficient
Difference = Salary_{Health Care} - Salary_{Financial}

Theorems

Ordinary Least Squares (OLS)
Interpretation of Coefficients in Multiple Linear Regression

Suitable Grade Level

Undergraduate