To address this problem, we need to fit two regression models:

Model 1: Incorporates the number of hours worked and client type as independent variables.

The general regression model can be expressed as:

$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \epsilon$

Where:

• $y$: Net profit earned from the client
• $x_1$: Number of hours spent working with the client
• $x_2$: Dummy variable for "manufacturing" (1 if manufacturing, 0 otherwise)
• $x_3$: Dummy variable for "service" (1 if service, 0 otherwise)
• $\beta_0$: Intercept
• $\beta_1, \beta_2, \beta_3$: Coefficients for the independent variables

Model 2: A second-order model to check for improvements in fit.

The second-order model adds interaction terms and quadratic terms. It can be expressed as:

$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \beta_4 x_1^2 + \beta_5 (x_1 \cdot x_2) + \beta_6 (x_1 \cdot x_3) + \epsilon$

Where:

• $x_1^2$: Square of the hours worked
• $x_1 \cdot x_2$: Interaction term between hours worked and manufacturing dummy
• $x_1 \cdot x_3$: Interaction term between hours worked and service dummy

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Steps to Solve:

1. Fit Model 1:

   • Use the data to estimate $\beta_0, \beta_1, \beta_2, \beta_3$.
   • Input these values into the regression equation.

2. Fit Model 2:

   • Include the additional second-order terms ($x_1^2$, $x_1 \cdot x_2$, $x_1 \cdot x_3$).
   • Estimate the coefficients for all terms.

3. Compare Models:

   • Use a goodness-of-fit metric such as adjusted $R^2$, $F$-statistic, or AIC to determine which model fits better.
   • Perform a hypothesis test (e.g., ANOVA) to check if the additional terms in the second-order model significantly improve the fit.

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If you can upload the data, I can directly calculate and fit these models for you using regression analysis tools. Would you like to proceed with this?

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Relative Questions:

1. What method should be used to interpret the significance of regression coefficients in this context?
2. How do dummy variables for categorical predictors affect regression analysis?
3. Why is a second-order model considered for improving the fit in regression analysis?
4. What is the difference between adjusted $R^2$ and $R^2$, and why is it important here?
5. How can interaction terms in a regression model reveal relationships between predictors?

Tip:

Always check the residual plots and perform diagnostics (e.g., VIF for multicollinearity) when fitting regression models to ensure validity of assumptions.