Part 1: Regression Model

The regression model provided is:

$\hat{y} = 1717.2 + (23.5)x_1 + (-441.9)x_2 + (-2307.1)x_3$

where:

• $x_1$: Hours spent working with the client
• $x_2$: Dummy variable for service clients ($x_2 = 1$ if service client, $x_2 = 0$ otherwise)
• $x_3$: Dummy variable for governmental clients ($x_3 = 1$ if governmental client, $x_3 = 0$ otherwise)

Part 2: Coefficient Interpretation

1. $x_1$ (Hours spent):

   • Coefficient: $23.5$
   • Interpretation: The average change in profitability for a one-unit increase in hours worked, holding the client type constant, is $23.5.

2. $x_2$ (Service client):

   • Coefficient: $-441.9$
   • Interpretation: The average difference in profitability between service clients and manufacturing clients (baseline group), holding hours constant, is $-441.9$.

3. $x_3$ (Governmental client):

   • Coefficient: $-2307.1$
   • Interpretation: The average difference in profitability between governmental clients and manufacturing clients (baseline group), holding hours constant, is $-2307.1$.

Descriptions of the Model

• The intercept ($1717.2$) represents the average profitability of manufacturing clients (baseline group) when no hours are spent working with the client.
• The positive coefficient of $x_1$ suggests that more hours spent working with the client leads to increased profitability, regardless of client type.
• The negative coefficients of $x_2$ and $x_3$ indicate that both service and governmental clients are less profitable compared to manufacturing clients, holding hours constant, with governmental clients showing the largest reduction in profitability.

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Clarification Questions

Would you like further clarification or examples? Here are some related questions to consider:

1. How do we determine the baseline category in a regression with dummy variables?
2. What is the impact of multicollinearity in a regression model like this one?
3. How would the interpretation change if hours were interacted with client type?
4. Can we test whether the coefficients for service and governmental clients are significantly different from each other?
5. What assumptions must be satisfied for this regression model to be valid?

Tip: When interpreting dummy variables in regression, always compare to the baseline category (in this case, manufacturing clients). This helps contextualize the results.