**QUESTION**

**Pay Raise**

**Prompt**: The board of directors at a large corporation wants to base their division managers’ pay raises on the profit performance of their respective divisions. They have asked you to evaluate the performance and raises at other companies and propose a formula for calculating the percentage increase in base pay based on the percentage change in the division’s profit. You collected information from 50 divisions at similar companies and performed a linear regression on the percentage change in the division profits vs. the percentage change in the manager’s salary.

**Response Parameters**

**ANSWER**

**What is the regression equation from the Summary Output? Is this a useful model? How do you know.**

Ans: 𝑦̂=2.2899+0.9513𝑥

The tool automatically performs a test of hypothesis to determine if the slope of the population regression model (𝛽1) is equal to zero. 𝐻0: 𝛽1=0

If 𝛽1 is equal to zero, then the predicted y value would be a constant and there would be no value in knowing the regression equation. The p-value for this test is less than 0.0000. Therefore, we can reject the null hypothesis and we know the model is useful.

**Are the assumptions of regression satisfied? How did you verify them?**

Ans: There are four assumptions we need to verify, before we try to use the regression model.

Linearity – Is there a linear relationship between the dependent and independent variables? This assumption is checked by looking at the scatter plot of the original data. The scatter plot shows a relatively strong positive linear relationship.

Independence – Are the errors (residuals) independent of each other? This assumption is checked by looking at the residuals plot. The residuals plot does not show any patterns or trends. This indicates the errors are independent.

Normality – Are the residuals normally distributed around the regression line? This assumption is checked by looking at the Normal Probability Plot of the residuals. A straight line indicates the residuals are normally distributed. Our plot shows fairly linear.

Equal variance – Is the spread of the residuals approximately the same across the range of the dependent variable? This assumption is checked by looking at the residuals plot. The spread appears fairly equal through out.

All the four assumptions are satisfied. These assumptions are fairly robust. We should use care when using the regression model.

**Does change in division profit appear to be a good predictor for the manager’s pay raise? Why do you think that?**

Ans: The coefficient of correlation (R2) is the percentage of the change in y (increase in manager’s salary) that can be explained by the change in x (increase in profits). The R2 value for these two variables is 0.9321. About 93% of the increase in manager’s salary can be explained by the increase un profits).

The increase in profits apprears to be a fair predictor of the increase in manager’s salary, but there are other factors that are probably equally important.

**One of your company’s divisions had a –0.51 percent change in profits last year, while another had a 20 percent increase. What is the predicted percentage change in salary for these two division managers?**

Ans: If we start with the regression equation and plug -0.51 in for x we can calculate the predicted percentage change in salary for the first employee.

𝑦̂=2.2899+0.9513x

𝑦̂=2.2899+0.9513*(-0.51)

𝑦̂=1.8

He should have a change of 1.8%.

We should not use this regression model to predict the second employee’s change. The model covered profit percent change from -2 to 5. A change of 20% is too far outside the range of the model.

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