Making Predictions About Linear Regression
It’s easy to get caught up in all the calculations of regression. Always remember that understanding and interpreting your results is just as important as calculating them!
A building contractor examines the cost of having carpentry work done in some of his buildings in the current year. He finds that the cost for a given job can be predicted by this equation:
y = $50x + $65
Here, y is the cost of a job (in dollars), and x is the number of hours a job takes to complete. So the cost of a given job can be predicted by a base fee of $65 per job plus a cost of $50 per hour. Assume that the scatter plot and correlation both indicate strong linear relationships.
Sample questions

What is the predicted cost for a job that takes 4.75 hours to complete?
Answer: $302.50
To figure out the predicted cost of a job, use the equation y = $50x + $65, replacing x with the given number of hours to complete the job. In this case, x = 4.75, so y = $50(4.75) + $65 = $302.50.

How much more money do you predict a job taking 3.75 hours to complete will cost, as compared to a job taking 3.5 hours to complete?
Answer: $12.50
You can solve this problem in two ways.
First, the slope measures the change in cost (Y) for a given change in the number of hours (X). So you can simply calculate the change in hours (3.75 – 3.50 = 0.25), and then multiply by slope (50) to get the difference in cost, (0.25)(50) = $12.50.
Second, you can calculate the costs based on both number of hours, and then take the difference. So substitute x = 3.75 (hours) into the equation, and substitute x = 3.50 (hours) into the equation, calculate their y values (costs), and subtract. So you have
y = $50(3.75) + $65 = $252.50
y = $50(3.50) + $65 = $240.00
Subtract these two values to get $252.50 – $240.00 = $12.50.
This means the job is predicted to cost $12.50 more if the hours increase from 3.50 to 3.75.

Suppose that in a different city, a similar equation predicts carpentry costs, but the intercept is $75 (the slope remains the same). What is the predicted cost for a job taking 2 hours in this city?
Answer: $175
If the intercept is $75 while the slope remains the same, the new equation for predicting costs will be y = 50x + $75.
In this case, x = 2, so y = $50(2) + $75 = $175.
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