How do you find these residuals

Question

Data for a local hospital is displayed in a scatter plot. The graph shows the relationship between the length of a person's stay in the hospital in days, x, and the amount owed for the hospital bill. The line of best fit for the data is y = 1,899.66x + 852.81.

a. Find the residuals for the three patients who were in the hospital for 6 days, (6, 14320), (6, 7900), and (6, 13998).

b. Compare the residuals for the three patients. How are they similar? How are they different? What does the information about the residuals for the three patients tell you about their hospital bills.

Answer 1

Residuals are calculated by taking the actual value minus the predicted value.

For the point (6, 14320) the "actual value" is 14320 (as in $14.320). The predicted value is found by plugging in x = 6 into the line y = 1899.66x + 852.81:

1899.66(6) + 852.81 = 12250.77

So the residual is 14320 - 12250.77 = 2069.23

For the point (6, 7900) the actual is 7900 but the predicted (as found above) is 12250.77 so residual is 7900 - 12250.77 = -4350.77

For the point (6, 13998), the actual is 13998 and the predicted is 12250.77 so the residual is 13998 - 12250.77 = 1747.23

You can think about the "predicted value" as the average a person would expect to pay if they stayed in the hospital for "x" days. So if your cost for staying those number of days was more, you are above the average (or expected). If your cost was less, then you are below the average (or expected).