Motorboats often have tachometers instead of speedometers. The tachometer measures the engine speed in revolutions per minute. This can be used to determine the speed of the boat. The data in the table gives the average speed (in knots) of a boat for several engine speeds (in hundreds of revolutions per minute, or RPMs). Find the equation of a polynomial function that best models the data. Use the coefficient of determination to verify your choice.

Engine Speed
9
11
13
15
17
19
21.5
Boat Speed
6.43
7.61
8.82
9.86
10.88
12.36
15.24

*Note: You do not need to turn on the STAT PLOT to find the regression equation or the coefficient of determination. To find a regression equation only, omit steps 1, 2, 3, and 7.

Step 1: Turn on the STAT PLOT; press 2nd, Y=, ENTER, ENTER.

Step 2: Clear the functions; press Y= then, CLEAR for each function that needs deleting.

Step 3: Select values for the viewing window; press WINDOW.

Step 4: Go to the lists; press STAT, ENTER.

Step 5: Clear the lists; press the up arrow until the column heading is highlighted then, CLEAR, ENTER. Repeat for all columns that have unneeded data.

Step 6: Enter the data. (L1 = Engine speed; L2 = Boat speed.)

Step 7: View the data; press GRAPH.

Step 8: Choose the regression model; press STAT, right arrow.

Step 9: Select the linear regression; press 4, ENTER. Repeat Steps 8 and 9 for quadratic and cubic regression.

Compare the R2 values. Cubic regression has the highest R2; therefore, it is the best model for this data.

Step 10: Place the cubic regression into the function area; press STAT, right arrow, 6, VARS, right arrow, ENTER, ENTER, ENTER.

Step 11: View the scatter plot and regression model; press GRAPH.

In standard form, the cubic regression, rounded to the five decimal places, is as follows:

y = 0.00475x3 - 0.19432x2 + 3.13175x - 9.52995