Pre-Calc Notes 9/5

(preresiduals)

The Line of Best Fit

Last week we were spending time finding the errors between actual data and predicted values from a line of best fit. We were introduced to finding the errors by using our calculators. We are going to compare a line of best fit that we do versus a linear regression on the calculator to which one is the better fit. Below is data that is somewhat linear.

Year	1978	1979	1980	1981	1982	1983	1984	1985	1986	1987
Cost/Day ($)	194	217	245	284	327	369	411	460	501	539

Step 1: Using graph paper and a ruler, create a scattergram of the data listed above.

Step 2: Find the line of best fit using your ruler and calculating the slope and y-intercept using two pts from the line. Do your calculations in the space provided and indicate your equation.

Your Equation:____________________________

Step 3: To find the calculator’s equation: enter the data into the lists of your calculator. Put the years (abbrevated to 78, 79, etc) in L₁ and the Cost in L₂. Turn your stat plots on. Do a linear regression of the data: For the TI-83s, press STAT and arrow over once to CALC and go down to where it says LinReg (ax + b). Press Enter to bring the command to the home screen. Then do 2^nd 1 (for L₁) “,” above the 7 key, 2^nd 2 for (L₂ ) “,” VARS, arrow once to Y-VARS, Function, Enter, and 1 for Y₁. The command should look like LinReg (ax+b) L₁, L₂,Y₁ on your screen. Then press Enter and the equation with appear both on you screen and in Y₁.

For the TI-82s, do a LinReg (ax +b) by pressing Stat, arrow once to Calc, and go down to

LinReg(ax +b), press Enter. Once you have your equation on your screen. Go to Y= and place your cursor in the Y₁ spot. Press VARS, 5 for statistics, arrow to the right twice, and 7 for RegEQ. This will place the equation in Y₁.

Modfy your window and graph. Indicate the calculator’s equation in the space provided.

Calc Equation:____________________________

Step 4: Put your equation in Y₂.

Step 5: Go to your lists. Highlight the space where L₃ is located. We want the cost per day that is predicted by calculator’s equation. To do this, go to VARS, arrow over once, ENTER, and ENTER to get Y₁ to appear in the header. Enter ( 2^nd 1 for L₁ )….it should look like Y₁(L₁) in your calculator and press ENTER. All the predicted values of the cost should appear in L₃. Study the values in L₃ and describe in sentences what is happening mathematically.

_________________________________________________________________________

We are going to find both the mean absolute value and the mean squared values as we did in our activity last Friday.

Step 6: To find errors, we first have to find the error between the observed data and the predicted data. To do that, we have to subtract the predicted data from the observed data. So at the top of L₄, enter the equation “L₂ - L₃”. Note:Use the ALPHA “ keys. The quotes make the equation hold when you manipulate a different list. The errors should appear.

In L₅, we want the absolute value of the errors so they all become positive. Go to the top of L₅, press Enter and type in “abs(L₄)”

In L₆, we want the squared errors so they all become positive. Go to the top of L₆, press Enter and type in “L₄²”

To keep this all straight, I have placed a table with your six lists below. Write above the lists what data is in each one:

L1

L2

L3

L4

L5

L6

Step 7: Find the mean absolute error and the root mean squared error for the calculator’s equation.

mean absolute error ______________ root mean squared error _________________

Step 8: Do the same process above with your equation in Y₂. This requires that you change the formula in L₃ to “Y₂(L₁)”. The rest of the lists should change if you used the ALPHA “ keys.

Find the mean absolute error and the root mean squared error for your equation.

mean absolute error ______________ root mean squared error _________________

The line of best fit is the line with the smallest value for the sum of the squares of the errors. Which line has the line of best fit and why?