Math 147 Name: Linear Models

On April 9th, 1994 my son, Zane, was born. Following is a series of data points relating his weight to his age. Our goal is to find a linear equation that can be used to describe the weight of Zane as a function of time. Date Weight 4/9 4 lb. 5 oz 4/25 5 lb. 0.5 oz 6/6 8 lb. 2 oz 8/10 11 lb. 14 oz 10/25 14 lb. 13 oz 2/10 18 lb. 8.5 oz

a. First we must manipulate the data so that we can graph it easily. Let us set April 9th as t = 0. Now we need to convert each date to the number of days since April 9th. To make the weight easier to work with, let’s convert each to ounces. Remember that there are 16 ounces in one pound. Date Days since 4/9 Weight Weight in oz 4/9 0 4 lb. 5 oz

4/25 5 lb. 0.5 oz

6/6 8 lb. 2 oz

8/10 11 lb. 14 oz

10/25 14 lb. 13 oz

2/10 18 lb. 8.5 oz

b. Using the linear regression capabilities of your calculator, find the best fit line for the data.

c. Write a sentence (or two) explaining what the slope of the line has to do with my son.

d. Write a sentence (or two) explaining what the y-intercept of the line has to do with my son?

e. Using the slope and intercept, write an equation for the weight of my son as a function of time.

f. How according to your equation, how much should Zane weigh on his first birthday? (Convert this value to pounds and ounces)

g. Approximate when Zane weighed 10 pounds, using your equation. (Convert your answer to the actual date)

h. If our equation holds true, how much will Zane weigh on his 21st Birthday? Considering the build of his father, what does this result imply about the long-range accuracy of our model? Linear Regression on the TI 82 and TI 83

A. Entering the Data We must first enter the data into the calculator. Push the STAT button. You should get an option to edit your data list. You may first want to clear the list of any old values that are in there. There should be an option to do this also showing. It is abbreviated ClrStat on the 81 and ClrList on the TI-82. On the 82 you also need to tell it which lists to clear. Now go back to the Edit Data/List feature. The screen on the 81 lists a series of x and y values. Enter the day as x values and the weight as y values. For the 82, you should enter all the day values in one list and all the weight values in the second list. When you are done, choose QUIT

B. Viewing the Data First adjust the viewing window. For our problem set Xmin = 0, Xmax = 365, Xscl = 30, Ymin = 0, Ymax = 320, Yscl = 32. Any equations you have entered will also show up, so clear them while you are at it.

TI 81: From the STAT menu choose DRAW and SCATTER. Press ENTER to see the points you entered plotted.

TI 82: Choose STAT PLOT above the Y = button. Press Enter to select PLOT1. Turn the plot on, select the first Type that is listed, and make sure that the Xlist and Ylist are correct. When you are done press GRAPH.

C. Calculating a "best fit line" using Linear Regression.

TI - 81: Push STAT, then choose LinReg from the CALC menu. Press ENTER again to get the calculator to execute the command. You will be shown values for a and b in the equation Y = a + bX. Note that b is the slope and a is the y-intercept; a change from what we are used to.

TI - 82: Push STAT, then choose LinReg(ax+b) from the CALC menu. Press Enter again to get the calculator to execute the command. The calculator shows you values for a and b, and where they go in the equation.

That number r is called the correlation coefficient. It is a measure of how well the line fits the data. For a perfect fit, the absolute value of r would be one. For a completely worthless fit, r would be close to zero.

D. To see both the plot and the graph of the best fit line, Press Y= then VARS. For the 81, under the menu LR you will find RegEQ. Select this to paste the linear regression equation into the Y= menu. For the 82, you will find RegEQ in the Statistics variables, under the heading EQ. Once that is completed, press GRAPH to see both the line and the points. (Note to 81 users, your points will be cleared. you must re-display them following part B above with the exception of clearing the equation Y1.

Note to All: Linear regression is only one of several methods for finding the "best fit" line. The TI - 82 will also find the best fit Median-Median line (Med-Med on the STAT CALC menu).