1. 10/18/2015 V. J. Motto 1 Chapter 1: Models V. J. Motto MAT 112 Short Course in Calculus Data Sets and the “STAT” Function

2. Finding Equations using TI-83 Enter the points using the Statistics options. Produce a scatter plot. Use the Statistics functions to calculate the “curve of best fit” – linear, quadratic, cubic, etc. Judge whether this curve is the best possible relationship for the data. 10/18/2015V. J. Motto 2

3. Example 1 -- Two Points Find the equation for the line passing through (2, 3) and (4, 6) Solution 1. We need to enter the point values. 2. Press the STAT key. 3. Then from the EDIT menu select EDIT and enter the points as shown to the right. 10/18/2015V. J. Motto 3

4. Example 1 (continued) Scatter Plot Making a Scatter Plot 1. Press 2 nd + Y= keys  Stat Plot 2. Select Plot1 by touching the 1 key 3. Press the Enter key to turn Plot1 on. 10/18/2015V. J. Motto 4

5. Example 1 (continued) The Graph When you press the Graph key, you get the graph show below. Go to the Zoom menu and select the “9:ZoomStat” option. 10/18/2015V. J. Motto 5

6. Example 1 (continued) Setup We are going to use the Statistics functions to help us find the linear relationship. But first we need to turn the diagnostic function on. 1. Press 2 nd and 0 (zero) keys to get to the catalog. 2. Now press the x -1 -key to get the d-section. 3. Slide down to the DiagnosticOn and press the Enter key. 4. Press the Enter key again. 10/18/2015V. J. Motto 6

7. Example 1 (continued) The Equation We will use the Statistics functions to help us find the linear relationship. 1. Press the STAT button. 2. Now slide over to the “Cal” menu. 3. Choose option 4: LinReg(ax+b) 4. Press the Enter key twice. 10/18/2015V. J. Motto 7

8. Example 1 (continued) The Analysis From the information we know the following: The linear relationship is y = 1.5x + 0 or y = 1.5x Since r = 1, we know that the linear correlation is perfect positive. Since r 2 = 1, the “goodness of fit” measure, tells us that the model accommodates all the variances. 10/18/2015V. J. Motto 8

9. Comments on r r is the Linear Correlation coefficient and -1 ≤ r ≤ +1 If r = -1, there is perfect negative correlation. If r = +1, there is perfect positive correlation Where the line is drawn for weak or strong correlation varies by sample size and situation. 10/18/2015V. J. Motto 9

10. Comments on r 2 r 2 is often referred to as the “goodness of fit” measure. It tells us how well the model accommodates all the variances. For example, if r 2 = 0.89, we might say that the model accommodates 89% of the variance leaving 11% unaccounted. We would like our model to accommodate as much of the variance as possible. 10/18/2015V. J. Motto 10

11. Example 1 (continued) The graph 1. Press the STAT button. 2. Now slide over to the “Cal” menu. 3. Choose option 4: LinReg(ax+b) 4. Press the Enter key once. 1. Press the VARS key. 2. Slide over to Y-vars menu. 3. Select 1:Function 4. Select Y1 by press the Enter key 5. Press the Graph key to see the graph 10/18/2015V. J. Motto 11

12. Example 1 (continued) Graph 10/18/2015V. J. Motto 12

13. Additional Comments You can use this technique for modeling other functions 4:LinReg(ax + b) - linear regression 5:QuadReg – quadratic regression 6:CubicReg – cubic regression 7:QuartReg – quartic regression 8:LinReg(a + bx) – reverse linear regression 9:LnReg – natural logarithmic regression 0:ExReg – exponential regression 10/18/2015V. J. Motto 13

14. More to say R 2 or r 2 – the goodness of fit variable is an important consideration for all the models Some models (Exponential models) require manipulation of the data. 10/18/2015V. J. Motto 14