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Chapter 1: Data Sets and Models

V. J. MottoMAT 112 Short Course in Calculus

Models using a Calculator

The Process consists of the following steps:

Enter the points.Produce a scatter plot.Discover the “curve of best fit” – linear,

quadratic, cubic, exponential or ....Judge whether this curve is the best

possible relationship for the data.

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The TI-83/84 Calculator Solution

We will explore how to produce a scatter plot using the TI-83/84 calculator.

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For the TI-83/84 Calculator

Find the equation for the line passing through (2, 3) and (4, 6)

Solution1. We need to enter the

point values.

2. Press the STAT key.

3. Then from the EDIT menu select 1:EDIT and enter the points as shown to the right.

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Example 1 (continued) Scatter Plot

Making a Scatter Plot

1. Press 2nd + y= keys Stat Plot

2. Select Plot1 by touching the 1 key

3. Press the Enter key to turn Plot1 on.

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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.

This is an important step --- looking at the graph helps decide what model we should use.

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Statistical Analysis for TI-83/84

Now let’s use the Statistical Functions of the calculator to find our model

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On-Time Setup

This is a one-time operation unless your reset your calculator.

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 2nd and 0 (zero) keys to get to the Catalog.

2. The calculator is in the alpha-mode. So pressing 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.

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Example 1 (continued) The Model

We will use the Statistics functions to help us find the linear relationship.

1. Press the STAT button.

2. Now slide over to the CALC menu.

3. Choose option 4: LinReg(ax+b)

4. We need to add the y1 variable so we can graph the function. Press the VARS key; choose Y-VARS, then FUNCTION. Then select the y1 variable/

5. Press the Enter key twice to execute the subroutine.

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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.

Press the GRAPH key

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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.

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Comments on r2 - Goodness of fit

r2 is often referred to as the “goodness of fit” measure. It tells us how well the model accommodates all the variances.

For example, if r2 = 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.

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The TI-89 Solution

We will explore how to produce a scatter plot using the TI-89 calculator.

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For the TI-89 Calculator

Find the equation for the line passing through (2, 3) and (4, 6)

The Solution

1. We need to enter the point values.

2. Begin at the HOME screen and select the APPS button.

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For the TI-89 Calculator (continued)

1. Select Data for the table type, and for Variable enter “d”. Then press Enter.

2. The data table will appear empty. Enter your values as shown.

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For the TI-89 Calculator (continued)

The Plot Set upPress F2 from the Data/Matrix Editor window to get to the Plot Setup Window; then press F1 to define the plot.Enter the values shown.Then press the green diamond and F3 to produce the graph.

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For the TI-89 Calculator (continued)

Now press F2 and 9:Zoom get a better view of the data.

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For the TI-89 Calculator

Using Statistical Analysis for the TI-89 calculator to find the model.

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For the TI-89 Calculator

Statistical AnalysisGo back to the Data/Matrix Editor by pressing Home and then APPS.Then choose Current to use the matrix you already defined.

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For the TI-89 Calculator (continued)

For Calculation Type, use the arrow to select LinReg. (Obviously, if you wanted to perform a power regression, select PowerReg, etc.). Press Enter to save your selection.

Identify which column has the x and which has the y variable.

Then instruct the calculator to store the regression equation (RegEQ) by using the arrow. Here, y1(x) is the location where we have chosen to store the equation.

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For the TI-89 Calculator (continued)

Press Enter to view the equation of the model.

Press the green diamond and F3 to view the graph.

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The Analysis Again

From the information we know the following:The linear relationship is y = 1.5x + 0 or y =

1.5x Since corr = 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.

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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

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More to say

R2 or r2 – the goodness of fit variable is an important consideration for all the models

Some models (Exponential models) require manipulation of the data. We will discuss this when these examples arise.

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