Chapter 1: Data Sets and Models

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Chapter 1 Data Sets and Models

• V. J. Motto
• MAT 112 Short Course in Calculus

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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.

3
The TI-83/84 Calculator Solution

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

4
For the TI-83/84 Calculator

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

• Solution
• We need to enter the point values.
• Press the STAT key.
• Then from the EDIT menu select 1EDIT and enter
  the points as shown to the right.

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Example 1 (continued) Scatter Plot

• Making a Scatter Plot
• Press 2nd y keys ? Stat Plot
• Select Plot1 by touching the 1 key
• 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 9ZoomStat
  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 lets 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.
• Press the STAT button.
• Now slide over to the CALC menu.
• Choose option 4
  LinReg(axb)
• 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/
• 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
• We need to enter the point values.
• Begin at the HOME screen and select the APPS
  button.

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For the TI-89 Calculator (continued)

• Select Data for the table type, and for Variable
  enter d. Then press Enter.
• The data table will appear empty. Enter your
  values as shown.

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For the TI-89 Calculator (continued)

• The Plot Set up
• Press 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 9Zoom 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 Analysis
• Go 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.

22
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
• 4LinReg(ax b) - linear regression
• 5QuadReg quadratic regression
• 6CubicReg cubic regression
• 7QuartReg quartic regression
• 8LinReg(a bx) reverse linear regression
• 9LnReg natural logarithmic regression
• 0ExReg 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.