Fitting Linear Functions to Data

1
Fitting Linear Functions to Data

• Lesson 1.6

2
Cricket Chirps Temp.

• Your assignment was tocount cricket chirpsand
  check the temperature
• The data is saved and displayed on a spreadsheet
• Your science teacher wantsto know if you can
  find alinear equation to more orless match the
  data

3
Problems with Data

• Real data recorded
• Experiment results
• Periodic transactions
• Problems
• Data not always recorded accurately
• Actual data may not exactly fit theoretical
  relationships
• In any case
• Possible to use linear (and other) functions to
  analyze and model the data

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Fitting Functions to Data
Temperature Viscosity (lbssec/in2)
160 28
170 26
180 24
190 21
200 16
210 13
220 11
230 9

• Consider the data given by this example
• Note the plot ofthe data points
• Close to beingin a straight line

5
Finding a Line to Approximate the Data

• Draw a line by eye
• Note slope, y-intercept
• Statistical process (least squares method)
• Use a computer programsuch as Excel
• Use your TI calculator

6
You Try It
Weight Calories
100 2.7
120 3.2
150 4.0
170 4.6
200 5.4
220 5.9

• From Exercise 2, pg 65
• Enter data into data matrix ofcalculator
• APPS, 6, Current, Clear contents

7
Using Regression On Calculator

• Choose F5 for Calculations
• Choose calculationtype (LinReg for this)
• Specify columns where x and y values will come
  from

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Using Regression On Calculator

• It is possible to store the Regression EQuation
  to one of the Y functions

9
Using Regression On Calculator

• When all options areset, press ENTER andthe
  calculator comesup with an equation approximates
  your data

Note both the original x-y values and the
function which approximates the data
10
Using the Function

• Resulting function
• Use function to find Caloriesfor 195 lbs.
• C(195) 5.24This is called extrapolation
• Note It is dangerous to extrapolate beyond the
  existing data
• Consider C(1500) or C(-100) in the context of the
  problem
• The function gives a value but it is not valid

Weight Calories
100 2.7
120 3.2
150 4.0
170 4.6
200 5.4
220 5.9
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Interpolation

• Use given data
• Determine proportionaldistances

Weight Calories
100 2.7
120 3.2
150 4.0
170 4.6
195 ??
200 5.4
220 5.9
x
25
0.8
30
Note This answer is different from the
extrapolation results
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Interpolation vs. Extrapolation

• Which is right?
• Interpolation
• Between values with ratios
• Extrapolation
• Uses modeling functions
• Remember do NOT go beyond limits of known data

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Correlation Coefficient

• A statistical measure of how well a modeling
  function fits the data
• -1 corr 1
• corr close to 1 ? high correlation
• corr close to 0 ? low correlation
• Note high correlation does NOT imply cause and
  effect relationship

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Assignment

• Lesson 1.6
• Page 48
• Exercises1, 3, 5, 7