1. The Least Squares Line
Lesson 1.3

2. Problems with Data Real data recorded Problems In any case …
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

3. Fitting Functions to Data
Year
Death Rate
1910
84.4
1920
71.2
1930
80.5
1940
73.4
1950
60.3
1960
52.1
1970
56.2
1980
46.5
1990
36.9
2000 34
Consider the data given by this example
Note the plot of the data points
Close to being in a straight line

4. Finding a Line to Approximate the Data
Draw a line “by eye”
Note slope, y-intercept
Statistical process (least squares method)
Use a computer program such as Excel
Use your TI calculator
Use Geogebra

5. Simultaneous linear equations
Least Squares Line
Note text development of analyzing the data, pg 32, 33 for Y = mx + b, we can find m and b
Coefficient of Correlation
Simultaneous linear equations

6. 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

7. Regression on the Calculator
Example: Calories per min, and weight
Enter data into data matrix of calculator
APPS, 6, Current, Clear contents
Weight
Calories
100
2.7
120
3.2
150
4.0
170
4.6
200
5.4
220
5.9

8. Using Regression On Calculator
Choose F5 for Calculations
Choose calculation type (LinReg for this)
Specify columns where x and y values will come from

9. Using Regression On Calculator
It is possible to store the Regression EQuation to one of the Y= functions

10. Using Regression On Calculator
When all options are set, press ENTER and the calculator comes up with an equation approximates your data
Note both the original x-y values and the function which approximates the data

11. Using the Function Resulting function:
Use function to find Calories for 195 lbs.
C(195) = 5.24 This 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

12. Note : This answer is different from the extrapolation results
Interpolation
Use given data
Determine proportional “distances”
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

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

14. Assignment
Lesson 1.3
Page 41
Exercises 7 – 15 odd