The Least Squares Line Lesson 1.3

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

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

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

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

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

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

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

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

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

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

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

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

Assignment Lesson 1.3 Page 41 Exercises 7 – 15 odd