Do Now

• Please add your height (inches) and shoe size to the chart at the back of the classroom.

• If you do not know your height – use the measuring center (blue tape on the wall by the whiteboard).

Do Now

• Please add your height (inches) and shoe size to the appropriate chart (girls on one, boys on the other) on the sides of the classroom.

• If you do not know your height – use the measuring center (blue tape on the wall by the door).

Chapter 2: Functions and Models

Lesson 3: The Line of Best Fit

Mrs. Parziale

Important Terms• observed values: data collected from sources such as experiments or surveys

• predicted values or expected values: points predicted by a linear model

• errors or deviation: differences between the observed and predicted values of the dependent variable (y).

• line of best fit: line with the smallest value for the sum of the squares of the errors.

• method of least squares: process used to find the line of best fit

• center of gravity: a point (x, y) determined from the mean of the observed x-values and the mean of the observed y-values. This point will fall on the line of regression.

Example 1:

• Consider the following set of data:

• (a) Find the line of best fit: ________________________

x y2 104 76 68 510 1

Complete the following table:

x Observed y Predicted y Error Error2

2 104 76 68 510 1

Sum:

11.8y x

(c) Find the predicted value at x = 5. Is this interpolation or extrapolation? (d) Find the predicted value at x = 12. Is this interpolation or extrapolation? (e) Find the coordinate points of the center of gravity?

EX 2: Line using two points (2,10) & (8,5):

x Observed y Predicted y Error Error2

2 104 76 68 510 1

5 35

6 2y x

-- How does the sum of the squares of errors for this line compare to the line generated by the calculator in example 1? -- Which line is the better fit?

x Observed y Predicted y Error Error2

2 10 104 7 8.336 6 6.678 5 510 1 3.33

Sum

Line of Best Fit Activity

• Find the linear model that represents the line of best fit for the boys and girls data separately. (follow instructions on the lesson guide)

• What is the equation of the line? ___________• What is the r-value? ____________• How would you describe the strength of this line

based upon the r-value?• What is another method you could use to support

your finding? Explain the process.

• Use the lines to find Mrs. Parziale’s and Mrs. Skidmore’s shoe sizes.

• According to the Guinness Book of World Records, the tallest person who ever lived was Robert Wadlow, who was 8 ft, 11 inches tall. What shoe size would you predict for him?

• Is this an example of interpolation or extrapolation? • What do you think will happen if we combine all of

the data into one data set?

Line of Best Fit Activity

Example 2: In August 1990, Consumer Reports listed the following average

prices per pound for four grades of raw hamburger: Regular (72.5% lean) $1.57 Round (85% lean) $ 2.38 Chuck (80% lean) $1.85 Sirloin (92.5% lean)

$2.93

(a) Find an equation for the line of best fit through these points. Round to the nearest thousandth.

(b) Find the center of gravity for the data.

(c) Is the center of gravity on the regression line?

Closure

The table below gives the percent of total plastic in the U.S. that was used and then recycled, for specified years from 1985 to 1994.

• Let x = number of years after 1900• Let y = percent• Calculate the line of best fit.• Is the center of gravity on the line of

regression?

Years (after 1900)

Percent

85 0.9

89 1.7

91 2.6

93 3.5

94 4.5