CHAPTER 8LEAST SQUARES LINE

IMPORTANT TERMS – WRITE THESE DOWNWE WILL WORK ON THE DEFINITIONS AS WE GO!

• Model

• Linear Model

• Predicted value

• Residuals

• Least Squares

• Regression Line

• Slope

• Intercept

• Extrapolation

• Lurking Variable

LINEAR MODELS

• Remember the Equation of a Line (Algebra 1)?

• y = mx + b

*** Where m = slope (Slope = Rise/Run)

****b = the y-intercept (where the line crosses the y-axis)

• Y=mx + b is also known as the Slope intercept form of a line

• For each of the following

a) Identify the slope

b) Identify the intercept, and

c) Find the value of y for x = 10

• 1. y = 3x – 5

• 2. y = -17 + 2x

• 3. y = 13 – 0.25x

• 4. y = 7 + 4x

LINEAR MODELS• We can actually use a line to summarize all the data in scatterplots

• It may not go through each and every point on the scatterplot

• We want to find the line that somehow comes closer to all the points than any other line.

• This line will be called the Line of Best Fit

• ŷ = a + bx

• **This time, b is the SLOPE

• A is the Y-INTERCEPT

• And is the PREDICTED VALUE

LINEAR MODELS

• ŷ = a + bx b = slope

a = y-intercept

The hat over the y indicates that something has been predicted

***CALL IT “Y HAT”

EXAMPLE

• Cost of a pizza in dollars, c, modeled by the equation,

ĉ = 8 + 1.5t, where t is the number of toppings.

• The intercept suggests that the cost of a pizza with no toppings is $8

• The slope suggests that the cost increases by $1.50 for each topping.

• Find the predicted cost of a pizza when you order 5 toppings.

For each model described, explain what the slope and intercept mean:• a. ĉ = 25 + 2w, where c is the cost of shipping a package (in dollars) and

w is the weight of the package (in pounds)

Answer: The shipping on a package will cost a $25.00 plus $2 per pound.

• b. f = 40 + ¼ c Where f is temperature in degrees Fahrenheit and c is the number of chirps a cricket makes in 1 minute.

• c. p = 15 + 0.1m where p is your cell phone plan’s monthly charge (in dollars) and m is the number of minutes you used.

• d. ĉ = 11- 0.5h, where c is how tall a candle is (in inches) after it has been burning for h hours.

Answers to the previous slide’s problems:

b) Temperature will increase ¼ of a degree for every minute a cricket chirps. When there is no chirping, the temperature will be 40 degrees.

c) The monthly plan has a base charge of $15, plus an additional charge of $0.10 for every minute of call time.

d) The original height of a candle is 11 inches. Once it has been lit, it will shorten by ½ inch per hour.

RESIDUALS

• In a scatter plot, we can’t draw a line through all the points, BUT…we can draw the line of BEST FIT.

• Some of the points in the scatter plot might be above the line and some below.

• This line represents the predicted value and is written with a hat over the value.

• The difference between the true value and the line’s predicted value is called the residual.

• Residual = observed value – predicted value

or

• A negative residual means the predicted value ()

too large!

BK Broiler chicken sandwich with 30 grams of protein should have 36 grams of fat when, in fact, it actually has only 25 grams of fat.

• The BK broiler chicken’s residual would be

y – ŷ = 25 – 36 = -11 g of fat.

• The actual fat content is 11 grams less than the model predicts for a typical BK menu item with 30 grams of protein.

1. For a certain sandwich, the model predicts 15 grams of fat but it actually has 17 grams of fat. What’s the residual?

2. For another sandwich the model predicts 40 grams of fat and we see that the residual is -8 grams. What is the actual fat content of this sandwich?

3. There’s one BK sandwich that has a residual of 0. Explain what that means.