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AP3.2NotesWEB.notebook

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3.2Calculating the Equation of the

Least-Squares Line

vocabexample

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ObjectivesToday we will...

- Interpret the slope and y intercept of a least squares regression line

- Use the LSRL to predict y for a given x

- Explain the dangers of extrapolation


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Regression LineA line that describes how a response variable (y) changes as an explanatory variable (x) changes. We often use a regression line to predict the value of y for a given value of x

A regression line requires that we have an explanatory and response variable, unlike correlation

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Interpreting a Regression Line

y = a + bx

y is the predicted value

b is the slope

a is the y-intercept


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y versus yy is a prediction

y is the response variable value for an observation

You must put the hat on y when talking about a prediction - don't let him miss out on the partay!!

AP TIP

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Used HondasThe following scatterplot shows the number of miles driven (in thousands) and advertised price (in thousands) for 11 used Honda CR-Vs from the 2002-2006 model years. The scatterplot is y = 18773 - 86.18 x

(a) Interpret the slope and y intercept of the regression line.

(b) Predict the price of a used CR-V with 50,000 miles

(c) Predict the price of a used CR-V with 250,000 miles. How confident are you on this prediction?


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Used Hondas(a) Interpret the slope and y intercept of the regression line.

The slope b = -86.18 tells us that the predicted price of a used 2002-2006 Honda CR-V goes down by $86.18 for each addition 1,000 miles driven. The y intercept a = 18,773 tells us the predicted price of a used 2002-2006 Honda CR-V with 0 miles (Good Luck finding one!)

(b) Predict the price of a used CR-V with 50,000 milesThe predicted price of a CR-V with 50,000 miles is $14,464.

(c) Predict the price of a used CR-V with 250,000 miles. How confident are you on this prediction?

-$2772 - We are not confident because it is too far out and you can't sell a car for negative dollars

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Slope & Y Intercept

You must talk about the slope and y intercept as the predicted values!

AP TIP


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Extrapolationthe use of a regression line for prediction far outside the interval of values of the explanatory variable x used to obtain the line

usually not accurate

use caution if extrapolating, we rarely go far beyond the data that is displayed

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Track & Field The equation of the least-squares regression line for the sprint time and long-jump distance data is predicted long-jump distance = 305 - 27.6 (sprint time).

(a) Interpret the slope.

(b) Does it make sense to interpret the y-intercept? Explain.


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Track & Field The equation of the least-squares regression line for the sprint time and long-jump distance data is predicted long-jump distance = 305 - 27.6 (sprint time).

(a) Interpret the slope.The slope b = -27.6 tells us that the predicted long jump distance will decrease by 27.6 inches for every 1 second longer that the predicted sprint time is.

(b) Does it make sense to interpret the y-intercept? Explain.It does not make sense to interpret the y-intercept because we would never have a predicted sprint time of 0 seconds, as this is impossible.

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Residualresidual = observed y - predicted y

= y - y


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Least-Squares Regression Line

the line that makes the sum of the squared residuals as small as possible

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Homework

3.2 (Day 1) Assignment


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BellworkThe regression line shown in the figure is predicted fat gain = 3.505 - 0.00344(Nonexercise Activity)

Identify the slope and y intercept and interpret each value in context.

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Least-Squares Line

vocabexample


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ObjectivesToday we will...

- Calculate and interpret residuals

- Use technology to find a least-squares regression line

- Construct and interpret residual plots to assess if a linear model is appropriate

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RECAP

What is the equation for a linear regression?

How do we find the residual?


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Residual

residual = y - y

Think: Actual - Predicted (AP)

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Honda CR-V'sy = 18773 - 86.18 x

(a) What is the predicted price for a car with 86,000 miles?

(b) Calculate and interpret the residual for a Honda CR-V with 86,000 miles and an asking price of $10,998.


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Honda CR-V'sy = 18773 - 86.18 x (a) What is the predicted price

for a car with 86,000 miles?y = $11361.52

(b) Calculate and interpret the residual for a Honda CR-V with 86,000 miles and an asking price of $10,998.

residual = -$363.52This means that the actual

price is lower than the predicted price by about $364.

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Is the LSRL resistant to outliers?

Demo

http://bcs.whfreeman.com/tps4e/?raxsts=c1379369187609&uid=0&rau=0#628644__666392__


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LSRL on the calc

*must turn diagnostics ON

body weight 120 187 109 103 131 165 158 116back pack 26 30 26 24 29 35 31 28 weight

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McDonalds

(a) Calculate the equation of the least-squares regression line using technology. Make sure to define variables! Create a scatterplot on your calculator and graph the LSRL through it.

(b) Interpret the slope and y intercept in context

(c) Calculate and interpret the residual for the Big Mac, with 45g of carbs and 29g of fat


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McDonalds(a) Calculate the equation of the least-squares regression line using technology. Make sure to define variables! Create a scatterplot on your calculator and graph the LSRL through it.

y = -40.2 +1.71x (b) Interpret the slope and y intercept in context

The slope, b = 1.71, tells us that for an increase of 1g carb, we would predict an increase of 1.71g of fat.

The y intercept, a = -40.2, tells us that for 0g carb, there would be a predicted -40.2g fat (impossible).(c) Calculate and interpret the residual for the Big Mac, with 45g of carbs and 29g of fat

residual = -7.75g fat ; This means that the Big Mac has 7.75 fewer grams of fat then we would have predicted.

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How do we know a line is the right model to use?


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Redsidual Plota scatterplot of the residuals against the explanatory variable

helps us assess how well a regression line fits the data (FORM)

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Redsidual Plot

The residual plot should show no obvious pattern

The residuals should be relatively small in size

In order for a linear model to be appropriate...


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Calculator - Residual Plot

L1 = carbs (g)L2 = fat (g)L3 = predicted values *type in LSRL using L1 in place of x*L4 = L2 - L3

*Turn off plot 1 and set plot 2 to be L1 as x variable and L4 as y variable

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Honda's Construct and interpret a residual plot for the

Honda CR-V data


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Honda's Construct and interpret a residual plot for the

Honda CR-V data

There is no clear pattern, so a linear model is appropriate

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Homework

3.2 (Day 2)


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BellworkConstruct and interpret a residual plot for the

Honda CR-V data

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BellworkConstruct and interpret a residual plot for the

Honda CR-V data

There is no clear pattern, so a linear model is appropriate


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Least-Squares Line

vocabexample


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Objectives

Today we will...- Use r2 to see how well the line fits the data- Use the standard deviation of the residuals s to assess how well the line fits the data- Find the slope and intercept of a LSRL from the means and standard deviations of x and y

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Standard Deviation of the Residuals

Tells us how far the predictions are off, on average, from the actual values

s tells us the approximate size of a "typical" or "average" prediction error (residual)


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Standard Deviation of the Residuals

or find the SD using the residual list and 1-variable statistics in your calculator

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Honda's Calculate and interpret the standard deviation of the residuals

for the Honda CR-V data.


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Regression Equation

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Regression Equation

AP Formula sheet uses b1 for slope and b2 for y-intercept

Regression line always passes through (x , y)


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Fat GainSome people don't gain weight when they overeat. Perhaps fidgeting and other "nonexercise activity" (NEA) explains why - some people may spontaneously increase NEA when fed more. Researchers deliberately overfed 16 healthy young adults for 8 weeks. They measured fat gain (in kilograms) as the response variable and change in energy use (in calories) from activity other than deliberate exercise as the explanatory variable. The mean and the standard deviation of the 16 changes in NEA are x = 324.8 calories and sx = 257.66 calories. For the 16 fat gains, y = 2.388kg and sy = 1.1389 kg. The correlation between fat gain and NEA change is r = -0.78.

Find the equation of the LSRL

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Coefficient of Determination (r2)

another measure to tell us how well the LSRL predicts values of the response variable y

r2 represents our percent of variation accounted for

the fraction of variation in the values of y that is accounted for by the LSRL of y on x.


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Coefficient of Determination (r2)

Interpret: The percent of variation that is accounted for is....

If r2 = 1 then all of the variation in y is accounted for by the linear relationship with x

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Body Weight vs. Backpack Weight

Calculate and interpret the coefficient of determination (r2) for the data of body weight vs. back pack weight.

63% of the variation is accounted for.


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Homework

3.2 (Day 3) Assignment

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BELLWORK

1. For the least-squares regression of fat gain on NEA, r2 = 0.606. Give an interpretation of this value in context.

2. A linear regression for hiker's back pack weights (y) and their actual weight (lbs) = x is performed. For this regression, s = 2.27. Interpret this value in context.


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Least-Squares Line

vocabexample


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Objectives

- Identify the equation of a LSRL from computer output

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RECAP!


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Does it matter which variable you put with x and which you put with y?

Correlation?

Linear Regression?

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Which of the following has the highest correlation?

Trick question! They are all the same.

Always graph and check FORM before making calculations!


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How do Outliers affect....

Correlation?

Least-Squares Regression Line?

Standard Deviation of Residuals?

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How is...

r related to r2?

r2 related to s?r2 and s both measure how well the LSRL fits the data

r2 is unitless (standardized measure)s takes the units of the response variable

neither address FORM


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Computer Output

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Computer Output


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Mentos and Diet CokeWhen Mentos are dropped into a newly opened bottle of Diet Coke, carbon dioxide is released from the Diet Coke very rapidly, causing the Diet Coke to be expelled from the bottle. Will more Diet Coke be expelled when there is a larger number of Mentos dropped in the bottle? Two statistics students, Brittany and Allie, decided to find out. Using 16 ounce (2 cup) bottles of Diet Coke, they dropped either 2, 3, 4, or 5 Mentos into a randomly selected bottle, waited for the fizzing to die down, and measured the number of cups remaining in the bottle. Then, they subtracted this measurement from the original amount in the bottle to calculate the amount of Diet Coke expelled (in cups).

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Mentos and Diet CokeOutput from a regression analysis is shown below.

(a) What is the equation of the least-squares regression line? Define any variables you use.


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(b) Interpret the slope of the least-squares regression line.

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(c) What is the correlation?


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(d) Is a linear model appropriate for this data? Explain.

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(e) Would you be willing to use the linear model to predict the amount of Diet Coke expelled when 10 mentos are used? Explain.


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(f) Calculate and interpret the residual for bottle of diet coke that had 2 mentos and lost 1.25 cups.

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(g) Interpret the values of r2 and s.


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(h) If the amount expelled was measured in ounces instead of cups, how would the values of r2 and s be affected? Explain.

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Mentos and Diet Coke(a) What is the equation of the least-squares regression line? Define any variables you use.

(b) Interpret the slope of the least-squares regression line.

(c) What is the correlation?


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Mentos and Diet Coke(d) Is a linear model appropriate for this data? Explain.

(f) Calculate and interpret the residual for bottle of diet coke that had 2 mentos and lost 1.25 cups.

(g) Interpret the values of r2 and s.

(h) If the amount expelled was measured in ounces instead of cups, how would the values of and s be affected? Explain.

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What Stuck With You?

reading computer outputr 2 and s

least square

regression lines

interpreting slope and y-int

residualsusing calculator

extr

apol

atio

n


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Homework

3.2 (Day 4)

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BellworkHow well does the number of beers a person drinks predict his or her blood alcohol content (BAC)? Sixteen volunteers with an initial BAC of 0 drank a randomly assigned number of cans of beer. Thirty minutes later, a police officer measured their BAC. Least-squares regression was performed on the data. A scatterplot with the regression line added, a residual plot, and some computer output from the regression are shown below.

(a) Write the equation of the LSRL(b) Find the correlation. How do we know if it's positive or negative?


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FRAPPY!

used hondasjandrejko.weebly.com/uploads/4/9/7/7/4977448/ap-3.2-notes-web.pdf · used 2002-2006...

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