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Chapter 3: Describing Relationships Chapter 3: Describing Relationships

Chapter 3: Describing Relationships - PowerPoint Presentation

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Chapter 3: Describing Relationships - PPT Presentation

32 LeastSquares Regression Where we are headed Correlation and Regression Applet Scatterplot draw line of best fit LSRL About that line IF one variable can be used to explain or predict the other we can summarize the relationship with a regression line ID: 357390

gain line nea fat line gain fat nea regression variable predict change cal find data explanatory scatterplot person lsrl

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Slide1

Chapter 3: Describing Relationships

3.2: Least-Squares RegressionSlide2

Where we are headed…

Correlation and Regression Applet

Scatterplot, draw line of best fit, LSRL!Slide3

About that line…

IF

one variable can be used to explain or predict the other, we can summarize the relationship with a regression line.

A regression line is a 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

.Slide4

About that line…

Figure 3.7 on page 165 is a scatterplot of the change in

nonexercise

activity (cal) and measured fat gain (kg) after 8 weeks for 16 healthy young adults.

The plot shows a moderately strong, negative, linear association between NEA change and fat gain with no outliers.

The regression line predicts fat gain from change in NEA.

When

nonexercise

activity = 800 cal, our line predicts a fat gain of about 0.8 kg after 8 weeks.Slide5

How we find it…

Least-Squares Regression Line

: The LSRL of

y

on

x

is the line that makes the sum of the squared residuals (vertical distance from a point to the line) as small as possible.Slide6

How we find it…

Definition:

Equation of the least-squares regression line

We have data on an explanatory variable

x

and a response variable

y

for

n

individuals. From the data, calculate the means and standard deviations of the two variables and their correlation. The least squares regression line is the line

ŷ

=

a + bx read as “y hat” with

slope

and

y

intercept

Still the predicted value of y when x=0

See your formula sheet!

Slide7

How we find it…

Some people don’t gain weight even when they overeat. Perhaps fidgeting and other “

nonexercise

activity” (NEA) explains why. The mean and standard deviation of 16 changes in NEA are

calories and

calories. For the 16 fat gains, the mean and standard deviation are

kg and

kg. The correlation between fat gain and NEA changes is

.

Find the equation of the LSRL for predicting fat gain from NEA change.

 Slide8

How we find it…

Or, if we had the data:

NEA change (

cal

)

-94

-57

-29

135

143

151

245

355Fat gain (kg)

4.5

3.0

3.7

2.7

3.2

3.6

2.4

1.3

NEA change (

cal

)

392

473

486

535

571

580

620

690

Fat gain

(kg)

3.8

1.7

1.6

2.2

1.0

0.4

2.3

1.1Slide9

What it means…

Fat Gain and NEA:

Interpret the slope in context:

Interpret the

y

-intercept in context:

 Slide10

What it means…

Fat Gain and NEA:

Predict the fat gain for a person whose NEA increases by 400

cal

when she overeats.

 

Predict the fat gain for a person whose NEA decreases by 500

cal

when she overeats.Slide11

Extrapolation

We can use a regression line to predict the response

ŷ

for a specific value of the explanatory variable

x

. The accuracy of the prediction depends on how much the data scatter about the line.

While we can substitute any value of

x

into the equation of the regression line, we must exercise caution in making predictions outside the observed values of

x.

Definition:

Extrapolation

is the use of a regression line for prediction far outside the interval of values of the explanatory variable

x

used to obtain the line. Such predictions are often not accurate.

Don’t make predictions using values of x that are much larger or much smaller than those that actually appear in your data.Slide12

So Try It Out…

Use your scatterplot for “Case of the Missing Cookies” to:

Interpret the scatterplot.

Find the equation of the LSRL.

Interpret the slope and y-intercept in the context of the problem.

Predict the height for a person whose hand span is 5 cm. For a person whose hand span is 25 cm. How reliable are these predictions?