Notebook File Chapter 8

8/10/2019 Notebook File Chapter 8

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The least squares regression line(LSRL)

Recall from yesterday:

Scatterplot:

Direction

Form

Strength

Unusual features

quantitative variables

does not imply

causation

'r' applies tolinearassociations only

a 'bent' association

may be made linear by

re-expressing the data

After standardization


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Linear Regression Line

Equation 'models' the relationship allowing us to predict y

values for any given x value

y = 34x +310 where y = rent with hotel

x = spaces from 'go'

(39, 2000)

y = 34(39)+ 310

y = 1636

'hat' means predicted

Linear Regression Line

Residual

residual = y - y

residual = 2000 - 1636

= 364

if residual > 0 then observed y value is greater than predicted

value

i.e. the actual data point is abovethe regression line

residual = observed - predicted

Calculating Slope of Least Squares Regression line

Recall: r = .83

for explanation, see

Math Box on page 175

Moving away from 'standardized' equation to actual

equation

This is nice, but it means we have to change all of our data to z-

scores, solve equation and then convert z-scores back to regular

data but this is too cumbersome!

slope=

y-intercept


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Revisiting yesterday's problem

a =

b =

Remember: the

least squares

regression line

always passes

through the

center of the

data (x, y)


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QUOTATION OF THE DAY

"Thank God the research didn't find that novels increased toothdecay or blocked up your arteries."

LOUISE ERDRICH,

the novelist, on studies finding that after reading literary fiction people performedbetter on tests measuring empathy, social perception and emotional intelligence.

To find the equation of a regression line,

use the slope

and the point

If a scatterplot satisfies what conditions, we can discusscorrelation and regression line?

no outliers

no discernible pattern besides being linear

both variables quantitative


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Important notes about the LSRL

Before you use the LSRL, you must look at the

scatterplot to make sure the relationship is

linear

Be sure to distinguish between actual data values and

predicted values. Use words like 'predicted' and 'on

average'

The slope of the LSRL has the same sign as r, the coorelation

coefficent

Be able to answer questions, such as:

What is the equation of the LSRL?

What is the interpretation of the slope and y-intercept?


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The answer? Residuals

How welldoes the line fit the data?

How?Create a scatterplot of

the residuals vs. the x-values.

How appropriate is our line in makingpredications?

What is it about R2?

R2: the coefficient of determination

Shows how good one term is at predicting

another. The closer r2is to 1, the better the

statistical chance that the input x will correctly

produce term y

If r square is .80, then 80% of the variation in the values of y is explained

by its linear relationship with the values of x

Residuals

Registrations

400 600 800 1000

0

-10

10


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Residuals (continued)

A good residual plot - one that confirms the

appropriate use of the LSRS - should :

have similar scatter throughout

have no interesting features

should have no direction of shape

Residuals (continued)

A good residual plot - one that confirms the

appropriate use of the LSRS - should :

se = standard deviation

of the residuals

A small Seimplies residuals that are closely centered

about the mean

The mean of the residuals = 0

the units of the residuals are the same units of the y-values

Step 1: Check that the two variables arequantitative

Check that the association islinear

Check for presence of outliers

Before you find the best-fit-line:

Step 2: Do the math

Find the least squares line

r

r2


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State the conclusion

What does the y-intercept tell you? What does R2 tell you?

se = standard deviation of the residuals


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Finally: check theresidual plotto make sure that the linear model is

appropriate

Note: you can not work backwards;

You cannot use the linear regression line to predict x


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r(sy/sx).

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Notebook File Chapter 8