15 Power Point Part 2

8/7/2019 15 Power Point Part 2

1/12

Inference for Regression:Inference for Regression:Chapter 15Chapter 15 Part 2Part 2


2/12

Is your number up?Is your number up?

Often there is no linear relationshipOften there is no linear relationshipbetween two variables. For example, thebetween two variables. For example, the

sum of the last four digits of a personssum of the last four digits of a persons

phone number and the number of letters inphone number and the number of letters intheir full name are not associated.their full name are not associated.


3/12

Lets Try It!Lets Try It!

LetLet xxbe the sum of the last four digits in ofbe the sum of the last four digits in of

your phone number andyour phone number and yybe the numberbe the number

of letters in your full name (first, middle,of letters in your full name (first, middle,and last.) Lets make a scatterplot of ourand last.) Lets make a scatterplot of our

data and then compute the equation of thedata and then compute the equation of the

regression line.regression line.


4/12

Although our graph shows little or no correlation,Although our graph shows little or no correlation,

the chances are excellent thatthe chances are excellent that bb11 isnt exactlyisnt exactly

equal to 0. Even when the true slope,equal to 0. Even when the true slope, 11, is 0,, is 0,the estimate,the estimate, bb11, will usually turn out to be, will usually turn out to be

different from 0.different from 0.

In such cases, the estimated slope is notIn such cases, the estimated slope is not

significant and differs from 0 simply becausesignificant and differs from 0 simply because

cases were picked at random. If another classcases were picked at random. If another class

did this activity, the value ofdid this activity, the value ofbb

11 would probablywould probablynot be 0 either and would probably be differentnot be 0 either and would probably be different

from ourfrom ourbb11 too.too.


5/12

Whats the Difference?Whats the Difference?

A significance test for a regression slopeA significance test for a regression slope

asks, Is that trend real, or could theasks, Is that trend real, or could the

numbers come out the way they did bynumbers come out the way they did by

chance? The key question is How far ischance? The key question is How far is

bb11 from 0 (or some other hypothesizedfrom 0 (or some other hypothesized

value ofvalue of11) in terms of the standard) in terms of the standarderror?error?


6/12

The test statistic for the slope is the differenceThe test statistic for the slope is the difference

between the estimated slope,between the estimated slope, bb11, and the, and the

hypothesized slope,hypothesized slope, 11, as measured in standard, as measured in standarderrors:errors:

If a linear model is correct and the nullIf a linear model is correct and the null

hypothesis is true, then the test statistic has ahypothesis is true, then the test statistic has a tt--

distribution withdistribution with nn -- 2 degrees of freedom.2 degrees of freedom.

1

11

bs

b

t

F

!


7/12

Mars Rocks!Mars Rocks!

Verify the test statistic. Then use theVerify the test statistic. Then use the dfdf totocheck thecheck the PP--value against the calculatorvalue against the calculator

and theand the tt--table.table.


8/12

tt--Test for SlopeTest for Slope

Generally you will use this test when you haveGenerally you will use this test when you have

the sample data that that show two variablesthe sample data that that show two variables

that appear to have a (positive or negative)that appear to have a (positive or negative)

linear association and you want to establish thatlinear association and you want to establish that

this association is real. That is, you want tothis association is real. That is, you want to

determine that the nonzero correlation you seedetermine that the nonzero correlation you see

didnt happen just by chancedidnt happen just by chance that therethat there

actually is a true linear relationship with aactually is a true linear relationship with a

nonzero slope and so knowing the value ofnonzero slope and so knowing the value ofxx isis

helpful in predictinghelpful in predicting yy..


9/12

tt--Test for SlopeTest for Slope

Name the test:Name the test:tt--

test for slopetest for slope

State the Hypotheses:State the Hypotheses:

HH00:: 11 = 0= 0

HHaa:: 11 0 0

(usually(usually -- but the test could be onebut the test could be one--sided and the hypothesizedsided and the hypothesizedvalue does not have to be 0.)value does not have to be 0.)

Check Conditions:Check Conditions:LL linear scatterplotlinear scatterplot

II independentindependent yyvaluesvalues

NENE normal distribution of the errorsnormal distribution of the errors

SS same spread around LSR linesame spread around LSR line

SS SRSSRS


10/12

Do the Math:Do the Math:

dfdf==nn -- 22 PP--ValueValue

Conclusion in Context.Conclusion in Context.

1

11

bs

bt

F

!


11/12


12/12

Documents

15 Power Point Part 2