Chapter 14Inference for Regression

AP Statistics14.1 – Inference about the Model14.2 – Predictions and Conditions

Two Quantitative Variables

• Plot and Interpret– Explanatory Variable and Response Variable– FSDD

• Numerical Summary– Correlation (r) – describes strength and direction

• Mathematical Model– LSRL for predicting bxay ˆ

Conditions for Regression Inference

• For any fixed value of x, the response y varies according to a Normal distribution

• Repeated responses y are Independent of each other

• Parameters of Interest: • The standard deviation of y (call it ) is

the same for all values of x. The value of is unknown t-procedures!

• Degrees of Freedom: n – 2

xy

Conditions for Regression Inference (Cont’d)

• Look at residuals: • residual = Actual – Predicted

• The true relationship is linear• Response varies Normally about the True

regression line• To estimate , use standard error about

the line (s)

Inference

• Unknown parameters: • a and b are unbiased estimators of the

least squares regression line for the true intercept and slope , respectively

• There are n residuals, one for each data point. The residuals from a LSRL always have mean zero. This simplifies their standard error.

,,

Standard Error about the Line• Two variables gives: n – 2 df (not n – 1)

• Call the sample standard deviation (s) a standard error to emphasize that it is estimated from data

• Calculator will calculate s! Thank you TI!

2

21 residualn

s

t-procedures (n - 2 df)

• CI’s for the regression slopestandard error of

the LSRL slope b is:

• Testing hypothesis of No linear relationship

• x does not predict y r = 0

bSEtb *

2)( xx

sSEb

0:0 HbSEbtstatistict :

• What is the equation of the LSRL?• Estimate the parameters• In your opinion, is the LSRL an appropriate

model for the data? Would you be willing to predict a students height, if you knew that his arm span is 76 inches?

and

•Construct a 95% CI for mean increase in IQ for each additional peak in crying

Scatter Plot and LSRL?Perform a Test of Significance

Checking the Regression Conditions

• All observations are Independent• There is a true LINEAR relationship• The Standard Deviation of the response

variable (y) about the true line is the Same everywhere

• The response (y) varies Normally about the true regression line

* Verifying Conditions uses the Residuals!