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Chapter 14
Inference for Regression
AP Statistics
14.1 – Inference about the Model
14.2 – 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
yˆ  a  bx
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:  y    x
• 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
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)
1
2
s
residual

n2
• Call the sample standard deviation (s) a
standard error to emphasize that it is estimated
from data
• Calculator will calculate s! Thank you TI!
t-procedures (n - 2 df)
• CI’s for the regression slope 
standard error of the
LSRL slope b is:
s
b SE 
b
2
(
x

x
)

b  t * SE
• Testing hypothesis of No linear relationship
H0 :   0
b
t  statistic : t 
SEb
• x does not predict y  r = 0
• What is the equation of the LSRL?
• Estimate the parameters  and 
• 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?
•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!