Transcript Chapter 14 - Inference for Regression

Chapter 14 - Inference for
Regression
Paul M. Wilson, Jr.
May 29, 2002
AP Statistics
Introduction - Chapter 3 All Over
Again!
• We use the linear regression model when a
scatterplot shows a linear relationship
between a quantitative explanatory variable
“x” and a quantitative response variable
“y”.
Do the following when given a
set of data:
• Make a scatterplot plotting the explanatory
variable “x” horizontally and the response
variable “y” vertically.
• Use a calculator to fit the LSRL to the data.
• Look for outliers and influential
observations.
• Calculate the correlation “r” and its square.
Equations to Remember in Linear
Regression
• Don’t forget the
equation for the
LSRL!
• The equation to
calculate resids is still
the same!
• We have a standard
error about the LSRL.
• Degrees of freedom!
• We have a confidence
interval for regression
slope.
• There are also
standard hypotheses
for no linear
relationship.
The Equations for Regression
Inference
• Equations for the LSRL, residuals, and the
standard error about the LSRL:
yˆ  a  bx
resids  y  yˆ
s  1 / n  2 ( y  yˆ )
2
Equations Continued
• We also have equations for degrees of
freedom and the confidence interval:
• We use n-2 degrees of freedom since we
have two variables to observe.
d . freedom  n  2

conf . int .  b  t SEb
The Null and Alternative
Hypotheses
• The null hypothesis
states that there is no
linear relationship and
is written in the form
of:
Ho : B  0
• The alternative states
that there is in fact
some linear
relationship and can be
written in one of 3
forms: H : B  0
a
Ha : B  0
Ha : B  0
3 Basic Assumptions for
Regression
• The true relationship is linear : look at the
scatterplot to check that the overall pattern
is roughly linear.
• The standard deviation of the response
about the true line is the same everywhere.
• The response varies normally about the true
regression line.
Quick Facts about the
Confidence Interval
• We use a confidence interval to estimate the
mean response.
• A confidence interval says that the interval
obtained is correct a certain percentage of
the time in repeated use.
• The confidence interval for the mean
response “u” is

yˆ  t SEuˆ
More advanced work in linear
regression can be found
throughout Chapter14!