AP Statistics Section 15 B
Estimating the Parameters
To estimate the true regression line  y    x ,
we will use the calculated least-squares
regression line yˆ  a  bx . The y-intercept, a, will
be an unbiased estimator of the true yintercept,  , and the slope, b, is an unbiased
estimator of the true slope,  .
The remaining parameter of the model is the standard
deviation,  , which describes the variability of the
response y about the true regression line. We will
estimate the unknown standard deviation  by a
sample standard deviation of the residuals (i.e. the
standard error about the least-squares line)
s
 yi  yˆ 
 residuals

n2
n2
2
2
L3  L RESID ENTER
Get L RESID through LIST
STAT CALC 1 - Var Stats : L 3 ENTER
To find s directly :
STAT TESTS LinRegTTest
Choose correct lists but nothing else matters at this point
The slope,  , of the true regression line is usually the most
important parameter in a regression problem. The confidence
interval for  has the familiar form: estimate  t  SEestimate . Because
b is our estimate, the confidence interval becomes __________.
b  t  SEb
In this expression, the standard error of the least-squares slope b
is
SEb 
s
 x  x 
2
When we do hypothsis testing :
b
test statistic : t 
SE b
b
SE b 
t
Find using LinRegTTest
and t* is the critical value for the t(n – 2) density curve with area
C between –t* and t*.
Example: Construct and interpret a 95% confidence
interval for the slope of the true regression line for the
crying baby/IQ scenario.
children
The population of interest is __________
Conditions:
No mention of whether that sample was a SRS. If not
results may not generalize to the population.
Relationship appears linear
Residuals appear approx normally distibuted but we
do have an outlier.
The residuals are fairly constant from small x to large x
Example: Construct and interpret a 95% confidence
interval for the slope of the true regression line for the
crying baby/IQ scenario.
SEb 
1.493
 .487
3.065
1.493  2.042(.487)
.95
.025
df  38  2  36
t   2.042
.499,2.487 
We are 95% confident that the mean IQ score increases by between
.499 and 2.487 points for each increase of 1 in the crying intensity.
The figure below shows the basic output for the crying study
from the regression command in the Minitab software package.
Regression Analysis
The regression equation is
IQ = 91.3 + 1.49 Crycount
Predictor
Constant
Crycount
S = 17.50
Estimates 
Coef
91.268
1.4929
StDev
8.934
0.4870
SEb
T
10.22
3.07
P
0.000
0.004
Test
Statistic
P  value