INFERENCES USING REGRESSION—ONE INDEPENDENT VARIABLE
A. Confidence Intervals
STANDARD ERROR*
Of the ESTIMATOR
INFERENCE TYPE
PARAMETER
ESTIMATOR
MODEL:
Is there a relationship between
X and Y?
B
b
sb  se
MEAN OF THE
CONDITIONAL
DISTRIBUTION OF Y
WHEN X=X0
Yμ | X  X0
Ŷ  a  bX0
sμ  se
Ŷ  a  bX0
Y| X  X0
A SPECIFIC VALUE OF Y
WHEN X=X0
1
 (X  X)2
1 (X0X)2

n  (X  X)2
2
1
sp  se 1  (X0X)
n  (X  X)2
CONFIDENCE
INTERVAL
B {b  tα sb}
Yμ | X  X0 {Ŷ  tα sμ }
Y | X  X0 {Ŷ  tα sp}
B. Hypothesis Testing: Is B actually non-zero? (Inference about the existence of a relationship between Y and X)
H0: B = 0 (Y is not related to X) for Ho B=1 or other values are also possible
H1: B > 0 (e.g., Y and X positively related) other alternatives such as B < 0 or B = 0 are also possible
The test statistic is
critical t value)
* Note that
b
to  bs- B  s
b
 (X  X)   X
2
(If the hypothesized value of B is 0) Compare the observed t to the appropriate t (the
b
2
2
 nX