2702
.
0
130
1562
13
4
.
44
130
8
.
514
13
2
2
2
1
x
x
n
y
x
xy
n
b
Example
713
.
0
10
2702
.
0
4154
.
3
1
0
x
b
y
b
i
i
x
2702
.
0
713
.
0
y
ˆ
𝑛 = 13
σ
? = 130
σ
? = 44.4
σ ?? = 514.8
σ ?
2
= 1562
σ ?
2
= 171.3

X
2702
.
0
713
.
0
Y
ˆ

Interpretation of the
Intercept,
b
0
•
b
0
is the estimated average value of Y when
the value of X is zero (if x = 0 is in the range of
observed x values)
–
Here, no sparrow with age 0, so b
0
= 0.5816 has
no logical interpretation, except that it is the part
on the Y-axis where the regression line passes
through
X
2702
.
0
713
.
0
Y
ˆ

Interpretation of the
slope,
b
1
•
b
1
measures the estimated change in the
average value of Y as a result of a one-unit
change in X
Here, b
1
= 0.2791 tells us that, on the average, the
wing length of sparrows increases by 0.2791 cm
per day
X
2702
.
0
713
.
0
Y
ˆ

Least Squares Regression Properties
•
The sum of the residuals from the least squares
regression line is 0
(
)
•
The sum of the squared residuals is a minimum
(minimized
)
•
The simple regression line always passes through
the mean of the y variable and the mean of the x
variable
•
The least squares coefficients are unbiased
estimates of
β
0
and
β
1
0
)
ˆ
(
y
y
2
)
ˆ
(
y
y