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where
X
0
= a selected value of
X
, and degrees of freedom for
t
equal the degrees of freedom for
residue mean square. In the example we had
ˆ
Y
=
0 13606
.
+
0 005962
.
X
Residualmean square
=
0.01118 with 60 degrees of freedom
n
=
62
X
=
49 .935
∑
=
2
59 397 6775
,
.
Y
= 0.303, and 95% confidence limits
So, if we pick
X
0
= 28 we have
(
)
2
XX
x
−
1
ˆ
0
Yt
±
(
Residualmean square)
1
+
+
∑
2
n
For other values of
X
0
we would get:
95% Limits
Lower
Y
X
0
Upper
8
0.184
0.139
0.229
49.1935
0.429
0.402
0.456
70
0.553
0.521
0.585
90
0.673
0.629
0.717
Note that these are confidence limits for the regression of
Y
on
X
(see Figure 7.4). They indi-
cate the limits within which the true mean of
Y
for a given
X
will lie unless a 1-in-20 chance has
occurred. The limits do not apply to a single predicted value of
Y
. The limits within which a single
Y
might lie are given by
0.80
0.60
Ŷ
= 0.13606 + 0.005962X
0.40
0.20
95-percent confidence limits
0
20
40
60
80
100
X
FIGURE 7.4
Confidence limits for the regression of
Y
on
X
.
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