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which represents the proportion of the total variability around y that is
explained by the regression line. The coeﬃcient of determination is the
proportion of total sample variability explained by the linear relationship
SSyy − SSE
and it is equal to the square of the simple linear coeﬃcient of correlation r:
r2 = 1 − SSE
SSyy (yi − yi )2
(yi − yi )2
1. A high correlation does not necessarily imply that a causal relationship
exists between x and y - only that a linear trend may exist;
2. A low correlation does not necessarily imply that x and y are unrelated
- only that x and y are not strongly linearly related.
Let we need to predict the future possible mean value of y for a speciﬁc
value of x, say xp .
The (1 − α)% conﬁdence interval for mean value of y at the point
x = xp is deﬁned as
y ± tα/2 · s ·
ˆ 1 (xp − x)2
where y = β0 + β1 xp , tα/2 is based on (n − 2) degree of freedom.
The (1 − α)% prediction interval for an individual new value of
y at the point x = xp is
y ± tα/2 · s ·
ˆ 1+ ¯
1 (xp − x)2
where y = β0 + β1 x, tα/2 is based on (n − 2) degree of freedom.
7 A. Zhensykbaev Regression Remark. The term "prediction interval" is used when the interval
formed is intended to enclose the value of a random variable. The term
"conﬁdence interval" is reserved for estimation of population parameters.
Example 4. Refer to example 1.
a. Find 95% conﬁdence interval for mean monthly sales when the appliance store spends $600 on advertaising.
b. Using 95% prediction interval predict the monthly sales for next
month, if $600 is spent on advertaising.
Solution. For $600 advertaising x = xp = 6 and
y = 0.7x − 0.1.
Then y = 0.7 · 6 − 0.1 = 4.1.
a. The 95% conﬁdence interval for the mean value of y at the point
x = xp = 6 is (α = 0.05)
4.1 ± 3.182 · 0.605 · 1 (6 − 3)2
= 4.1 ± 2.019.
10 Thus, the expected mean value of sales revenue will belong to the interval
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