Unformatted text preview: $6119] with probability P = (1 − α) = 0.95.
b. The 95% prediction interval for an individual new value of y at the
point x = xp = 6 is
4.1 ± 3.182 · 0.605 · 1+ 1 (6 − 3)2
+
= 4.1 ± 2.79.
5
10 Therefore, we predict with 95% conﬁdence that the sales revenue next
month will fall in the interval from $1310 to $6890.
Conﬁdence strip for the mean value of y .
The conﬁdence interval for the mean value of y gives the conﬁdence strip
for the graph
y = y + β1 (x − x) ± tα/2 · s ·
¯ˆ
¯ 8 ¯
1 (x − x)2
+
.
n
SSxx A. Zhensykbaev Regression For our case
y = 2 + 0.7(x − 3) ± 3.182 · 0.605 · 1 (x − 3)2
+
.
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"  Pic. 2 Residual analysis.
The goal is to know, whether the data indicate signiﬁcant departures
from the assumptions: εi (i = 1 : n) are independent and
Estimates for errors are
ˆ
ˆ
εi = yi − (β0 + β1 xi ).
ˆ
Residuals are εi (i = 1 : n).
ˆ
To verify our assumptions we have to use statistical tests: for normality
and independence (say, chisquare and Spearman’s rank test, runs, etc.).
Multiple linear regression.
Most practical applications use regression models which are more complex than the simple linear regression. A realistic probabilistic model for
monthly sales revenue includes not only the advertising expenditure but
also such factors as season, prices, etc. One may model such situations by
the multiple linear regression.
9 A. Zhensykbaev Regression The model is
k y = β 0 + β 1 x1 + · · · + β k xk + ε = βi xi + ε,
i=0 where y is the dependent random variable, βi are parameters (unkn...
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This note was uploaded on 02/11/2014 for the course MATH 1390 taught by Professor Christopherstocker during the Spring '13 term at Marquette.
 Spring '13
 ChristopherStocker
 Math, Linear Regression, Regression Analysis

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