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Unformatted text preview: ample 2. Refer to example 1.
n 1
SSyy = 26 − 102 = 6,
5 2
yi = 26,
i=1 SSE = 6 − 0.7 · 7 = 1.1, s2 = 1.1
= 0.3667,
5−2 s = 0.605. The test for slope 1 .
In view of four assumptions about ε2 the sampling distribution of the
ˆ
least squares estimator β1 will be normal with mean β1 and standard
deviation
σ
s
σβ1 = √
≈ sβ1 = √
.
ˆ
ˆ
SSxx
SSxx
ˆ
s ˆ is called the estimated standard error of the least squares slope β1 .
β1 Usually σ is unknown and appropriate test statistic is a tstatistic
t= ˆ
β
√1
.
s/ SSxx Let we’d like to test
H0 = {β1 = 0}.
Alternative hypothesis is
Onetailed test
Ha = {β1 < 0} (or Ha = {β1 > 0}) and reject region is
RR = {t < −tα } (or Ha = {t > tα }), where tα are based on (n − 2) degree of freedom and we can ﬁnd it using
Table VI (McClave and Benson).
Twotailed test
Ha = {β1 = 0}
and reject region is
RR = {t > tα/2 }.
5 A. Zhensykbaev Regression Example 3. Refer to example 1. Apply twotailed test with α = 0.05
for H0 = {β1 = 0}.
√
ˆ
S Sxx = 10 = 3.162
β1 = 0.7, s = 0.605,
ˆ
β
0.7 · 3.162
√1
=
= 3.66,
tα/2 = t0.025 = 3.182.
0.605
s/ SSxx
Since t = 3.66 > 3.182 = t0.025 , we reject H0 . Hence, our data don’t
contradict the linear dependence of y on x.
t= A 100(1 − α)% conﬁdence interval.
Another way to make inferences about the slope β1 is a conﬁdence
interval:
s
ˆ
β1 ± tα/2 · sβ1 ,
,
sβ1 =
ˆ
ˆ
SSxx
where tα/2 is based on (n − 2) degree of freedom.
In considered example 95% conﬁdence interval is
0.605
ˆ
β1 ± t0.025 · sβ1 = 0.7 ± 3.182 √
= 0.7 ± 0.61
ˆ
10
β1 ∈ (0.09, 1.31). Measures of strength of a relationship.
A bivariate relationship describes a relationship between two variables x
and y . A numerical measure of it is provided by the coeﬃcient of correlation.
Coeﬃcient of correlation r is a measure of the strength of the linear
relationship between x and y .
r= SSxy
.
S Sxx SSyy r=√ 7
= 0.904.
10 · 6 For considered case 6 A. Zhensykbaev Regression Coeﬃcient of determination r2 is another measure of the streng...
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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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