Testing Hypothesis about
Coefficients
n
Use ttests for individual coefficients
n
Shows if a specific independent variable is
relevant
n
Hypotheses:
n
H0: βj = 0 (no linear relationship)
n
H1: βj ≠ 0
(linear relationship does exist
between xj and y)
Ch. 1220
12.4
Evaluating Individual
Regression Coefficients
H0: βj
= 0 (no linear relationship)
H1: βj ≠ 0
(linear relationship does exist
between xi and y)
Test Statistic:
(
df = n – k – 1)
j
b
j
S
0
b
t

=
(continued)
Ch. 1221
Evaluating Individual
Regression Coefficients
Regression Statistics
Multiple R
0.72213
R Square
0.52148
Adjusted R Square
0.44172
Standard Error
47.46341
Observations
15
ANOVA
df
SS
MS
F
Significance F
Regression
2
29460.027
14730.013
6.53861
0.01201
Residual
12
27033.306
2252.776
Total
14
56493.333
Coefficients
Standard Error
t Stat
Pvalue
Lower 95%
Upper 95%
tvalue for Price is
t = 2.306, with
pvalue .0398
tvalue for Advertising is t = 2.855,
with pvalue .0145
(continued)
Ch. 1222
H0: βj = 0
H1: βj
≠
0
d.f. = 1521 = 12
α
= .05
t12, .025 =
2.1788
The test statistic for each variable falls in the
rejection region (pvalues < .05)
There is evidence that both Price and
Advertising affect pie sales at
= .05
From Excel output:
Reject H0 for each variable
Coefficients
Standard Error
t Stat
Pvalue
Price
24.97509
10.83213
2.30565
0.03979
Advertising
74.13096
25.96732
2.85478
0.01449
Decision:
Conclusion:
Reject H0
Reject H0
α
/2=.025

tα/2
Do not reject H0
0
tα/2
α
/2=.025
2.1788
2.1788
Example: Evaluating Individual
Regression Coefficients
Ch. 1223
Confidence Interval Estimate
for the Slope
Confidence interval limits for the population slope βj
Example:
Form a 95% confidence interval for the effect of
changes in price (x1) on pie sales:
24.975 ± (2.1788)(10.832)
So the interval is
48.576 < β1 < 1.374
j
b
α/2
1,
K
n
j
S
t
b


±
Coefficients
Standard Error
Intercept
306.52619
114.25389
Price
24.97509
10.83213
Advertising
74.13096
25.96732
where t has
(n – K – 1) d.f.
Here,
t has
(15 – 2 – 1) = 12
d.f.
Ch. 1224
Confidence Interval Estimate
for the Slope
Confidence interval for the population slope βi
Example:
Excel output also reports these interval endpoints:
Weekly sales are estimated to be reduced by between 1.37 to
48.58 pies for each increase of $1 in the selling price
Coefficients
Standard Error
…
Lower 95%
Upper 95%
Intercept
306.52619
114.25389
…
57.58835
555.46404
Price
24.97509
10.83213
…
48.57626
1.37392
Advertising
74.13096
25.96732
…
17.55303
130.70888
(continued)
Ch. 1225
Test on All Coefficients
n
FTest for Overall Significance of the Model
n
Shows if there is a linear relationship between
all
of the
X
variables considered together and
Y
n
Use F test statistic
n
Hypotheses:
H0: β1 = β2 = … = βk = 0
(no linear relationship)
H1: at least one
βi
≠ 0
(at least one independent
variable affects Y)
Ch. 1226
12.5
FTest for Overall Significance
n
Test statistic:
where F has
k
(numerator) and
(n – K – 1)
(denominator)
degrees of freedom
n
The decision rule is
1)
K
SSE/(n
SSR/K
s
MSR
F
2
e


=
=
α
1,
K
n
k,
0
F
F
if
H
Reject


Ch. 1227
FTest for Overall Significance
6.5386
2252.8
14730.0
MSE
MSR
F
=
=
=
Regression Statistics
Multiple R
0.72213
R Square
0.52148
Adjusted R Square
0.44172
Standard Error
47.46341
Observations
15
ANOVA
df
SS
MS
F
Significance F
Regression
2
29460.027
14730.013
6.53861
0.01201
Residual
12
27033.306
2252.776
Total
14
56493.333
Coefficients
Standard Error
t Stat
Pvalue
Lower 95%
Upper 95%
(continued)
With 2 and 12 degrees
of freedom
Pvalue for
the FTest
Ch. 1228
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 Spring '11
 JenishNazgul
 Statistics, Regression Analysis, R Square, Ch.