Page 370 - Fundamentals of Probability and Statistics for Engineers

Linear Models and Linear Regression 353

1=2
8 9
" # 1
n
< =
X 2
t n 2;
=2 b 2 x i x† : 11:44†
: ;
iˆ1
Similarly, significance tests about the value of can be easily carried out with
^
use of A as the test statistic.
An important special case of the above is the test of H 0 : ˆ 0 against
H 1 : 6ˆ 0. This particular situation corresponds essentially to the significance
test of linear regression. Accepting H 0 is equivalent to concluding that there is
no reason to accept a linear relationship between E Y and x at a specified
f g
significance level
. In many cases, this may indicate the lack of a causal
f g
relationship between E Y and independent variable x.
Example 11.5. Problem: it is speculated that the starting salary of a clerk is a
function of the clerk’s height. Assume that salary (Y ) is normally distributed and
its mean is linearly related to height (x); use the data given in Table 11.3 to test
f g
the assumption that E Y and x are linearly related at the 5% significance level.

Table 11.3 Salary, y (in $10 000), with height, x (in feet),

for Example 11.5
x 5.7 5.7 5.7 5.7 6.1 6.1 6.1 6.1
y 2.25 2.10 1.90 1.95 2.40 1.95 2.10 2.25
Answer: in this case, we wish to test H 0 : ˆ 0 against H 1 : 6ˆ 0, with
ˆ 0:05.
From the data in Table 11.3, we have

1
n X
n
" #" #
^
X
ˆ x i x† y i y† x i x† 2
iˆ1 iˆ1
ˆ 0:31;
t n 2;
=2 ˆ t 6;0:025 ˆ 2:447; from Table A.4;
" #
n n
1 X 2 X 2
b 2
ˆ y i y† ^ 2 x i x† ˆ 0:02;
n 2
iˆ1 iˆ1
n
X 2
x i x† ˆ 0:32:
iˆ1
According to Equation (11.44), we have
1=2
8 9
" # 1
n
< =
X 2
t 6;0:025 b 2 x i x† ˆ 0:61:
: ;
iˆ1

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