Page 18-55
Confidence intervals for the slope (Β) and intercept (A):
• First, we obtain t
n-2,
α
/2
= t
3
,
0.025
= 3.18244630528 (See chapter 17 for a
program to solve for t
ν
,a
):
• Next, we calculate the terms
(t
n-2,
α
/2
)⋅s
e
/√S
xx
= 3.182…⋅(0.1826…/2.5)
1/2
= 0.8602…
(t
n-2,
α
/2
)⋅s
e
⋅[(1/n)+x
2
/S
xx
]
1/2
=
3.1824…⋅√0.1826…⋅[(1/5)+3
2
/2.5]
1/2
= 2.65
• Finally, for the slope B, the 95% confidence interval is
(-0.86-0.860242, -0.86+0.860242) = (-1.72, -0.00024217)
For the intercept A, the 95% confidence interval is (3.24-2.6514,
3.24+2.6514) = (0.58855,5.8914).
Example 2
-- Suppose that the y-data used in Example 1 represent the
elongation (in hundredths of an inch) of a metal wire when subjected to a
force x (in tens of pounds). The physical phenomenon is such that we expect
the intercept, A, to be zero. To check if that should be the case, we test the
null hypothesis, H
0
: Α = 0, against the alternative hypothesis, H
1
: Α ≠ 0, at the
level of significance α = 0.05.
The test statistic is t
0
= (a-0)/[(1/n)+x
2
/S
xx
]
1/2
= (-0.86)/ [(1/5)+3
2
/2.5]
½
= -
0.44117. The critical value of t, for ν = n – 2 = 3, and α/2 = 0.025, can
be calculated using the numerical solver for the equation α = UTPT(γ,t)
developed in Chapter 17. In this program, γ represents the degrees of
freedom (n-2), and α represents the probability of exceeding a certain value
of t, i.e., Pr[ t>t
α
] = 1 – α. For the present example, the value of the level of
significance is α = 0.05, g = 3, and t
n-2,
α
/2
= t
3,0.025
. Also, for γ = 3 and α =
0.025, t
n-2,
α
/2
= t
3,0.025
= 3.18244630528. Because t
0
> - t
n-2,
α
/2
, we cannot
reject the null hypothesis, H
0
: Α = 0, against the alternative hypothesis, H
1
: Α
≠ 0, at the level of significance α = 0.05.
This result suggests that taking A = 0 for this linear regression should be
acceptable. After all, the value we found for a, was –0.86, which is relatively
close to zero.
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