preface to fifteenth edition

2.136 SECTION 2
The confidence limits for the slope are given by b t b , where the t-value is taken at the desired
confidence level and (N 2) degrees of freedom. Similarly, the confidence limits for the intercept
are given by a ts a . The closeness of xˆ to x i is answered in terms of a confidence interval for x 0
that extends from an upper confidence (UCL) to a lower confidence (LCL) level. Let us choose 95%
for the confidence interval. Then, remembering that this is a two-tailed test (UCL and LCL), we
obtain from a table of Student’s t distribution the critical value of t c (t 0.975 ) and the appropriate
number of degrees of freedom.
Example 14 For the best-fit line found in Example 13, express the result in terms of confidence
intervals for the slope and intercept. We will choose 95% for the confidence interval.
The standard deviation s Y/X is given by Equation 19, but first a supplementary table must be
constructed for the Y residuals and other data which will be needed in subsequent equations.
Ŷ (Y i Ŷ) (Y i Ŷ) 2 X 2
i
108.6 2.55 6.50 1.00
114.4 0.56 0.31 1.21
120.3 0.67 0.45 1.44
138.0 1.00 1.00 2.25
149.8 3.21 10.32 2.89
161.6 3.57 12.94 3.61
173.4 0.65 0.42 4.41
179.2 2.76 7.61 4.84
191.0 4.02 16.16 5.76
208.7 2.30 5.30 7.29
220.5 0.48 0.23 8.41
226.4 0.40 0.16 9.00
61.20 52.11
Now substitute the appropriate values into Equation 19 where there are 12 2 10 degrees of
freedom:
61.20
sX/Y
2.47
q 10
We can now calculate s b and s a from Equations 20 and 21, respectively:
and
Now, using a two-tailed value for Student’s t:
s Y/X
s b 1.07
p5.31
52.11
sa
2.47 2.23
q12(5.306)
b ts 58.91 2.23(1.07) 58.91 2.39
b
a ts 49.64 2.23(2.23) 49.64 4.97
a