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OCR for page 36

36
against n = Sample size for alternative x (before); and
m = Sample size for alternative y (after).
H1 : x - y 0
The probability of mistakenly accepting the null hypothe-
where sis is alpha (alpha is usually set to 5 percent to get a 95 percent
confidence level test). This is Type I error.
x = The mean travel time (or mean observation of some
There also is the chance of mistakenly rejecting the null hy-
other relevant measure, such as delay) for alternative pothesis. This is called Type II error and it varies with the dif-
x (before); and ference between the sample means, their standard deviation,
y = The mean for alternative y (after). and the sample size. (Analysts should consult standard statisti-
This is a two-sided t test with the following optimal rejec- cal textbooks for tables on the Type II errors associated with dif-
tion region for a given alpha (acceptable Type I error). ferent confidence intervals and sample sizes.)
1 1
x - y > t (1- /2);(n+m-2) s p + 4.5 What to Do If the
n m Null Hypothesis
where Cannot Be Rejected
x - y = The absolute value of the difference in the mean re- If the null hypothesis of no significant difference in the
sults for alternative x (before) and alternative y mean results for the before and after conditions cannot be re-
jected, the analyst has the following options:
(after);
sp = The pooled standard deviation;
1. Change to a more sensitive performance measure. For ex-
t = The Student's t distribution for a level of confidence
ample, delay will tend to be much more sensitive to im-
of (1 - alpha) and (n + m - 2) degrees of freedom;
provements than travel time because a large component of
n = Sample size for alternative x (before); and
travel time is often the free-flow travel time, which is un-
m = Sample size for alternative y (after). affected by most improvements.
(n - 1) sx2 + (m - 1) s 2
y
2. Increase the number of measurements made for the after
p =
s2 condition in the hopes that the variance of the mean will
(m - n - 2 )
decrease sufficiently to reject the null hypothesis.
3. Reduce the confidence level from 95 percent to a lower level
where
where the before and after conditions are significantly dif-
sp = Pooled standard deviation; ferent, and report the lower confidence level in the results.
sx = Standard deviation of results for alternative x (before); 4. Accept results that the improvement did not significantly
sy = Standard deviation of results for alternative y (after); improve conditions.