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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.