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Publication Number: FHWA-HRT-05-042
Date: October 2005

Safety Effects of Differential Speed Limits

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APPENDIX F: THEORETICAL CONSIDERATIONS IN THE COMPUTATION OF CONFIDENCE INTERVALS FOR THE 85TH PERCENTILE SPEED

It was informally suggested to the investigators that a question arises as to how to determine the confidence intervals for the 85th percentile speed.(28) The team considered this question and developed the following derivation. To determine confidence intervals associated with a mean speed, for example, investigators generally use the formula shown in figure 47:

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Figure 47. Equation. Formula to determine confidence intervals associated with mean speed.

Thus, for a sample of n=200 vehicles, a standard deviation of 4.82 km/h (3 mi/h), a mean speed of 80.5 km/h (50 mi/h), the 95 percent confidence interval becomes:

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Figure 48. Equation. Example of formula in figure 47.

Suppose for an instant that speeds are represented perfectly by the normal distribution, such that the 85th percentile speed is 85.4 km/h (53.11 mi/h). If the investigator applies this same equation to compute a 95 percent confidence interval for that 85th percentile speed, then the investigator computes the confidence interval as being:

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Figure 49. Equation. Confidence interval for 85th percentile speed.

If the investigator can assume that the central limit theorem still applies to the question of determining an 85th percentile speed, then figure 49 should still be applicable. However, it may be the case that the investigator cannot presume that the central limit theorem will hold at relatively small n; for example, it may be the case that at the upper tail of the normal distribution, the odds of observing a vehicle are relatively small. One way to explore the implications of this is to use the binomial distribution, where the investigator says that there are two groups of speeds: group 1 (the vehicles traveling below the 85th percentile speed) and group 2 (the vehicles traveling above the 85th percentile speed). Thus, assuming an observed 85th percentile speed of 85.4 km/h (53.11 mi/h), a vehicle has an 85 percent probability of being in group 1 and a 15 percent probability of being in group 2. The investigator can establish 95 percent confidence bounds for the binomial distribution using figure 50.

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Figure 50. Equation. Binomial distribution.

The interpretation of figure 50 is that while investigators may guess that about 0.8500 of the vehicles fall into group 1, they are 95 percent confident that at least 0.8005 of the vehicles fall into this group and that no more than 0.8995 of the vehicles fall into this group. The investigators can then map each of these proportions shown in figure 50 back to the normal distribution. For example, the researcher finds that the value on the normal distribution curve corresponding to a cumulative frequency of 0.8005 of the vehicles (assuming a mean of 80.5 km/h (50 mi/h) and a standard deviation of 4.8 km/h (3 mi/h)) is 84.5 km/h (52.53 mi/h). Similarly, the investigator finds that the value on the normal distribution curve corresponding to a cumulative frequency of 0.8995 of the vehicles is 86.6 km/h (53.84 mi/h). Thus, the 95 percent confidence bounds are 84.5 km/h (52.53 mi/h) to 86.6 km/h (53.84 mi/h) and shaded in table 33 for a sample size of 200.

It is apparent that the bounds from the binomial assumption (84.5 km/h (52.53 mi/h) to 86.6 km/h (53.84 mi/h)) are wider than the range based on the normal assumption in figure 49, presuming a sample size of n=200. Table 33 shows that as n increases, the range given by the normal assumption and the binomial assumption become very similar.

Table 33. 95 Percent confidence intervals for the 85th percentile speed.*

  n=30 n=200 n=2000
Lower Bound Upper Bound Lower Bound Upper Bound Lower Bound Upper Bound
Normal
Assumption
52.04 54.18 52.69 53.53 52.98 53.24
Binomial
Assumption
51.77 56.03 52.53 53.84 52.91 53.32

*Assumes a mean speed of 80.5 km/h (50 mi/h) and a standard deviation of 4.8 km/h (3 mi/h).

Examination of table 33 shows that if the "binomial assumption" based on figure 52 is correct, yet the investigator uses the normal assumption as was done in this study, then the error the investigator may make is to assume significant differences when, in fact, such differences are not significant. Fortunately, this study, which used the normal assumption, tended to not find significant differences in the 85th percentile speed, as shown in tables 9 and 11, except in the cases of Idaho where the p value was an extremely low 0.000. Thus, it appears that despite some theoretical imperfections in consideration of the 85th percentile speed, the use of the analysis of variance to detect significant differences was practically appropriate.

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