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Publication Number: FHWA-HRT-05-042
Date: October 2005
Safety Effects of Differential Speed Limits
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APPENDIX E. DETERMINING THE SAMPLE SIZE
Although the speeds of individual vehicles are not available, mean speeds from speed monitoring stations in Idaho are shown in table 31 below. For example, in 1991, vehicle speeds were monitored at 24 sites in Idaho, where each site measured an unknown number of vehicles. The mean speeds from these 24 sites were 104 km (64.66 mi/h).
Table 31. Summary Idaho data from speed sampling sites.
Test for Significant Differences Using the Sample Size as the Number of Sites
Visually, figure 1 generally shows an upward trend in speeds. Statistically, however, there is not always a significant different in mean speeds of the individual site means. For example, compare the two shaded rows that contrast 1991 and 1995. With Nx=24, Ny=36, and Ux - Uy=65.59 - 64.66, there is no statistically significant difference as measured by the equation in figure 45. The logical conclusion is that there is no statistically significant difference in the mean speed of speed sampling site means.
Test for Significant Differences Using the Sample Size as the Number of Vehicles
While the last sentence in the above paragraph is correct, it is not very meaningful in practice. The problem is that the sites shown in figure 45 (e.g., the 24 sites that comprise the initial 1991 year) are not individual speeds. Rather, they are means of individual vehicles. That is, the 24 sites from 1991 do not represent 24 vehicles. Rather, they represent far more vehicles. Unfortunately, there are two pieces of information that prevent researchers from performing an exact statistical test on the means of individual speeds:
Simulating the Speed Variances
To estimate the types of individual speed distributions that might give rise to those shown in figure 45, the speeds of individual vehicles were simulated for two hypothetical sites. Each site initially had 1,365 speeds generated within a predefined range. Because normality could not be guaranteed at the site, the individual vehicle speeds were generated in the form of a nonnormal distribution. Figure 46 shows a histogram for the individual speeds at site X and site Y, with each bin being 1.6 km/h (1 mi/h) in width.
This simulation showed standard deviations of 14.61 when 25 speeds were used and standard deviation of 14.31 when 1,365 speeds were used. Thus, for a normal distribution, the standard deviation would likely be lower.
Determining the Difference in Mean Speeds Needed to Show Significant Difference Based on the Number of Vehicles
Returning to figure 3 and using the larger standard deviation of 14.61, researchers may ask at what point a difference in mean speeds leads to a significant difference. Table 32 summarizes these results. With just 100 vehicles, for example, a difference of 3.94 mi/h is required to show a significant difference, whereas with 10,000 vehicles, the difference need be only 0.62 km/h (0.39) mi/hour.
Table 32. Sample sizes required to achieve significant differences.
Researchers can compare these data to the number of vehicles available for each State, summarized as follows:
Thus, if researchers were to base the sample size on the number of vehicles, the results for Indiana and Virginia would change from an insignificant difference to a significant difference, as noted in the body of the report.
This analysis considers only statistical significance, which is a different issue from practical significance. In fact, table 32 may be carried to its illogical extreme in the sense that a sample size of 100,000 would mean that a mean speed difference of about one tenth of one mi/h indicated a statistically significant difference. Yet, intuitively, researchers would argue that such a difference is not meaningful because the difference is so tiny that it cannot be readily observed. As researchers extend table 32 to a larger number of vehicles, there comes a point at which the test of statistical significance is not a useful indicator for what is being measured: whether a change in speeds has significant meaning. The question becomes, then, at what point does that occur? One way to address this issue is to realize that by using the number of sites as the sample size n, the researchers prevent themselves from reaching this point. In short, using the number of sites gives a practical way of ensuring that statistically significant differences have a practical meaning. Another way of summarizing this issue is to state that by using the number of sites as n, researchers raise the bar for the test of significant differences.