U.S. Department of Transportation
Federal Highway Administration
1200 New Jersey Avenue, SE
Washington, DC 20590
This section describes the steps to conduct a stratified random sampling of registrations.
Determining sample size. To draw statistical inference about whether a State is in compliance with proof of payment required under 23 U.S.C. 141(c), a sample must be drawn from the population. Outcomes for the sample drawn from the population are either that proof of payment is demonstrated, or proof of payment is not demonstrated. Such a population is known as a binomial distribution. The probability of proof of payment compliance is p, while the probability of noncompliance is (1-p) which could be stated as q. The mean of the population is then p * q. The sample drawn from the population, if drawn randomly, can provide the probabilities p and q without bias. Multiplying q times the total population (N) of heavy vehicles in a State can provide an estimate of the number of vehicles without proof of payment. An approximate confidence interval for the number of vehicles without proof of payment can be calculated by:
|Equation 1.0||q * N - 2 * √ p * q / n = L1|
|Equation 2.0||q * N + 2 * p √ pq / n = L2|
|Where||L1 = the lower limit and L2 is the upper limit with a 95 percent confidence interval q = the probability of noncompliance p = the probability of compliance N = the total population size n = the sample population.|
Dividing each of the limits by the population N will provide a 95 percent confidence interval for the percent of noncompliance.
The size of the sample required depends on several variables. For a confidence interval as described above, there must be 60 registrations in the sample size. The "2" is the t value associated with equation 1 and 2 above and a 95 percent confidence level and an unknown population size. The size of the difference between L1 and L2 is not limited and thus quite a wide range could exist. However, if you want to limit the confidence interval to a certain size, use this formula1:
|Equation 3.0||SS = Z2 * (p * q) / C2|
|Where||SS = Sample size Z = confidence level or Z statistic p = probability of compliance q = probability of noncompliance C = confidence interval|
The confidence level or Z statistic (for a known population) for a 95 percent confidence level is 1.96. The confidence interval is the desired percent difference you would like between the upper and lower limit as a fraction. For example, if you want a 3 percent range, C would be 0.03. For a binomial distribution, the sample size will depend upon the expected probability of compliance. For example, if a 95 percent confidence level is desired with a 3 percent confidence interval and the probability of compliance is 97 percent, the sample size is approximately 124. However, if the expected probability of compliance drops to 90 percent, the sample size jumps to 384. One additional adjustment can be made to the formula. If the population size is known, the sample size can be corrected by the formula:
|Equation 4.0||CSS = SS / (1 + (SS / N))|
|Where||CSS = corrected sample size SS = sample size from equation 3 N = known population size|
Drawing a random sample. Draw a random sample by obtaining a random number table or random number generator and generating a set of random numbers of the size of your required sample size, n. The numbers generated are the registrations that would be drawn from the population. For example, if five random numbers were generated for a population of 10,000, such as 433, 9,111, 17, 12, and 5,544, the 12th, 17th, 433rd, 5,544th, and the 9,111th observation would be pulled from the population.
Stratification. There are two approaches to selecting samples from stratified populations. The first is to draw the sample and place the registrations into each group or stratification. The second approach is to determine the appropriate groupings or categories and select from each group. For HVUT, several categories might be of interest: size of operation, exemptions by type, size of vehicle, and registrations that occur lawfully without proof of payment (e.g., registered within 60 days of purchase). There may be other categories depending upon the State's procedures for registering vehicles. Thus, categories of interest need to be determined for each State. Agree on the approach at the preliminary meeting between division and Washington office personnel.
In the first approach, if there are stratifications in the population that would be useful to know, the simplest approach is to draw the sample, determine the compliance rates for each of the population types, and weight the compliance rate by the percentage of the population represented by that group or category. For example, if compliance appears to be associated with the size of operation, the sample can be stratified by size, e.g., less than 25, 26 to 100, and 101 and greater registrations per owner. Any breakdown of ownership size can be used if it appears that compliance differs by a particular size of operation.
In the second approach, the groups that are of interest would be determined during compliance review preparation and agreed upon by the division and Washington office personnel at the preliminary meeting. Using the agreed-upon sample size, the registrations would be selected randomly from each category based upon the percentage of the total population represented by each category. In other words, if 90 percent of the population is HVUT registrations where proof of payment should be present, then 90 percent of the sample drawn should be from registrations where proof of payment should be found. The remaining 10 percent would be drawn from the exemption group. Further stratifications could be made within each group, for example by size or type of exemption. The selections are still randomly selected, but within each group. This approach provides more control over obtaining registrations for each category than does the first approach. This approach is especially useful if lower numbers of registrations are drawn for the sample population and there is a potential that registrations in a particular group may not be drawn when using a random selection methodology.
1 Sample size calculator: http://www.surveysystem.com/sscalc.htm. Date accessed: 1/24/2008.
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