2003 Motor Vehicle Occupant Safety Survey: Air Bags

APPENDIX A: PRECISION OF SAMPLING ESTIMATES*

Precision of Sample Estimates

The objective of the sampling procedures used on this study was to produce a random sample of the target population. A random sample shares the same properties and characteristics of the total population from which it is drawn, subject to a certain level of sampling error. This means that with a properly drawn sample we can make statements about the properties and characteristics of the total population within certain specified limits of certainty and sampling variability.

The confidence interval for sample estimates of population proportions, using simple random sampling without replacement, is calculated by the following formula:

Where:

 se (x) = the standard error of the sample estimate for a proportion; p = some proportion of the sample displaying a certain characteristic or attribute; q = (1 p); n = the size of the sample; z = the standardized normal variable, given a specified confidence level (1.96 for samples of this size).

The sample sizes for the surveys are large enough to permit estimates for sub-samples of particular interest. Table 56, on the next page, presents the expected size of the sampling error for specified sample sizes of 12,000 and less, at different response distributions on a categorical variable. As the table shows, larger samples produce smaller expected sampling variances, but there is a constantly declining marginal utility of variance reduction per sample size increase.

TABLE 19

Expected Sampling Error (Plus or Minus)
At the 95% Confidence Level
(Simple Random Sample)

Percentage of the Sample or Sub-Sample Giving
A Certain Response or Displaying a
Certain Characteristic for Percentages Near:

 Size of Sample or Sub-Sample 10 or 90 20 or 80 30 or 70 40 or 60 50 6,000 0.8 1.0 1.2 1.2 1.3 4,500 0.9 1.2 1.3 1.4 1.5 4,000 0.9 1.2 1.4 1.5 1.5 3,000 1.1 1.4 1.6 1.8 1.8 2,000 1.3 1.8 2.0 2.1 2.2 1,500 1.5 2.0 2.3 2.5 2.5 1,300 1.6 2.2 2.5 2.7 2.7 1,200 1.7 2.3 2.6 2.8 2.8 1,100 1.8 2.4 2.7 2.9 3.0 1,000 1.9 2.5 2.8 3.0 3.1 900 2.0 2.6 3.0 3.2 3.3 800 2.1 2.8 3.2 3.4 3.5 700 2.2 3.0 3.4 3.6 3.7 600 2.4 3.2 3.7 3.9 4.0 500 2.6 3.5 4.0 4.3 4.4 400 2.9 3.9 4.5 4.8 4.9 300 3.4 4.5 5.2 5.6 5.7 200 4.2 5.6 6.4 6.8 6.9 150 4.8 6.4 7.4 7.9 8.0 100 5.9 7.9 9.0 9.7 9.8 75 6.8 9.1 10.4 11.2 11.4 50 8.4 11.2 12.8 13.7 14.0

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NOTE:  Entries are expressed as percentage points (+ or -)

However, the sampling design for this study included a separate, concurrently administered over-sample of youth and young adults (age 16-39). Both the cross-sectional sample and the over-sample of the youth/younger adult population were drawn as simple random samples; however, the disproportionate sampling of the age 16-39 population introduces a design effect that makes it inappropriate to assume that the sampling error for total sample estimates will be identical to those of a simple random sample.

In order to calculate a specific interval for estimates from a sample, the appropriate statistical formula for calculating the allowance for sampling error (at a 95% confidence interval) in a stratified sample with a disproportionate design is:

where:

 ASE = allowance for sampling error at the 95% confidence level; h = a sample stratum; g = number of sample strata; Wh = stratum h as a proportion of total population; fh = the sampling fraction for group h the number in the sample divided by the number in the universe; s2h = the variance in the stratum h for proportions this is equal to ph (1.0 ph); nh = the sample size for the stratum h.

Although Table 19 provides a useful approximation of the magnitude of expected sampling error, precise calculation of allowances for sampling error requires the use of this formula. To assess the design effect for sample estimates, we calculated sampling errors for the disproportionate sample for a number of key variables using the above formula. These estimates were then compared to the sampling errors for the same variables, assuming a simple random sample of the same size. The two strata (h1 and h2) in the disproportionate sample were all respondents age 16-39 and all respondents age 40 and over, respectively. The proportion for the 16-39 year old stratum (w1) was 53.0 percent while the proportion for the 40 and over stratum (w2) was 47.0 percent.

As shown in Table 20, the disproportionate sampling increases the confidence interval by an average of 0.7 percent, compared to a simple random sample of the same size. This means the sample design slightly decreases the sampling precision for total population estimates, while increasing the precision of sampling estimates for the sub-sample aged 16-39 years old. Since the average difference in the confidence interval between the stratified disproportionate sample and a simple random sample is less than one percentage point, the sampling error table for a simple random sample will provide a reasonable approximation of the precision of sampling estimates in the survey.

     p= CONFIDENCE INTERVALS PERCENTAGE POINTS + AT 95% CONFIDENCE LEVEL TABLE 20 Design Effect on Confidence Intervals for Sample Estimates Between Disproportionate Sample Used in Occupant Protection Survey and a Proportionate Sample of Same Size 89.2% 0.77 0.78 1.3% 63.4% 1.21 1.23 1.7% 85.1% 0.94 0.94 ---- 33.1% 1.24 1.26 1.6% 82.7% 0.98 0.98 ---- 69.3% 1.15 1.16 .9% 63.9% 1.20 1.22 .9% 9.3% 0.73 0.72 -1.4% 23.6% 1.06 1.08 1.9% 17.2% 0.94 0.96 2.1% 74.1% 1.17 1.19 1.7% 76.5% 1.12 1.14 1.8% 8.6% 0.70 0.70 ---- 13.2% 0.84 0.81 -3.6% 48.0% 1.24 1.27 2.4% 0.7% *Total sample proportions using SRS formula Unless specified otherwise N=6180

Estimating Statistical Significance

The estimates of sampling precision presented in the previous section yield confidence bands around the sample estimates, within which the true population value should lie. This type of sampling estimate is appropriate when the goal of the research is to estimate a population distribution parameter. However, the purpose of some surveys is to provide a comparison of population parameters estimated from independent samples (e.g. annual tracking surveys) or between subsets of the same sample. In such instances, the question is not simply whether or not there is any difference in the sample statistics that estimate the population parameter, but rather is the difference between the sample estimates statistically significant (i.e., beyond the expected limits of sampling error for both sample estimates).

To test whether or not a difference between two sample proportions is statistically significant, a rather simple calculation can be made. The maximum expected sampling error (i.e., confidence interval in the previous formula) of the first sample is designated s1 and the maximum expected sampling error of the second sample is s2. The sampling error of the difference between these estimates is sd and is calculated as:

Any difference between observed proportions that exceeds sd is a statistically significant difference at the specified confidence interval. Note that this technique is mathematically equivalent to generating standardized tests of the difference between proportions.

An illustration of the pooled sampling error between sub-samples for various sizes is presented in Table 58. This table can be used to determine the size of the difference in proportions between drivers and non-drivers or other sub-samples that would be statistically significant.

Sample Size 4000 3500 3000 2500 2000 1500 1000 900 800 700 600 500 400 300 200 100 TABLE 21 Pooled Sampling Error Expressed as Percentages for Given Sample Sizes (Assuming P=Q) 14.1 10.0 7.1 5.9 5.1 4.7 4.3 4.0 3.8 3.6 3.5 3.0 2.7 2.5 2.4 2.3 2.2 14.1 10.0 7.1 5.9 5.2 4.7 4.3 4.1 3.8 3.7 3.5 3.0 2.7 2.6 2.4 2.3 14.1 10.0 7.2 5.9 5.2 4.7 4.4 4.1 3.9 3.7 3.6 3.1 2,8 2.7 2.5 14.1 10.0 7.2 6.0 5.3 4.8 4.5 4.2 4.0 3.8 3.7 3.2 2.9 2.8 14.2 10.1 7.3 6.1 5.4 4.9 4.6 4.3 4.1 3.9 3.8 3.3 3.1 14.2 10.2 7.4 6.2 5.5 5.1 4.7 4.5 4.3 4.1 4.0 3.6 14.3 10.3 7.6 6.5 5.8 5.4 5.1 4.8 4.7 4.5 4.4 14.4 10.4 7.7 6.5 5.9 5.5 5.2 4.9 4.8 4.6 14.4 10.4 7.8 6.6 6.0 5.6 5.3 5.1 4.9 14.5 10.5 7.9 6.8 6.1 5.7 5.5 5.2 14.6 10.6 8.0 6.9 6.3 5.9 5.7 14.7 10.8 8.2 7.2 6.6 6.2 14.8 11.0 8.5 7.5 6.9 15.1 11.4 9.0 8.0 15.6 12.1 9.8 17.1 13.9 19.8 50 100 200 300 400 500 600 700 800 900 1000 1500 2000 2500 3000 3500 4000 Sample Size