1998 SURVEY RESULTS
APPENDIX A
*PRECISION OF SAMPLE ESTIMATES
*Reprinted from:
Boyle, J. and K. Sharp. 1998 Motor Vehicle Occupant Safety Survey: Methodology Report. DOTHS809029. Washington DC: U.S. Department of Transportation, National Highway Traffic Safety Administration.
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:
var
(x) = z
Where:
var (x)

=  the expected sampling error of the mean of some variable, expressed as a proportion 
p

=  some proportion of the sample displaying a certain characteristic or attribute 
q

=  (1  p) 
z

=  the standardized normal variable, given a specified confidence level (1.96 for samples of this size). 
n

=  the size of the sample 
The sample sizes for the surveys are large enough to permit estimates for subsamples of particular interest. Table 5, on the next page, presents the expected size of the sampling error for specified sample sizes of 8,000 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.
Expected Sampling Error (Plus or Minus) At the 95% Confidence Level (Simple Random Sample) Percentage of the Sample or Subsample Giving 

Size of Sample or Subsample 
10 or 90  20 or 80  30 or 70  40 or 60  50 
8,000  0.7  0..9  1.0  1.1  1.1 
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 
NOTE: Entries are expressed as percentage points (+ or )
However, the sampling design for this study included a separate, concurrently administered oversample of youth and young adults (age 1639). Both the crosssectional sample and the oversample of the youth/younger adult population were drawn as simple random samples; however, the disproportionate sampling of the age 1639 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; 
w_{h}

=  stratum h as a proportion of total population; 
f_{h}

=  the sampling fraction for group h  the number in the sample divided by the number in the universe; 
s^{2}_{h}

=  the variance in the stratum h  for proportions this is equal to p_{h} (1.0  p_{h}); 
n_{h}

=  the sample size for the stratum h. 
Although Table 5 above 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 (h^{1} and h^{2}) in the disproportionate sample were all respondents age 1639 and all respondents age 40 and over respectively. The proportion for the 1639 year old stratum (w^{1}) was 45.7 percent while the proportion for the 40 and over stratum (w^{2}) was 54.3 percent.
As shown in Table 6 below, the disproportionate sampling increases the confidence interval by about 2 percent, compared to a simple random sample of the same size. This means that sample design introduces almost no measurable loss in sampling precision for total population estimates, while increasing the precision of sampling estimates for the target population aged 1639 years old. Since the difference in sampling precision between the stratified disproportion sample and a simple random sample is less than one tenth of percentage point in each case, the sampling error table for a simple random sample will provide a reasonable approximation of the precision of sampling estimates in the survey.
Design Effect on Confidence Intervals for Sample Estimates Between Disproportionate Sample Used in Occupant Protection Survey And a Proportionate Sample of Same Size 

PERCENTAGE POINTS + AT 95% CONFIDENCE LEVEL 

PROPORTIONATE SAMPLING* 
PROPORTIONATE SAMPLING 
CONFIDENCE INTERVALS ABOUT ESTIMATES 

USE NEW VARIABLES  
Driven in the past year 



Drunk alcohol in past year 



Always use safety belt 



Dislike seat belts 



Always use passenger belt (front) 



Favor (a lot) seat belt laws 



Secondary enforcement 



Ever ticketed by police for seatbelt 



Recall Crash dummies 



Ever injured in vehicle accident 



Drives a car for work almost every day 



Set a good example for others (reason for using seat belts) 



Driverside only Air Bag in vehicle 



Race: Black/African American 



Ethnicity: Hispanic 



Male/Female 



AVERAGE DIFFERENCE IN CONFIDENCE INTERVALS 

*Total sample proportions using SRS formula
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 which 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. Call the total sampling error (i.e., var (x) in the previous formula) of the first sample s1 and the total sampling error of the second sample s2. Then, the sampling error of the difference between these estimates is sd which is calculated as:
sd =
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 subsamples for various sizes is presented in Table 7. This table can be used to indicate the size of difference in proportions between drivers and nondrivers or other subsamples that would be statistically significant.
Pooled Sampling Error Expressed as Percentages For Given Sample Sizes (Assuming P=Q) 

Sample Size  
4000  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 
3500  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  
3000  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  
2500  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  
2000  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  
1500  14.2  10.2  7.4  6.2  5.5  5.1  4.7  4.5  4.3  4.1  4.0  3.6  
1000  14.3  10.3  7.6  6.5  5.8  5.4  5.1  4.8  4.7  4.5  4.4  
900  14.4  10.4  7.7  6.5  5.9  5.5  5.2  4.9  4.8  4.6  
800  14.4  10.4  7.8  6.6  6.0  5.6  5.3  5.1  4.9  
700  14.5  10.5  7.9  6.8  6.1  5.7  5.5  5.2  
600  14.6  10.6  8.0  6.9  6.3  5.9  5.7  
500  14.7  10.8  8.2  7.2  6.6  6.2  
400  14.8  11.0  8.5  7.5  6.9  
300  15.1  11.4  9.0  8.0  
200  15.6  12.1  9.8  
100  17.1  13.9  
50  19.8  
50  100  200  300  400  500  600  700  800  900  1000  1500  2000  2500  3000  3500  4000  
