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Immigration and Naturalization Service's Deferred Inspections at Airports
Report Number I-2001-29
September 2001


APPENDIX III

Statistical Sampling Model

The statistical sampling universe for our model was defined as INS deferred inspections (sampling units) at nine air ports of entry (POE) [redacted]. The deferrals issued at these airports represented 70 percent of all deferrals granted during the period covered by our audit. The defined universe contained 7,443 deferred inspections (sampling units), and our sample test results were projected only to our defined universe. Our model did not contain the deferred inspections at foreign airports or smaller domestic airports (representing 30 percent of deferred inspections), and we did not project specific results to those airports. Nevertheless, in our judgment, the policy recommendations contained in this report are appropriately addressed to the deferred inspection process taken as a whole because: (1) our coverage was substantial, and (2) even in the unlikely event that no errors existed in small or foreign airports, the statistically computed error rate would still be significant.

Because the number of deferred inspections varied widely from POE to POE, we used a stratified random sampling method to provide effective coverage of the units and to obtain precise estimates of the characteristics tested. Each unit was tested for multiple characteristics, which we term as discrete variables involving nominal measures. Thus each unit was tested for multiple variables. Statistical analysis was conducted on the test results of twelve variables. Two of these twelve variables were nested in another variable. Statistical estimates of the totals and the 95 percent confidence intervals were computed on all variables, except the nested variables.11 On the nested variables, the estimated number of items was computed using the formula (C) below. We present both lower and upper 95 percent confidence limits on the expected total number by using the formulas (A) and (B) given below.

From the universe of 7,443 units we tested a random sample of 725 units (a sampling fraction of 9.74 percent). The randomly sampled units were spread across all nine air POEs, which were considered for the sampling universe. The sample test results were projected to the universe of 7,443 deferred inspections.

The table below provides the details of the test results and projections for the random variables tested. Following the table are the mathematical model notations, and formulas used to compute the estimates of expected values and variances.

Test Results and Projections for the Random Variables Tested

Question (Variable Tested) Answer (Results of Test) Instances in the Random Sample of 725 % of Instances in the Random sample of 725 Projection at 95% Lower Confidence Limit 12 Projection at 95% Upper Confidence Limit 13
1. Did individual appear for the deferred inspection? YES50069.0%  
NO7910.9%652979
DOCUMENTATION LACKING14620.1%13111724
2. Did I-546 indicate that INS retained individual's documentation? YES38252.7%  
NO19827.3%17872245
DOCUMENTATION LACKING14520.0%13381688
3. Was onward office address accurate on I-546? YES41357.0%  
NO16823.2%14991908
DOCUMENTATION LACKING14419.9%13271678
4. Was date (day of the week) accurate on I-546? YES57679.4%  
NO172.3%93248
DOCUMENTATION LACKING13218.2%11971554
5. Was the time accurate on I-546? YES44261.0%  
NO699.5%565860
DOCUMENTATION LACKING14119.4%12941648
TNS7310.1%576857
6. Was the completed onward office copy of the I-546 on file at the POE? YES11215.4%  
NO61384.6%61476499
7. Is there evidence that "deterrents" were used? YES17223.7%  
NO33746.5%31853636
DOCUMENTATION LACKING21629.8%21212409
8. Was case listed on IO95 summary report? YES46263.7%  
NO26336.3%24542939
9. Did the I-546 specifically state what was needed to complete inspections? YES52872.8%  
NO537.3%399662
DOCUMENTATION LACKING14419.9%13261679
10. Was a Form I-94 issued to individuals for deferral at the POE? YES44961.9%  
NO27638.1%25943043
If form I-94 was issued, indicate the disposition as recorded in NIIS: ITEMS WITH "DE"18841.9%  
ITEMS WITH "DT"24755.0%ESTIMATED VALUE 398 
ALL OTHER ITEMS143.1%ESTIMATED VALUE 23 
If form I-94 was issued, was NIIS Updated? YES255.6%  
NO42494.4%ESTIMATED VALUE 684 

Mathematical Model Notations and Formulas Used to Compute the Estimates of Expected Values and Variances

The mathematical model notations, and formulas used to compute the unbiased estimates of expected values, totals, and the variances are as follows:

HThe number of strata
NhThe number of units in the stratum h, where
nhThe number of units sampled from the stratum h
= fhThe sampling fraction in the stratum h
yhijRandom variable j corresponding to ith sampled item from the hth stratum
yhijk = 1if the randomly selected unit i in the stratum h has the value of type k for random variable j
0 otherwise
nhjkThe number of units with value of type k for the variable j in the stratum h
nhjk
 Estimate of the total number of items of type k in the universe corresponding to the jth variable
phjk Sample proportion of hits of kth type of the jth variable in the hth stratum

                          

To compute the variance of the estimate the formulas used are as follows.

             

             

The 95% lower confidence limit on the estimate is given by

                 (A)

and 95% upper confidence limit on the estimate is given by

                (B)

To estimate rates in nested environment, we used the following formulas.

Let k2 type for a variable j2 is nested in the k1 type of the variable j1. Then the estimated number of occurrences of the type k2 for the variable i2 is given by the following formula.

                 (C)


Footnotes

  1. Nested random variables depend on the event occurrence of another random variable. As such, sample selection probabilities of the nested variables are not available. Consequently, we computed estimated values for the occurrences of the events of these variables assuming the event-occurring rate of the nested variables within the sample and the universe is similar.

  2. This is the most conservative projection at the 95% confidence level. In other words, the projected number of instances is at least that many in the universe. We use the most conservative projection throughout the body of the audit report.

  3. This projection is provided for information purposes. The projected number of instances is at most that many in the universe.


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