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?	YES	500	69.0%
		NO	79	10.9%	652	979
		DOCUMENTATION LACKING	146	20.1%	1311	1724
2.	Did I-546 indicate that INS retained individual's documentation?	YES	382	52.7%
		NO	198	27.3%	1787	2245
		DOCUMENTATION LACKING	145	20.0%	1338	1688
3.	Was onward office address accurate on I-546?	YES	413	57.0%
		NO	168	23.2%	1499	1908
		DOCUMENTATION LACKING	144	19.9%	1327	1678
4.	Was date (day of the week) accurate on I-546?	YES	576	79.4%
		NO	17	2.3%	93	248
		DOCUMENTATION LACKING	132	18.2%	1197	1554
5.	Was the time accurate on I-546?	YES	442	61.0%
		NO	69	9.5%	565	860
		DOCUMENTATION LACKING	141	19.4%	1294	1648
		TNS	73	10.1%	576	857
6.	Was the completed onward office copy of the I-546 on file at the POE?	YES	112	15.4%
		NO	613	84.6%	6147	6499
7.	Is there evidence that "deterrents" were used?	YES	172	23.7%
		NO	337	46.5%	3185	3636
		DOCUMENTATION LACKING	216	29.8%	2121	2409
8.	Was case listed on IO95 summary report?	YES	462	63.7%
		NO	263	36.3%	2454	2939
9.	Did the I-546 specifically state what was needed to complete inspections?	YES	528	72.8%
		NO	53	7.3%	399	662
		DOCUMENTATION LACKING	144	19.9%	1326	1679
10.	Was a Form I-94 issued to individuals for deferral at the POE?	YES	449	61.9%
		NO	276	38.1%	2594	3043
	If form I-94 was issued, indicate the disposition as recorded in NIIS:	ITEMS WITH "DE"	188	41.9%
		ITEMS WITH "DT"	247	55.0%	ESTIMATED VALUE 398
		ALL OTHER ITEMS	14	3.1%	ESTIMATED VALUE 23
	If form I-94 was issued, was NIIS Updated?	YES	25	5.6%
		NO	424	94.4%	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:

H	The number of strata
Nh	The number of units in the stratum h, where
nh	The number of units sampled from the stratum h
= fh	The sampling fraction in the stratum h
yhij	Random variable j corresponding to ith sampled item from the hth stratum
yhijk =		1	if the randomly selected unit i in the stratum h has the value of type k for random variable j
		0	otherwise
nhjk	The 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)

11. 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.

12. 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.

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