Debra P., a Minor, by Irene P., Her Mother and Next Friend, Plaintiffs v. Ralph D. Turlington, Individually and as Commissioner of Education

R. LANIER ANDERSON, III, Circuit Judge: A class consisting of Florida high school seniors who have failed a state-wide competency examination appeal findings entered by the district court on remand from Debra P. v. Turlington, 644 F.2d 397 (5th Cir. 1981) (Unit B). We affirm the decision of the district court. FACTS In 1978 the Florida legislature approved an amendment to the Educational Accountability Act of 1976, Fla.Stat. § 229.55 et seq., requiring Florida public school students to pass a functional literacy examination, the SSAT-II, in order to receive a state high school diploma. Fla.Stat. § 232.-246(1)(b) (1978). Shortly thereafter, present and future twelth grade students who had failed or would fail the test filed suit, challenging the constitutionality of using the test for diploma denials under the due process and equal protection clauses of the Fourteenth Amendment. They also challenged this use of the SSAT-II under Title VI of the Civil Rights Act of 1964, 42 U.S.C. § 2000d et seq. (1976), and the Equal Educational Opportunities Act (“EEOA”), 20 U.S.C. § 1703 (1976). 1 Central to their challenge was the argument that the diploma sanction had an unconstitutionally disproportionate impact on blacks. At the time of the 1979 hearing, after three test administrations, the failure rate of black students was approximately 10 times greater than that of white students. 2 *1407 The district court held that use of the SSAT-II for diploma denials violated the due process and equal protection clauses, Title VI, and the EEOA. Debra P. v. Turlington, 474 F.Supp. 244 (M.D.Fla.1979). The court enjoined the test’s use as a diploma sanction until the 1982-83 school year, but allowed the state to use the test in the interim for remediation, which the court found violated neither the Constitution nor statutes. The district court found that the SSAT-II’s content was valid, id. at 261, which would allow the state to use it as a diploma sanction after 1982. The district court issued the four-year injunction for two reasons. First, the court found that the examination violated the equal protection clause, Title VI, and the EEOA by perpetuating past discrimination against black students who had attended segregated schools for the first four years of their education. 474 F.Supp. at 250-57. The students in the high school class of 1983 would be the first to have attended physically integrated schools for all 12 years of their educational careers; thus, they would be the first students against whom the diploma sanction could be applied. Second, the court held that the test’s implementation schedule provided insufficient notice, in violation of the due process clause. Id. at 267. The court determined that because instruction of the skills necessary to complete the SSAT-II is a cumulative and time consuming process, id. at 264, four to six years should intervene between announcement of the test and implementation of the diploma sanction. Id. at 267. The class of 1983 would be the first with six years notice of the sanction and adequate instruction to take the test. On appeal, the former Fifth Circuit Court of Appeals upheld the district court’s injunction, but remanded for further findings on two issues that would affect use of the SSAT-II as a diploma sanction in 1983 and thereafter. First, the circuit court required the state to demonstrate on remand that the competency exam was “a fair test of that which is taught in [Florida’s] classrooms.” Debra P. v. Turlington, 644 F.2d 397, 408 (5th Cir.1981) (Unit B). The circuit court declared clearly erroneous the district court’s holding that the test’s content was valid; the record was “simply insufficient in proof that the test administered measures what was actually taught in the schools of Florida.” Id. at 405. Without such proof, use of the test as a diploma sanction would violate the due process and equal protection clauses. Id. at 404, 406. Second, the circuit court held that if the state was able to prove that the test tested what was “actually taught,” the state would then have to demonstrate either that the test’s racially discriminatory impact was not due to the present effects of past intentional discrimination, or that the test’s use as a diploma sanction would remedy those effects. Id. at 407-08. On remand, the district court tried the two issues separately. After trial on the first issue, the district court concluded that the state had met its burden of proving by a preponderance of the evidence that the competency examination is “instructionally valid,” i.e., a fair test of that which is taught in Florida’s schools. Debra P. v. Turlington, 564 F.Supp. 177, 186 (M.D.Fla. 1983). After an evidentiary hearing on the second issue, the court found that although vestiges of past segregation still exist to some extent, and although the test still has a racially discriminatory impact, there is no causal link between the disproportionate failure rate of black students and those present effects of past segregation. Id. at 188. The court found, moreover, that even if there were a causal connection, the defendants had carried their burden of showing that the diploma sanction would remedy those effects. Id. The propriety of these findings forms the basis for this appeal. Our opinion will refer more particularly to facts adduced at the hearings in the district court on remand. I. INSTRUCTIONAL VALIDITY Following remand, the Florida Department of Education commissioned IOX Assessment Associates, a private consulting *1408 firm, to design a study to determine whether Florida’s school districts teach the skills tested by the competency examination. IOX designed a four-part study. The first part of the study consisted of a teacher survey, which was distributed to all of Florida’s 65,000 teachers. Forty-seven thousand teachers responded to the survey. The survey asked whether the teacher had provided instruction during 1981-82 relating to the skills tested on the SSAT-II and if so, whether that instruction had been sufficient for a student to master the skills. The second part of the study was a district survey completed by all of Florida’s 67 school districts and 4 university laboratory schools. The survey requested the districts (1) to estimate in which of grades 2-12 students were taught the test skills and in which grade a majority of students would have mastered the skills, (2) to describe any major variations in instruction among schools in the district, (3) to describe any remedial programs specifically related to mastery of test skills, (4) to describe staff development activities related to teaching test skills, (5) to list instructional materials specifically used to teach test skills, and (6) to identify any program designed specifically to help students pass the test. The district survey instructed respondents to consider only the 1981-82 school year. The third part of the study was a series of site visits to verify the accuracy of the district reports. Each district was visited by at least one visitation team. The visitation teams were composed of one program auditor from the Department of Education and two administrators from school districts other than the one visited. After two days of interviewing teachers and reviewing instructional materials, a team would prepare a report of findings and impressions about the accuracy of the district report. The fourth portion of the study consisted of surveys given in a randomly selected school within each school district to students in an English or Social Studies class selected at random within the school by the visitation team. The survey asked the students whether they had been taught the skills tested by the competency exam at any point during their educational careers. In all, thirty-two hundred students completed this survey. At trial, the state introduced the study results, over the objection of the appellants that the results constituted inadmissible hearsay. The state also presented three expert witnesses who concluded on the basis of the data from the IOX study that Florida was teaching what it was testing. One expert was Dr. Popham, designer of the study, who relied in large part on a “Mastery Exposure Index” prepared by the Department of Education. The Index is a distillation of the survey data and purportedly demonstrates that each student is given an average of 2.7 opportunities over the length of his educational career to master test skills. Dr. Popham testified that one such opportunity would be enough to make the competency exam “instructionally valid” — a fair test of that which is taught. All three experts for the state testified that their conclusions of instructional validity were bolstered by the state’s extensive remedial efforts to help students who initially fail the test pass it before graduation. 3 Two experts testified on behalf of the appellants that the IOX study did not prove instructional validity. One expert opined that the survey was not a reliable indication of what had been taught to the Class of 1983 because it focused only on the 1981-82 school year, rather than the twelth graders’ entire school careers. The other expert testified that the survey demonstrated that in school districts with high rates of SSAT-II failure, students were not taught test skills. On the basis of the evidence presented at trial, particularly the evidence of remedial *1409 programs and efforts, the district court concluded that the competency exam is instructionally valid because “students are afforded an adequate opportunity to learn the skills tested on the SSAT-II before it is used as a diploma sanction.” 564 F.Supp. at 186. Appellants raise three objections to the district court’s finding that the SSAT-II is instructionally valid. A. Legal Standard First, the appellants argue that the district court applied an improper legal standard in determining the validity of the test’s content. They claim that the Fifth Circuit opinion required direct evidence that students were “actually taught” the subjects tested. In essence, they argue that the district court erred by considering circumstantial evidence. 4 We think that appellants read the prior opinion of this court too restrictively. The opinion gives no indication that in requiring that the state prove “that the test covered things actually taught in the classroom,” 644 F.2d at 405, the panel meant to limit the proof to direct evidence of classroom activities. 5 In the absence of such an indication, the normal rule that evidence, whether direct or circumstantial, may be considered if relevant, see Fed.R.Evid. 402, should apply. Contrary to appellants’ contentions, the district clearly understood the question posed to it by the panel opinion: “Put simply, the task assigned to this Court by the Court of Appeals was to find out if Florida is teaching what it is testing.” 564 F.Supp. at 180. B. Factual Findings Appellants challenge on several grounds the district court’s factual finding that students are afforded an adequate opportunity to learn the SSAT-II skills. In examining these contentions, we bear in mind that the district court’s findings of fact and inferences drawn therefrom are clearly erroneous only when “the reviewing court on the entire evidence is left with the definite and firm conviction that a mistake has been committed.” United States v. Gypsum Co., 333 U.S. 364, 395, 68 S.Ct. 525, 541, 92 L.Ed. 746 (1948). 1. The Focus on the 1981-82 School Year The appellants object that the IOX study, relied on by the state’s experts and the district court, is invalid because it evaluated only the 1981-82 school year. They argue that the most the study can show is that in 1981-82 Florida was providing instruction in grades 2-12 that, considered cumulatively, would prepare a student for the SSAT-II. Thus, they maintain that the study proves that the Class of 1983 was taught the SSAT-II skills only if one assumes that what was taught in Florida schools in 1981-82 has been taught in Florida schools since 1971. Appellants argue that such an assumption is contradicted by the district court’s own 1979 findings that “there is evidence that certain skills were not taught in Florida public schools,” that instruction prior to 1977 “was neutral and devoid of the present objectives,” and that results of the 1977 test “reflect the obvious inadequacy of the prior instruction in the standard objectives.” 474 F.Supp. at 263, 264. It was partially on the basis of these findings that the court in 1979 issued its *1410 four-year injunction against use of the test as a diploma sanction. Id. at 267. The testimony of Dr. Popham, the designer of the IOX study and expert witness for the state, seems to support the appellants’ argument that the study proves instructional validity as to the Class of 1983 only if one assumes that the same instruction was offered since 1971. He testified that “definitive, conclusive evidence” as to what the twelfth graders in the Class of 1983 had been taught could have been obtained only by tracking them back to second grade, 40 R. 73, but that this would have required prohibitively “elaborate and expensive mechanisms.” 41 R. 222. Therefore, he devised a “cross-sectional” study of grades 2-12 from which “we could draw reasonable inferences regarding what had transpired at the earlier grade levels.” 40 R. 73. Popham concluded that such inferences would be reasonable because there had not been “dramatic modifications in instruction, certainly [not] after the introduction of this legislation.” 40 R. 74. Popham had no evidence to support this conclusion, other than his own judgment that “most of the basic reading skills and mathematic skills have probably always been emphasized in Florida schools in most grades.” 42 R. 286 (emphasis added). Another of Florida’s expert witnesses, Dr. Robert Gagne, also based his opinion that then current twelfth graders had been taught SSAT-II skills on the assumption that the skills taught in 1981-82 had been taught “over a period of years.” 46 R. 707. In addition, however, he relied on the extensive remedial programs Florida offers to students who initially fail the SSAT-II. Id. One of appellants’ experts, Dr. Robert Linn, challenged the assumption that instruction had been the same throughout the twelfth graders’ careers as it was in 1981-82. He testified that institution of the SSAT-II in 1976 would have created a change in the emphasis the Florida teachers placed on test skills. 45 R. 533. The district court did not directly address the charge that the study was invalid because it demonstrated only what was taught in grades 2-12 in 1981-82. However, the court, by emphasizing evidence of Florida’s remedial efforts, indicated its view that the evidence of such efforts compensated for any such deficiency in the IOX study. To the extent that the IOX study relies on an inference that what was taught in 1981-82 was taught as far back as 1971, we acknowledge some concern, given the district court's 1979 findings and the abysmal results of the first SSAT-II administration in the fall of 1977. 6 Even Dr. Pop-ham’s testimony indicated his belief that enactment of the 1976 Educational Accountability Act prompted a dramatic modification in instruction. 40 R. 74. On the other hand, there was a clearly reasonable inference that the instruction revealed in the 1981-82 survey was substantially the same as the post-Act instruction, i.e., beginning in 1977. Our question about the pre-1977 instruction undermines to some extent our confidence in portions of the IOX study, such as the teacher survey and the Master Exposure Index relied on by Dr. Popham. However, we are not convinced that our doubts concerning the pre-1977 inference would seriously affect our conclusion. There was evidence that “the bulk of the [SSAT-II] instruction occurs in the latter years” of a child’s educational career, which in this case would be the years since 1977. 41 R. 211 (testimony of Dr. Popham). Moreover, other portions of the study do not rely on the questionable inference at all. For instance, the evidence of high school remedial programs as they relate to the Class of 1983 and subsequent classes, and the results of the student survey are not dependent on any inference of what transpired before 1977. Florida’s remedial efforts are extensive. The state provides funds to each school district to be used solely for providing re *1411 medial instruction to students who need additional .assistance in mastering basic skills. Fla.Stat.Ann. § 236.088 (West Supp.1983). Students have five chances to pass the SSAT-II between 10th and 12th grades, and if they fail, they are offered remedial help. Students may also elect to remain in school for an additional year on a fulltime or parttime basis to receive, at the state’s expense, “special instruction designed to remedy [their] identified deficiencies.” Fla.Stat.Ann. § 232.246(4) (West Supp.1983). 7 If they then pass the SSATII, they are awarded their diplomas. The state’s experts testified at length about the breadth and effectiveness of the state’s remedial efforts. 8 All agreed that the efforts were substantial and bolstered a finding of instructional validity. Dr. Robert Gagne testified “that with suitable instructional programs, and I see a good deal of evidence that those exist, that the skills which may be initially failed on the test, let’s say, in 10th grade ... could readily be learned by the next administration of the test or some next administration of the test in any case.” 46 R. 675. Significantly, appellants’ experts, Dr. Linn and Dr. Calfee, did not examine in any detail the district reports that described the remedial programs. 45 R. 573 (testimony of Dr.. Linn); 48 R. 895-97 (testimony of Dr. Calfee). 9 Dr. Linn did not examine responses from remedial teachers because they were not available to him. He conceded that such responses “would be very desirable information to have for this instructional validity question.” 45 R. 603. 10 The results of the student survey are also impressive. Ninety to ninety-nine percent of the students surveyed statewide said they had been taught the test skills. 41 R. 186. Dr. Popham found this “very powerful evidence” of instructional validity. Id.. 11 The remedial efforts and the student survey, in addition to the evidence that most instruction is provided in the later school years, persuade us that there was adequate evidence to support the district court’s finding of instructional validity, notwithstanding our reservations about the inference concerning the pre-1977 instruction. We are also persuaded in this regard by the SSAT-II pass rate in the Class of 1983. After four of five test administrations, 99.84 percent of the class had passed the communications portion of the test and 97.23 percent had passed the math portion. 564 F.Supp. at 180. Appel *1412 lants’ expert Dr. Calfee in fact conceded the instructional validity of the communication section of the test, 49 R. 1011, as did appellants’ counsel at oral argument before this court. 2. Dr. Calfee’s Correllation Dr. Robert Calfee, an expert witness for the appellants, testified that the results of the IOX teacher survey when compared with pass rates in the different school districts demonstrate a correllation between the amount of instruction reported by teachers and performance on the test. Cal-fee found that students failed the test in greatest numbers in districts where teachers reported teaching the fewest test skills. 47 R. 811. Dr. Calfee concluded that the SSAT-II is not instructionally valid as to some districts. The state’s experts testified on the basis of the results of the same teacher survey that students in all of Florida’s districts were given an opportunity to learn the SSAT-II skills before the end of 12th grade. The district court chose to credit the state’s experts over Dr. Calfee. “It is settled law that the weight to be accorded expert opinion is solely within the discretion of the judge sitting without a jury ____ In the ultimate analysis, the trier of fact is the final arbiter as between experts whose opinions may differ____” Pittman v. Gilmore, 556 F.2d 1259, 1261 (5th Cir.1977). Accord Ludlow Corp. v. Textile Rubber and Chemical Co., 636 F.2d 1057, 1060 (5th Cir.1981). We think that the district court was within its discretion in crediting the state’s experts over Dr. Calfee. 12 '3. Failure to Focus on Students Who Failed The appellants object that the state offered no direct and specific evidence that students who never passed the SSAT-II were taught test skills. Appellants assert that these students should have been the focus of the inquiry into instructional validity. None of the experts at trial suggested that an instructional validity study must focus on students who have failed the test. The experts conceded that there are no accepted educational standards for determining whether a test is instructionally valid. Under these circumstances, we reject the unsupported assertion that the state study does not prove that the SSATII is a fair test of that which is taught because it did not focus specifically on students who had failed the test. 4. Inferences Drawn by the Trial Court The appellants argue that the district court was clearly erroneous in inferring that skills were actually taught from circumstantial evidence presented at trial. Such evidence included evidence of inclusion of the skills in the curriculum, inclusion of the skills by individual teachers in their course of instruction, the existence of remedial efforts, and evidence from the student survey that students remembered receiving instruction in the skills. We cannot say that this inference is clearly erroneous. As our discussion above makes clear, we believe the evidence at trial was such that the district court could reasonably infer that the students were actually taught what was necessary to pass the exam. C. Admissibility of the Study Results The appellants argue that the trial judge improperly admitted the IOX study results into evidence. *1413 “The trial judge has broad discretion in the matter of admission or exclusion of evidence and will not be overturned absent an abuse of this discretion.” ADP-Financial Computer Services v. First National Bank, 703 F.2d 1261, 1266 (11th Cir.1983) (citing Salem v. U.S. Lines Co., 370 U.S. 31, 82 S.Ct. 1119, 8 L.Ed.2d 313 (1962)). The district court ruled that the study results, although hearsay, were admissible under Fed.R.Evid. 803(24), 13 which provides in part: The following are not excluded by the hearsay rule, even though the declarant is available as a witness: (24) Other exceptions. A statement not specifically covered by any of the foregoing exceptions but having equivalent circumstantial guarantees of trustworthiness, if the court determines that (A) the statement is offered as evidence of the material facts; (B) the statement is more probative on the point for which it is offered than any other evidence which the proponent can procure through reasonable efforts; and (C) the general purposes of these rules and the interests of justice will best be served by admission of the statement into evidence____ Appellants maintain that the study results do not satisfy two conditions of 803(24). First, they argue that the results were not more probative on the point for which they were offered than any other evidence that the proponent could procure through reasonable efforts. The appellants assert that their experts’ testimony demonstrated that there were less expensive methods of obtaining information about instructional validity that would have more directly demonstrated that Florida is teaching what it is testing. Dr. Popham testified, however, that he had considered and rejected these methods as either impractical, prohibitively expensive, or less reliable. See, e.g., 40 R. 53, 81-82; 41 R. 229-30. Given this expert testimony, the district court was within its discretion in determining that the results were more probative than other available evidence. Second, the appellants claim that the results did not possess sufficient circumstantial guarantees of trustworthiness. They claim that the trustworthiness of the results is questionable because the study respondents were aware of the purpose of the survey, the study was conducted by the state itself, and instructions to respondents placed pressure on them to give answers favorable to the state. It is true that study respondents were aware of the study’s purpose and that the state conducted this study. However, the study did possess guarantees of trustworthiness. For instance, site visitors verified the accuracy of the district reports, and site visitors were told that other visiting teams might audit their reports. 41 R. 163. Dr. Popham testified that steps were taken to remove pressure from teachers to respond positively to the teacher survey: responses were anonymous, teachers were encouraged to answer honestly, and all teachers, not just math and language arts teachers, were surveyed to remove even the perception that the state was seeking only positive responses. 40 R. 73; 41 R. 100. The district court relied on steps taken to safeguard trustworthiness in determining that the study was trustworthy. 564 F.Supp. at 182 n. 7. The district court also noted that appellants’ expert Dr. Cal-fee “had no problem with the study procedures.” Id. Under these circumstances, the district court was within its discretion in determining that this study was sufficiently trustworthy to qualify for admission under 803(24). We therefore find no error in the- admission of the IOX study results. See also Jellibeans, Inc. v. Skating Clubs of Georgia, Inc., 716 F.2d 833, 844-45 (11th Cir. 1983) (consumer survey admissible in trade name case; “alleged technical deficiencies affect the survey’s weight and not its admissibility”); C.A. May Marine Supply *1414 Co. v. Brunswick Corp., 649 F.2d 1049, 1055 n. 10 (5th Cir.1981) (consumer survey in wrongful dealership termination case inadmissible only because irrelevant; “[i]f the inadequacies in the survey had been technical, such as the format of the question or the manner in which it was the survey [sic] was taken, those shortcomings would have borne on the weight of the evidence, not its admissibility”); Holiday Inns, Inc. v. Holiday Out, 481 F.2d 445, 447 (5th Cir.1973) (consumer survey admissible in trade name case; “format of the questions and the manner of conducting the survey” go to weight of evidence). II. “VESTIGES” OF PAST DISCRIMINATION The prior Fifth Circuit opinion in this case held that if, upon remand, the state demonstrated that the SSAT-II is a fair test of that which is taught, the trial court should examine the relationship between continuing “vestiges” of past intentional segregation and the test’s racially discriminatory impact. Use of the test as a diploma sanction would be permissible only if the state satisfied the test set forth in McNeal v. Tate County School District, 508 F.2d 1017 (5th Cir.1975), by demonstrating either (1) that the disproportionate failure of blacks was not caused by the present effects of past intentional segregation, or (2) that the use of the test as a diploma sanction would remedy those effects. 644 F.2d at 407. 14 Following the trial of this issue on remand, the district court concluded that vestiges of past purposeful segregation still exist to some extent in Florida schools. The district court found that some individual teachers have racial biases that impair their teaching abilities. 564 F.Supp. at 187. The court also noted the plaintiffs’ evidence “that black students are still being suspended more often than white students, that several counties still do not have black administrators, that racial stereotypes still persist, and that blacks are being assigned to EMR [educable mentally retarded] classes more readily than whites.” Id. The court also noted the undisputed fact that the SSAT-II failure rate among black students is still disproportionately high. Fifty-seven percent of the students in the Class of 1983 who have failed the test are black, even though blacks constitute only 20% of the class. 564 F.Supp. at 186; 50 R. 20-21. However, the court found on the basis of the evidence at trial that there is no causal *1415 link between the present effects of past school segregation and the disproportionate failure rate. 564 F.Supp. at 188. The district court found, moreover, that even if there were a causal connection, the state had proven that the test will help remedy those effects. Id. The appellants challenge these last two factual findings as clearly erroneous. A. The Finding of No Causal Link The district court based its finding that there is no causal link between effects of past discrimination and the SSAT-II’s disproportionate impact partly on the testimony of Dr. Barbara Lerner, an expert witness for the state. The court credited her testimony that the existing vestiges of school segregation do not affect performance on the SSAT-II. Dr. Lerner testified that other factors, such as the educational background of a student’s parents, class size, attendance, and amounts of homework, more directly relate to student performance. The district court also based its finding on evidence that black students in counties where there are black administrators have not performed appreciably better on the SSAT-II than black students in counties where there are no black authority figures, and that more than 90% of black students have passed the test. Finally, the court credited testimony of appellants’ expert Dr. Na’im Akbar that many of the impediments to learning faced by black students arise outside the classroom. We think that this was sufficient evidence upon which to base a finding that whatever vestiges of discrimination remain do not cause the disproportionate failure rate among black students. We are particularly impressed by the very high percentages of blacks who in fact have passed the test. Ninety-nine and one-half percent of the black members of the Class of 1983 passed the communications portion of the test, and ninety-one percent passed the mathematics portion. According to appellants’ experts, vestiges of past intentional segregation operate statewide and therefore presumably touch all black students. The fact that ninety-one to ninety-nine percent of the black students nonetheless pass the SSAT-II is strong evidence that the vestiges do not cause blacks to fail the test. 15 The appellants vigorously argue that their evidence demonstrates that racial stereotypes, which the district court found still persist to some extent in Florida schools, cause teachers to have lower expectations of black students, which negatively affects the performance of those students. Dr. Lerner’s testimony disparaged this lower expectation theory. Dr. Lerner did not suggest that lower teacher expectations do not result in lower performance. However, she testified that numerous studies demonstrate that teachers very quickly correct initial expectations in light of their actual experience with students. 52 R. 232. Appellants contend that the studies relied on by Dr. Lerner were done outside the context of desegregation. Dr. Lerner testified, however, that actual experience will correct initial expectations based on race, except in the case of profoundly bigoted persons. 53 R. 275. Although Dr. Lerner’s experience with the problems of desegregation in Florida may have been less than desirable, the district court was entitled to credit her testimony, which was also corroborated by the high pass rate among black students indicating that any lower expectations among Florida teachers did not have a significant effect. Although the evidence before the district court was far from conclusive, after a careful review of the record, we are convinced *1416 that the inferences and conclusions drawn by the district court are reasonable and supported by the evidence. We conclude that the district court’s findings on causation were not clearly erroneous. 16 B. The Finding that the Diploma Sanction Remedies Vestiges The district court also found that, even assuming there was a causal link between vestiges and disparate impact, the state had satisfied the second prong of the McNeal test by demonstrating that use of the SSAT-II as a diploma sanction would remedy the vestiges of past intentional segregation. The district court relied, again, on Dr. Lerner’s testimony. Dr. Lerner testified that the best way to encourage student performance is by setting objective standards and creating a “climate of order” to motivate students. Dr. Lerner cited studies showing that such a climate is responsible for greater student achievement in Catholic schools. Dr. Lerner concluded that the use of the SSAT-II as a diploma sanction will “necessarily remedy” vestiges of past segregation by contributing to objective standards and a “climate of order” that motivates students. 52 R. 243. This evidence supports the district court’s finding that the diploma sanction is needed to remedy the present effects of past segregation in Florida’s schools. The district court finding is also supported by another fact alluded to by the district court, and of which we take judicial notice, namely, that the diploma sanction will motivate teachers and administrators, as well as students. Although the sanction is to deny the student the diploma, diploma denial reflects adversely on the teachers and administrators of the school system responsible for the student’s education. We think it is clear that teachers and administrators will work to avoid this stigma, thus tending to remedy any lingering lower expectations on the part of teachers. The remarkable improvement in the SSAT-II pass rate among black students over the last six years demonstrates that use of the SSAT-II as a diploma sanction will be effective in overcoming the effects of past segregation. Appellants argue that the improvement has nothing to do with diploma sanctions because the test has not yet been used to deny diplomas. However, we think it likely that the threat of diploma sanction that existed throughout the course of this litigation contributed to the improved pass rate, and that actual use of the test as a diploma sanction will be equally, if not more, effective in helping black students overcome discriminatory vestiges and pass the SSAT-II. Thus, we affirm the finding that use of the SSAT-II as a diploma sanction will help remedy vestiges of past discrimination. We acknowledge a heavy sense of discomfort over the unfairness if discriminatory vestiges (albeit much attenuated) have in fact caused a student to fail the SSAT-II. 17 However, that apparent unfairness would be outweighed by the demonstrated effect of the diploma sanction in remedying the greater unfairness of functional illiteracy. CONCLUSION We affirm the district court’s findings (1) that students were actually taught test skills, (2) that vestiges of past intentional segregation do not cause the SSAT-II’s disproportionate impact on blacks, and (3) that use of the SSAT-II as a diploma sanc *1417 tion will help remedy the vestiges of past segregation. Therefore, the State of Florida may deny diplomas to students (beginning with the Class of 1983) who have not yet passed the SSAT-II. AFFIRMED. Footnotes [1] For a fuller discussion of this litigation, see Debra P. v. Turlington, 644 F.2d 397, 400-02 (5th Cir.1981) (Unit B). [2] After three test administrations, 20.049% of the black students in the Class of 1979 had not passed the test, compared with 1.9% of the white students in that class. 644 F.2d at 401. As noted later in this opinion, the pass rate of all students had improved dramatically by the time of the 1983 hearings on remand. [3] As discussed infra pp. 1410-1411, students first take the SSAT-II in 10th grade and have four more opportunities to pass the test before graduation. [4] Parts of what the appellants attack as an "improper legal standard" are properly characterized as the inferences the district court drew from circumstantial evidence. For example, the appellants argue that the district court employed an "improper legal standard” by inferring that test skills were "actually taught” on the basis of evidence that test skills are included in the curriculum and that a majority of teachers realize they should teach the test skills. This is an attack on the district court’s finding of fact, not on the legal standard it employed. We address the court’s finding of fact in the next section. [5] In fact, a passage in the opinion stating that the test would be invalid if "it tests matters outside the curriculum,” 644 F.2d at 406, indicates just the opposite, i.e., that proof of the contents of the curriculum would be important and that proof need not be limited to direct proof of classroom activities. [6] Seventy-eight percent of the black students and twenty-five percent of the white students taking the test in 1977 failed one or more sections of the test. 644 F.2d at 401. [7] Students who have obtained a certificate of completion, awarded to students who have satisfied all requirements for graduation except passing the SSAT-II, may also take adult education classes aimed at remediating SSAT-II skills. Fla.Stat.Ann. § 228.072 (West Supp.1983). They may then retake the SSAT-II and receive diplomas if they pass. A student who has received a certificate of completion may also take the SSAT-II after leaving school even if he or she is not enrolled in adult education classes. 564 F.Supp. at 185 n. 16. [8] 41 R. 206-07 (testimony of Dr. Popham); 44 R. 454-77 (testimony of Dr. Henderson); 46 R. 663-64; 674-80; 707-08 (testimony of Dr. Gagne). [9] Dr. Calfee thought the district reports were "too indirect” to prove instructional validity. 48 R. 897. [10] Appellants argue that the existence of remedial Instruction has no significance because the state produced no specific evidence about what was taught in the remedial classes. It is true that the district surveys did not contain questions about what skills were taught in remedial programs. However, Dr. Gagne testified that lists of remedial materials contained in the district reports demonstrated that the remedial programs concentrated on SSAT-II skills. 46 R. 677. Moreover, Dr. Popham thought it was clear that SSAT-II skills would be taught in remedial classes specifically designed to help students pass the SSAT-II. 42 R. 263. We cannot disagree with the district court’s significant reliance on the remedial programs. [11] Appellants’ expert Dr. Linn questioned the results of the student survey because it was not strictly random. 45 R. 545. Schools to be visited by survey teams were selected by a random sampling technique and the teams were instructed to choose a class in the school at random and administer the survey to it. Dr. Linn was concerned that the second step, because it depended on the discretion of the survey team, would not produce random results. However, he had no evidence that this skewed the results of the survey. 45 R. 591. We find the survey results persuasive. [12] The district court itself elicited from Dr. Cal-fee that his study did not look at which skills teachers from a district reported teaching but only how many. 48 R. 845. Thus, if two teachers each reported teaching eight of thirteen skills, Calfee’s analysis assumed that they each taught the same eight skills, whereas they might have taught all thirteen between the two of them. Moreover, Dr. Calfee testified that in the case of six or seven school districts his correllation between teacher responses and pass rates completely broke down. 48 R. 851. Dr. Calfee attributed this to “noise in the data," id. at 850, although he also said he did not understand why this happened: "Here, I scratched my head.” Id. at 853. [13] Alternatively, the district court admitted the study results under Fed.R.Evid. 703. Because we hold here that the study results were admissible under Rule 803(24), we need not evaluate this alternative ground. [14] McNeal v. Tate County School District held that ability grouping that has a racially discriminatory impact "may nevertheless be permitted in an otherwise unitary system if the school district can demonstrate that its assignment method is not based on the present results of past segregation or will remedy such results through better educational opportunities." 508 F.2d at 1020. Appellants contend that McNeal requires in all cases as a threshhold matter that a school system be "unitary” in the sense that its operation has eliminated all vestiges of discrimination. We disagree. An absolute requirement that there be no present effects of past discrimination would be incompatible with both prongs of McNeal. Proof that the present effects of past discrimination did not cause the disparate impact, i.e., the first McNeal prong, and proof that the test or procedure at issue would remedy such vestiges of past discrimination, i.e., the second prong, both assume the continued presence of vestiges of past discrimination. In McNeal, the school district, simultaneous with its court-ordered transition from a dual or segregated to a unitary system, employed an ability grouping plan that resulted in substantial classroom segregation. The language of the McNeal opinion on which plaintiffs rely barred the use of ability grouping until the school "district can show that steps to bring disadvantaged students to peer status have ended the educational disadvantages caused by prior segregation." 508 F.2d at 1021. In the instant appeal, however, Florida dismantled its dual system in 1971, and the diploma sanction at issue will not be implemented until the students taking the test shall have attended all 12 of their school years after 1971. Although the district court acknowledged the continued existence of some vestiges of past discrimination in its 1979 decision, and that such vestiges continued "to some extent” at the time of the 1983 hearing, the court found that such vestiges "did not significantly affect the educational opportunities of black students who started school after 1971.” 7 R. 1542; 564 F.Supp. at 187; 474 F.Supp. at 252, 256. Accord Debra P. v. Turlington, 644 F.2d at 407 (approving the application of the two-prong McNeal test in the instant case). [15] We also think that Florida's remedial efforts play a role in severing any possible link between vestiges and poor performance by black students. Even assuming that discriminatory vestiges contributed to a student’s initial failure to pass the SSAT-II, it seems that the state's extensive remedial programs would enable the student to pass the test before graduation despite the effects of discrimination. It seems unlikely that remedial programs designed specifically to help students who have failed the SSAT-II would be infected by the same vestiges that exist to some extent in other parts of the school system. [16] The appellants also argue that the district court employed the wrong legal standard in determining that there was no causal link between present effects of past intentional segregation and the SSAT-II’s disproportionate impact. We do not understand the basis for this claim. The district court was instructed to determine on remand whether present effects caused the disproportionate impact. This is precisely the question the district court examined and answered. There was no legal error. [17] For purposes of analysis under the second prong of McNeal, we have assumed arguendo that there may be a causal link between vestiges and SSAT-II failure by black students. Of course, we have already found in Section II.A, supra, that the state demonstrated that there is no such link.