Plaintiffs claim that defendants Coronet Insurance Company (Coronet) and Elston Claims Service (Elston) violated the Racketeering Influenced and Corrupt Organization (RICO) statutes, 18 U.S.C. §§ 1961-1965, by using polygraph tests as a means for denying insurance claims. Plaintiffs contend that Coronet and Elston defrauded policyholders when they indicated that polygraph tests were required to receive payment on automobile theft and vandalism claims and denied a high percentage of claims based on reported failures or refusals to take the tests. The district court denied plaintiffs’ motion for class certification and granted summary judgment in favor of the defendants Coronet and Elston. The plaintiffs appeal. We affirm.
I. FACTS
We rely on the plaintiffs’ version of the facts as we must defer to the nonmoving party when reviewing a grant of summary judgment. Spring v. Sheboygan Area School Dist., 865 F.2d 883, 886 (7th Cir.1989). Coronet sells insurance coverage for automobile vandalism, theft, and larceny. Elston, an insurance adjusting company that receives 95% of its business from Coronet, processes the automobile theft or vandalism claims of Coronet’s insured.
Prior to 1981, when notified that a claimant’s car had been stolen or vandalized, Elston would obtain an initial report of the loss from the claimant, review the police report, and inspect damage to the auto if it was recovered. If there were discrepancies in the information it obtained, Elston would request that the claimant make a statement under oath. In 1981 that procedure changed. Coronet instructed Elston to ask all claimants to undergo polygraph examinations, even before Elston received a police report regarding the automobile theft or damage. Plaintiffs testified in deposition that Elston representatives told them over the telephone that they must take the polygraph tests if they wanted to be paid.
*956Within forty-eight hours of a reported theft or vandalism claim, Elston would also mail to the claimant a letter that stated:
In order to expedite settlement of your claim we request that you submit to a polygraph test. Please contact this office to set up a time convenient for you. Your cooperation in this matter would be most appreciated.
The letter later read:
In order to speed up investigation and settlement of your claim, we request that you submit to a polygraph test.
If you do not submit to a polygraph test, you will be required to appear and give a statement under oath, in the presence of a court reporter.
The above is authorized and required under the terms of your policy.
The polygraph examiner testified that it was his practice to obtain a signed consent form from each claimant before he administered a polygraph test. Of the 867 claimants who agreed to take the polygraph tests from 1983 to 1987, Coronet’s records indicate that 440 passed the exam, 422 were told that they failed the test, and 5 had inconclusive results. According to the plaintiffs, immediately after the polygraph examiner reported that a claimant failed the test, Coronet would deny the claim based solely on the test results.
Approximately half of the claimants refused to take a polygraph test. An Elston claims adjuster testified in deposition that Coronet paid only 25-40% of the claims of those who refused to take the polygraph test. This testimony contradicts Coronet’s computer records, which show that, from 1983 to 1987, 2,630 claimants reported stolen vehicles. Of those claims, the records show that 613 persons were not paid.1 Based on these records, Coronet contends that it paid 77% of the claims submitted from 1983 to 1987.
The plaintiffs include four individuals and two couples from whom Coronet withheld indemnity because they refused to take a polygraph test or because they allegedly failed the test. Francisco Marcial and Carretha Williams reported that their cars were stolen and were informed that they must take a polygraph test to receive indemnity. Both refused and both were not paid. Odessa Graham, Thomas Triplett, and James and Linda Young notified Coronet that their cars were stolen or vandalized, were told that they would not be paid unless they took a polygraph test, and were subsequently denied their claims because they allegedly failed the test. One couple, Jolanta and Witold Nytko, stated that when they reported their lost vehicle to Coronet, they were told that their claim would be “speeded up” if they took, a polygraph test. They were denied indemnity because Witold reportedly failed the examination.
Plaintiffs sought certification as a class action. Coronet and Elston moved for summary judgment. On October 25, 1988, the district court denied plaintiffs’ motion for class certification and granted Coronet’s and Elston’s motion for summary judgment. Plaintiffs filed notice of appeal on October 31, 1988.
II. ANALYSIS
A. Class Certification
The plaintiffs first challenge the district court’s denial of their motion for class certification. They argue that the class consists of all persons who purchased automobile insurance from Coronet, made claims under the theft or vandalism provisions of their policy, were “compelled” to take polygraph examinations “as a precondition to payment of the loss,” were reported to have failed the examination or refused to take the examination, and had their claims denied “without any further investigation or other basis for denial.” The district court refused to certify this class because it held that the plaintiffs had not satisfied *957the numerosity requirement of Rule 23 of the Federal Rules of Civil Procedure and that issues specific to individual plaintiffs would predominate over class issues.
We will not reverse a district court’s refusal to certify a class unless the district court abused its discretion. See First Interstate Bank, N.A. v. Chapman & Cutler, 837 F.2d 775, 781 (7th Cir.1988). Rule 23(a)(1) of the Federal Rules of Civil Procedure permits certification if the class is “so numerous that joinder of all members is impracticable.” As the district court acknowledged, plaintiffs are not required to specify the exact number of persons in the class, Vergara v. Hampton, 581 F.2d 1281, 1284 (7th Cir.1978), cert. denied, 441 U.S. 905, 99 S.Ct. 1993, 60 L.Ed.2d 373 (1979), but cannot rely on conclusory allegations that joinder is impractical or on speculation as to the size of the class in order to prove numerosity, Valentino v. Howlett, 528 F.2d 975, 978 (7th Cir.1976).
Plaintiffs contend that the class consists of approximately 400-600 persons but have failed to establish that 400-600 persons would have legitimate claims. They derive this class size by taking the number of persons who were denied indemnity and reportedly failed the polygraph tests (422) as the minimum number of class members and the total number of claimants that Coronet’s computer records indicate were denied claims from 1983 to 1987 (613) as the maximum number in the class. Plaintiffs assume, however, that Elston representatives told all policyholders that they must take the polygraph examination to receive payment. Plaintiffs also assume that every person who failed the test and was told that Coronet had denied their claim was rejected by Coronet solely on the basis of the test results. We cannot tell from the record whether those assumptions are correct. Furthermore, plaintiffs themselves have questioned the reliability of Coronet’s computer records by pointing out the discrepancy between the figures in the computer records and the testimony of Frank Gomez, the claims adjuster who testified that 60-75% of the claimants who refused to take polygraph tests were denied indemnity. In short, plaintiffs have only speculated as to the number of persons who may have legitimate claims against Coronet. The district court did not abuse its discretion in finding that plaintiffs failed to prove numerosity.
Acknowledging, however, that a nu-merosity problem can be resolved later in a suit, see Vergara, 581 F.2d at 1284, the district court also found that factual issues regarding representations made to the plaintiffs would supersede issues that the potential class shared. See Fed.R.Civ.Proc. 23(b)(3) (class may be certified if “the court finds that the questions of law or fact common to the members of the class predominate over any questions affecting only individual members”); Travis v. City of Chicago, No. 86 C 1133, slip op. at 3-4, 1987 WL 8175 (N.D.Ill. Mar. 16, 1987) (court refuses to certify because individual issues dominate class issues). To prove that Coronet “compelled” claimants to take the polygraph test, the district court held that plaintiffs would have to explain what Elston representatives told them about the tests. Plaintiffs, on the other hand, argue that letters from Elston uniformly informed plaintiffs that polygraph tests were required.
We agree with the district court that, in order to determine whether plaintiffs were compelled to take the tests, we need to know what Elston said to each claimant. As we will discuss in greater detail later, Elston’s letters to the plaintiffs regarding the polygraph tests may have implied that the plaintiffs should take polygraph tests to hasten Coronet’s response to their claims, but the letters did not, standing alone, compel plaintiffs to take the tests. In order to prove that they were compelled, plaintiffs must establish that Elston orally represented that they were required to take the polygraph examinations. Plaintiffs’ testimony in deposition indicates that what Elston told each person varied. Compare Record at 67, Plaintiffs’ Statement of Genuine Issues of Material Fact, Exhibit C (Deposition testimony of Francisco Marcial) (claims adjuster told Marcial that he must take the test before he would be paid) with *958Exhibit E (Deposition testimony of Witold Nytko) (claims adjuster told Nytko that taking polygraph test would “speed up the case”). Because of the importance of these representations and variances in what was said to the plaintiffs, the district court did not abuse its discretion in denying certification.
Contrary to plaintiffs’ contention, this finding does not reverse the burden of proof in an insurance claim or delve into plaintiffs’ state of mind. Plaintiffs are raising a RICO claim, not a contract claim based on the insurance policy. The presumption in favor of the insured in interpreting an insurance contract does not apply to a certification question in a RICO action. Likewise, a determination as to whether claimants were “compelled” to take the polygraph test does not necessarily require a court to analyze plaintiffs’ state of mind. Objective proof would suffice. Claimants who testified that Elston representatives told them that they must take a polygraph test to receive payment would not be obligated to explain their state of mind. In comparison, claimants who testified that Elston told them that polygraph testing would hasten their claim along would need to provide additional evidence of compulsion. (Claimants at that point may want to bring in evidence of their state of mind.) In either case, however, the court would not necessarily examine the plaintiffs’ state of mind.
B. The Alleged RICO Violation
Plaintiffs claim that Coronet’s and El-ston’s activities violate 18 U.S.C. §§ 1961-1968, the RICO statute. To support a RICO claim, plaintiffs must prove that El-ston and Coronet received income from a pattern of racketeering activity and used the income in an enterprise engaged in interstate commerce. 18 U.S.C. § 1961. Plaintiffs claim that the racketeering activity consisted of wire and mail fraud in violation of 18 U.S.C. §§ 1341, 1343.
To prove mail and wire fraud, plaintiffs must show that defendants knowingly schemed to defraud plaintiffs of their property and used the mail or wire services in furtherance of the scheme. See United States v. Wellman, 830 F.2d 1453, 1461 (7th Cir.1987); United States v. Shelton, 669 F.2d 446, 454-55 (7th Cir.1982), cert. denied sub. nom. Bledsoe v. United States, 456 U.S. 934, 102 S.Ct. 1989, 72 L.Ed.2d 454 (1982). “The scheme need not be fraudulent on its face, but must involve some sort of fraudulent misrepresentations or omissions calculated to deceive persons of ordinary prudence and comprehension.” Wellman, 830 F.2d at 1462.
While plaintiffs have presented strong arguments as to why the use of a polygraph test is unfair, and possibly illegal under state law, their explanation of how Coronet’s and Elston’s activities fit within the mail and wire fraud statutes is muddled. They allude to two overlapping theories: Coronet and Elston knew that polygraph testing is suspect but induced claimants to take polygraph tests in order to defraud a certain percentage of claimants of their rightful indemnity. Coronet and Elston defrauded plaintiffs of their claims by intentionally misrepresenting to plaintiffs that they must take the polygraph test when in fact plaintiffs were not obligated to take the test.2
A district court must grant summary judgment when there is no genuine issue as to any material fact and the moving party is entitled to judgment as a matter of *959law. Fed.R.Civ.P. 56(c). In our review of a grant of summary judgment, we are to draw all reasonable inferences regarding the facts in the light most favorable to the nonmoving party. Donald v. Polk County, 836 F.2d 376, 379 (7th Cir.1988). If the nonmoving party, however, fails to establish the existence of an element essential to his case and on which he bears the burden of proof, summary judgment in favor of the moving party is proper. Davis v. Warden, Joliet Correctional Inst., 867 F.2d 1003, 1017 (7th Cir.1989). See also Celotex Corp. v. Catrett, 477 U.S. 317, 322, 106 S.Ct. 2548, 2552, 91 L.Ed.2d 265 (1986). The nonmoving party cannot simply rely on its pleadings. Beard v. Whitley County REMC, 840 F.2d 405, 410 (7th Cir.1988). If the moving party demonstrates to the court that the nonmoving party’s evidence is insufficient to establish an essential element of the nonmoving party’s claim, and the nonmoving party cannot muster sufficient evidence to make out its claim, a trial would be useless and the moving party is entitled to summary judgment as a matter of law. Celotex Corp., 477 U.S. at 331, 106 S.Ct. at 2556.
1. Fraudulent Use of Polygraph Testing
Plaintiffs first claim that Coronet and Elston represented to plaintiffs that their claims would be paid when, in fact, Coronet intended to deny a predetermined percentage of the claims through the use of a devise it knew to be inherently unreliable — the polygraph test — and pressured claimants to take polygraph tests in order to guarantee a certain number of failed claims. Coronet and Elston protest that plaintiffs never developed this theory at the district court level. The district court also noted that plaintiffs’ argument was vague but stated, “it would seem that there could be no deception absent there being more to the scheme than merely unjustified reliance on polygraph results.” We acknowledge that plaintiffs’ claim has metamorphosed through the course of this lawsuit but, because plaintiffs are the nonmovants, we will construe their argument as favorably as possible.
Plaintiffs present the following facts: Some studies indicate that polygraph tests are inherently unreliable. Coronet used polygraph testing as a matter of routine, even though its polygraph examiner was aware of the studies that criticized polygraph testing. Elston representatives told plaintiffs that they must or should take the polygraph test to receive payment on their claims. Half of the claimants appeared to take the test even though they were not legally or contractually obligated to take it. Coronet denied the claims of approximately half the people who took the test based solely on the test results. One insurance adjuster testified that Coronet paid only 25-40% of the claims of those who refused to take the test.
Plaintiffs have not established that Coronet and Elston engaged in fraud. Although ordinarily proof of fraud requires a showing of a “false statement of material fact,” coverage under the mail and wire fraud statutes is read broadly and extends beyond the limits of common law fraud. United States v. McNeive, 536 F.2d 1245, 1247 (8th Cir.1976). Cf. United States v. Anderson, 809 F.2d 1281, 1284 (7th Cir.1987) (defendant who engaged in bribery activities convicted under mail fraud statute). As the Supreme Court noted in Sedima, S.P.R.L. v. Imrex Co., 473 U.S. 479, 488, 105 S.Ct. 3275, 3280-81, 87 L.Ed.2d 346 (1985), however, a civil RICO action may be brought against defendants who are indictable for one of the various federal crimes listed under the RICO statutes. Plaintiffs’ proof of fraud, unchallenged, could not lead to an indictment of Coronet or Elston. Cf. McNally v. United States, 483 U.S. 350, 107 S.Ct. 2875, 2881, 97 L.Ed.2d 292 (1987) (“There are no constructive offenses; and before one can be punished, it must be shown that his case is plainly within the statute.”).
Plaintiffs’ case against Coronet and El-ston is riddled with significant loopholes. Plaintiffs have not established that Coronet intended to defraud plaintiffs of their claims. See United States v. Mascio, 774 F.2d 219, 221 (7th Cir.1985) (“Intent is a critical element of the mail fraud offense.”). They present no proof that Coro*960net officials were aware of the studies that criticized polygraph testing. Although plaintiffs repeatedly emphasize that polygraph tests are inherently unreliable, the record shows that courts, the State of Illinois, and persons interested in polygraphy disagree on the issue. See Marsh v. Lake Forest Hospital, 166 Ill.App.3d 70, 73-74, 116 Ill.Dec. 612, 614-15, 519 N.E.2d 504, 506-07, appeal denied, 121 Ill.2d 571, 122 Ill.Dec. 439, 526 N.E.2d 832 (1988) (“these cases [discussing polygraph tests] represent a recognition by the supreme court that polygraph results may be useful for some purposes other than as evidence at trial”); Moskos v. National Ben Franklin Ins. Co., 60 Ill.App.3d 130, 134, 17 Ill.Dec. 389, 393, 376 N.E.2d 388, 392 (1978) (polygraph test results held admissible as evidence of good faith); Ill.Rev.Stat. ch. 111, ¶ ¶ 2401-2432 (Smith-Hurd 1985) (Illinois licensing procedures for polygraph examiners). Coronet may innocently believe that polygraph tests are reliable. Plaintiffs have not demonstrated that Coronet officials knew polygraph testing was unreliable or presented any basis for a jury to conclude that Coronet purposely used what they knew to be an inherently unreliable device to avoid claims.
Plaintiffs have not established that El-ston fraudulently administered the polygraph tests or falsely reported test results. Plaintiffs have not alleged that the polygraph examiner intentionally falsified his reports, nor have they provided this court with any statements regarding the truthfulness of answers that they gave on the polygraph examinations.
Furthermore plaintiffs want us to accept that Coronet’s practices are so extreme that they can only be explained as fraudulent. We have no way of gauging, however, whether the number of claims that Coronet rejected is disproportionate with those denied by other insurance companies. Plaintiffs have not indicated that Coronet’s profitability increased when it began to use polygraph tests or provided statistics regarding other insurance companies who deal with this type of insurance. Both parties at oral argument agreed that Coronet insures persons considered high risk because they have unfavorable histories or live in high risk areas of the community. In order to determine whether Coronet and Elston committed fraud, we need clear evidence that Coronet and Elston engaged in practices that were deliberately fraudulent. Although plaintiffs’ counsel has characterized the results of the polygraph exams as “shocking,” plaintiffs must do more than assert their disbelief to withstand summary judgment.3
2. Representations Made to Plaintiffs
Plaintiffs have asserted that, in addition to relying on polygraph tests, Coronet represented to claimants in violation of the law and the insurance contract that they must take the polygraph tests. Plaintiffs stated that Elston representatives told them that polygraph tests were required to receive payment.
The Illinois Department of Insurance has issued Regulation 919.60(d) that states:
No company shall require any insured to submit to a polygraph or other similar type of examination as a condition precedent to payment of a claim. The use of examinations under oath, sworn statements or similar procedures authorized under the terms of the applicable insurance contracts shall not be restricted.
Illinois courts have held that polygraph test results are inadmissible as substantive evidence in court, Marsh, 166 Ill.App.3d at *96173-74, 116 Ill.Dec. at 614-15, 519 N.E.2d at 506-07, but are admissible to show an insurance company acted in good faith in denying an insured’s claim, Moskos, 60 Ill.App.3d at 134, 17 Ill.Dec. at 392, 376 N.E.2d at 391. Under Illinois law then, Coronet may request a claimant to take a polygraph test but may not require a claimant to take the test as a condition for payment of the claim. Coronet may also use the polygraph test to show its good faith in denying a claim but may not present the polygraph test results as evidence that claimant presented a fraudulent claim.
It appears from the allegations that Coronet may have violated Illinois law, but not that Coronet and Elston committed fraud. As the district court carefully explained,
While it would be a violation of Regulation 919.60(d) to impose such a requirement [that plaintiffs take a polygraph test], [plaintiffs] did not testify that they were told Illinois law or the insurance policy required the polygraph examination. Therefore, their deposition testimony only supports a contract or insurance law claim, not a claim for fraud. Defendants were not being deceptive; under Coronent’s (sic) practices — at least those practices shown by plaintiffs’ evidence— the claims would indeed be denied if the request for a polygraph examination was refused.
Record at 77. If Coronet told plaintiffs that they must take a polygraph test in order to receive payment, Coronet violated state law. A violation of state law, however, does not necessarily indicate that the actions were also fraudulent. Cf. North Barrington Dev., Inc. v. Fanslow, 547 F.Supp. 207, 210 (N.D.Ill.1980) (the purpose of RICO is not to transform state law violations into federal violations, but to prevent interference with free competition). Coronet did not misstate its policy or represent that its policy was in conformity with Regulation 919.60(d). It did not attempt to conceal its reliance on polygraph tests. Cf. McNeive, 536 F.2d at 1252 (court found that mail fraud was not evident because defendant did not misrepresent the existence or the nature of the gratuity practice). Even when coupled with Coronet’s practice of denying insurance claims based on test results, these representations do not rise to the level of a RICO violation.
Likewise, the letters that Coronet sent to claimants do not fraudulently compel the insured to take the polygraph tests. The insurance contract that covered plaintiffs’ claims states that “the insured shall cooperate with the company.” The second letter, the more emphatic of the two letters, notified claimants that if they refused to take the polygraph test, they would be obligated to make a sworn statement regarding the loss. This letter alone did not mandate that claimants take the test. It requested, albeit in words intended to persuade claimants to take the test, that plaintiffs cooperate by taking the polygraph test. It provided an alternative, however, to those who refused to take the test. The letters did not misrepresent the law or thé terms of the insurance contract, which allowed Coronet to elicit plaintiffs’ “cooperation” and obtain a sworn statement of loss from a claimant. Because the letters did not compel claimants to take the polygraph tests or misrepresent the terms of the contract or Illinois law, they cannot support a RICO claim.
III. CONCLUSION
Although plaintiffs may have a viable claim under state law against Coronet and Elston, they have not established that Coronet and Elston engaged in mail or wire fraud. Plaintiffs have not established that defendants fraudulently intended to use polygraph testing to deny legitimate claims. They did not establish that by using polygraph testing Coronet and Elston schemed to defraud claimants. Finally they did not establish that Coronet and Elston deceived or attempted to deceive plaintiffs through false representations. While we do not condone Coronet’s practices, plaintiffs have failed to establish that those practices come within the parameters of the RICO statute. We AFFIRM.
. Assuming that Coronet would deny all claims made by persons who reportedly failed the polygraph tests, the computer records indicate that 191 persons who refused to take polygraph tests were denied indemnity from 1983 to 1987. In comparison, the claims adjuster’s testimony suggests that Coronet refused to pay approximately 1,058 to 1,322 persons who refused to take the polygraph test.
. Plaintiffs also asserted at the district court level that they were fraudulently induced to pay insurance premiums. The district court held that plaintiffs had failed to prove that Coronet intended to defraud plaintiffs at the time it induced them to enter into the insurance contract. According to the district court, the undisputed facts showed that about 70-77% of the automobile claims were paid and that this did not indicate an intent to defraud plaintiffs of their premiums. Although plaintiffs vacillate between attacking the 70-77% figure and adopting it to support their position, they do not challenge on appeal the district court’s decision with regard to the insurance premiums. We conclude that plaintiffs have waived their argument with regard to the insurance premiums. Even if the plaintiffs did not intend to waive the argument, we agree with the district court that there was no proof that Coronet was trying to defraud plaintiffs of their premiums.
. Plaintiffs have at certain points in their brief, and at oral argument, relied upon Coronet’s computer records indicating Coronet pays on 77% of its theft and vandalism claims. The district court also thought that plaintiffs had conceded that 70-77% were paid. Plaintiffs, in their reply brief, however, attack the figure and embrace the testimony of the claims adjuster who stated that in his estimation only 25-40% of the claims of those who refused to take the test were paid (meaning that the 77% figure is faulty). Although plaintiffs should decide exactly which facts they adopt as reliable and stick with them, we need not remand based on this seeming dispute about the facts. Accepting the claims adjuster’s testimony (the most disfavora-ble position for Coronet), we find that plaintiffs have not established that Coronet is intentionally rejecting valid claims for indemnity.