United States v. William Rea, Getty Terminals Corp., and John Pabone, Getty Terminals Corp., William Rea, John Pabone, and John Quock

KEARSE, Circuit Judge: Defendants William Rea, Getty Terminals Corp. (“Getty”), and John Pabone appeal from judgments entered in the United States District Court for the Eastern District of New York, following a jury trial before Leonard D. Wexler, Judge, convicting each on one count of conspiracy to defraud the United States, in violation of 18 U.S.C. § 371 (1988) (count 1), and three counts of tax evasion, in violation of 26 U.S.C. § 7201 (1988) (counts 2-4). Getty was sentenced principally to pay fines totaling $400,000. Rea was sentenced principally to (1) a three-year term of imprisonment on count 1, of which he was to serve two months in prison and the remainder on probation; and (2) three concurrent three-year terms of probation on counts 2-4, to be served concurrently with the sentence imposed on count 1. Pabone was sentenced principally to (1) a four-year prison term on count 1, and (2) three five-year terms of probation on counts 2-4, to be served concurrently with each other but consecutively to the sentence imposed on count 1. Rea and Pabone were also ordered to satisfy such tax liabilities as might be determined by the Probation Department. On appeal, defendants contend principally that the trial court erred in various evidentiary rulings and that the trial evidence was insufficient to support their convictions. For the reasons below, we affirm the judgments of conviction. I. BACKGROUND At all pertinent times, Getty was a producer of gasoline, and Rea was the head of its supply and distribution department. Pabone was the president and owner of J & J Petroleum Co., Inc. (“J & J”), a gasoline wholesaler. The conduct at issue, which took place in 1985 unless otherwise indicated, occurred in the context of the federal system of imposing excise taxes on sales of gasoline. With certain exceptions, a producer of gasoline, upon the sale of gasoline to a nonproducer, is required to pay the federal excise tax. In practice, this tax is charged to and collected from the purchaser. A producer of gasoline that sells gasoline to another producer is not required to pay the government such a tax. See 26 C.F.R. § 48.4083-1(a) {“In general. Gasoline may be sold tax free by a producer or importer of gasoline to other producers of gasoline_”). A wholesale distributor is considered a producer within the meaning of § 48.4083-1(a) if it has obtained an Internal Revenue Service (“IRS”) Certificate of Registry (Form 637) (“637 License”); see 26 C.F.R. § 48.4082-1(d)(2) (distributor is considered a producer only with respect to gasoline it sells on or after the date on which it is issued a 637 License). The government’s case at trial was presented through documentary evidence *1211 and the testimony of a variety of witnesses, including Charles Sarowitz, who had served as J & J’s accountant and negotiated purchases of gasoline for J & J, and persons who during the pertinent period had worked as clerical employees of Getty in connection with the invoicing of gasoline sales. Taken in the light most favorable to the government, the evidence painted the following general picture. A. The Tax Evasion Scheme In May 1985, J & J was seeking a new supplier of gasoline. Sarowitz, who performed business functions for J & J on instructions from Pabone, saw Rea’s name on a list of wholesalers he had been given, and he telephoned Rea to set up a meeting. Shortly thereafter, in late May or early June, Sarowitz and Pabone met with Rea at Getty’s offices (the “May/June meeting”). Sarowitz told Rea that J & J wanted to purchase gasoline from Getty. He also informed Rea that J & J did not hold a 637 License. At some point during this meeting, which lasted some 30-40 minutes, Getty senior vice president Alvin Smith stopped in to introduce himself; he expressed the view that Sarowitz and Pabone were probably “a bunch of bootleggers,” i.e., persons who sell gasoline without paying the required excise taxes. Sarowitz and Pabone denied that they were bootleggers. At this meeting, Rea agreed that Getty would become a supplier of gasoline to J & J. Shortly thereafter, on June 4, Sarowitz and Pabone met with John Quock, president and owner of Tun Yung Fuel Oil Corp. (“Tun Yung”). Tun Yung held a valid 637 License. At that meeting, Quock agreed to allow J & J to use Tun Yung’s 637 License in order to avoid paying the nine-cent-per-gallon federal excise tax that Getty would ordinarily charge on J & J’s purchases. It was agreed at that meeting that these federal taxes would not be paid. In return for allowing J & J to use Tun Yung’s 637 License, Quock was to receive half of the tax money J & J saved; he was to receive his payment not in cash but in gasoline drawn from J & J’s accounts at Getty’s terminals. Pabone later told a J & J salesman that J & J was in a position to sell gasoline at very competitive prices because it was using Tun Yung’s 637 License to buy gasoline from Getty tax-free. On June 7, J & J began to make purchases from Getty, buying barges of gasoline. J & J paid Getty for the gasoline by means of a wire transfer in accordance with instructions Sarowitz received from Rea. On its invoices for the barges of gasoline, Getty listed Tun Yung as the purchaser. Sa-rowitz testified that he did not place the order for the June 7 barge purchase, and he was not sure who had done so. There was no testimony as to who instructed Rea that Tun Yung should be invoiced on this first sale to J & J. Getty’s billing procedure called for Rea to prepare for each sale a bulk transaction sheet called a “BT,” which typically included the name of the purchaser, the quantity, type, and price of the gasoline being purchased, the method of payment, and an indication of whether federal excise tax was to be charged. The information shown on the BTs was then transferred to “deal sheets” prepared by Rea or employees working under him in the supply and distribution department. Copies of each deal sheet would then be sent to other Getty departments, e.g., the tax, inventory, and billing departments. Using the deal sheets given them, employees in the billing department would charge tax only on those sales for which the BT, prepared by Rea, had indicated that taxes should be charged. On the J & J sales, Rea’s BTs indicated that Tun Yung should be invoiced and that no tax should be charged. Accordingly, Getty neither billed nor paid federal excise taxes on any of the barge sales to J & J. On July 3, Sarowitz again met with Rea, this time to discuss the possibility of J & J’s purchasing gasoline not by the barge, but directly from Getty’s various terminal facilities (called “in-place” sales). Sarowitz and Rea agreed that thereafter, Sarowitz or Pabone would order in-place gasoline, that payments for the gasoline would be wired to Getty by J & J, and that though the in-place gasoline was being sold to J & *1212 J, and J & J was receiving the gasoline, these sales would be invoiced to Tun Yung. Pabone eventually executed a personal guarantee of payment on sales invoiced to Tun Yung. Further, though the Getty invoices were to list Tun Yung as the purchaser, Sarowitz instructed Rea that any orders from Quock with respect to the J & J gasoline were not to be followed. In addition, though the invoices were to show Tun Yung’s name and address, Sarowitz instructed Rea to mail the invoices directly to J & J. As discussed in greater detail in Part II.A. below, Sarowitz testified, over objection, that Rea “had to” have known he was being asked to show Tun Yung as the purchaser on the invoices so that the nine-cent-per-gallon federal excise tax could go unpaid. The agreed in-place sales to J & J began shortly after the July 3 meeting, but the initial transactions were invoiced to J & J, not Tun Yung. Upon discovering this, Sa-rowitz immediately telephoned Rea and reminded him that the in-place sales to J & J were to be invoiced to Tun Yung. Rea promised to “take care [of] it.” Getty also made out invoices to Westwind, another company owned by Quock, but one that did not have a 637 License. Sarowitz again contacted Rea, who again promised to take care of having the invoices made out to Tun Yung. Thereafter, the invoices on the in-place sales to J & J showed Tun Yung as the purchaser. As to the earlier invoices to J & J or Westwind, despite the fact that neither entity held a 637 License, federal excise taxes were not charged because none of the BTs prepared by Rea indicated that taxes should be charged. Getty did not pay excise taxes on these sales. The purchases by J & J with the Getty invoices listing Tun Yung as purchaser continued until the fall. During hurricane “Gloria” in September 1985, J & J “over- pulled” — i.e., overdrew — its inventory account at Getty by some $1,680,000 worth of gasoline. The extent of the overpull was the subject of dispute and the matter was negotiated at some length by J & J, represented by Sarowitz in consultation with Pa-bone by telephone, and Getty, represented by several officials, including its credit manager Robert Vislay, its treasurer, senior vice president Smith, and another vice president. Eventually, agreement was reached on the amount of J & J’s indebtedness to Getty and on a payment schedule to be met by J & J. Despite J & J’s concession of its indebtedness for the overpull, and despite the understanding of the Getty officials that the gasoline had been over-pulled by J & J from J & J’s own account, this gasoline was nonetheless invoiced to Tun Yung, and federal excise taxes were neither charged nor paid. After settlement of the overpull dispute, Getty’s tax department became involved in the billing of J & J’s purchases and Getty decided that thereafter, even if Tun Yung were invoiced, excise taxes should be charged. As a result, Pabone instructed Sarowitz to make J & J’s gasoline purchases elsewhere. Before the scheme ended, J & J had purchased more than 11 million gallons of gasoline from Getty, on which, at nine cents a gallon, $1,019,136.96 in federal excise taxes were due but unpaid. B. The Investigation and the Prosecution The Getty-J & J dealings came under investigation by federal authorities in early 1987. Rea, questioned by an IRS agent, gave a version of the events that was at odds with the evidence described above. He stated that in the spring of 1985 he had agreed to sell to Pabone on condition that the sales (a) would be on a prepaid basis, and (b) would include all state and federal taxes because J & J had no state exemption certificate or 637 License. Rea said he could not explain the Getty invoices to Tun Yung on the sales to J & J and said he was unaware that J & J was using Tun Yung’s 637 License. He said this billing procedure must have been a mistake attributable to Getty credit manager Vislay. In early 1987, J & J received a federal grand jury subpoena for records concerning its transactions with Getty. Pabone had the documents moved from their normal location to the basement of the build *1213 ing and then caused the boiler in the basement to be opened, allowing water to damage the documents. When he learned that the government wanted the documents despite the water damage, he instructed Sa-rowitz to go through the records prior to turning them over and to extract material that might be incriminating so that it could be destroyed. Eventually, Getty, Rea, Pabone, and Quock were charged in a four-count indictment. Count 1 charged each defendant with conspiring to defraud the United States of excise taxes on the gasoline sold to J & J and invoiced to Tun Yung, in violation of 18 U.S.C. § 371. Counts 2, 3, and 4 charged each defendant with tax evasion, in violation of 26 U.S.C. § 7201, with respect to the excise taxes due for the second, third, and fourth quarters, respectively, of 1985. The government’s evidence at trial painted the general picture outlined above. None of the individual defendants testified at trial. Rea introduced the testimony of three character witnesses. Pabone introduced, inter alia, testimony designed to deflect the government’s contention that he had intentionally sought to destroy incriminating evidence, as well as the testimony of an attorney hired in 1986 who said he had discussed with Pabone J & J’s federal excise tax obligations. Getty called two witnesses, including a petroleum industry expert who testified, inter alia, that it was a common industry practice for a nonparty to a petroleum contract to pick up from the seller, and to pay the seller for, gasoline that had been ordered by another company. The jury convicted all of the defendants on all counts, and appellants were sentenced as indicated above. Only Rea, Getty, and Pabone have appealed. II. DISCUSSION On appeal, Rea and Getty contend principally (1) that the trial court erred in allowing Sarowitz to testify that Rea must have known he was participating in a tax evasion scheme, and (2) that the evidence was insufficient to support a conclusion that they knowingly entered into the charged scheme intending to evade the payment of excise taxes. In addition, Rea contends that the court erred in excluding from evidence the results of a polygraph examination he had taken, and Pabone contends that the district court erred in (a) denying a pretrial motion to sever his case from that of Quock, (b) excluding from evidence testimony concerning certain statements he had made, and (c) failing to conduct an eviden-tiary hearing on certain contested factual matters in connection with sentencing. Though we conclude that Sarowitz’s testimony with respect to Rea’s knowledge was improperly admitted, we find in that error no basis for reversal, and we conclude that the other contentions are meritless. A. The Testimony that Rea “Had To” Know that the Reason for Invoicing Tun Yung Was To Permit the Evasion of Taxes The state of Rea’s knowledge was an important factor in this prosecution. Counts 2-4 charged violations of 26 U.S.C. § 7201, which provides, in pertinent part, that [a]ny person who willfully attempts in any manner to evade or defeat any tax imposed by this title or the payment thereof shall, in addition to other penalties provided by law, be guilty of a felony.... Willfulness, as used in such a statute, means the “ ‘voluntary, intentional violation of a known legal duty.’ ” Cheek v. United States, — U.S. -, 111 S.Ct. 604, 610, 112 L.Ed.2d 617 (1991); see also United States v. Gurary, 860 F.2d 521, 523 (2d Cir.1988) (“In the context of [26 U.S.C. § 7206(2) ], ‘willfully’ means an intentional violation of a known legal duty.”), cert. denied, 490 U.S. 1035, 109 S.Ct. 1931, 104 L.Ed.2d 403 (1989). Count 1 charged conspiracy to evade the payment and impede the collection of federal excise taxes by the invoicing of Tun Yung on sales to J & J, and since the essence of any conspiracy is agreement, the government was required to present evidence from which it could reasonably be inferred that the defendant in question knew of the illegal venture and *1214 knowingly joined and participated in it. See, e.g., United States v. Labat, 905 F.2d 18, 21 (2d Cir.1990); United States v. Villegas, 899 F.2d 1324, 1338 (2d Cir.), cert. denied, — U.S. -, 111 S.Ct. 535, 112 L.Ed.2d 545 (1990); United States v. Sanchez Solis, 882 F.2d 693, 696 (2d Cir.1989). In order to prove conspiracy, the government need not present evidence of an explicit agreement; proof of a tacit understanding will suffice. United States v. Beech-Nut Nutrition Corp., 871 F.2d 1181, 1191 (2d Cir.), cert. denied, 493 U.S. 933, 110 S.Ct. 324, 107 L.Ed.2d 314 (1989); United States v. Rubin, 844 F.2d 979, 984 (2d Cir.1988). The coconspirators need not have agreed on the details of the conspiracy, so long as they have agreed on the essential nature of the plan, United States v. Bagaric, 706 F.2d 42, 63 (2d Cir.), cert. denied, 464 U.S. 840, 104 S.Ct. 133, 78 L.Ed.2d 128 (1983); and their goals need not be congruent, so long as they are not at cross-purposes, United States v. Beech-Nut Nutrition Corp., 871 F.2d at 1192; United States v. Heinemann, 801 F.2d 86, 92 n. 1 (2d Cir.1986), cert. denied, 479 U.S. 1094, 107 S.Ct. 1308, 94 L.Ed.2d 163 (1987). The defendant’s knowledge of the conspiracy and participation in it with the requisite criminal intent may be established through circumstantial evidence. See, e.g., United States v. Villegas, 899 F.2d at 1338-39; United States v. Tutino, 883 F.2d 1125, 1129 (2d Cir.1989), cert. denied, 493 U.S. 1081, 110 S.Ct. 1139, 107 L.Ed.2d 1044 (1990); United States v. Young, 745 F.2d 733, 762 (2d Cir.1984), cert. denied, 470 U.S. 1084, 105 S.Ct. 1842, 85 L.Ed.2d 142 (1985). A defendant need not have joined a conspiracy at its inception in order to incur liability for the unlawful acts of the conspiracy committed both before and after he or she became a member. United States v. Blackmon, 839 F.2d 900, 911 (2d Cir.1988); United States v. Guillette, 547 F.2d 743, 751 (2d Cir.1976), cert. denied, 434 U.S. 839, 98 S.Ct. 132, 54 L.Ed.2d 102 (1977). At the trial of the present case, Sarowitz was allowed to testify that Rea “had to” know that the reason he was asked to list Tun Yung on the invoices as the purchaser of the gasoline on sales to J & J was to permit the evasion of the federal excise taxes on those sales. Rea and Getty contend that Sarowitz should not have been allowed to give his opinion as to Rea’s state of mind, because (a) knowledge was an ultimate issue in the case, (b) the opinion was not “helpful” to the jury, as that term is used in Fed.R.Evid. 701, and (c) the testimony was unduly prejudicial because the remainder of the government’s evidence failed to prove knowledge. We find no merit in the first and third of these contentions, and, though we agree with the second, we conclude that the error does not require reversal. 1. Admissibility in Principle The contention that Sarowitz’s opinion as to Rea’s knowledge should have been excluded from evidence as a matter of principle, on the ground that the state of Rea’s knowledge was an ultimate issue in the case, is' untenable. In what has been described as “a sharp departure in theory, if not in practice, from the common law,” 2 S. Saltzburg & M. Martin, Federal Rules of Evidence Manual 1 (5th ed. 1990), the Federal Rules of Evidence allow the court to permit a lay witness, in certain circumstances, to testify in the form of an opinion. Thus, Rule 701 provides as follows: If the witness is not testifying as an expert, the witness’ testimony in the form of opinions or inferences is limited to those opinions or inferences which are (a) rationally based on the perception of the witness and (b) helpful to a clear understanding of the witness’ testimony or the determination of a fact in issue. Fed.R.Evid. 701. In addition, Rule 704(a) states that the fact that lay opinion testimony may bear on one of the ultimate issues in the case is not, of itself, a ground for exclusion: Except as provided in subdivision (b), testimony in the form of an opinion or inference otherwise admissible is not objectionable because it embraces an ultimate issue to be decided by the trier of fact. Fed.R.Evid. 704(a). Since neither Rule 701 nor Rule 704(a) limits the subject matter of lay opin *1215 ion testimony, there is no theoretical prohibition against allowing lay witnesses to give their opinions as to the mental states of others. See generally 3 J. Weinstein & M. Berger, Weinstein’s Evidence 11701[02], at 701-19 to 701-21 (1991). Accordingly, these Rules do not, in principle, bar a lay witness from testifying as to whether a defendant in a criminal prosecution had the requisite knowledge. See, e.g., United States v. Fowler, 932 F.2d 306, 312 (4th Cir.1991); United States v. Ruppel, 666 F.2d 261, 269-70 (5th Cir. Unit A), cert. denied, 458 U.S. 1107, 102 S.Ct. 3487, 73 L.Ed.2d 1369 (1982); United States v. Smith, 550 F.2d 277, 281 (5th Cir.), cert. denied, 434 U.S. 841, 98 S.Ct. 138, 54 L.Ed.2d 105 (1977). We note that subdivision (b) of Rule 704, which limits Rule 704(a), expressly forbids opinion testimony by an expert as to the state of mind of a defendant in a criminal case where that mental state is an element of the crime: No expert witness, testifying with respect to the mental state or condition of a defendant in a criminal case may state an opinion or inference as to whether the defendant did or did not have the mental state or condition constituting an element of the crime charged or of a defense thereto. Such ultimate issues are matters for the trier of fact alone. Fed.R.Evid. 704(b). Though the last sentence of this subdivision, if read literally, could bé understood to bar even opinions of lay witnesses on the ultimate issue of the state of mind of a defendant in a criminal case, such an interpretation is contraindicated for several reasons. First, the legislative history of subdivision (b), which was added in 1984, indicates that the drafters and Congress focused only on problems with regard to expert witnesses, not lay witnesses. See, e.g., S.Rep. No. 98-225, 98th Cong., 2d Sess. 230 (1983), reprinted in 1984 U.S.Code Cong. & Admin.News 3182, 3412 (“The purpose of this amendment is to eliminate the confusing spectacle of competing expert witnesses testifying to directly contradictory conclusions as to the ultimate legal issue to be found by the trier of fact.”). Second, the last sentence of Rule 704(b) cannot stand alone; its reference to “[s]uch” issues makes it unintelligible without reference to the first sentence, which by its terms is directed only toward expert witnesses. Third, if the drafters had intended that the last sentence be read as applying to lay witnesses as well as expert witnesses, there would have been no need to include the word “expert” in the first sentence. We conclude that Rule 704(b) limits only expert testimony, not lay testimony, and that there was no general principle barring Sarowitz’s testimony as to his opinion of Rea’s knowledge. 2. The Rational-Basis and Helpfulness Preconditions to Admissibility Under Rule 701 ' Rule 701’s authorization of lay opinion testimony was adopted because “[witnesses often find difficulty in expressing themselves in language which is not that of an opinion or conclusion.” Fed.R.Evid. 701 Advisory Committee Note on 1972 Proposed Rules. But while such testimony is not totally excluded, there are two express preconditions to its admissibility: first, the opinion evidence may be allowed only if the court finds that the opinion is “rationally based on” the witness’s own perceptions; and second, it is to be allowed only if the court concludes that it will be “helpful” to a clear understanding of the witness’s testimony or the determination of a fact in issue. See Fed.R.Evid. 701; see also Fed. R.Evid. 704(a) (opinion testimony on ultimate issue must be “otherwise admissible”). The rational-basis requirement “is the -familiar requirement of first-hand knowledge or observation.”, Fed.R.Evid. 701 Advisory Committee Note on 1972 Proposed Rules; see Fed.R.Evid. 602 (“A witness may not testify to a matter unless evidence is introduced sufficient to support a finding that the witness has personal knowledge of the matter.”). Rule 701’s helpfulness requirement is designed to provide “assurance[ ] against the admission of opinions which would merely tell the jury what result to reach.” Fed.R.Evid. 704 Advisory Committee Note on 1972 Proposed Rules. Thus, if “attempts are made to introduce meaningless assertions which *1216 amount to little more than choosing up sides, exclusion for lack of helpfulness is called for by [Rule 701].” Fed.R.Evid. 701 Advisory Committee Note on 1972 Proposed Rules. When the issue is a party’s knowledge, which is perhaps a more easily fathomed state of mind than, for example, intent or motivation, we suspect that in most instances a proffered lay opinion will not meet the requirements of Rule 701. There are a number of objective factual bases from which it is possible to infer with some confidence that a person knows a given fact. These include what the person was told directly, what he was in a position to see or hear, what statements he himself made to others, conduct in which he engaged, and what his background and experience were. When a witness has not identified the objective bases for his opinion, the proffered opinion obviously fails completely to meet the requirements of Rule 701, first because there is no way for the court to assess whether it is rationally based on the witness’s perceptions, and second because the opinion does not help the jury but only tells it in conclusory fashion what it should find. See, e.g., United States v. Whitworth, 856 F.2d 1268, 1284-85 (9th Cir.1988) (opinion that defendant had requisite knowledge based simply on “[c]ommon sense” did not meet Rule 701 requirements), cert. denied, 489 U.S. 1084, 109 S.Ct. 1541, 103 L.Ed.2d 846 (1989); see also United States v. Phillips, 600 F.2d 535, 538-39 (5th Cir.1979) (opinion, with no testimony as to objective bases, that defendant “indicate[d]” he “understood” held arguably inadmissible and, in any event, wholly insufficient to establish knowledge). On the other hand, when a witness has fully described what a defendant was in a position to observe, what the defendant was told, and what the defendant said or did, the witness’s opinion as to the defendant’s knowledge will often not be “helpful” within the meaning of Rule 701 because the jury will be in as good a position as the witness to draw the inference as to whether or not the defendant knew. Cf. Krueger v. State Farm Mutual Automobile Insurance Co., 707 F.2d 312, 316-17 (8th Cir.1983) (in wrongful death action, testimony as to whether motorist had enough time to avoid hitting decedent was properly excluded where witness had described the distances, speeds, and conditions of which he was aware to jury in sufficient detail to communicate his perceptions adequately; opinion was therefore not necessary for an understanding of his testimony and hence not “helpful”). Lay opinion testimony will probably be more helpful when the inference of knowledge is to be drawn not from observed events or communications that can be adequately described to the jury, but from such factors as the defendant’s history or job experience. See, e.g., United States v. Fowler, 932 F.2d at 312 (Department of Defense officials properly allowed to give opinion that a person with defendant’s experience in the department would know rules forbidding giving certain documents to contractors); United States v. Smith, 550 F.2d at 281 (witness properly allowed to testify to her belief that defendant who ran a federally funded program understood certain federal regulations). In addition, even those lay opinions that pass Rule 701’s dual test of admissibility may be excluded by the court under Fed.R.Evid. 403 if the court determines that the admission of the opinion will be cumulative or a waste of time, or that its helpfulness is substantially outweighed by the danger of unfair prejudice to the party opposing admission of the evidence. See generally Hogan v. American Telephone & Telegraph Co., 812 F.2d 409, 411 (8th Cir.1987); Fed.R.Evid. 704 Advisory Committee Note on 1972 Proposed Rules. The questions of whether the proffered opinion is rationally based on the witness’s knowledge and whether it will be helpful are, like all preconditions to the admissibility of evidence, determinations to be made by the trial judge. See generally Fed.R.Evid. 104(a); United States v. Stanchich, 550 F.2d 1294, 1297-98 (2d Cir.1977). Some preconditions to admissibility, of course, are principally factual and permit the court to admit proposed testimony *1217 “subject to ... the introduction of evidence sufficient to support a finding of the fulfillment of the condition.” Fed.R.Evid. 104(b); see, e.g., United States v. Stanchich, 550 F.2d at 1297 (upholding conditional admission of coconspirator statement pursuant to Fed.R.Evid. 801(d)(2)(E), subject to the introduction of evidence that statement was in furtherance of the conspiracy). We view the first Rule 701 precondition, i.e., the requirement that a witness’s opinion have a rational basis in his own perceptions, as one that in theory would be amenable to this kind of treatment. On the other hand, we view the question of whether the witness’s opinion will be “helpful” within the meaning of Rule 701 as a legal matter that must be determined by the court before it may allow the opinion to be heard by the jury. Given the fact that often the same evidence is pertinent to both determinations, we think it will be rare that the court can safely accept lay opinion testimony “subject to connection.” Application of these principles to the present case gives us considerable pause, in part because at the time of the admission of Sarowitz’s opinion as to Rea’s knowledge there had been little indication of a rational basis for that opinion, and more importantly because, though eventually an arguably adequate foundation was aired in the evidence, the opinion was not helpful within the meaning of Rule 701. The opinion was elicited by the government during its examination of Sarowitz with respect to his July 3, 1985 meeting with Rea. Following the Sarowitz-Pabone-Rea meeting in late May or early June, J & J had been purchasing barges of gasoline. At some point, however, Pabone told Sarowitz that J & J wanted to make in-place purchases from the Getty terminals, and Sarowitz accordingly arranged the second meeting with Rea. Asked to describe the July 3 meeting, which was not attended by Pabone, Sarowitz testified as follows: Q. What was said by whom? A. I don’t remember which one of us said it. Q. All right. What do you recall being said during that conversation? A. That the in place sales would be invoiced to Tun Yung and sold to J & J. Q. Did you discuss how the gas would be ordered? A. Yes. Q. What was said? A. That either myself or Mr. Pabone would order the gasoline. Q. Was anything said with regard to payment? A. Payment would be wire transferred. Q. Why was Tun Yung going to be invoiced for this gasoline[?] Mr. Abramowitz [attorney for Rea]: Objection. The Court: I’ll allow it. A. So that the 9 cents wouldn’t be paid. Q. Which 9 cents are you talking about? A. Federal excise tax. Q. Did Bill Rea know that? Mr. Rooney [attorney for Getty]: Objection. The Court: Overruled. You can answer it. If you can. A. He had to. Q. Why do you say that[]? Mr. Abramowitz: I object and move to strike. The Court: Sustained. (Trial Transcript (“Tr.”) 67-68.) This description of the July 3 meeting itself did not establish a proper foundation for the witness’s conclusory opinion that Rea must have known that the purpose of invoicing Tun Yung was to evade taxes. Further, there was little in the trial record prior to the quoted passage to provide the requisite foundation. Sarowitz was the government’s first witness; he had described his own background, noting that he had been trained as an accountant, serving Pabone in that capacity since 1983, and had become involved in ordering gasoline for J & J at Pabone’s direction. As to the gasoline industry, Sarowitz had testified generally that a company holding a 637 License is required to pay federal excise taxes when it sells to a company that has no such license. He testified that J & J never had a 637 License, and he illustrated the *1218 tax requirements by testifying that if Getty sold gasoline to J & J it would collect the taxes from J & J and remit them to the government; and if Getty sold gasoline to Tun Yung, it would not be required to remit any tax to the government and hence would not charge the tax. Sarowitz had also described the May/ June meeting with Getty, but only in very general terms. His description had been simply that he told Rea that J & J was interested in purchasing gasoline from Getty; that Smith stopped in to introduce himself and opined that Sarowitz and Pabone were bootleggers; that Sarowitz and Pa-bone denied that they were bootleggers; and that Rea agreed that Getty would sell gasoline to J & J. There was nothing in this testimony to suggest that Rea was given any indication by Sarowitz or Pabone at the May/June meeting that he would be participating in a tax evasion scheme. Though Smith, dropping in at the May/ June meeting, suggested that Sarowitz and Pabone might be bootleggers, Sarowitz and Pabone had denied it. Indeed, the June 4 meeting, which Rea had not attended and at which Pabone and Quock agreed that J & J would use Tun Yung’s 637 License to evade taxes, had not yet occurred when Sarowitz and Pabone first met with Rea. Further, though Sarowitz eventually testified that he had told Rea at the first meeting that J & J had no 637 License, the testimony that he had ever informed Rea of that fact came only after he was allowed to testify that Rea “had to” know that the invoicing of Tun Yung was part of a scheme to evade taxes. As to the purchases of gasoline prior to the July 3 meeting, Sarowitz’s testimony had likewise been sketchy. He said he had not placed the first order and did not know who had done so. And he had given no testimony as to who instructed Rea that Tun Yung should be invoiced on this first sale to J & J or what Rea had been told. Thus, when the court allowed Sarowitz to give his opinion that Rea had to know, it did so in the context of a record in which there had been no testimony that Rea had been told that Pabone and Quock had agreed to a tax evasion scheme, no testimony that Rea had been told that J & J would be using Tun Yung’s 637 License, and no testimony that Rea had been informed that J & J did not have its own 637 License. The record, when the opinion was offered, did not permit the court to conclude that the opinion was based on Sarowitz’s perceptions. It did not suffice for the court, presented with an objection to the question “Did Bill Rea know that,” to permit the witness to answer “[i]f you can.” The question, which was objectionable on a technical basis because in form it called for a factual answer on a matter that the witness could not know as a matter of fact, was permissibly treated by the court as a question calling instead for the witness’s opinion; but the court in effect gave the witness carte blanche to answer if he simply had an opinion. It was instead the responsibility of the court itself to determine whether the witness could properly answer the question in light of the conditions imposed by Rule 701. The record does not indicate that the court did so. Further, once the court allowed the witness to respond with his opinion and chose not to strike the answer, it was an abuse of discretion to preclude testimony as to the basis for the opinion. Adequate exploration was especially important in the present case, since Rea did not attend the key meeting between Pabone and Quock, and later testimony from Sarowitz provided ammunition to oppose a finding of rational basis. For example, Sarowitz later testified that he never told Rea about Pabone’s agreement with Quock to split the evaded taxes; he was not sure whether or not Tun Yung’s name had even been mentioned at the first meeting with Rea. Sarowitz also later testified that he never told Rea that he wanted Tun Yung invoiced in order to permit J & J to avoid paying taxes; nor did he offer Rea anything that Rea could have construed as a bribe to induce him to engage in wrongdoing. Further, Sarowitz apparently could not testify to any statement made by Rea in connection with any federal excise tax requirement. Finally, *1219 after Sarowitz gave his opinion, he testified that in connection with the first in-place purchases after the July 3 meeting, Rea in fact did not have Tun Yung invoiced; rather he had J & J itself and another unlicensed company invoiced, acts that were more likely to reveal than conceal that a taxable event had occurred. Nonetheless, other post-opinion evidence appears to have provided a rational basis for Sarowitz’s opinion. That evidence included Sarowitz’s testimony that he several times told Rea that J & J had no 637 License, and that despite having received that information Rea prepared BTs that ensured that J & J would not be charged excise taxes on sales to J & J even when it was invoiced as the purchaser. If all of this evidence had been presented before Rea was permitted to state his opinion, and the court at that point had found the rational basis precondition satisfied, we would not have viewed that finding as clearly erroneous. Thus, though it was inappropriate for the court to admit the Sarowitz opinion on the record as it stood when the opinion was offered, we conclude that the lack-of-foundation problem was ultimately cured. The question of whether Sarow-itz’s testimony could properly be found helpful, however, is a different matter. The trial court gave no indication that it considered the question of whether this testimony would be helpful to the jury, and we cannot conclude that the opinion passed the helpfulness test since, given all the evidence, it would appear that the jury was perhaps even more capable than Sarowitz— on the basis of his own perceptions — of inferring whether or not Rea knew he was participating in a tax evasion scheme. For example, the government introduced into evidence records kept by Rea (which there was no evidence Sarowitz had seen), and evidence as to Rea’s extensive experience in the petroleum industry (of which Sarow-itz may or may not have been aware, having simply selected Rea’s name from a list someone had given him). Further, there was eventually expert testimony that it was unusual to invoice one company when the gasoline was being ordered, picked up, and paid for by another. Thus, in summation, the government did not rely on Sarow-itz’s opinion but instead argued that the pertinent knowledge was disclosed by Rea’s own records, which revealed that Rea knew J & J was the real purchaser of the gasoline, and by his common sense based on 25 years’ experience in the industry. It argued that this evidence compelled the inference that Rea knew the taxes were due and were not being paid, stating that Sarowitz didn’t need to tell Mr. Rea that they were evading the taxes. You don’t need to tell someone who has been in the gasoline business for 25 years how to invoice gasoline, how to use a 637. You don’t have to tell somebody that has been in the gasoline business for 25 years that you could use somebody else’s 637 to invoice when somebody else is paying for the gasoline. Bill Rea knew. (Tr. 853-54.) Thus, by the conclusion of the presentation of evidence, the jury knew what Sa-rowitz had told Rea and what Rea had done, and in addition it possessed more evidence as to Rea’s industry experience and as to the contents of Rea’s records than Sarowitz indicated he possessed by the time of the July 3 meeting. In all the circumstances, the Sarowitz testimony that Rea “had to” have known that he was to invoice Tun Yung in order to evade payment of excise taxes was not “helpful” within the meaning of Rule 701. It did no more than instruct the jury as to what result it should reach on the issue of knowledge. We conclude that the admission of this opinion was therefore an abuse of discretion. 3. Harmless Error The conclusion that Sarowitz should not have been allowed to give his opinion does not end our analysis, however, for only an error that affects substantial rights is a basis for reversal. A defendant has a right to a fair trial, but not necessarily to a perfect one, see Delaware v. Van Arsdall, 475 U.S. 673, 681, 106 S.Ct. 1431, *1220 1436, 89 L.Ed.2d 674 (1986); United States v. Pisani, 773 F.2d 397, 402 (2d Cir.1985), reh’g denied, 787 F.2d 71 (2d Cir.1986), and an error does not warrant reversal if it is harmless, Fed.R.Crim.P. 52(a). An abuse of discretion in applying the prerequisites of Rule 701, permitting a witness to give opinion testimony that is not helpful to the jury, is not an error of constitutional magnitude. The test for determining whether non-constitutional error is harmless is “what effect the error had or reasonably may be taken to have had upon the jury’s decision.” Kotteakos v. United States, 328 U.S. 750, 764, 66 S.Ct. 1239, 1247, 90 L.Ed. 1557 (1946); United States v. Frank, 494 F.2d 145, 161 n. 19 (2d Cir.), cert. denied, 419 U.S. 828, 95 S.Ct. 48, 42 L.Ed.2d 52 (1974). An erroneous ruling on the admissibility of evidence is harmless if the appellate court can conclude with fair assurance that the evidence did not substantially influence the jury. See Kotteakos v. United States, 328 U.S. at 764-65, 66 S.Ct. at 1247-48; United States v. Pedroza, 750 F.2d 187, 197 (2d Cir.1984), cert. denied, 479 U.S. 842, 107 S.Ct. 151, 93 L.Ed.2d 92 (1986); United States v. Check, 582 F.2d 668, 684 (2d Cir.1978); United States v. Ruffin, 575 F.2d 346, 360 (2d Cir.1978). To say that the erroneously admitted testimony did not substantially influence the jury we are not required to conclude that it could not have had any effect whatever; the error is harmless if we can conclude that that testimony was “unimportant in relation to everything else the jury considered on the issue in question, as revealed in the record,” Yates v. Evatt, — U.S.-, 111 S.Ct. 1884, 1893, 114 L.Ed.2d 432 (1991) (stating a test for error of constitutional dimension). Applying these standards and reviewing the record as a whole, we can conclude with fair assurance that the jury did not regard the Sarowitz opinion as important and was not influenced by it. First, the evidence was ample that Rea knew J & J was the purchaser. J & J’s desire to purchase directly from Getty was, according to Sarowitz’s testimony, the sole topic of discussion at the first meeting between J & J and Getty. Rea knew J & J was paying for the gasoline; from the outset he gave Sarowitz instructions on making payments. At the July 3 meeting, it was specified that the gasoline would be ordered by Pabone or Sarowitz, and thereafter Rea received daily calls with respect to J & J’s orders from Sarowitz identifying himself as “Charlie from J & J.” Thus, Rea knew the gasoline was ordered by J & J, paid for by J & J, and picked up by J & J. Further, after the July 3 meeting, Rea was instructed to send the Tun Yung invoices not to Tun Yung but to J & J; and he was instructed that Tun Yung was not to be allowed to pull gasoline from the J & J account without explicit permission from J & J. Second, Rea had been told on several occasions that J & J had no 637 License. Rea had 25 years’ experience in the industry, and we believe there was no chance that the jury might infer either (1) that Rea did not know that Getty was obligated to pay the excise tax on sales to J & J, or (2) that he did not know of the normal industry practice to pass excise tax charges on to the unlicensed purchaser. Further, the jury was entitled to infer that Rea, when interviewed by the authorities in connection with this matter, gave a false exculpatory statement when he said he had agreed to sell to J & J only on condition that it pay excise taxes; this statement was inconsistent with the documents prepared by Rea, and the jury was entitled to infer from its falsity a consciousness of guilt on Rea’s part. In short, the evidence was compelling that Rea knew or must have known that on sales to J & J, tax was due. Though Rea and Getty claim that they believed Tun Yung itself was collecting the excise taxes from J & J and paying them to the government, Sarowitz testified that while the scheme was ongoing he never told anyone at Getty that J & J had paid Tun Yung the taxes. Third, though Rea and Getty argue strenuously that Rea’s causing J & J and Westwind, rather than Tun Yung, to be invoiced on the first sales after the July 3 meeting proves that he did not know of the tax evasion scheme, this argument is not *1221 compelling. Its principal flaw is that even on the sales to J & J on which Rea caused Getty to invoice J & J itself, thereby failing to use the cover of the Tun Yung 637 License, Rea saw to it that no excise tax was charged. Thus, the evidence was strong that, while knowing J & J was the purchaser and knowing that J & J had no 637 License, Rea prepared BTs that caused Getty not to collect the required federal taxes from J & J. Fourth, as to the invoicing of Tun Yung on the sales to J & J, Getty’s own industry expert testified that it was not a normal industry practice to have gasoline ordered, picked up, and paid for by one company and invoiced to another. It is hardly a leap to infer that when Sarowitz insisted that Tun Yung be listed as the purchaser on these J & J sales, Rea, with his quarter century of industry experience, must have known that this abnormal practice was merely a coverup facet of the plan to evade payment of excise taxes on the J & J sales. Thus, the properly admitted evidence was more than sufficient to permit the jury to infer that Rea must have known that he was participating in a tax evasion scheme. In sum, in light of the properly admitted evidence discussed here (see also Part II.B. below), the jury could easily infer that, though Rea may not have been told of the specific profit-splitting agreement between Pabone and Quoek, Rea knew that if Getty charged J & J the required excise tax, J & J would not buy from Getty, and he was ready, willing, and able to, and did, preparé documentation that would prevent the charging of such taxes. The record as to the summations and the pace and focus of the jury’s deliberations reveals no basis for believing the jury was, instead, influenced in the least by Sarowitz’s opinion. The government did not once mention that opinion in its summation. The jury, which plainly did not regard this as a particularly close case, reaching its verdict in less than a day, gave no indication that it was focusing on the opinion at all. Some two hours into its deliberations, the jury asked to have testimony reread, initially requesting Sarowitz’s testimony only with respect to the first meeting. It thereafter requested his testimony with regard to the second meeting; it did not single out any part of that testimony. It also asked to have reread the testimony of two, other witnesses. After the rereading of all of the testimony, the jury deliberated only four more hours before returning verdicts of guilty as to all four defendants on all four counts. We conclude that the Sarowitz opinion did not substantially influence the jury and that the error of its admission must be considered harmless. B. The Sufficiency of the Evidence as to Intent Rea and Getty also argue that the evidence was insufficient to support their convictions because it did not sufficiently establish, inter alia, their intent to join or further the tax evasion scheme. In challenging the sufficiency of the evidence presented at trial to support a jury’s verdict, a defendant bears a heavy burden. United States v. Maldonado-Rivera, 922 F.2d 934, 978 (2d Cir.1990), cert. denied, — U.S. -, -, 111 S.Ct. 2858, 115 L.Ed.2d 1025, 1026 (1991); United States v. Rubin, 844 F.2d at 983; United States v. Losada, 674 F.2d 167, 173 (2d Cir.), cert. denied, 457 U.S. 1125, 102 S.Ct. 2945, 73 L.Ed.2d 1341 (1982). The government’s proof need not preclude every reasonable hypothesis that is consistent with innocence, United States v. Chang An-Lo, 851 F.2d 547, 554 (2d Cir.), cert. denied, 488 U.S. 966, 109 S.Ct. 493, 102 L.Ed.2d 530 (1988), and in reviewing a sufficiency challenge, we must view the evidence in the light most favorable to the government, Glasser v. United States, 315 U.S. 60, 80, 62 S.Ct. 457, 469, 86 L.Ed. 680 (1942), viewing piécés of evidence not in isolation but in conjunction, United States v. Maldonado-Rivera, 922 F.2d at 978; United States v. Geaney, 417 F.2d 1116, 1121 (2d Cir.1969), cert. denied, 397 U.S. 1028, 90 S.Ct. 1276, 25 L.Ed.2d 539 (1970), and drawing all possible inferences in the government’s favor, United States v. Bagaric, 706 F.2d at 64. Matters of the choice between competing inferences, the credibility of the witnesses, and the weight of the evidence are within *1222 the province of the jury, and we are not entitled to second-guess the jury’s assessments. See, e.g., United States v. Roman, 870 F.2d 65, 71 (2d Cir.), cert. denied, 490 U.S. 1109, 109 S.Ct. 3164, 104 L.Ed.2d 1026 (1989); United States v. Rubin, 844 F.2d at 983; United States v. Stratton, 779 F.2d 820, 828 (2d Cir.1985), cert. denied, 476 U.S. 1162, 106 S.Ct. 2285, 90 L.Ed.2d 726 (1986). Getty and Rea have not met their burden here. Getty contends that the government failed to prove that it had the specific intent to evade federal excise taxes, arguing (a) that since it was invoicing Tun Yung, which had a 637 License, Getty genuinely believed it had no excise tax obligation, (b) that the fact that J & J was the actual purchaser did not establish Getty’s knowledge of that fact or its specific intent to evade taxes, (c) that even establishment of Getty’s knowledge that J & J was the actual purchaser would not prove that Getty intended that the taxes be evaded, and (d) that it was entitled to assume that Tun Yung or J & J would pay the excise taxes. The principal flaws in these arguments are that they would have us view the record piecemeal and in the light most favorable to Getty. The isolated fact that Tun Yung had a 637 License did not mean that Getty’s mere listing of Tun Yung as the purchaser on invoices exonerated Getty from paying the taxes. The producer is liable for the tax if it sells to a non-producer that has no 637 License. The liability is its own; the fact that the unlicensed customer may say that it will pay the tax, or may say that a licensed third party will pay the tax, does not relieve the producer of its tax liability. Certainly the producer cannot escape that tax liability by pretending that gasoline it knows it has sold to an unlicensed customer has really been sold to a licensed entity. And proof of such a pretense will ordinarily support a jury finding that the producer willfully attempted to evade or defeat the payment of the required tax. In support of its challenge to the sufficiency of the evidence as to its knowledge, Getty places substantial reliance on the premise that it was a common industry practice, according to its expert witness, for a nonparty to a petroleum contract to pick up from the seller, and to pay the seller for, gasoline that was ordered by another company. Getty points to BTs and deal sheets for the June sales, which listed Tun Yung as the buyer and Quock as the “contact person” for the buyer. For example, the deal sheet that had been prepared with respect to the June 7 sale to J & J read in part as follows: Sold To: Tun Yung Oil Corp. Contact person: John Quock Tel.: (212) 406-0100 Special instructions or conditions: Tun Yung to w.t. funds 6-7-85 There are several defects in Getty’s argument. First, though it could have been inferred from the information listed on the BTs and deal sheets that, as Getty argues, the gasoline had in fact been ordered by Tun Yung, the jury was also free to infer that, in light of J & J’s just having come to see Getty and having spent 30-40 minutes telling Getty that J & J wanted to purchase gasoline directly from Getty, Getty knew these June sales were in fact sales to J & J and that it was required on those sales to pay excise tax. That inference was supported by the trial evidence that this deal sheet was not an accurate record of the transaction since, though the deal sheet indicated that Tun Yung was to wire transfer (“w.t.”) the funds, it was in fact J & J that paid for the gasoline, wiring the money to Getty in accordance with Rea’s instructions. The jury was also free to infer that other aspects, such as who placed the initial order, were likewise not reflective of the actual events. Further, even if one credited Getty’s assertion that it believed the early sales were upon orders placed by Tun Yung, it is clear that the orders after the July 3 meeting were placed by J & J. Sarowitz testified that it was agreed at that meeting that either he or Pabone would place the orders, and testified that thereafter he called Rea daily or even several times a day. Thus, as *1223 of early July at the latest, the gasoline was not only picked up and paid for by J & J; it was also ordered by J & J. Tun Yung’s role was simply to be listed on the invoice as the purchaser. Getty’s expert testified that it was not a normal industry practice to have gasoline ordered, picked up, and paid for by one company and invoiced to another. Finally, in addition to the evidence discussed in Part II.A.3. above with respect to the knowledge of Rea, there was evidence that high-ranking Getty officials knew that J & J was the purchaser and that Getty knew J & J had no 637 License. The record shows that after the September ov-erpulls by J & J, several Getty officials, including its treasurer, secretary, credit manager, and two vice presidents, became involved in securing payment. The record does not suggest that they looked to Tun Yung for payment, despite the fact that all of this gasoline had been invoiced to Tun Yung; rather, they demanded payment from J & J. Further, even on the over-pulled gasoline, which it knew it had sold to J & J, Getty did not charge excise taxes. One of the Getty officials involved in the overpull negotiations was Smith, who at the very first meeting had tagged J & J as a bootlegger. From all of this evidence, the jury could easily infer that Getty knew that J & J had ordered the gasoline and was the buyer; that Getty knew J & J was unlicensed but was willing nonetheless not to charge J & J the excise taxes; and that Getty deliberately engaged in the abnormal practice of invoicing Tun Yung in order to conceal the fact that it had made sales on which excise taxes were due. The uniformity of Getty’s practice of billing and paying no federal taxes on any gasoline picked up and paid for by J & J, no matter who ordered it, permitted the jury to infer that from the very first meeting at which Getty agreed to sell to J & J, Getty knew it was in fact selling to J & J. Getty also argues that the government failed to prove that it had the intent to evade taxes because there was no evidence that it stood to receive any benefit from the tax evasion scheme or that Rea intended to benefit Getty. Getty argues that it saved nothing by not paying the tax because if it had believed it was liable it would have, in accordance with normal practice, passed the tax on to its customer. In a related vein, Rea argues that he had no stake in and did not receive any benefit from the scheme, emphasizing the evidence that while Pabone and Quock divided more than $1 million in evaded taxes between themselves, they did not give him so much as a bottle of liquor. Again these arguments would have us view bits of evidence in isolation. We are required instead to review the jury’s verdict against the larger backdrop of the record as a whole. The government presented evidence that prior to 1985, Getty’s wholesale gasoline business had been negligible. In February 1985, Getty hired the experienced Rea to build up its wholesale department. Rea was Getty’s only trader involved in bulk transfers. Though Tun Yung may have purchased gasoline from Getty at some time in the past, in 1985, prior to the May/ June meeting, it was not purchasing from Getty. After that meeting, however, Getty’s sales invoiced to Tun Yung burgeoned. Getty terminal managers told a J & J employee during the summer of 1985 that they had never done a large volume of business until J & J came along. In June and July, the sales to Tun Yung amounted to more than 21% of Getty’s gasoline sales in New York State to entities not affiliated with Getty; in August that figure rose to more than 38%; and in September it rose to more than 54%. As Getty’s sales to Tun Yung increased, Getty’s total sales increased proportionately; in 1985 sales invoiced to Tun Yung constituted more than 7% of Getty’s total sales nationwide. By the end of the scheme in the fall of 1985, Getty had sold more than 11 million gallons of gasoline to J & J invoiced to Tun Yung. After the scheme ended in the fall of 1985, when Pabone told Sarowitz to buy elsewhere as a result of Getty’s decision to begin charging excise taxes, the sales invoiced to Tun Yung quickly dropped to zero. The jury was entitled to infer from this evidence that Getty profited handsome *1224 ly from the increased sales generated by its participation in the scheme. A similar inference as to Rea’s interest in the scheme was available from the size of the bonus Getty paid Rea in 1985. Rea was hired in February of that year at an annual salary of $100,000. His assignment was to increase Getty’s wholesale sales. At the end of 1985, Getty’s incentive compensation committee voted him, in addition, a $90,000 bonus. The committee had no established criteria for determining the amount of bonuses, and it was permissible to infer that the sales to J & J, accounting for more than 7% of Getty’s national sales, were the impetus for Rea’s large bonus. Plainly Rea played an integral role in attainment of the conspiracy’s objectives, for without an employee at Getty ensuring that J & J was not charged excise tax, J & J would have been so charged; and if Tun Yung rather than J & J were not invoiced on these sales, it would have been clearer from the face of the documentation that the sales to J & J were taxable events. The evidence indicated that if J & J had been charged excise taxes, it would not have bought from Getty, for when Getty eventually decided that the taxes would be charged, it lost J & J as a customer. The jury could well have inferred that Rea was pursuing his interest in keeping J & J as a customer in order to maintain and increase the level of wholesale gasoline sales that he had generated and to earn a large bonus for doing so effectively what he had been hired to do. In sum, while the record does not indicate that Rea and Getty received a portion of the excise taxes evaded by the charged scheme, the evidence adduced at trial was sufficient to permit the jury to infer that both Rea and Getty knew of the scheme, had an interest in its continuation, and benefited from it in other ways. C. Other Arguments Defendants also challenge several of the district court’s evidentiary or procedural rulings, including its rejection of Rea’s attempt to introduce the results of a polygraph test he had taken, the denial of Pa-bone’s motion to be tried separately from Quock, the exclusion of certain out-of-court statements of Pabone offered at trial by Pabone, and sentencing Pabone without an evidentiary hearing into his net worth. We have considered all of defendants’ challenges and have found them to be without merit. Only those mentioned warrant discussion. 1. The Exclusion of the Rea Polygraph Results Rea contends that the district court improperly denied his motion for an in limine ruling permitting him to introduce at trial the results of a polygraph examination to which he had voluntarily submitted. We find no error. This court has intimated in the past that the results of polygraph examinations are not admissible in this Circuit, see, e.g., United States v. Bortnovsky, 879 F.2d 30, 35 (2d Cir.1989) (compiling cases). The district court stated that it was not prepared to disregard Second Circuit authority and that it did not believe polygraph tests were sufficiently reliable to warrant the admission of the results in evidence. We see no abuse of discretion in this ruling. Even assuming that such test results are not per se inadmissible, the record before us does not indicate that the results of Rea’s test were sufficiently reliable or sufficiently relevant to warrant admission. 2. Pabone’s Contentions Pabone contends first that the district court erred in denying his motion for a severance of his trial from that of Quock based on Pabone’s expectation that his defense would be inconsistent with that of Quock. Severance on the ground of inconsistent defenses is required only when the jury, in order to believe the core testimony offered on behalf of one defendant, must necessarily disbelieve the testimony offered on behalf of his codefendant, United States v. Serpoosh, 919 F.2d 835, 838 (2d Cir.1990); United States v. Potamitis, 739 F.2d 784, 790 (2d Cir.), cert. denied, 469 U.S. 918, 105 S.Ct. 297, 83 L.Ed.2d 232 (1984), or when the jury will infer that both *1225 defendants are guilty solely because of the conflict, United States v. Serpoosh, 919 F.2d at 838. A motion for severance is committed to the discretion of the district court, and its denial will not be disturbed unless the defendant demonstrates that the denial caused him substantial prejudice amounting to a miscarriage of justice. United States v. Villegas, 899 F.2d at 1346; United States v. Potamitis, 739 F.2d at 790. Pabone has not carried this burden here. In his pretrial severance motion, Pabone argued that his defense would be that J & J had relied on Tun Yung to pay the excise taxes and to pass them on to J & J in a proper fashion; he did not indicate what he expected Quock’s defense to be. The district court denied the motion without prejudice, indicating that it would grant Pabone a mistrial if it turned out that the joint trial was unduly prejudicial. At trial, Quoek’s defense was in the nature of a general denial, as he urged the jury simply to find that he “was not a party to any conspiracy.” Since there was no logical impediment to the jury's acceptance of both defendants’ positions, on the theory that Pa-bone had thought Tun Yung would pay the excise taxes and bill J & J and that Quock had failed to pay the taxes simply as a result of negligence rather than conspiracy, there is no basis for reversal. Pabone also contends that the trial court erred in precluding him from introducing at trial certain statements that he here attributes to Quock. Our review of the record reveals that the statements in question were not statements of Quock but statements made by Pabone. To the extent that Pabone offered proof of his own prior statements for the truth of the matters asserted, they were hearsay, see Fed.R.Evid. 801(c); to the extent that the statements were offered instead only to show that Pabone made them, he has not persuaded us that the fact of their utterance had any relevance. Pabone’s contention that the government, by presenting evidence of certain of Pabone’s statements, “opened the door” for him to introduce evidence of other statements by him, is meritless. The concept of “opening the door,” or “curative admissibility,” gives the trial court discretion to" permit a party to introduce otherwise inadmissible evidence on an issue (a) when the opposing party has introduced inadmissible evidence on the same issue, and (b) when it is needed to rebut a false impression that may have resulted from the opposing party’s evidence. See, e.g., United States v. Brown, 921 F.2d 1304, 1307 (D.C.Cir.1990); United States v. Whitworth, 856 F.2d at 1285; 1 J. Wig-more, Evidence § 15 (Tillers rev. 1983). To be admissible under the doctrine of curative admissibility, the evidence that allegedly opened the door must in fact have been inadmissible. Properly admitted evidence does not open the door to inadmissible evidence. United States v. Brown, 921 F.2d at 1307. Here, the testimony elicited by the government was not inadmissible. The statements of Pabone when offered by the government were, by definition, not hearsay. See Fed.R.Evid. 801(d)(2)(A). Hence their admission did not open the door to the introduction by Pabone of his own hearsay or irrelevant statements. Finally, Pabone contends that he is entitled to a remand for resentencing because the district court failed to hold an evidentiary hearing into the government’s assertions that he had ties to organized crime and had previously been incarcerated for civil contempt, and because the probation department did not pass on to the court all of the information it had obtained from Pabone as to his financial condition. These contentions have no merit. Though it appears that the financial information given the district court was indeed incomplete, there is no indication in the record that the scantiness of the financial records before the court adversely affected the sentence imposed in any way. Rather, counsel represented that Pabone had provided the authorities with ample financial information (of which counsel stated he had a copy that he had neglected to bring with him to the sentencing hearing), and the court appeared to accept that representation. *1226 As to the government’s allegations that Pabone had ties to organized crime, while counsel mentioned United States v. Fatico, 579 F.2d 707 (2d Cir.1978), cert. denied, 444 U.S. 1073, 100 S.Ct. 1018, 62 L.Ed.2d 755 (1980), he did not appear to request an evidentiary hearing. Rather, he suggested that the court could take judicial notice that the organized crime families referred to by the government did not accept as associates men of Pabone’s ethnic background, and he appears to have conceded that a photograph introduced at trial was an indication that Pabone did have such ties. As to the government’s representation that Pabone had previously been held in contempt of court, counsel made no comment whatever. We see no basis for resen-tencing. CONCLUSION We have considered all of defendants’ arguments on these appeals and have found in them no basis for reversal. The judgments of conviction are affirmed.