Defendants, the owner and two employees of The Training Center (TTC), a vocational school in Dearborn, Michigan, were convicted of defrauding the Department of Education by falsifying or directing the falsification of records the school was required to maintain for its students to receive financial aid. Mahmoud Younis, TTC’s owner, and Muna Jaber, were convicted of Conspiracy to Defraud the United States (18 U.S.C. § 371), Wire Fraud (18 U.S.C. § 1343), and Student Financial Aid Fraud (20 U.S.C. § 1097). Sahar Younes was convicted of Student Financial Aid Fraud. The three challenge their convictions on several grounds, and Younis challenges various aspect of his sentence. We affirm the convictions and affirm in part and reverse in part Younis’s sentence.
I. Factual and Procedural History
Factual History. TTC was a for-profit vocational school in Dearborn, Michigan, that, beginning in the early 1990s, participated in federal financial aid programs. Federal financial aid is granted in the form of direct tuition payments to qualifying schools. To remain qualified, schools must follow certain procedures and be regularly accredited by an approved accrediting body. To be accredited, TTC was required, among other things, to ensure the eligibility of its students to enroll, to enroll them in courses for which financial aid was available (for example, travel and computer, but, importantly, not English as a Second Language (ESL)), to submit to periodic independent audits, and to maintain on file records showing student eligibility. Eligibility records can be a copy of each student’s high school diploma, a GED certificate, a certification that the student had completed high school abroad, or a certificate that the student had passed an “Ability to Benefit” (ATB) test.
Younis owned TTC during the relevant period. According to the allegations, Younis, Younes, and Jaber falsified or directed the falsification of information in student records necessary to receive financial aid. In particular, TTC employees (1) supplemented student files with fake high school diplomas, GED scores, or ATB test scores (even when the students might have possessed proper documentation at home), and (2) indicated that students had enrolled in travel and computer classes when in fact they had enrolled in ESL. The government also argued — though, as discussed below, not in the indictment — that Younis sent false reports to TTC’s accrediting body (Accrediting Commission of Career Schools and Colleges of Technology (ACCSCT)) concerning attendance at TTC advisory committee meetings.
The government alleged that TTC employees typically falsified student files on Saturdays in the weeks prior to the ACCSCT audits. Lorilynn Coggins, the person charged with monitoring TTC’s financial aid compliance, gave lists of incomplete student files to Younis, TTC administrator Walid Alsabbagh,1 or TTC director of financial aid, Linda Johnson. Linda Johnson’s assistant from October 1998 until July 1999 was defendant Muna Jaber. Coggins, Alsabbagh, or Younis would then instruct other employees to complete the files. “Depending on how much time [was available],” an employee testified, the employees would either contact the students *306to obtain accurate information, or simply “come up with information on [their] own.”
The government also alleged fraudulent misclassification of ESL students as enrolled in courses eligible for aid. Specifically, the government identified several teachers who taught ESL courses, including defendant Sahar Younes, but whose students were recorded as enrolling in travel or computer courses. Younes testified that she taught only ESL courses, but admitted leaving blank the course descriptions on attendance sheets that eventually listed her as teaching travel or computer.
Procedural History. The original indictment charged Younis, Younes, and Jaber with conspiring to defraud the government by falsifying or directing the falsification of student information in violation of 18 U.S.C. § 371.2 The indictment also charged wire fraud, in violation of 18 U.S.C. § 1343,3 and student financial aid fraud, in violation of 20 U.S.C. § 1097.4
The government filed a second indictment after the district court dismissed the first count for failing to state an offense within the five-year statute of limitations and the third count for insufficient specificity. The Second Superseding Indictment (SSI), the operative document for this appeal, repeated the three charges, but added background information regarding the dates and amounts of loan disbursements received by TTC and the general dates of the overt acts supporting the conspiracy. The SSI supplemented the list of overt acts with allegations that, on or around particular dates, Younis submitted reports and attendance sheets to ACCSCT and that Jaber certified financial aid forms for several students. Defendants renewed their earlier objections, but the district court upheld the SSI.
*307Several spreadsheets cataloging students whose files contained false documentation were admitted at trial as summary-evidence under Fed.R.Evid. 1006. Defendants objected to the spreadsheets as argumentative — specifically, for including the government’s conclusion that certain student files contained “false” or “fraudulent” information — and the court had those columns removed from the spreadsheets. The court, over defendants’ objection, then admitted the spreadsheets as a summary of other evidence in the record. The defendants renewed their objection and also objected to the spreadsheets as being prejudicial and inaccurate, insisting that some students whose files were listed as containing fraudulent documentation were nonetheless eligible for aid.
Younis was sentenced to a 41-month term of imprisonment and ordered to pay restitution of $793,225 and a fine of $125,000. Jaber was sentenced to three years probation, the first six months to be served in home confinement, and ordered to pay a $300 special assessment and restitution of $5000. Younes was sentenced to two years probation, the first six months to be served in home confinement, and ordered to pay a $100 special assessment and restitution of $63,193.
II. Analysis
A. Trial Errors
All defendants urge four types of trial error: insufficiency of the indictment, abuse of discretion in denying a bill of particulars, abuse of discretion in admitting the spreadsheets, and refusal to grant a motion to acquit for insufficient evidence. Additionally, Jaber alone urges prosecutorial misconduct, and Younes alone argues that the district court abused its discretion by failing to admit her polygraph test.
1. Notice in the Indictment
Younis’s first argument, joined by both Jaber and Younes, is that the indictment was insufficiently specific to provide the notice to the defendants required by the Sixth Amendment. The sufficiency of an indictment is reviewed de novo. United States v. Superior Growers Supply, Inc., 982 F.2d 173, 177 (6th Cir.1992).
In general, an indictment is constitutionally adequate “if it, first, contains the elements of the offense charged and fairly informs a defendant of the charge against which he must defend, and, second, enables him to plead an acquittal or conviction in bar of future prosecutions for the same offense.” Id. at 176 (citing Hamling v. United States, 418 U.S. 87, 117, 94 S.Ct. 2887, 41 L.Ed.2d 590 (1974)); see United States v. Douglas, 398 F.3d 407, 413 (6th Cir.2005) (reciting the twin goals of providing notice of the current charge and allowing the defendant to plead double jeopardy for a subsequent charge on the same conduct). “An indictment will usually be sufficient if it states the offense using the words of the statute itself, as long as the statute fully and unambiguously states all the elements of the offense.” Superior Growers, 982 F.2d at 177 (citation omitted). However, the recitation of statutory language “ ‘must be accompanied with such a statement of the facts and circumstances as will inform the accused of the specific offense, coming under the general description with which he is charged.’” Id. (quoting Hamling, 418 U.S. at 117-18, 94 S.Ct. 2887). An indictment must assert “facts ... which, if proved, would establish prima facie the defendant’s commission of [the] crime.” Id. at 177 (citation omitted).
Count One charged the defendants with conspiracy to defraud the government. 18 U.S.C. § 371. “The gist of the offense of criminal conspiracy ... is agreement among conspirators to commit an *308offense attended by an act of one or more of the conspirators to effect the object of conspiracy.” United States v. Falcone, 311 U.S. 205, 210, 61 S.Ct. 204, 85 L.Ed. 128 (1940). “Those having no knowledge of the conspiracy are not conspirators,” even if one such person furthers the conspiracy’s object. Id. at 210-11, 61 S.Ct. 204. “The elements of a section 371 conspiracy to defraud are: (1) an agreement to accomplish an illegal objective against the United States; (2) one or more overt acts in furtherance of the illegal purpose; and (3) the intent to commit the substantive offense, i.e., to defraud the United States.” Douglas, 398 F.3d at 413 (6th Cir.2005). The overt act need not be itself a crime. Braverman v. United States, 317 U.S. 49, 53, 63 S.Ct. 99, 87 L.Ed. 23 (1942).
The indictment here provides sufficient notice of the conspiracy charge and would support a later double-jeopardy defense. It alleges that Younis, Younes, and Jaber had an agreement (“knowingly and willfully conspired together”) with unlawful intent (“to defraud an agency of the United States”) and committed various overt acts (including “falsifying] and directing] the falsification of records” as well as various lawful acts) in furtherance of the conspiracy. The indictment thus alerted the three defendants that each could defend on the ground that (1) he or she was not party to the conspiracy, (2) the conspiracy had no unlawful purpose, or (3) the alleged overt acts did not occur. The SSI provided particular dates for several of the alleged overt acts and general dates for all others. In doing so, the indictment provided a sufficient bar to subsequent suits. The government could not again charge the defendants with conspiring to defraud the Department of Education between July 1995 and August 1999 and furthering such a conspiracy through the articulated acts.
Yet Younis contends that the indictment is insufficiently specific because it “contains no description of any agreement between [the] defendants, apart from its conclusory quotation of the statute” and because “Count One contains no section describing the goal of the conspiracy, its method or means, or its membership ... [and] does not allege the date of the agreement, its scope, [or] the role of any participants.” Younis cites no authority holding that an indictment must describe these features; he instead devotes substantial attention to Superior Growers, supra, in which an indictment was dismissed, not for its technical defects, but rather on substantive grounds for failing to allege that the defendants possessed the “requisite criminal intent” for the crime charged. 982 F.2d at 178. The case did not concern the relevant question here, whether allegations of clearly criminal conduct were sufficiently specific to provide the defendant the degree of notice required by the Constitution.
More relevantly, though still unpersuasively, Younis also relies on the 1888 case of United States v. Hess, 124 U.S. 483, 8 S.Ct. 571, 31 L.Ed. 516 (1888), which held that while an indictment may follow statutory language, “it is not sufficient that the indictment shall charge the offence in the same generic terms as in the [statute]; [for a] crime is made up of acts and intent; and these must be set forth in the indictment with reasonable particularly of time, place, and circumstances.” Id. at 487-88, 8 S.Ct. 571. The indictment here did not parrot the statute: whereas section 371 speaks simply of “eonspir[ing] to commit any offense against the United States, or to defraud the United States, ... in any manner or for any purpose,” (emphasis added) the indictment charged that the defendants “falsified or directed the falsification of records and violated regulations [a particular method] in order to continue *309accreditation and to receive funds from the Department of Education [a particular object].” And as for the time, date, and circumstances of the alleged conduct, the indictment specified a month, or in some cases a day, when each of the overt acts occurred. This information provided the defendants with sufficient detail to mount a defense, and it was sufficiently particular to bar a subsequent suit concerning the same conduct. Hess requires no more. See id. at 488, 8 S.Ct. 571.
Indeed, Younis seeks more detail in the indictment than is required to prove a conviction. “To establish a conspiracy, the government need not show an explicit agreement, but merely that the defendant knew the object of the conspiracy and voluntarily associated himself with it to further its objectives.” United States v. Jamieson, 427 F.3d 394, 402 (6th Cir.2005) (quotation marks omitted). And the “membership” of the conspiracy need not be alleged, because an indictment may be sufficient without identifying the co-conspirators. See United States v. Phibbs, 999 F.2d 1053, 1083 (6th Cir.1993) (“[I]t is the grand jury’s statement of the existence of the conspiracy agreement rather than the identity of those who agree which places the defendant on notice of the charge he must be prepared to meet.” (quoting United States v. Piccolo, 723 F.2d 1234, 1239 (6th Cir.1983))).
Younis also takes issue with the indictment’s use of the passive voice in the list of overt acts, suggesting that the Constitution requires those who commit the overt acts be named. See, e.g., Indictment, J.A. 84-86, “G.E.D. certificates and transcripts were falsified”; “[ATB] Tests were falsely made”; “High School Diplomas were falsified.” But an indictment need not name which defendant committed which overt acts. All that must be proved for a conviction is that “one of the conspirators ... knowingly committed at least one overt act charged in the indictment ... [and] that such overt act was knowingly done in furtherance of some object or purpose of the conspiracy as charged.” United States v. Kraig, 99 F.3d 1361, 1368 (6th Cir.1996) (citation omitted). Paragraph B of the indictment alleged that the defendants (actively) “falsified or directed the falsification of’ various documents for the purpose of receiving funds from the Department of Education; it then introduced the list of overt acts with the statement that “[t]he following actions were taken in this regard.” (Emphasis added.) This sufficed to allege that at least one of the defendants was responsible for either the commission of or the direction of each of the alleged overt acts.
Counts Two and Three pass constitutional muster because they recite the statutory elements of wire fraud and student financial aid fraud and incorporate the overt acts alleged under Count One. As noted by the court in Superior Growers, 982 F.2d at 177, the statutory elements by themselves would not provide sufficient notice of the charged conduct. Counts Two and Three thus obtain an acceptable level of specificity only by incorporating the overt acts from Count One. Just as with Count One, we find that those overt acts were sufficiently specific to bar a subsequent suit and to allow the defendants to defend.
The last matter regarding the indictment concerns the allegation that Younis sent advisory committee meeting minutes to ACCSCT. The indictment charged that, as one of the overt acts, Younis “submitted reports to the ACCSCT.” At trial, government witness and advisory committee member George Saba testified that a report sent by Younis to ACCSCT falsely asserted his attendance at a committee meeting. The government referred to the *310falsification in its closing argument, and the jury requested to view the supporting exhibits during its deliberations. Younis contends that this alleged falsification was beyond the scope of the indictment (because the indictment said nothing about the submission of false reports to ACCSCT) and that, as demonstrated by the jury’s interest, he was prejudiced by its introduction.
Even assuming that the allegation concerning the committee minutes was beyond the scope of the indictment, Younis cannot show that he was prejudiced by the error. In particular, it is hard to see how the falsity of the report sent to ACCSCT, as opposed to the mere fact of it being sent (as alleged in the indictment), secured his conviction. As noted, a conspiracy conviction requires the finding of an agreement with an unlawful purpose and an act furthering it. To have convicted Younis, the jury had to find first an agreement and an unlawful purpose (which we believe to have been supported by other evidence). To have then found that he sent false reports to ACCSCT, it would have found both that they were false and that he sent them. And because the sending of even accurate reports would have furthered the conspiracy, we believe Younis would have been convicted without the additional allegation of the reports’ falsity.
2. Bill of Particulars
Younis, joined by Younes and Jaber, argues briefly that the district court abused its discretion by refusing to order the government to provide a bill of particulars. See United States v. Rey, 923 F.2d 1217, 1222 (6th Cir.1991). “Proof of abuse of discretion requires a showing of actual surprise at trial and prejudice to the defendant’s substantial rights by the denial.” Id. “An abuse of discretion exists only when the reviewing court is firmly convinced that a mistake has been made.” United States v. Salisbury, 983 F.2d 1369, 1375 (6th Cir.1993).
We hold that the district court did not abuse its discretion by refusing to order a bill of particulars. As will be seen below, the most crucial testimony in the government’s conspiracy case was employees’ accounts that Younis and Jaber directed the falsification of files leading up to the audits, not the truth or falsity of any particular files. The allegations that the defendants engaged in such conduct appeared on the face of the indictment, and the employees who testified were not fringe players, but in most cases familiar TTC employees who had also been arrested and who had executed plea agreements with the government. That they would testify to Younis and Jaber’s instructions to them was highly predictable, without a bill of particulars. Thus, we are not “firmly convinced that a mistake has been made” by the district court’s refusal to grant one. Id.
Younis’s citations do not alter our conclusion. In United States v. Dempsey, 733 F.2d 392 (6th Cir.1984), this court simply remarked (albeit approvingly) that the district court had granted a bill of particulars; it did not articulate circumstances requiring one. In United States v. Bortnovsky, 820 F.2d 572 (2d Cir.1987), the court ordered a bill of particulars where the government failed to provide the dates of burglaries that the defendants had allegedly faked for insurance claims. Here, the indictment provided the general dates of the falsifications, and the defendants had sufficient notice of their suspected roles in the fraud. And in Salisbury, the issue was “not that the prosecution failed to set out a sufficiently detailed factual basis to support its charges, but that the statute [at issue] fail[ed] to provide sufficient guid*311anee to notify defendant that her activities were [proscribed].” 983 F.2d at 1375.
3. Admission of Spreadsheets
Younis next contends the district court abused its discretion by admitting several spreadsheets as summary evidence. “The admission of summary charts is a matter within the discretion of the district court, whose decisions in such matters will be upheld absent an abuse of discretion.” United States v. Bray, 139 F.3d 1104, 1109 (6th Cir.1998) (quotation omitted).
Fed.R.Evid. 1006 provides:
The contents of voluminous writings, recordings, or photographs which cannot conveniently be examined in court may be presented in the form of a chart, summary, or calculation. The originals, or duplicates, shall be made available for examination or copying, or both, by other parties at [a] reasonable time and place. The court may order that they be produced in court.
This court has identified five preconditions to the admission of summary charts under the rule:
The documents ... must be so voluminous that they cannot conveniently be examined in court by the trier of fact. That is, the documents must be sufficiently numerous as to make comprehension difficult and inconvenient....
[T]he proponent of the summary must also have made the documents available for examination or copying, or both, by other parties at [a] reasonable time and place....
[T]he proponent of the summary [must] establish that the underlying documents are admissible in evidence. Thus, if the underlying documents are hearsay and not admissible under any exception, a chart or other summary based on those documents is likewise inadmissible....
[A] summary document must be accurate and nonprejudicial. This means first that the information on the document summarizes the information contained in the underlying documents accurately, correctly, and in a nonmisleading manner. Nothing should be lost in the translation.
[A] summary document must be properly introduced before it may be admitted into evidence.
Bray, 139 F.3d at 1109-10 (quotation marks omitted). Younis contends that the admission of the spreadsheets violated the fourth condition by being argumentative, misleading, and unreliable. Younis also objects that the spreadsheets contain hearsay, but he does not develop the argument.
Younis contends that the spreadsheets were impermissibly argumentative because they contained the government’s conclusions about the falsity of particular documents and because they did not attempt to summarize the underlying documents as a whole. Younis fails, however, to identify anything in particular in the spreadsheets that is argumentative. Previous versions of the spreadsheets featured column headings such as “Start Date Falsified” and labeled several entries as “FRAUD,” which the district judge ordered redacted. Yet, the revised versions, the only relevant ones here, contained no such conclusions. Consider Government Exhibit 562D, which listed students whose records indicated that they enrolled in “travel,” “travel/night,” or “travel/computer” courses with Sahar Younes, who (as noted) taught only English courses. Younis would characterize such information as conclusive of fraud and thus argumentative. But the entries were not conclusive because the information summarized also appeared in the student records provided to the jury *312(per Bray’s second requirement). The spreadsheets included no new information; they consolidated — but did not modify— information that was already admitted and available to the jury.
Regarding the spreadsheets’ capacity to mislead the jury, Younis’s urges that “[t]he charts’ only significance was as lists setting forth the government’s conclusion that the files listed contained fraud, in the amounts enumerated.” Yet all of the government’s exhibits were intended to support its conclusion that fraud occurred, including the spreadsheets summarizing those exhibits. Younis does not argue that the spreadsheets excluded information that would have undercut the fraud inference, and therefore, his argument that the spreadsheets were misleading must fail.
Finally, Younis is wrong to suggest that the spreadsheets were unreliable. Younis notes that “four students listed as ineligible testified they had legitimate high school diplomas or GEDs” and that “a number of students whose files contained falsified attendance sheets ... in fact had attended class and completed the program.” But these facts do not undermine the only claim made by the spreadsheets: that the files TTC maintained for those students contained falsified high school diplomas, GEDs, or attendance sheets.
4. Sufficiency of the Evidence
Younis, Jaber, and Younes each allege the insufficiency of evidence supporting their convictions. “When reviewing the sufficiency of the evidence to support a criminal conviction, the relevant question is whether, after viewing the evidence in the light most favorable to the prosecution, any rational trier of fact could have found the essential elements of the crime beyond a reasonable doubt.” United States v. Caraway, 411 F.3d 679, 682 (6th Cir.2005) (quotation marks omitted). We consider each defendant in turn.
Younis. Younis contends that the evidence was insufficient to show that he knew of or directed the fraud at TTC. He does so primarily by challenging the credibility of employees Linda Johnson, Greg Moore, and Don Williams, each of whom testified that Younis directly instructed them to falsify documents. Younis recognizes that witness credibility is a jury question, but insists that this court should disregard Johnson’s testimony because “where testimony is patently incredible or defies physical realities, it may be rejected by the court, despite the jury’s evaluation.” Id.
The problem for Younis is that nothing in Johnson, Moore, or Williams’ testimony was “patently incredible” or “defie[d] physical reality.” None testified to anything self-contradictory or implausible. And the fact that other witnesses did not testify that Younis directed them to falsify student records cannot be a basis for rejecting what Younis purportedly told Johnson, Moore, and Williams.
Jaber. Sufficient evidence also supported the conclusion that Jaber “knew the object of the conspiracy and voluntarily associated [herjself with it to further its objectives.” United States v. Jamieson, 427 F.3d 394, 402 (6th Cir.2005).
Jaber told Department of Education investigator Maureen Pawlak that she “knew” documents were “reconstructed” and “believed” others to be falsifying them. (J.A. 816.) At the time, Jaber was working in TTC’s financial aid department with Linda Johnson, understood the eligibility requirements for financial aid, and processed the lists of incomplete files. In her reply brief, Jaber argues that some of Pawlak’s testimony on cross-examination suggests that although Jaber believed it possible that documents were being falsi*313fied, Jaber lacked knowledge of the wrongdoing. This is a plausible reading of Pawlak’s testimony, but so too would be the conclusion that Pawlak believed Jaber to possess actual knowledge of the fraud. Drawing all inferences in the government’s favor, as we must, we find sufficient evidence that Jaber had actual knowledge.
Recall as well that Johnson testified Jaber was “involved” in the falsifying of documents. (J.A. 773.) Jaber’s only response is to question Johnson’s credibility and to assert that Johnson’s testimony was not based on personal knowledge and lacked factual support. Such matters are for the fact-finder or are within the discretion of the district court.
Finally, Maria Neal, employed by TTC to do various administrative duties between January 1999 and September 1999, testified that Jaber directed her to fill in blank attendance sheets and to make the files of ESL students reflect enrollment in computer or travel classes. In response, Jaber points out that Neal, on cross-examination, noted that she did not view herself as falsifying anything when she filled in the forms. But this is irrelevant because Jaber, not Neal, is the defendant, and it was reasonable for the jury to conclude that Jaber, in instructing Neal to falsify the forms, appreciated the fraud she was perpetrating.
Younes. Last, sufficient evidence supports Younes’s conviction for student financial aid fraud.
Though she taught only ESL, Younes admitted to Department of Education investigator Steven Hunt that she signed attendance records that falsely represented that her students had taken travel or computer courses. Younes argues that this did not demonstrate her knowledge or participation in the fraud because, in the conversation with Hunt, she noted that she neither dated her signature nor completed the portion of the form that included the course descriptions. Rather, she insists that she simply left the forms partially blank and passed them along to Cassandra Smith. Smith, however, testified that Younes was occasionally present when Smith completed the forms. On this evidence, we believe that a rational juror could infer that Younes had sufficient knowledge of the fraud to support her conviction.
5. Prosecutorial Misconduct
Jaber alone argues for reversal of her conviction on the basis of prosecutorial misconduct. She contends that, in closing arguments, “[t]he prosecutor misstated the law, commented on Ms. Jaber’s failure to testify and argued prejudicial, uncorroborated, unadmitted and unreliable evidence to the jury.” Jaber’s trial counsel failed to object at the time, so we review the trial court’s conduct for plain error. See United States v. Combs, 369 F.3d 925, 938 (6th Cir.2004).
To establish plain error in a claim of prosecutorial misconduct, Jaber “must show (1) the existence of an error, (2) that is plain, (3) that substantially affects the rights of the defendant, and (4) that substantially affects the fairness of and integrity of the proceedings.” United States v. Owens, 426 F.3d 800, 806 (6th Cir.2005). Jaber must show that “the prosecutorial misconduct was so especially flagrant that it constitutes plain error.” Id. at 807 (quotation marks omitted). Flagrancy is to be determined “by examining four factors: (1) whether the statements tended to mislead the jury or prejudice the defendant, (2) whether the statements were isolated or among a series, (3) whether the statements were deliberate, and (4) the total strength of the evidence against the accused.” Id. Under this standard, the prosecutor’s re*314marks do not warrant remand or reversal of Jaber’s conviction.
Jaber first challenges the prosecutor’s statement that “Unless you hear otherwise, it is reasonable to assume that somebody signs a documents [sic], it’s their signature. It purports that’s what it is.” (J.A. 940.) Jaber contends that the prosecutor thus misstated Fed.R.Evid. 901 (requiring authentication or identification of documents), “improperly attempted] to shift the burden of proof to Ms. Jaber to prove her innocence,” and improperly referenced Jaber’s right not to testify. In response, the government argues that the statement could be read as referring not retrospectively to the defendant’s burden at trial, but prospectively to what the defense might say in closing. Because this reading is plausible, the error is not “plain.” As the government also notes, Jaber’s trial counsel conceded the authenticity of the signatures in question, which precludes the error from “substantially affecting] the fairness of and integrity of the proceedings.” Owens, 426 F.3d at 806.
Jaber next challenges the prosecution’s mention of two exhibits that she says were not admitted into evidence. The government contends, however, that the exhibits, numbers 534E and 534F, were admitted into evidence because the prosecution initially moved for “admission of 534A through F,” laid a foundation for some of the subparts, and then — generally, it contends — “move[d] for admission.” (J.A. 843.) But this argument rings hollow because the prosecutor was addressing only 534A when he made the second motion, and it is fair to conclude that the prosecutor erred by mentioning the exhibits in closing.
We need not, however, consider whether this error was plain because it did not “substantially affect[] the rights of the defendant [or] the fairness of and integrity of the proceedings.” Owens, 426 F.3d at 806. The evidence against Jaber was “otherwise strong.” United States v. Forrest, 402 F.3d 678, 686 (6th Cir.2005). According to Pawlak, Neal, and Johnson, Jaber either knew of or was involved with the fraud, and it is undisputed that Jaber worked closely with Johnson when fraudulent activities were occurring. We thus find the prosecutor’s improper identification of Jaber as “Director of Admissions” and further suggestion that she knew of the “ACCS[C]T report” (J.A. 945) insufficiently prejudicial to warrant reversal on plain-error review.
6. Failure to Admit Younes’s Polygraph Test
Younes further contends that the district court abused its discretion by refusing to admit the results of a unilaterally obtained polygraph test that Younes hoped would bolster her denials of guilt and by refusing to hold an evidentiary hearing about the results. We review such decisions for abuse of discretion. United States v. Thomas, 167 F.3d 299, 308 (6th Cir.1999).
“The district court is not obligated to hold a Daubert hearing, and in the absence of a hearing, [this court] must review the record to determine whether the district court erred in its assessment of the relevance and reliability of the expert testimony.” Clay v. Ford Motor Co., 215 F.3d 663, 667 (6th Cir.2000). In this case, the district court found that a hearing was unnecessary because Younes did not meet her burden of establishing the reliability of the proposed testimony. It also found that the evidence should be excluded under Fed.R.Evid. 403 because its probative value was outweighed by the risk of unfair prejudice.
*315In refusing to grant a hearing or admit the polygraph evidence, the district court did not abuse its discretion. The Supreme Court noted in United States v. Scheffer, 523 U.S. 303, 309, 118 S.Ct. 1261, 140 L.Ed.2d 413 (1998), that “the scientific community remains extremely polarized about the reliability of polygraph techniques,” and Younes offers no new evidence suggesting otherwise. Moreover, this court takes the view that polygraph test results are generally disfavored and should be viewed with great skepticism. Thomas, 167 F.3d at 308. In this case, the fact that the test was one-sided and conducted without the government’s agreement further militates against admissibility.
B. Sentencing Errors
Younis contends that the district court erred in calculating the loss amount used in his sentencing calculation, in ordering $793,225 in restitution, and in imposing a $125,000 fine.
1. Amount of Loss
Younis’s offense level under the now-advisory Sentencing Guidelines is a function of the amount of loss caused by the fraud. After lengthy argument on the issue, the district court determined the loss to be $793,225. Relying on the spreadsheets the government presented at trial and sentencing, the court described this amount as the “intended loss with regard to all the dummy up of the files.” (J.A. 1003.) Those spreadsheets reflected government disbursements for: (1) students with fraudulent documents in their files, (2) students enrolled in certain classes ineligible for government aid, and (3) students whose attendance records had been falsified — a total of $866,000. When Younis then challenged the attendance figures, the court reduced the figure to $793,225. That sum then led the district court to apply former Sentencing Guideline § 2F1.1 (1998) (providing for the use of the greater of “actual loss” and “intended loss”) and add nine levels to the base offense level because it found the loss to be greater than $500,000 but less than $800,000.
The government must prove facts used for sentence enhancement under the advisory Guidelines by a preponderance of the evidence. United States v. Bernal-Aveja, 414 F.3d 625, 627 (6th Cir.2005); United States v. Treadway, 328 F.3d 878, 884 (6th Cir.2003). The district court’s loss calculation is reviewed for clear error. United States v. Orlando, 363 F.3d 596, 600-01 (6th Cir.2004). The amount of loss “need not be determined with precision. The court need only make a reasonable estimate of the loss, given the available information.” U.S.S.G. § 2F1.1 app. note 8; see United States v. Kohlbach, 38 F.3d 832, 835 (6th Cir.1994).
Younis argues that the court erred in determining his sentence based on intended loss “when the students’ eligibility for benefits meant actual loss was not possible.” Younis relies on United States v. Watkins, 994 F.2d 1192, 1196 (6th Cir. 1993), and United States v. Khan, 969 F.2d 218, 221 (6th Cir.1992), for the proposition that “the intended loss must have been possible to be deemed relevant.” 994 F.2d at 1196. Younis maintains that some of the students on the spreadsheet were actually eligible for financial aid because, for example, some students graduated from high school or had equivalent degrees, even though their files at TTC were supplemented by false documentation. Those students’ eligibility, according to Younis, makes it impossible for the government to have suffered a loss (since the students could have received the money anyway), and that impossibility means that the dis*316trict court erred by including these sums in the “intended loss.” Younis also points to a 2003 amendment to the Guidelines’ commentary providing that, in cases involving government benefits, loss is “not less than the value of benefits obtained by unintended recipients or diverted to unintended uses, as the case may be.” U.S.S.G. 2B1.1 app. note 3 (2003). The application note provides an example: “[I]f the defendant was the intended recipient of food stamps having a value of $100 but fraudulently received food stamps having a value of $150, loss is $50.” Id.
Younis’s argument fails because, as clarified by a 2001 amendment to the Guidelines’ commentary, impossible losses are to be included in intended loss. Prior to 2001, the Sixth Circuit’s rule in Khan and Watkins was oft-criticized in other circuits. See, e.g., United States v. Miller, 316 F.3d 495, 501-02 (4th Cir.2003) (collecting cases); United States v. Studevent, 116 F.3d 1559, 1562 (D.C.Cir.1997); United States v. Coffman, 94 F.3d 330, 336-37 (7th Cir.1996). In response, the 2001 amendment “clarif[ied] that intended loss ‘includes intended pecuniary harm that would have been impossible or unlikely to occur.’ ” United States v. Anderson, 353 F.3d 490, 505 n. 13 (6th Cir.2003); see United States v. McBride, 362 F.3d 360, 374 (6th Cir.2004) (“Amendment 617 resolved a circuit split regarding the meaning of ‘intended loss’ by clarifying that the definition reached ‘unlikely or impossible losses ... because their inclusion better reflects the culpability of the offender.’ ”); Miller, 316 F.3d at 503 (“[Although mindful of the pitfalls of using subsequent amendments to interpret prior legislation, we note that the Sentencing Commission has recently agreed that the majority interpretation of § 2F1.1 best conforms with the objectives of the Guidelines, and amended the definition of loss to clarify this point.”). Clarifying amendments to the Guidelines are to be followed even when they diverge from circuit precedent. See United States v. Snoddy, 139 F.3d 1224, 1229-31 (8th Cir.1998); United States v. Caballero, 936 F.2d 1292, 1298 (D.C.Cir.1991) (“In light of this clarification, we must reassess [United States v. Williams, 891 F.2d 921 (D.C.Cir.1989) ] and reinterpret section 3B1 so that our interpretation comports with the language of the Guidelines.”).5
Indeed, Younis’s argument would fail even in the absence of the clarifying amendment. His argument that the government could not lose money on loans to eligible students for whom TTC maintained false documentation assumes that eligibility was the sole requirement for government funding. But to qualify for benefits does not entitle one to them, and the government was in fact defrauded when it disbursed funds to students whose applications or documentation were deficient. To borrow from the food stamp example found in the Guidelines, suppose an individual qualified to receive $100 in food stamps procures $100 in food stamps under a false identity. Under Younis’s logic, no loss would occur. But the government is entitled to award benefits on the basis of proper documentation; it need not, in a case like this, show that it would have turned doum, various candidates for benefits if the documentation it received from them were accurate. Accordingly, because the government loans catalogued on the spreadsheets were awarded on the basis of illegitimate documentation, the *317amount of those loans constituted a possible loss to the government.
We hold that the district court did not apply the wrong law or commit clear error in determining that the intended loss in this case was the full amount on the spreadsheets.
2. Restitution
Younis next argues that the district court erred by ordering restitution based on intended loss. This court reviews a restitution award for abuse of discretion. United States v. Dunigan, 163 F.3d 979, 981 (6th Cir.1999). A fair reading of the transcript reveals that the court awarded restitution based on its earlier finding of the amount of intended loss:
The Court: ... Restitution in the amount of — what is the amount Mr. Wiedenmeyer?
Prob. Officer: Your Honor, you did rule on loss amount, which was $793,225.
Mr. White: That does include the intended lost portion.
The Court: Right. I understand. The Court will impose restitution in the amount of $793,225....
(J.A. 1003.) The government responds to Younis’s argument by again arguing that it “showed, through reliable evidence, that roughly $800,000 was paid to TTC” in reliance on fraudulent student files. As discussed above, the court might have reasonably found this figure to be the actual loss amount, but it never purported to do so. Younis’s counsel pointed out to the district court that the. amount the probation officer quoted was the “intended loss,” and the court agreed.
Although the Sixth Circuit has not addressed this issue directly, other circuits have invariably concluded that restitution may not be based on intended loss. See United States v. Rhodes, 330 F.3d 949, 953 (7th Cir.2003) (“While for sentencing purposes ‘loss’ is defined as the greater of either the ‘actual’ or the ‘intended’ amount lost due to the fraud, for restitution purposes the statute implicitly requires that the restitution award be based on the amount of loss actually caused by the defendant’s offense.” (citation omitted)); United States v. Tucker, 217 F.3d 960, 963 (8th Cir.2000) (“[T]he inquiry is actual loss; intended loss is irrelevant.”); United States v. Germosen, 139 F.3d 120, 130 (2d Cir.1998) (“Application Note 7 to § 2F1.1 of the Sentencing Guidelines allows for a loss calculation based on ‘intended loss,’ whereas 18 U.S.C. § 3663(a)(1)(A) requires a showing of actual loss for purposes of restitution.”); United States v. Messner, 107 F.3d 1448, 1455 (10th Cir.1997) (holding that the district court abused its discretion by awarding damages based on intended loss); see also United States v. Finkley, 324 F.3d 401, 404 (6th Cir.2003) (remanding where the government conceded that the district court erred by awarding restitution in the amount of intended loss instead of actual loss). We hold that the district court erred by awarding restitution based on “intended loss,” and remand the matter to the district court.
3. Fine
Younis’s final argument is that the district court erroneously ordered him to pay a $125,000 fine. The parties agree that the Guideline’s fine range was $7,500 to $75,000. Younis contends that the district court, which in its statement of reasons found “no reason to depart” from the Guidelines range (J.A. 1384), calculated the fine amount based on the presentence report’s recommendation of an offense level of 28 (fine range of $12,500 to $125,000) rather than an offense level of 22, the level *318the court used at sentencing. The government maintains that the district court reasonably used its sentencing discretion to impose a fíne outside the Guidelines range.
A fine is part of the sentence. See United States v. Ukomadu, 236 F.3d 333, 339 (6th Cir.2001). Even post-Booker, most other circuits require sentencing judges to provide explanations for sentences that fall outside the Guidelines range. See, e.g., United States v. Montes-Pineda, 445 F.3d 375, 380 (4th Cir.2006) (“District courts are obligated to explain their sentences, whether those sentences are within or beyond the Guidelines range, although they should especially explain sentences outside this range.” (quotation omitted)); United States v. Robinson, 435 F.3d 699, 701 (7th Cir.2006) (“If the judge is inclined to impose a sentence outside the advisory guidelines range, ... the judge must explain why the sentence imposed is appropriate in light of the statutory factors specified in § 3553(a).”); United States v. Lewis, 424 F.3d 239, 244 (2d Cir.2005) (noting that Booker “left unimpaired” § 3553(c)’s requirement that district courts state specific reasons for imposing sentences outside the Guidelines). In this circuit, “we encourage the sentencing judge to explicitly state his reasons for applying particular Guidelines, and sentencing within the recommended Guidelines range, or in the alternative, for choosing to sentence outside that range.” United States v. Jones, 399 F.3d 640, 650 (6th Cir.2005); see also United States v. Davis, 458 F.3d 491, 495-96 (6th Cir.2006) (discussing the duties of district judges post-Booker). Particularly because the district judge’s comments suggest that he did not believe himself to be departing from the appropriate Guidelines range, we remand for determination of the appropriate fine.
III. Conclusion
We affirm Younis’s, Jaber’s, and Younes’s convictions: first, because the indictment was sufficiently specific and because the district court did not abuse its discretion in refusing to grant a bill of particulars; second, because the district court did not abuse its discretion in admitting the spreadsheets as summary evidence; and third, because the evidence was sufficient. We also reject Jaber’s claim of prosecutorial misconduct and Younes’s claim that the district court erred in refusing to admit the polygraph results and in refusing to grant an evidentiary hearing regarding the results.
We affirm the district court’s use of the spreadsheet amounts in calculating Younis’s offense level. We remand for the district court to base its restitution order on actual loss and to reassess Younis’s fine.