Wyatt Henderson appeals his conviction and sentence after a jury trial for: (1) use of excessive force under color of law in violation of 18 U.S.C. § 242; (2) submitting a misleading and incomplete report of the incident to his supervisor with the intent to hinder, delay, or prevent the communication of information to a law enforcement officer relating to the commission of a federal offense in violation of 18 U.S.C. § 1512(b)(3); and (3) providing a false statement of material fact to an FBI agent in violation of 18 U.S.C. § 1001.
The charges against Henderson stem from accusations that he unlawfully pistol-whipped an arrestee, Christopher Grant, while a corporal with the Charlotte County, Florida, Sheriffs Department and then falsified the report of the incident. On appeal, Henderson asserts that the district court made various evidentiary errors that entitle him to a new trial, erroneously excluded police officers from jury selection in violation of his Sixth Amendment right to a grand and petit jury selected from a group representing a fair cross-section of the community, and erred in sentencing him to 87 months imprisonment.
I. BACRGROUND
Christopher Grant had been targeted by the Charlotte County Sheriffs Department’s Vice and Organized Crime Component (VOCC) for selling marijuana. VOCC officers set up an undercover sting operation to arrest Grant, attempting to apprehend him in a parking lot. However, Grant fled the area in his minivan before officers could detain him.
Henderson, who had been stationed in an unmarked car within view of the parking lot, pursued and stopped Grant on the side of the road. Stopping his car almost parallel to Grant’s van, Henderson pointed his service weapon at Grant and ordered him out of his car and onto his knees with his hands in view. Grant testified that he complied, exiting the car and kneeling on the ground with his hands on his head.1 Detective Reith Bennett, who had arrived at the scene shortly after Henderson, testified that Henderson approached Grant with his gun still in his right hand, placed a knee on Grant’s back and using his substantial weight advantage, “rode him to the ground.” On the way down, Grant’s chin struck the pavement.2 At this time, Grant testified that he saw a “black object” coming towards his head, and was forcefully struck in the jaw with a force he compared to that of a “mack truck.” Detective Bennett corroborated this testimony, stating that once Grant was prone and offering no resistance, he saw Henderson’s gun arm move to strike Grant. In contrast, Henderson testified that he put his gun in his car before approaching Grant because *1297he was not wearing a holster and did not want to physically confront Grant with a loose weapon. The defense also introduced evidence asserting that any injuries Grant suffered were the result of this initial impact with the pavement..
After the incident, both at the scene of the arrest and back at the police station, detectives testified that Henderson made incriminating statements to the effect that he had in fact struck Grant with his pistol. Notably, Detectives Bennett and Jack Collins stated that when informed that he was to write a report about what happened at the scene, Henderson threw a cellular telephone across the room and replied, “Jesus Christ, you can’t pistol-whip anybody any more,” or something to that effect, and said that he needed to wipe DNA off of his gun.
Detectives Bennett and Collins further testified that Henderson told other VOCC officers-his subordinates-not to include details of the arrest in their own police reports. Detective Bennett also testified that Henderson told him to deny that he had struck Grant. Allegedly fearing retaliation by Henderson, his boss, Bennett omitted any statement about a pistol-whipping. Bennett also testified that Henderson was very angry when he discovered Bennett was filing a supplemental report about the incident, and accused him of betrayal. Ultimately, one detective submitted a report directly to Henderson's supervisor stating that Grant questioned officers after his arrest about being pistol-whipped.
Henderson’s report, on the other hand, stated that “no force” had been used in the arrest, and made no mention of striking Grant. In a subsequent interview with an FBI agent investigating Grant’s allegations of excessive force, Henderson represented that he had thrown his gun into his car before approaching Grant, forming the basis of the charge that Henderson made a false statement “material” to the investigation.
Henderson’s arguments on appeal relate to evidentiary rulings, jury selection and sentencing and we address each in turn.
II. EVIDENTIARY ISSUES
We review evidentiary rulings for an abuse of discretion. Morro v. City of Birmingham, 117 F.3d 508, 513 (11th Cir.1997), cert. denied, 523 U.S. 1020, 118 S.Ct. 1299, 140 L.Ed.2d 465 (1998). However, basing an evidentiary ruling on an erroneous view of the law constitutes an abuse of discretion per se. Conroy v. Abraham Chevrolet-Tampa, Inc., 375 F.3d 1228, 1232 (11th Cir.2004).
Henderson argues that six evidentiary rulings entitle him to a reversal of his conviction: (1) excluding evidence of a domestic dispute involving Detective Collins, a witness for the government, which, Henderson asserts, would have shown Collins’ bias against him; (2) permitting Collins to testify that he found Grant’s version of the incident to be credible; (3) admitting the opinion of Grant’s oral surgeon as to the cause of his facial injuries, when the government failed to give timely notice designating-her as an expert'witness; (4) excluding polygraph evidence favorable to Henderson; (5) excluding testimony by an expert on police procedure offered by Henderson; and (6) admitting evidence that the sheriff who had originally hired Henderson had been removed for misconduct. He also claims that even if these errors, viewed separately, were harmless, their cumulative effect was not, and entitles him to a new trial.
1. Bias 'Evidence Concerning Detective Collins
At trial, Henderson sought to show that Collins was biased against him because Henderson had transferred Collins *1298from the VOCC to an undesirable traffic assignment. The court admitted evidence offered by Henderson, through the testimony of another officer, that Collins was transferred in part because he had been leaking proprietary VOCC information to his former supervisors. Henderson’s counsel then asked the officer if there were additional reasons for Collins’ transfer. The district court sustained the objection to the question and Henderson argues that this constitutes reversible error.
The government argues, as a threshold issue, that we may only review Henderson’s objection to the exclusion of this “bias” evidence for plain error because Henderson never made an offer of proof to the trial court about what the excluded evidence would have shown. Collins v. Wayne Corp., 621 F.2d 777, 781 (5th Cir.1980).3 Under Federal Rule of Evidence 103(a)(2), the objecting party must make an offer of proof to the court, or else show that the substance of the excluded evidence was apparent from the context of the proceeding, to preserve an objection to a ruling excluding evidence. Fed.R.Evid. 103(a)(2); United States v. Quinn, 123 F.3d 1415, 1420 (11th Cir.1997). Here, the government notes, there is no indication that the district court knew the nature of the evidence since Henderson did not identify it until a post-trial motion and his initial appellate brief. After trial, Henderson claimed that he wanted to present evidence that Collins was also dismissed because he had engaged in a domestic dispute with his wife that involved his service weapon. He argues that he could not have made an offer of proof at trial because the district judge did not permit bench conferences, requiring parties to reserve issues they did not wish to discuss before the jury to a time when the jurors were not required to be present in court. Moreover, he indicates that the district judge also forbade “speaking objections,” where the objecting party explains the basis for its objection, allowing the objecting party to voice only a very abbreviated basis for its complaint.
The record contains no evidence that Henderson made any effort to advise the court at any time during trial of the nature of the evidence sought to be elicited, something that Henderson’s counsel also admitted at oral argument. But, in any event, exclusion of this bias evidence was not an abuse of discretion even under a preserved error standard. Federal Rule of Evidence 403 allows a judge to exclude relevant evidence “if its probative value is substantially outweighed by the danger of unfair prejudice, confusion of the issues, or misleading the jury, or by considerations of undue delay, waste of time, or needless presentation of cumulative evidence.” Fed.R.Evid. 403. Even if relevant, the excluded bias evidence posed a danger of unfair prejudice that substantially outweighed any probative value it contained. Henderson had already admitted bias evidence regarding Collins’ transfer, making this evidence less probative of bias and somewhat cumulative. Moreover, evidence about Collins’ domestic dispute posed a disproportionate risk of unfairly inflaming the jury’s emotions and sidetracking the trial regarding the irrelevant question of whether or not Collins engaged in spousal abuse. See United States v. Hands, 184 F.3d 1322, 1328 (11th Cir.1999) (stating that impeachment evidence of violent spousal abuse is “particularly likely to incite a jury to an irrational decision” and excluding it under Rule 403) (internal quotation marks and citation omitted).
*1299 2. Detective Collins’ Statement that He Found Grant’s Account Credible ■
On direct examination, Collins testified about his initial reports regarding Grant’s arrest and stated that he initially thought Grant’s injury occurred when his chin hit the ground. He also testified to Henderson’s multiple, incriminating comments in the days after the arrest, including Henderson’s statement that “you can’t pistol-whip anybody any more.” On cross-examination by Henderson’s counsel, Collins was asked if he considered Henderson’s explanation that Grant was injured as part of the take-down during arrest to be credible at the time he prepared his report of the incident. Collins responded affirmatively. On redirect, when the government explored Collins’ initial views about the incident, Collins explained that he initially believed Henderson because many arrestees claim that they have been harmed by an officer. When asked whether, notwithstanding his initial views, there came a time when he found Grant’s claim to be credible, Collins also answered affirmatively. Henderson argues that because a determination of witness credibility is the sole province of the jury, this testimony rendered his trial fundamentally unfair in violation of his due process rights.
The Federal Rules of Evidence preclude a witness from testifying as to the credibility of another witness. Rule 608(a) restricts a party from attacking or supporting a witness’ credibility save through evidence “referring] only to character for truthfulness or untruthfulness.” Fed.R.Evid. 608(a). See also Snowden v. Singletary, 135 F.3d 732, 739 (11th Cir.1998) (“Witness credibility is the sole province of the jury.”).
However, in this case, the government was not attempting to substitute Detective Collins’ judgment about Grant’s credibility for the jury’s through the contested testimony. Rather, it sought to respond to questions on cross-examination that attempted to discredit Collins because, even though Grant told Collins at the scene of the arrest that someone had hit him with a pistol, Collins did not note Grant’s accusations on police reports he compiled and filed shortly after the arrest. Henderson’s cross-examination elicited Collins’ testimony that Henderson’s account of what happened at the scene seemed credible to Collins at the time and Grant’s accusations did not sound believable. On redirect, the government sought to establish why Collins had changed his mind, eliciting Collins’ explanation that although he did not believe Grant at first, he came to believe him later because of Henderson’s behavior in the days after the arrest. It was only after observing Henderson’s behavior and comments in the days following the incident that Collins felt that Grant might be telling the truth, and therefore decided to take action. The testimony that Collins found Grant’s story to be credible was not offered to prove Grant’s or Henderson’s truthfulness as witnesses; instead, it explained why Collins waited to report Henderson’s potential misconduct.
Although such testimony might be improper if the government had attempted to use this line of questioning as an indirect way of bolstering Grant’s credibility or attacking that of Henderson, cf. United States v. Samara, 643 F.2d 701, 705 (10th Cir.1981), that is not the case here. The district court did not abuse its discretion by permitting Collins’ testimony.
S. Testimony of Dr. Patricia Scott, Oral Surgeon
To establish that Grant sustained additional injuries to his jaw from the alleged pistol-whipping, the government offered the testimony of an oral surgeon, Dr. Patricia Scott. The government initially *1300planned to offer Dr. Scott as an expert witness, but failed to provide proper notice pursuant to a pretrial scheduling order, and offered her as a fact witness instead. Dr. Scott was an oral surgeon and Grant’s treating physician who had examined him after the arrest. She testified that Grant’s jaw had a hairline fracture and, over Henderson’s objection, opined that it resulted from a blow to the left side of his face. Henderson argues that this testimony offered an impermissible lay opinion pursuant to Fed.R.Evid. 701.
Non-expert (or lay) witnesses may only testify to 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, and (c) not based on scientific, technical, or other specialized knowledge within the scope of Rule 702.” Fed.R.Evid. 701. Subsection (c) was added in 2000, in an attempt to rein in the admission of expert testimony under the guise of lay opinion. Fed.R.Evid. 701, Advisory Committee’s note to 2000 amendment (noting that the amendment was aimed at “eliminatfing] the risk that the reliability requirements set forth in Rule 702 will be evaded through the simple expedient of proffering an expert in lay witness clothing.”).
In this case, it is certainly arguable that although Dr. Scott was the treating physician, her opinion regarding the cause of Grant’s injuries was not helpful to a clear understanding of her decision making process, nor did it pertain to Grant’s treatment. See Davoll v. Webb, 194 F.3d 1116, 1138 (10th Cir.1999) (“A treating physician is not considered an expert witness if he or she testifies about observations based on personal knowledge, including the treatment of the party.”); Weese v. Schukman, 98 F.3d 542, 550 (10th Cir.1996) (commenting that doctor’s lay opinions “were based on his experience as a physician and were clearly helpful to an understanding of his decision making process in the situation.”). Dr. Scott did not need to determine how Grant was injured to treat him in this case. Her diagnosis of the injury itself, that Grant’s jaw was fractured, would be permissible lay testimony, but her statement about the cause of the injury was, as she admitted, a “hypothesis.” And the ability to answer hypothetical questions is “[t]he essential difference” between expert and lay witnesses. Asplundh Mfg. Div. v. Benton Harbor Eng’g, 57 F.3d 1190, 1202 n. 16 (3d Cir.1995) (quoting Teen-Ed Inc. v. Kimball Int’l, Inc., 620 F.2d 399, 404 (3d Cir.1980)); see also Daubert v. Merrell Dow Pharmaceuticals, Inc., 509 U.S. 579, 592, 113 S.Ct. 2786, 125 L.Ed.2d 469 (1993).
However, assuming that the admission of this statement was error, we apply the harmless error standard to erroneous evidentiary rulings. 28 U.S.C. § 2111 (2005); Fed.R.Evid. 103(a); Fed.R.Crim.P. 52(a), United States v. Frazier, 387 F.3d 1244, 1266 n. 20 (11th Cir.2004) (en banc). When a trial judge has erroneously admitted evidence in a criminal prosecution, we ask whether the error “[had] a substantial influence on the outcome of a case or [left] grave doubt as to whether they affected the outcome of a case.” Id. (citing Kotteakos v. United States, 328 U.S. 750, 764-65, 66 S.Ct. 1239, 1248, 90 L.Ed. 1557 (1946)) (internal quotation marks omitted).
Because we do not harbor a grave doubt that the jury would have changed its verdict on any of the three charges had it not heard Dr. Scott’s statement, we find any error harmless. The statement at issue was not integral to the prosecution’s case, as the government offered significant non-medical testimony from other witnesses supporting the conclusion that any such injury was indeed caused by unlawful use of force. See Hands, 184 F.3d at 1331-33 *1301(holding evidentiary error prejudicial when the government’s case rested in significant part on the inadmissible evidence). It received no mention in the opening statement, and an oblique reference at worst during closing argument. See id. at 1332 (prosecution’s use of improper evidence in closing argument can exacerbate potential prejudice caused by its unlawful admission); United States v. Beale, 921 F.2d 1412, 1432 n. 18 (11th Cir.1991). The defense also countered this statement with expert testimony from a well-credentialed forensic pathologist. Cf. United States v. Lankford, 955 F.2d 1545, 1552-53 (11th Cir.1992) (finding that defense’s inability to rebut prosecution’s expert testimony relevant in holding exclusion of defense’s own expert testimony not harmless).
More important by far was Dr. Scott’s unobjected-to testimony that the left side of Grant’s jaw indeed suffered a fracture. The fact that Grant was injured on the left side of his jaw, and not the doctor’s speculation as to how it occurred, was the critical point in the government’s case because the government introduced significant evidence linking any injury in that area of Grant’s body to unlawful use of force by Henderson. In fact, the prosecution need not even have proved that Grant’s jaw was fractured in order to secure Henderson’s conviction; it only had to prove that Henderson’s alleged strike injured Grant in some way. In particular, evidence of Grant’s injury combined with Detective Bennett’s eyewitness account of seeing Henderson strike Grant with a black object in his hand and Detective Bennett’s and Collins’ testimony about Henderson’s subsequent statements, would have been sufficient in and of itself to support a guilty verdict for unlawful use of force. See Frazier, 387 F.3d at 1266-67 (noting that presence of other substantial evidence regarding the defendant’s guilt supported finding of harmless error).4 Also, the prosecution raised enough questions about Henderson’s own account of Grant’s arrest that a jury could reasonably have disbelieved his story of what happened. Cf. United States v. Bennett, 848 F.2d 1134, 1139 (11th Cir.1988) (“[A] defendant’s implausible explanation may constitute positive evidence in support of a jury verdict.”). For the foregoing reasons, we find no reversible error.
A Exclusion of Henderson’s Polygraph Evidence
Before trial, Henderson made a motion in limine to admit the results of two polygraph examinations, one .performed at the request of the FBI and another performed by a private examiner at Henderson’s request. Both tests returned inconclusive results after an initial battery of questions, and indicated that Henderson responded truthfully to a second round of questioning. A magistrate judge recommended that the district court deny the motion because the evidence did not satisfy the requirements of Daubert v. Merrell Dow Pharmaceuticals, Inc., 509 U.S. 579, 113 S.Ct. 2786, 125 L.Ed.2d 469 (1993), and was also inadmissible under Fed.R.Evid. 403. The district court adopted the magistrate’s recommendations and excluded the evidence. We find no abuse of discretion.
United States v. Piccinonna, 885 F.2d 1529 (11th Cir.1989) (en banc) restricts the use of polygraph evidence in this Circuit to only two contexts. A district court may admit polygraph evidence when the parties *1302stipulate in advance as to the test’s circumstances and scope of its admissibility, or “to impeach or corroborate the testimony of a witness at trial.” United States v. Gilliard, 133 F.3d 809, 811-12 (11th Cir.1998) (citing Piccinonna, 885 F.2d at 1536 (en banc)). However, this rule does not “preempt[ ] or limit[ ] in any way the trial court’s discretion to exclude polygraph expert testimony on other grounds under the Federal Rules of Evidence.” Id. (citing Piccinonna, 885 F.2d at 1536 (en banc)).
It is the second category which concerns us here. To admit polygraph evidence to impeach or corroborate a witness’ testimony, the proponent must satisfy three conditions. Piccinonna, 885 F.2d at 1536. First, it must provide adequate notice to the opposing party that it will offer polygraph evidence. Id. Second, the opposing party must be given a “reasonable opportunity to have its own polygraph expert administer a test covering substantially the same questions.” Id. Third, the Federal Rules of Evidence for the admissibility of corroboration or impeachment testimony will govern the proffer. Id.
We find that even if the admission of the polygraph evidence was proper to corroborate Henderson’s testimony under Piccinonna, the district court did not abuse its discretion in excluding it under Rule 702, as interpreted by Daubert, which requires expert scientific evidence to be both reliable and relevant pursuant to Rule 702. Daubert, 509 U.S. at 589, 113 S.Ct. 2786. Thus, the evidence must: (1) constitute scientific knowledge; and (2) assist the trier of fact to understand the evidence or to determine a fact at issue. Id. at 589-91, 113 S.Ct. 2786. The scientific knowledge question requires the trial court to consider the theory or technique upon which the testimony is based in light of at least five factors:
(1) whether the theory or technique can be and has been tested;
(2) whether the theory or technique has been subjected to peer review and publication;
(3) the known or potential rate of error for that theory or technique;
(4) the existence and maintenance of standards controlling the theory or technique’s operation; and
(5) whether the theory or technique has attained general acceptance within the relevant scientific community.
Id. at 593-94, 113 S.Ct. 2786; Gilliard, 133 F.3d at 812. In determining whether the evidence appropriately assists the trier of fact, the Daubert Court underlined the enhanced importance and role Fed.R.Evid. 408 plays in excluding overly prejudicial evidence, because “[ejxpert evidence can be both powerful and quite misleading because of the difficulty in evaluating it.” Daubert, 509 U.S. at 595, 113 S.Ct. 2786 (quoting Weinstein, Rule 702 of the Federal Rules of Evidence is Sound; It Should Not Be Amended, 138 F.R.D. 631, 632 (1991)).
The magistrate judge excluded Henderson’s proffered polygraph evidence as not constituting scientific knowledge and as being insufficiently relevant. She analyzed all five Daubert factors in coming to her first conclusion, finding that polygraph techniques were subject to peer review and publication but concluding that none of the four remaining factors supported admissibility.
First, she found that the theories of polygraphy at issue could not adequately be tested. Polygraphs monitor the subject’s physical responses to questioning. United States v. Scheffer, 523 U.S. 303, 313, 118 S.Ct. 1261, 140 L.Ed.2d 413 (1998) (plurality opinion). An examiner then interprets the physiological data and opines about whether the subject was lying. Id. *1303(plurality opinion).5 The magistrate concluded that while the physical responses recorded by a polygraph machine may be tested, “there is no available data to prove that those specific responses are attributable to lying.” Second, she remarked that the error rate for polygraph testing “is not much more reliable than random chance and does not meet the stricter standards of scientific methods required by Rule 702 and Daubert.” Third, she noted that despite the presence of standards regulating polygraphers, all compliance is self-imposed, and that effective countermeasures existed to defeat accurate results. Finally, she determined that polygraphy did not enjoy general acceptance from the scientific community, relying both on disagreement between federal Circuit courts and Justices of the U.S. Supreme Court over the effectiveness of polygraphs, and from one of Henderson’s own witnesses. The district court approved and adopted the report and recommendation in its entirety.
Even if we presume that Henderson sought to admit the polygraph evidence solely to corroborate Henderson’s testimony at trial (and there is no indication from his motion in limine that such was the case), its exclusion was not an abuse of discretion under Rule 702. The Supreme Court itself has clearly indicated that reasonable judges can disagree over the reliability of polygraph methodology. Scheffer, 523 U.S. at 309, 118 S.Ct. 1261 (majority .opinion) (holding that per se exclusion of polygraph testimony did not violate the Sixth Amendment) (“[TJhere is simply no consensus that polygraph evidence is reliable. To this day the scientific community remains extremely polarized about the reliability of polygraph techniques.”); id. at 312, 118 S.Ct. 1261 (majority opinion) (“Although the degree of reliability of polygraph evidence may depend on a variety of identifiable factors, there is simply no way to know in a particular case whether a polygraph examiner’s conclusion is accurate, because certain doubts and uncertainties plague even the best polygraph exams.”); see also Piccinonna, 885 F.2d at 1535 (en banc) (“[Pjolygraphy is a developing and inexact science[.]”). We .also note that we have yet to hold that exclusion of polygraph evidence at trial was an abuse of discretion under Piccinonna.6 The trial court did not abuse its discretion in excluding the evidence under Rule 702.7 See *1304Gilliard, 133 F.3d at 814-15. We need not reach whether Rule 403 would also have permitted exclusion.
5. Exclusion of Henderson’s Expert on Police Procedure
Next, Henderson argues that the district court violated his fundamental right to present witnesses in his own defense by excluding expert testimony opining that Henderson’s alleged action of throwing his gun into his car before arresting Grant was appropriate and the best decision under the circumstances. In support of this opinion, the expert was to have provided statistics about the number of officers injured with their own weapons and testimony that officers are trained to keep their service weapons out of reach during a ground fight.
The issue in this case was not whether it was proper police procedure for an officer to place his service weapon out of reach before engaging a suspect in a physical confrontation, but whether or not Henderson actually did so. Federal Rule of Evidence 702 restricts expert testimony to evidence that “will assist the trier of fact to understand the evidence or to determine a fact in issue.” Fed.R.Evid. 702. There was thus no abuse of discretion in excluding the evidence, and no infringement upon the defendant’s constitutional rights. See Taylor v. Illinois, 484 U.S. 400, 410, 108 S.Ct. 646, 98 L.Ed.2d 798 (1988) (“The accused does not have an unfettered right to offer testimony that is incompetent, privileged, or otherwise inadmissible under standard rules of evidence.”); Frazier, 387 F.3d at 1271.
6. Admission of Evidence about the Dismissal for Misconduct of the Sheriff Who Had Hired Officer Henderson
Henderson argues that the district court impermissibly allowed the government to introduce evidence that the sheriff who had hired Henderson was removed for fraud after Grant’s arrest but before trial. Although he does not cite any specific rule of evidence upon which to base his argument, we infer from Henderson’s argument that this evidence established his “guilt by association” and from the cases he cites that Fed.R.Evid. 403 and 404(b) are the appropriate foundations. Rule 404(b) prohibits admitting evidence of “other crimes, wrongs, or acts” to “prove the character of a person in order to show action in conformity therewith.” Fed.R.Evid. 404(b). However, the government proffered this evidence not to show Henderson’s bad character and his actions in conformity therewith, but instead to rehabilitate the credibility of Detective Bennett, whom Henderson had impeached for having been disciplined by the sheriffs office during this sheriffs tenure. Nor did the district court abuse its discretion in *1305admitting this evidence pursuant to Fed.R.Evid. 403. The information did little to prejudice Henderson given the context in which the government elicited the testimony, and was of probative value in rehabilitating Detective Bennett. .
III. EXCLUSION OF LAW ENFORCEMENT OFFICERS FROM THE GRAND AND PETIT JURY POOLS
The 1968 Jury Selection and Service Act, 28 U.S.C. § 1863, requires each U.S. District Court to “devise and place into operation a written plan for random selection of grand and petit jurors that shall be designed to achieve the objectives of [the Act].” 28 U.S.C. § 1863(a) (2005). The Act also requires the plan to bar
members of the fire or police departments of any State, the District of Columbia, any territory or possession of the United States, or any subdivision of a State, the District of Columbia, or such territory or possession.. .public officers in the executive, legislative, or judicial branches of the Government of the United States, or of any State, the District of Columbia, any territory or possession of the United States, or any subdivision of a State, the District of Columbia, or such territory or possession, who are actively engaged in the performance of official duties.
Id. § 1863(b)(6) (2005).
On the other hand, a criminal defendant enjoys a Sixth Amendment right to an impartial jury trial, which includes a right to the presence of a fair cross-section of the community on venire panels, or lists from which grand and petit juries are drawn. Taylor v. Louisiana, 419 U.S. 522, 526-29, 95 S.Ct. 692, 42 L.Ed.2d 690 (1975) (petit jury); United States v. Terry, 60 F.3d 1541, 1544 (11th Cir.1995) (grand and petit juries). To show a prima facie violation of this right, a defendant must prove:
(1) that the group alleged to be excluded is a “distinctive” group in the community;
(2) that the representation of this group in venires from which juries are selected is not fair and reasonable in relation to the number of such persons in the community; and
(3) that this underrepresentation is due to systematic exclusion of the group in the jury-selection process.
Duren v. Missouri, 439 U.S. 357, 364, 99 S.Ct. 664, 58 L.Ed.2d 579 (1979); Terry, 60 F.3d at 1544. The state may rebut a prima facie showing that its exemption or qualification violates a defendant’s Sixth Amendment rights if “it may be fairly said that the jury lists or panels are representative of the community,” and that “a significant state interest is manifestly and primarily advanced by those aspects of the jury selection process...that result in the disproportionate exclusion of a distinctive group.” Duren, 439 U.S. at 367-68, 99 S.Ct. 664. This Circuit has held that the exclusion of jurors to the extent 28 U.S.C. § 1863(b)(6) requires is constitutional, reasoning that even if a defendant could prove that a plan under § 1863(b)(6) established a prima facie violation, the state could rebut that prima facie case under the Du-ren standard. Terry, 60 F.3d at 1543-44.
The Middle District of Florida adopted the provisions of 28 U.S.C. § 1863(b)(6) in their entirety in its Plan for the Qualification and Random Selection of Grand and Petit Jurors (“Jury Plan”). However, the Middle District’s Juror Questionnaire Form exempts federal law enforcement officers, a group that the Jury Plan and § 1863(b)(6) do not exempt. Moreover, it is undisputed that the pattern and practice in the Middle District of Florida is to exempt anyone with arrest powers, including part-time law enforcement officers and even private law enforcement *1306officers. In support of his Sixth Amendment argument at trial, Henderson indicates that at least 0.55% of the estimated eligible juror pool in the Ft. Myers Division of the Middle District is exempt as law enforcement officers.
Although the exemption here is somewhat broader than the exemption considered in Terry, we believe that it is still constitutional, even assuming that Henderson can show a prima facie Sixth Amendment violation. The Middle District’s exemption does not reach much further than the exemption in Terry. Even relying on Henderson’s figures, federal, part-time, and private law enforcement officers would only compose a fraction of the 0.55% of eligible jurors in the pool. It thus “may be fairly said that the jury lists or panels are representative of the community.” Moreover, “a significant state interest is manifestly and primarily advanced” by this somewhat broader exemption, particularly with respect to federal law enforcement officers. The exemption of individuals in part-time and private law enforcement also “manifestly and primarily” advances the same significant state interest of “allowing police officers to perform their duties without the interruption of jury service,” although not to the same extent. Thus, the Middle District’s Jury Plan did not violate Henderson’s constitutional rights.
Nor can Henderson prevail on his statutory argument that the Jury Plan exceeds the scope of the Act. A criminal defendant must show more than a “technical violation” of the Act to obtain relief. United States v. Tuttle, 729 F.2d 1325, 1328 (11th Cir.1984), cert. denied, 469 U.S. 1192, 105 S.Ct. 968, 83 L.Ed.2d 972 (1985). Rather, he must show a “substantial failure to comply” with its provisions, which means the violation must have “a significant impact on the composition of an average jury.” Id. We have held that underrepresentation by an average of 1.4 persons on a 23-person jury panel does not constitute a sufficiently significant impact. United States v. Goff, 509 F.2d 825, 826-27 (5th Cir.), cert. denied, 423 U.S. 857, 96 S.Ct. 109, 46 L.Ed.2d 83 (1975). Similarly, United States v. Hawkins, 661 F.2d 436 (5th Cir. Unit B Nov.1981),8 held that underrepre-sentation by two persons per 23-juror pool did not meet this standard. Id. at 443. And in Tuttle, underrepresentation of an average of 0.96 jurors from a 12-person pool was held insufficient. 729 F.2d at 1328-29. Since Henderson’s figures state that law enforcement officers as a whole compose only 0.55% of the population in the division, the additional sweep of the Jury Plan in this case excludes, on average, less than one juror per 23-person panel. This does not constitute a significant enough impact on the composition of an average jury to violate the Act.
IV. SENTENCING ERRORS
Henderson makes several challenges to his sentence, including an argument made for the first time on appeal that Blakely v. Washington, 542 U.S. 296, 124 S.Ct. 2531, 159 L.Ed.2d 403 (2004), and United States v. Booker, — U.S. -, 125 S.Ct. 738, 160 L.Ed.2d 621 (2005), render his sentence unlawful. Because Henderson’s Blakely/Booker claim entitles him to resentenc-ing, we do not reach his other claims.
Other than the fact of a prior conviction, any fact that increases the penalty for a crime beyond the prescribed statutory maximum must be submitted to *1307a jury, and proved beyond a reasonable doubt. Apprendi v. New Jersey, 530 U.S. 466, 490, 120 S.Ct. 2348, 147 L.Ed.2d 435 (2000). In Blakely, the Supreme Court held “that the ‘statutory maximum’ for Ap-prendi purposes is the maximum sentence a judge may impose solely on the basis of the facts reflected in the jury verdict or admitted by the defendant.” Id., 124 S.Ct. at 2537. And in Booker, it applied this holding to sentences imposed under the United States Sentencing Guidelines. Id., 125 S.Ct. at 749 (“As the dissenting opinions in Blakely recognized, there is no distinction of constitutional significance between the Federal Sentencing Guidelines and the Washington [sentencing] procedures at issue in that case.”). Accordingly, it held that “[a]ny fact (other than a prior conviction) which is necessary to support a sentence exceeding the maximum authorized by the facts established by a plea of guilty or a jury verdict must be admitted by the defendant or proved to a jury beyond a reasonable doubt.” Id. at 756. In light of this holding, the Booker Court also held the mandatory sentencing guidelines unconstitutional, and excised two parts of the Sentencing Reform Act to cure the constitutional defects: 18 U.S.C. §3553(b)(1), making the guidelines result binding on the sentencing court; and § 3742(e), requiring de novo review of sentences on appeal. Id. at 764. Thus, the remainder of the Sentencing Reform Act now satisfies constitutional requirements. Id. Both Booker holdings apply retroactively to cases on direct review. Id. at 769 (`We must apply today's holdings-both the Sixth Amendment holding and our remedial interpretation of the Sentencing Act-to all cases on direct review.").
Because Henderson raised his Blakely/Booker challenge for the first time on appeal, we review it for plain error. United States v. Rodriguez, 398 F.3d 1291, 1297-98 (11th Cir.2005). Under that standard, we may only grant the defendant relief when, at the time of appeal, there is: (1) error, (2) that is plain, and (3) that affects substantial rights. United States v. Cotton, 535 U.S. 625, 631, 122 S.Ct. 1781, 152 L.Ed.2d 860 (2002). Once these three requirements are satisfied, we have the discretion .to provide relief if (4) the error seriously affects the fairness, integrity, or public reputation of judicial proceedings. Id.
Henderson’s sentence is in error because it violates Booker. In this case, the district court found, by a preponderance of the evidence, that the illegal use of force involved an aggravated assault with a firearm, that Henderson used the firearm with intent to cause bodily injury, and that the aggravated assault resulted in bodily injury. Combined with other enhancements and his criminal history category of I, the guidelines provided for a sentencing range of 87-108 months. The judge imposed a sentence of 87 months, but remarked that she “[thought] that probably under the circumstances, that’s a little high...but that’s what the guidelines call for and that’s what I’m obligated to sentence him.”
This 87-month sentence exceeded the maximum authorized by the facts established by the jury verdict, and facts necessary to support that sentence-namely, those supporting the fi~rearm~re1ated enhancements-were not found by a jury beyond a reasonable doubt or admitted by the defendant, and did not constitute prior convictions. T~hus, under Booker, Henderson's sentence violated his Sixth Amendment rights. See Rodriguez, 398 F.3d at 1298.
This constitutional Booker error also qualifies as "plain," because "where the law at the time of trial was settled and clearly contrary to the law at the time of appeal-it is enough that the error be `plain' at the time of appellate consider*1308ation.” Johnson v. United States, 520 U.S. 461, 467-68, 117 S.Ct. 1544, 137 L.Ed.2d 718 (1997); see also Rodriguez, 398 F.3d at 1299.
The error also affected Henderson’s substantial rights, because the district judge imposed the lowest permissible sentence under the mandatory guidelines and stated that she thought the sentence was too high, but was bound by the guidelines in any case. An error that “affects substantial rights” is one that “affected the outcome of the district court proceedings.” Cotton, 535 U.S. at 632, 122 S.Ct. 1781 (quoting United States v. Olano, 507 U.S. 725, 734, 113 S.Ct. 1770, 123 L.Ed.2d 508 (1993)). The defendant bears the burden of proving that the error affected his substantial rights. Olano, 507 U.S. at 734, 113 S.Ct. 1770. To discharge this burden, the Supreme Court held that the defendant must establish a “reasonable probability that, but for the error,” the outcome would have been different. United States v. Dominguez Benitez, 542 U.S. 74, 124 S.Ct. 2333, 2340, 159 L.Ed.2d 157 (2004). A reasonable probability of a different result is one that is “sufficient to undermine confidence in the outcome of the proceeding.” Id. (internal quotation marks omitted) (citing Strickland v. Washington, 466 U.S. 668, 694, 104 S.Ct. 2052, 80 L.Ed.2d 674 (1984), and United States v. Bagley, 473 U.S. 667, 682, 105 S.Ct. 3375, 87 L.Ed.2d 481 (1985)). In the Booker context, we therefore ask whether there is a reasonable probability that the district court would impose a lesser sentence on remand under an advisory guidelines scheme than the unconstitutional sentence it imposed under the mandatory guidelines scheme. Rodriguez, 398 F.3d at 1300-01.
At the very least, when the sentencing judge expressed an opinion that the required guidelines sentence was too severe, and, notably, sentenced the defendant to the lowest possible sentence in the guidelines range, as the judge did in Henderson’s case, the defendant has met his burden of proving prejudice. United States v. Shelton, 400 F.3d 1325, 1332-33 (11th Cir.2005). He also has met the fourth prong of the plain-error test because the district court expressed a desire to impose a lower sentence than the guidelines permitted, and after Booker the district court has discretion to do so. See id. at 1333-34.
Because the district court committed plain error by using facts it found by a preponderance of the evidence to enhance Henderson’s sentence beyond the maximum authorized by the facts established by the jury’s verdict, we vacate Henderson’s sentence and remand this ease to the district court for resentencing in accordance with Booker.
AFFIRMED in part, VACATED in part, and REMANDED.
. Henderson testified that he ordered Grant to lie down and that Grant refused to comply.
. The parties agree that this initial take-down was lawful, and that any injury that Grant suffered from his contact with the pavement does not form the basis of the charges against Henderson.
. In Bonner v. City of Prichard, 661 F.2d 1206, 1209 (11th Cir.1981) (en banc), the Eleventh Circuit adopted as binding precedent all Fifth Circuit decisions handed down prior to the close of business on September 30, 1981.
. We stress, however, that we review the record de novo when conducting a harmless error analysis, unlike our review of sufficiency of the evidence challenges, in which we view witness credibility in the light most favorable to the government. Hands, 184 F.3d at 1330 n. 23.
. There are multiple methods of questioning that examiners employ, some more accepted than others. See generally Gilliard, 133 F.3d at 812-15. It is unclear which types of questioning were used here, but the answer in this case is ultimately immaterial to the district court’s use of discretion in excluding the evidence, given the district court’s findings and the Supreme Court's statements in Scheffer. See infra.
. However, we have held that prosecutorial suppression of exculpatory polygraph evidence admissible under Florida law constituted a Brady violation. Jacobs v. Singletary, 952 F.2d 1282, 1287-89 (11th Cir.1992).
. The dissent argues that, in determining that the district court did not abuse its discretion by excluding Henderson's polygraph evidence on the basis that it did not satisfy the Daubert factors and was thus insufficiently reliable under Fed.R.Evid. 702, we have "reversed the holding of Piccinonna that polygraph evidence is sufficiently reliable to be generally admissible at trial in this circuit.’’ Post at 1310 (Hill, J., dissenting). The dissent therefore contends that when a district court excludes polygraph evidence "for not satisfying the Daubert factors,” its decision is based on an erroneous view of Piccinonna and thus constitutes an- abuse of discretion per se. Post at 1310 (Hill, J., dissenting).
But the dissent overstates the scope of the en banc Court's holding in Piccinonna. Picci-nonna predates Daubert, and its holding is specifically restricted to the admissibility of polygraph evidence under the now-defunct Frye "general acceptance test,” which prior to Piccinonna raised a per se bar to admission of polygraph evidence in this Circuit. Piccinonna, 885 F.2d at 1531-32 (en banc). To quote the era banc Court, "Our holding states merely that in the limited circumstances delineated *1304above, the Frye general acceptance test does not act as a bar to admission of polygraph evidence as a matter of law.’’ Id. at 1536. We were also careful to note that ''[n]either of [the] two modifications to the per se exclusionary rule should be construed to preempt or limit in any way the trial court’s discretion to exclude polygraph expert testimony on other grounds under the Federal Rules of Evidence.” Id. (emphasis added); see also Gilliard, 133 F.3d at 812.
Daubert, of course, held that "Frye has been superseded,” 509 U.S. at 589 n. 6, 113 S.Ct. 2786, and outlined five factors that guide a district court’s inquiry, pursuant to Fed. R.Evid. 702, into whether proffered scientific expert testimony is sufficiently reliable. Id. at 589-95, 113 S.Ct. 2786. Because Piccinon-na's holding concerned only the admissibility of polygraph evidence under the Frye "general acceptance test,” it does not preclude a district court from excluding polygraph evidence for not satisfying the Daubert factors. In fact, this Circuit has already held that a district court did not abuse its discretion when it excluded polygraph evidence for failure to satisfy the Daubert factors. Gilliard, 133 F.3d at 814-15.
. In Stein v. Reynolds Securities, Inc., the Eleventh Circuit adopted as binding precedent all decisions of Unit B of the former Fifth Circuit handed down after September 30, 1981. 667 F.2d 33 (11th Cir.1982).