This is a consolidated criminal appeal of the convictions and sentences of seven former Miami police officers. In the first trial, Appellants Oscar Ronda (“Ronda”), Jesus Agüero (“Agüero”), Arturo Beguiris-tain (“Beguiristain”), and Jorge Castello (“Castello”) were convicted of conspiracy to obstruct justice, in violation of 18 U.S.C. §§ 1512(b)(3) and 371, and obstruction of justice, in violation of 18 U.S.C. § 1512(b)(3). The jury hung with respect to Appellants Jorge Garcia (“Garcia”), Israel Gonzalez (“Gonzalez”), and Jose Quintero (“Quintero”).
Upon retrial, Appellants Garcia, Gonzalez and Quintero were convicted of conspiracy to obstruct justice, in violation of 18 U.S.C. §§ 1512(b)(3) and 371. The second jury also found Appellants Garcia and Gonzalez guilty of obstruction of justice, in violation of 18 U.S.C. § 1512(b)(3), and perjury before a federal grand jury, in violation of 18 U.S.C. § 1623.
Appellants challenge their convictions on numerous grounds and appeal their sentences under United States v. Booker, 543 U.S. 220, 125 S.Ct. 738, 160 L.Ed.2d 621 (2005). After review and oral argument, we affirm each Appellant’s convictions and sentences.
*1277I. FACTS
Between 1995 and 1997, the seven Appellants were members of the Miami Police Department, where they worked for the Street Narcotics Unit and/or the Crime Suppression Team. Appellants were responsible for the detection of firearms violations, drug trafficking, and street-level crimes against property and individuals.
Each Appellant was involved in at least one of four police shootings. These four shootings are referred to by^the name of the location where the shooting occurred: 1-395, 43rd Street, Coconut Grove, and N.W. 7th Court.1 The evidence at trial showed that Appellants illegally fabricated evidence to make the shootings appear justified, by planting guns at the scenes of the shootings and/or by making false and misleading statements to investigators.
We outline the evidence of the shootings and Appellants’ efforts to mislead the ensuing investigations.
A. 1-395 Shootings
On November 7, 1995, Appellants Agüe-ro, Beguiristain, Gonzalez, and Garcia were engaged in plainclothes activity in downtown Miami. They were accompanied by non-party Officers Sampson, Mer-volion, and Hames. Additional officers arrived at the scene later, as described below.
(1) Officers Kill Two Suspects
The officers noticed a car occupied by four black men and suspected the men were involved in a recent “smash and grab” robbery. Seeing the car on the on-ramp near 1-395, Appellant Beguiristain rammed into it with his patrol car. This disabled the suspects’ car and forced the car onto 1-395.
Two of the suspects, Derrick Wiltshire (‘Wiltshire”) and Antonio Young (“Young”), exited the car and began to flee. The two climbed over the highway guardrail, dangled themselves down from the rail, and dropped onto Miami Avenue, where they began to run from the scene.
As Young and Wiltshire fled, Appellants Agüero and Gonzalez and non-party Officers Mervolion and Hames shot them repeatedly.2 Young died on Miami Avenue of multiple gunshot wounds in the back. Though Wiltshire had been shot, he fled to a nearby alley, chased by Officers Davis and Bell, two additional police officers who arrived at the scene quickly after hearing of the car crash and chase. Wiltshire died in the alley from multiple gunshot wounds after a brief struggle with Officer Bell.
At trial, Officers Mervolion and Hames (who both pleaded guilty to obstructing justice and testified for the government) testified that neither Young nor Wiltshire was armed and that the suspects never fired upon the officers. Non-party Officers Davis, Bell, and Sampson also testified that neither Young nor Wiltshire had a gun and that they did not see a gun near either victim’s body immediately after the shootings. At least two civilian witnesses observed that neither Young nor Wiltshire appeared to be armed. No guns were found in the victims’ car.
*1278 (2) Officers Plant Guns
Additional officers arrived at the scene after the shootings, including Appellants Beguiristain and Quintero. Despite the fact that neither Young nor Wiltshire had been armed, Beguiristain and Quintero each informed their superiors that they had found a gun near the suspects. Appellant Beguiristain handed over a gun he claimed to have found next to Young’s body.3 Appellant Quintero produced a gun that he claimed to have found in the alley where Wiltshire died.4
Evidence at trial, in particular the testimony of Officers Mervolion and Hames, revealed that the two guns “found” near the bodies of Young and Wiltshire were in fact planted by the officers in order to justify the shootings of Young and Wilt-shire. Officer Hames testified that immediately after the shooting, Appellant Be-guiristain informed him that Appellant Quintero was on his way with a gun. Officer Mervolion testified that either Appellant Agüero or Appellant Garcia had said that he was “going to the station,” and that Mervolion had understood the speaker to mean that he intended to obtain a weapon to plant at the scene.
(S) Officers Make False Statements
As a matter of department policy, the homicide unit of the Miami Police Department investigates all deaths involving police officers and all shootings by police officers, whether or not the shooting victim dies. Similarly, the Miami Police Department’s internal affairs unit (rather than its homicide unit) investigates every situation where a Miami police officer discharges his weapon but the shot does not strike anyone. Appellants, as Miami police officers, knew that the 1-395 shootings would be investigated.
According to Officers Mervolion and Hames, the day after the shootings, Appellant Agüero organized a lunch meeting so that the officers could “get on the same page” before giving statements to investigators. Along with Officers Mervolion and Hames, Appellants Agüero, Beguiristain, Garcia, and Gonzalez attended this meeting. Officer Hames testified at trial that at the meeting, Hames proposed to the officers that they all state that Young and Wiltshire were holding guns in their right hands when they jumped from the expressway and fled from arrest.
The Miami Police Department’s homicide unit did in fact open an investigation on the day of the 1-395 shootings. On November 8, 1995, the day after the shootings, Appellants Beguiristain and Quintero each gave a sworn statement to Miami Police Department investigators. In his sworn statement, Appellant Beguiristain stated that when he approached Young after Young had been shot, Beguiristain found him “in pain” and “moving around.” Beguiristain swore that he found a nine millimeter pistol lying next to Young, and that Beguiristain picked it up “[bjecause *1279the guy was still moving.” In his sworn statement, Appellant Quintero stated that after arriving at the scene of the 1-395 shootings, he entered the alley where Wilt-shire had scuffled with Officer Bell and found a gun under a nearby bush.
Appellants Agüero, Garcia, and Gonzalez gave sworn statements to Miami Police Department investigators on November 14, 1995. In their sworn statements, Appellants Agüero and Garcia asserted that the 1-395 suspects killed by the officers were armed with guns in their right hands and that the officers did not fire upon the suspects until the suspects aimed their guns at the officers. In his sworn statement, Appellant Gonzalez stated that he saw one of the fleeing suspects holding a gun in his right hand.5
As is standard practice in the Miami Police Department, the homicide unit’s report on the 1-395 shootings was forwarded to the Florida State Attorney’s Office. The State Attorney’s Office initiated a judicial inquest into the deaths of Young and Wiltshire.6 During the course of that investigation, some of the officers involved in the 1-395 shootings gave sworn depositions in Florida state court.
On April 2, 1996, Appellants Beguiris-tain and Quintero gave sworn testimony in the state judicial proceeding. In his sworn testimony, Appellant Beguiristain testified that after Young had been shot on November 7, 1995, Beguiristain found a nine millimeter pistol lying next to Young, and that Beguiristain picked it up because “[Young] was still moving around” and “I didn’t want him to grab the gun.” In his sworn testimony, Appellant Quintero averred that he found a “blue steel Browning” pistol in the alley where Officer Bell had struggled with Wiltshire.
In 1998, the State of Florida charged Jerry Miller, one of the 1-395 suspects who was not shot by the officers, with armed battery and aggravated assault of a police officer. On October 30,1998, Appellant Garcia gave a sworn deposition in Miller’s criminal case in Florida state court. In his sworn testimony, Appellant Garcia reiterated that Young and Wiltshire had guns.
In 2000, Wiltshire’s estate initiated a civil suit in federal district court in the Southern District of Florida, charging the City of Miami and the individual officers with wrongful death. In that lawsuit, Appellant Gonzalez gave a sworn deposition on August 10, 2000. In his deposition, Appellant Gonzalez testified that he saw what he believed to be a handgun in one of the fleeing suspects’ right hand.
B. USrd Street Shooting
On April 13, 1996, a number of Miami officers chased suspect Steven Carter (“Carter”) after a woman’s purse was snatched. Appellants Agüero, Beguiris-tain, Garcia, and Quintero were involved in the chase, as were non-party Officers Mer-volion, Acuna and Williams.7 During the chase, Appellant Agüero fired his weapon without striking Carter. Officer Mervo-lion eventually found and arrested Carter where he was hiding in a shed, unarmed.
*1280According to Officer Mervolion’s testimony at trial, Appellant Beguiristain, after the arrest, told Mervolion that they would need “that gun,” alluding to a weapon Appellants Agüero and Quintero had stolen during an unrelated arrest. At Appellant Aguero’s request, Officer Mervolion had kept the stolen gun in his police car, and he handed it over to Appellant Beguiris-tain to plant. After planting the gun under a tree, Appellant Beguiristain used his police radio to announce that he had “found” a gun on the scene. Appellant Agüero then directed the investigating officer to the gun, where Appellant Beguiris-tain had hidden it under some leaves.
On April 13, 1996,' Appellant Agüero gave a sworn statement to investigators stating that during the chase, Carter turned in his direction and pointed a gun at him. On the same day, Appellant Be-guiristain gave a sworn statement in which he claimed that after Carter was arrested, he found the gun underneath the pile of leaves.
C. Coconut Grove
On June 26, 1997, Appellants Agüero, Beguiristain, Castello, and Garcia, along with non-party Officers Jacobo, Acuna, and Mervolion, were conducting a drug sting near the Cocowalk mall.
(1) Officer Castello Shoots Unarmed Man
While patrolling near the Cocowalk mall, Appellant Castello and non-party Officer Jacobo drove up to a nearby convenience store in their patrol car. When they arrived, the officers observed two homeless men, Rick Simms (“Simms”) and Daniel Hoban (“Hoban”), drinking beer outside the store. Simms and Hoban got into a minor scuffle. Appellant Castello and Ja-cobo got out of their car, yelling to Simms and Hoban to “drop it.” Appellant Castel-lo fired three shots, hitting Hoban in the leg.
According to the testimony of the Fire Chief, a paramedic, a passerby and a police officer, all of whom arrived at the scene shortly after the shooting, Hoban was unarmed and there was no weapon on the ground. Hoban’s walkman radio was lying near him, which may have been in his hand, leading Appellant Castello to believe Hoban was armed.
(2) Officers Plant Gun
Appellant Ronda, who was working a drug bust nearby, ran to the scene and arrived minutes after the shooting. Appellant Agüero and Officer Acuna arrived about fifteen minutes after the shooting, along with many other officers and emergency response professionals.
Before arriving, Officer Acuna radioed Appellant Beguiristain to ask if he needed “anything from the locker.” At trial, Officer Mervolion testified that he saw Appellant Agüero plant a handgun under a car at the scene of the shooting. Appellant Ronda then “discovered” this gun, fifteen minutes after the shooting and despite the fact that none of the others present at the scene had seen a gun until then. Later examination revealed Appellant Aguero’s fingerprint on the side of the gun, even though the gun was supposedly not discovered or touched by Appellant Agüero.8
(8) Officers Make False Statements
The Miami Police Department immediately initiated an investigation of the Coconut Grove incident. As part of the investigation, Appellant Ronda gave sworn *1281statements to state law enforcement officers on July 15, 1997, and July 25, 1997. In both sworn statements, Appellant Ronda averred that within minutes of the shooting of Hoban, he ran to the scene and saw a “blue steel gun” on the ground near Hoban.
Appellant Beguiristain gave sworn statements to investigators on July 15, 1997, and July 16, 1997. In both sworn statements, Appellant Beguiristain testified that he found a “blue steel gun” underneath a car near Hoban while Hoban was being treated by paramedics for the gunshot wound.
Appellant Castello gave a sworn statement to investigators on September 30, 1997. In his sworn statement, Appellant Castello testified that he saw a “blue steel gun” in Hoban’s hand when he shot Ho-ban, and that he saw a “blue steel gun” on the ground immediately after shooting Ho-ban.
II. PROCEDURAL HISTORY
A. Federal Investigation
The state investigations into the shooting incidents were eventually joined by a parallel federal investigation into whether the officers had broken federal law by violating the constitutional and civil rights of the shooting victims. In late September 1997, two months after the June 26, 1997 Coconut Grove shooting, Randall A. Glass, an Agent in the Civil Rights Squad of the Federal Bureau of Investigation (“FBI”), opened an FBI investigation into the shooting based on media and newspaper reports of the incident. Specifically, the FBI investigated whether the shooting represented a potential violation of Ho-ban’s federal constitutional and civil rights by local, state and federal officials. See 18 U.S.C. §§ 241 and 242.9 The FBI investigation also sought to uncover whether Appellant Castello had shot Hoban unnecessarily and whether he and other officers had then planted evidence to make the shooting appear justified.
Within less than a month, the FBI’s investigation expanded to include the I-395, 43rd Street, and N.W. 7th Court shootings. During the course of its investigation, the FBI requested and received from the Miami Police Department the entire case files from the Department’s investigations of all four shooting incidents. These files included transcripts of all the sworn statements made by the Appellants over the preceding years.
A federal grand jury was empaneled in the Southern District of Florida to investi*1282gate whether any of the officers had broken federal law. As part of the investigation, Appellant Garcia testified before the federal grand jury on August 9, 2001. In his sworn grand jury testimony, Appellant Garcia stated that he had no doubt that two of the 1-395 suspects were carrying guns, and that the officers fired at the suspects only after they pointed their guns at the officers.
On August 16, 2001, Appellant Gonzalez testified before the federal grand jury. When asked about one of the suspects fleeing the 1-395 scene, Appellant Gonzalez testified, “I believe that he was armed.” When asked if he recalled where he saw the weapon, Appellant Gonzalez replied, “in his right hand.” When asked if he was able to see the gun as the suspect was running, Appellant Gonzalez replied, ‘Tes.” Appellant Gonzalez was also asked, “[y]ou just saw him carrying the weapon as he was running, right?,” to which Gonzalez responded, “Might.”
B. Indictment
On September 6, 2001, the grand jury returned a superseding indictment against the seven Appellants, along with four other officers. This indictment was the relevant one for the first trial, at which Appellants Ronda, Agüero, Beguiristain and Castello were convicted. Subsequent to the verdict in the first trial, on May 30, 2003, a second superseding indictment was issued with respect to Appellants Garcia, Gonzalez, and Quintero. The second superseding indictment charged Garcia, Gonzalez and Quintero with the same counts as had been charged in the earlier indictment. However, the second superseding indictment focused solely on the 1-395 shootings and did not raise any allegations with respect to the other shootings. Because the differences between the two indictments are not material to most of the issues on appeal, we refer to them collectively as “the indictment” wherever the distinction is irrelevant.
Count One of the first superseding indictment charged all seven Appellants with conspiracy to obstruct justice, in violation of 18 U.S.C. § 1512(b)(3) (obstruction of justice)10 and 18 U.S.C. § 371 (conspiracy to commit any federal offense).11 The government grouped all four shooting incidents as a single related conspiracy to hinder investigation of the shootings themselves; the officers were not charged with four separate conspiracies. Count One of the second superseding indictment charged Garcia, Gonzalez and Quintero with a similar conspiracy to obstruct justice, also in violation of 18 U.S.C. §§ 1512(b)(3) and 371. Count One of the second superseding indictment, however, focused solely on the 1-395 shootings.
The remaining counts in the indictments charged Appellants, with the exception of Appellant Quintero, with specific acts of obstruction of justice, in violation of 18 U.S.C. § 1512(b)(3). Specifically, Counts Six through Ten of the first superseding indictment charged Appellants Beguiris-tain, Ronda and Castello with specific acts *1283of obstructing justice by making false and misleading statements to investigators in relation to the Coconut Grove shooting.12 Count Eleven charged Officer Acuna and Appellants Ronda, Agüero, Beguiristain, and Castello with obstructing justice by planting or aiding in the planting of a gun at the site of the Coconut Grove shooting. The remaining counts of the first superseding indictment either pertained only to Appellants Garcia, Gonzalez or Quinte-ro, or did not result in convictions, and therefore those counts are not discussed further.
In addition to the Count One conspiracy charge, the second superseding indictment charged Appellants Garcia and Gonzalez with one count each of obstruction of justice, in violation of 18 U.S.C. § 1512(b)(3), and with one count each of perjury before a federal grand jury, in violation of 18 U.S.C. § 1623. These counts all related to the 1-395 shootings and investigation.
C. Convictions
The first trial of all eleven officers began on January 6, 2003, and lasted for over two and a half months. After twenty days of deliberation, on April 9, 2003, the first jury convicted four defendants — Appellants Ronda, Agüero, Beguiristain and Castel-lo — of conspiracy to obstruct justice, in violation of 18 U.S.C. §§ 1512(b)(3) and 371, and obstruction of justice, in violation of 18 U.S.C. § 1512(b)(3).13 The first jury hung with respect to four defendants: Officer Acuna and Appellants Garcia, Gonzalez, and Quintero. The remaining three defendants, who were involved only in the N.W. 7th Court shooting, were acquitted.
Prior to re-trial, the government dismissed the charges against Acuna.
Appellants Garcia, Gonzalez, and Quinte-ro were then re-tried between February 23, 2004, and April 1, 2004. The second jury convicted Appellants Garcia, Gonzalez and Quintero of conspiracy to obstruct justice, in violation of 18 U.S.C. §§ 1512(b)(3) and 371. The second jury also convicted Appellants Garcia and Gonzalez of obstruction of justice, in violation of 18 U.S.C. § 1512(b)(3), and perjury before a federal grand jury, in violation of 18 U.S.C. § 1623.
The district court sentenced Appellant Beguiristain to 27 months’ imprisonment; Appellant Agüero to 37 months’ imprisonment; and Appellants Castello and Ronda each to 13 months’ imprisonment. The district court sentenced Appellants Garcia, Gonzalez and Quintero each to 16 months’ imprisonment.
III. APPEAL OF CONVICTIONS
On appeal, all seven Appellants contend that (1) the government failed to prove that their misleading conduct was directed at federal officials; (2) the district court abused its discretion by declining to give a jury instruction on Florida’s fleeing felon *1284statute; and (3) the evidence was insufficient to support their convictions for participating in an overarching conspiracy. Appellant Gonzalez also argues that his perjury and obstruction of justice convictions were not supported by sufficient evidence. Finally, Appellants Garcia, Gonzalez and Quintero assert that they should be retried or acquitted because the jury in the second trial was tainted by external evidence. We address each of these arguments in turn.14
A. Federal Nexus
All seven Appellants challenge their convictions for conspiracy to obstruct justice by contending that the government failed to show a sufficient “federal nexus” between their misleading conduct and the investigation they allegedly obstructed. We review the elements of the particular obstruction-of-justice offense in this case and then explain why Appellants’ argument lacks merit.
Count One charged Appellants with conspiring to obstruct justice in violation of 18 U.S.C. §§ 1512(b)(3) and 371. While § 1512(b) sets forth different types of obstruction-of-justice crimes, the only one at issue here is defined in § 1512(b)(3). In relevant part, 18 U.S.C. § 1512(b)(3) defines obstruction of justice as “engaging] in misleading conduct toward another person, with intent to ... hinder, delay, or prevent the communication to a law enforcement officer or judge of the United States of information relating to the commission or possible commission of a Federal offense ....” 18 U.S.C. § 1512(b)(3). Thus, to obstruct justice in violation of § 1512(b)(3), a defendant must knowingly and willfully (1) engage in misleading conduct toward another person, (2) with the intent to hinder, delay or prevent the communication of information to a federal official, (3) about the commission or the possible commission of a federal crime. United States v. Veal, 153 F.3d 1233, 1253 (11th Cir.1998), cert. denied, 526 U.S. 1147, 119 S.Ct. 2024, 143 L.Ed.2d 1035 (1999).
More specifically, Count One of the indictment charged that by planting the guns and lying about the shootings, the defendants conspired to hinder the communication of information about the shootings to federal officials.15 The “possible commission of a federal crime” named in the indictment was the possibility that the police shootings violated the victims’ federal rights. See 18 U.S.C. § 241 (making it unlawful where “two or more persons conspire to injure, oppress, threaten, or intimidate any person ... in the free exercise or enjoyment of any right or privilege secured to him by the Constitution or laws of the United States, or because of his having so exercised the same”); 18 U.S.C. § 242 (making it a federal crime for any person acting under color of law to deprive a citizen of his constitutional or civil rights). In other words, the government asserted that by planting the guns and *1285lying under oath, the officers intentionally hindered any federal investigation into whether the shootings violated 18 U.S.C. §§ 241 and 242.16
(1) United States v. Veal
Appellants’ “federal nexus” argument focuses on the second required element of 18 U.S.C. § 1512(b)(3), the intent to hinder a federal investigation. Appellants contend that their false statements and misleading actions were exclusively directed at state investigators and were never intended to be communicated to federal officials. As such, Appellants argue that the government failed to demonstrate the “federal nexus” necessary to uphold their convictions.
Appellants’ “federal nexus” argument is indistinguishable from the same claim rejected by this Court in Veal, 153 F.3d at 1250. The defendants in Veal were four Miami police officers who beat a suspect to death and then lied to state investigators about the beating. The four officers were convicted of violating 18 U.S.C. § 1512(b)(3) based on their false statements. On appeal, the Veal defendants argued that although they misled state investigators, they nonetheless could not be convicted of violating § 1512(b)(3) because they lacked the required intent for their misleading statements to reach federal officials. Veal, 153 F.3d at 1247-48.
In Veal, we rejected this “lack of federal nexus” argument and concluded that § 1512(b)(3) does not require a specific intent to mislead federal officials. Id. at 1252. Rather, “[f]or violation of § 1512(b)(3), it is sufficient if the misleading information is likely to be transferred to a federal agent.” Id. at 1251 (emphasis in original); see also 18 U.S.C. § 1512(g) (stating a prosecution under § 1512 does not require proof of any state of mind regarding whether the officials or proceedings that received the misleading information were federal officials or proceedings). Even though the Veal defendants had misled only state investigators, there existed “the possibility or likelihood that their false and misleading information would be transferred to federal authorities irrespective of the governmental authority represented by the initial investigators.” Veal, 153 F.3d at 1251-52 (emphasis in original).
The instant case is indistinguishable from Veal. The guns Appellants planted and the misleading statements Appellants made were directed at the state investigators who began the investigation of the police shootings. Regardless, as detailed below, this misleading information was likely to be transferred to federal investigators, and thus Appellants violated § 1512(b)(3). Indeed, the evidence at trial showed that Appellants’ misleading information was not only “likely” to be transferred to federal investigators, it in fact was transferred to federal investigators.
(2) Likelihood of Federal Investigation
As testimony at both trials indicated, as a matter of department policy, the homicide unit of the Miami Police Department investigates all deaths involving police officers and all shootings by police officers, whether the shooting victim dies or not. Similarly, the Miami Police Department’s internal affairs unit investigates every situation where a Miami police officer discharges his weapon but the gunshot does not strike anyone. Indeed, in the first trial, Captain David Rivero of the Miami Police Department testified that after any incident in which a Miami police officer draws and fires his weapon, the Miami *1286Police Department initiates a “massive investigation.”17 Thus, even before Appellants planted the guns at the scenes of the shootings, Appellants knew that any police shooting incident would be investigated by state officers. Indeed, Appellants’ knowledge that the shootings assuredly would be investigated was precisely what motivated them to plant evidence and to conspire to lie about the shootings.
More importantly for purposes of resolving the § 1512(b)(3) legal issue here, federal investigation into the police shootings was a distinct probability from the very onset of Appellants’ misleading conduct. According to Captain Rivero, the investigation by state officials into any shooting frequently involves consultation with federal officials, in particular the U.S. Attorney’s Office. Captain Rivero testified specifically that when the Miami Police Department investigates a police shooting, the sworn statements by the shooting officers are often forwarded to federal law enforcement agencies. According to Captain Rivero, “[i]f it is a questionable shooting, we will contact the [FBI] and we will have them look at it to see if there is anything federally that they can pursue.” Captain Rivero testified that “[a]ny time we have an officer that discharges his weapon and the shooting is questionable, we reach out to the [FBI] and we let them look through the case, and that is a standard practice.”18
Likewise, FBI Agent Glass explained that the FBI’s Civil Rights Squad investigates potential violations of federal constitutional and civil rights by local, state and federal officials.19 See 18 U.S.C. §§ 241 and 242. Agent Glass testified that the Civil Rights Squad regularly conducts investigations of allegations of excessive force by state officials, and has on a number of occasions investigated actions by police officers in South Florida.
In the instant case, the federal investigation that was distinctly probable at the time of the shootings unsurprisingly began not long thereafter. Captain Rivero testified that he personally provided the FBI with the entire case files from the Miami Police Department’s investigations of all four shooting incidents. Agent Glass averred that in late September 1997 — two months after the Coconut Grove shooting on June 26, 1997 — the FBI opened an investigation into the Coconut Grove shooting. Shortly thereafter, the FBI’s investigation expanded to include all of the shooting incidents described in Appellants’ trials.
As we explained in Veal, “federal jurisdiction under § 1512(b)(3) is based on the federal interest of protecting the integrity of potential federal investigations by ensuring that transfers of information to federal law enforcement officers and judges relating to the possible commission of federal offenses be truthful and unimpeded.” *1287Veal, 153 F.3d at 1250 (emphasis added). Section 1512(b)(3) “does not depend on the existence or imminency of a federal case or investigation but rather on the possible existence of a federal crime and a defendant’s intention to thwart an inquiry into that crime.” Id. (emphasis in original).
Under the standard articulated in Veal, the government proved the federal nexus required for the § 1512(b)(3) convictions in this case. From the time the Appellants entered the conspiracy to mislead investigators, a federal investigation into the shootings was not only possible, it was highly probable.
(8) Arthur Andersen v. United States
Although Veal clearly controls the instant case, Appellants contend that the recent Supreme Court decision in Arthur Andersen LLP v. United States, 544 U.S. 696, 125 S.Ct. 2129, 161 L.Ed.2d 1008 (2005), requires that we revise the standard set forth in Veal and overturn Appellants’ convictions. We disagree. Contrary to Appellants’ argument, Arthur Andersen involved a different criminal statute and does not implicitly overrule Veal.
In AHhur Andersen, the Supreme Court reversed a corporation’s conviction for obstruction of justice pursuant to 18 U.S.C. § 1512(b)(2), the subsection that precedes 18 U.S.C. § 1512(b)(3), the statute at issue here.20 Specifically, the accounting firm Arthur Andersen was convicted of one count of violating 18 U.S.C. § 1512(b)(2)(A) and (B) by corruptly persuading its employees to withhold testimony and documents from “an official proceeding,” in violation of 18 U.S.C. § 1512(b)(2)(A), and to destroy documents and other records with intent to impair their integrity or availability for use in “an official proceeding,” in violation of 18 U.S.C. § 1512(b)(2)(B). Arthur Andersen, 544 U.S. at 702, 125 S.Ct. at 2134. The Supreme Court reversed those convictions, finding in part that the district court’s jury instructions led the jury to believe it did not have to find “any nexus between the persuasion to destroy documents and any particular [official federal] proceeding.” Id. at 707, 125 S.Ct. at 2136 (punctuation omitted). The Supreme Court held that to be found guilty of violating 18 U.S.C. § 1512(b)(2), “[a] knowingly corrupt persuader cannot be someone who persuades others to shred documents ... when he does not have in contemplation any particular official proceeding in which those documents might be material.” Id. at 708, 125 S.Ct. at 2137 (punctuation omitted) (emphasis added).
*1288Appellants insist that Arthur Andersen’s holding with respect to 18 U.S.C. § 1512(b)(2) applies with equal force to their convictions under 18 U.S.C. § 1512(b)(3). Appellants assert that at the time they misled state investigators, they were not aware of any ongoing federal proceeding investigating their conduct, and in fact no such federal investigation existed. Appellants extrapolate that, as in Arthur Andersen, they cannot be found criminally liable where there was no official federal investigation until after their charged conduct took place.
Appellants’ attempt to analogize to Arthur Andersen is unavailing. Arthur Andersen interpreted and applied only § 1512(b)(2), which explicitly requires that the acts of obstruction relate to “an official proceeding.” Unlike § 1512(b)(2), § 1512(b)(3) makes no mention of “an official proceeding” and does not require that a defendant’s misleading conduct relate in any way either to an “official proceeding” or even to a particular ongoing investigation. See 18 U.S.C. § 1512(b)(3); see also United States v. Byrne, 435 F.3d 16, 24 (1st Cir.2006). There is simply no reason to believe that the Supreme Court’s holding in Arthur Andersen requires that we graft onto § 1512(b)(3) “an official proceeding” requirement based on statutory language in § 1512(b)(2) that does not appear in § 1512(b)(3).
As we already noted in Veal, the federal nexus required under § 1512(b)(2) is distinct from that required under § 1512(b)(3). Veal, 153 F.3d at 1249-50. Unlike the stricter “an official proceeding” requirement that appears in § 1512(b)(2), § 1512(b)(3) requires only that a defendant intended to hinder, delay, or prevent communication to any “law enforcement officer or judge of the United States.” Id. at 1248; 18 U.S.C. § 1512(b)(3). This distinction was critical to our decision in Veal that § 1512(b)(3) requires only “the possible existence of a federal crime and a defendant’s intention to thwart an inquiry into that crime.” Veal, 153 F.3d at 1250 (emphasis in original).
As we explained in Veal, § 1512(b)(3) “criminalizes the transfer of misleading information which actually relates to a potential federal offense .... ” Veal, 153 F.3d at 1252 (emphasis in original). The Veal Court correctly emphasized that § 1512(b)(3) does not require that a federal investigation be initiated nor that an official proceeding be ongoing. Id. Because Veal already distinguished between § 1512(b)(2) and (b)(3), Veal’s interpretation of § 1512(b)(3) is in no way altered by Arthur Andersen’s reading of § 1512(b)(2). See also Byrne, 435 F.3d at 24 (finding Arthur Andersen irrelevant to § 1512(b)(3)). In short, Veal still controls Appellants’ convictions.
B. Jury Instruction on Fleeing Felon Statute
Appellants next contend that their convictions must be overturned because the district court failed to give a jury instruction concerning Florida’s fleeing felon statute. See Fla. Stat. § 776.05. In relevant part, § 776.05 provides that a police officer is justified in using force against a fleeing felon in various circumstances, as follows:
A law enforcement officer ... is justified in the use of any force ... (3) When necessarily committed in arresting felons fleeing from justice ..., and (a) The officer reasonably believes that the fleeing felon poses a threat of death or serious physical harm to the officer or others; or (b) The officer reasonably believes that the fleeing felon has committed a crime involving the infliction or threatened infliction of serious physical harm to another person.
Id. Appellants contend that § 776.05 authorizes all of the police shootings in this *1289case, and that the lawfulness of the shootings themselves constitutes a complete defense to the obstruction-of-justice charges.
The district court denied Appellants’ request for a jury instruction on this fleeing felon statute. Instead, the district court decided that it would be more appropriate to introduce the fleeing felon statute itself into evidence. Accordingly, prior to closing statements in the first trial, the district court briefly reopened the case to allow defense counsel to read the fleeing felon statute to the jury. The district court then took judicial notice of the statute and provided the jurors with written copies of the statute. The district court followed the same procedure in the second trial, allowing the defense to read the fleeing felon statute into evidence, taking judicial notice of it, and providing copies to the jury.
The district court admitted the statute into evidence in order to ensure that the jury could consider Appellants’ arguments that they had no motive to plant guns and to lie about the shootings because the shootings were arguably justified by Florida’s fleeing felon statute. Appellants, of course, did not object to the statute being admitted into evidence and read to the jury, and they do not raise any issue on appeal about that procedure. Rather, Appellants’ sole argument is that the statute should have also been part of the district court’s jury instructions.
The government first argues that Appellants waived this issue in the district court. The government posits that Appellants effectively consented, or at least did not adequately object, to the district court’s decision to publish the fleeing felon statute as evidence in lieu of giving a jury charge based on the statute. When the district court, during the first trial, proposed to publish the statute rather than issue a jury instruction, Appellant Beguiristain’s counsel replied, “[a]s long as [the fleeing felon statute] comes before the jury, I don’t think we have a technical objection.” The district court specifically proposed that “I reopen to allow judicial notice of [the fleeing felon statute], and that can be read [to the jury] .... But I don’t think it is appropriate just to put [the fleeing felon statute] in the instructions as something that stands alone because it doesn’t relate to anything.” Appellant Beguiristain’s counsel responded, “[w]e have no objection to that technique, your honor.” After this compromise was worked out, counsel for the remaining Appellants raised no objections.
Likewise, when the fleeing felon statute was discussed prior to closing statements in the second trial, Appellant Gonzalez’s counsel stated, “[w]ith regard to the judicial notice [of the fleeing felon statute], Judge, we’ve agreed the way it was done last year [in the first trial] is the same way we would like it this year.” According to the government, counsel for Appellants Garcia and Quintero raised no objections and thereby also effectively consented.
In response, Appellants argue that the government quotes part of the record out of context, ignoring other sections where Appellants preserved their objection to the district court’s failure to instruct the jury on the fleeing felon statute. For example, Appellants point to the fact that they agreed to the district court’s final jury instructions with the caveat of “without waiving our prior objections.” We need not resolve the waiver issue, because even assuming Appellants adequately objected, Appellants have failed to establish any reversible error based on the district court’s not giving the requested instruction.21
*1290Appellants’ primary basis for their requested jury charge is, essentially, that because the shootings were authorized under Florida’s fleeing felon statute, the shootings could not have constituted violations or possible violations of federal civil rights law.22 Appellants contend that if the shootings themselves were lawful, even assuming Appellants hindered the investigation into the shootings, Appellants nonetheless did not violate § 1512(b)(3), because they did not “hinder, delay, or prevent the communication ... of information relating to the commission or possible commission of a Federal offense.” 18 U.S.C. § 1512(b)(3) (emphasis added).23
The fatal flaw in Appellants’ argument is that even if the police shootings themselves were ultimately determined to be lawful, the Appellants still violated the law by obstructing the investigations into the shootings. Essentially, Appellants’ jury instruction argument presumes that it is lawful to obstruct justice so long as the obstructed investigation ultimately “wouldn’t have found anything criminal anyway.” This misconstrues the meaning of obstruction of justice and ignores that § 1512(b)(3) criminalizes the obstruction of an investigation into either the commission of a federal offense or an investigation into the possible commission of a federal offense. 18 U.S.C. § 1512(b)(3).
As we said in Veal, “[b]y its wording, § 1512(b)(3) does not depend on the existence or imminency of a federal case or investigation but rather on the possible existence of a federal crime and a defendant’s intention to thwart an inquiry into that crime.” Veal, 153 F.3d at 1250 (emphasis in original). The fabrication of evidence to mislead federal investigators violates § 1512(b)(3) whether or not the potential federal investigation would have uncovered sufficient evidence to prove that a federal crime was actually committed. See United States v. Cobb, 905 F.2d 784, 790 (4th Cir.1990) (under § 1512(b)(3), “proof of an actual commission of a federal offense is not a necessary prerequisite to, or an essential element of, the crime of obstruction of justice”); United States v. Applewhaite, 195 F.3d 679, 687 (3d Cir.1999) (“[I]f the investigation or prosecution a defendant tries to hamper turns out to be federal, the witness is guilty of tampering with a federal witness even if the prosecution is unable to establish the facts necessary to establish a violation of federal law.”); United States v. Baldyga, 233 F.3d 674, 681 (1st Cir.2000) (“Section 1512(b)(3) does not require that the defendant be convicted of the federal offense .... [T]he dispositive issue is the federal character of the investigation, not guilty *1291verdicts on any federal offenses that may be charged.”).
Thus, given the nature of the § 1512(b)(3) charges being tried and all other factual circumstances of this particular case, we conclude that Appellants have shown no reversible error due to the lack of this jury instruction.
C. Conspiracy Convictions
Appellants Ronda, Agüero, Beguiristain and Castello also contend that the government failed to prove the overarching conspiracy charged in Count One.24 Specifically, Appellants argue that the government failed to prove that they were involved in a single conspiracy to plant guns in all four shootings and to hinder the investigation into all four shootings. Relatedly, Appellants assert that the district court erred by declining to sever the first trial into separate trials concerning each shooting. These arguments fail.
The government’s proof of an overarching conspiracy to hinder investigation into the shootings was overwhelming. Multiple Appellants participated in more than one of the cover-ups, highly indicative that the officers were involved in a conspiracy that transcended any particular shooting incident. Likewise, the officers’ immediate access to “throw-down” guns to plant at multiple shooting scenes demonstrated planning, pre-meditation, and an ongoing scheme to obstruct justice. Voluminous additional evidence confirmed that all Appellants took part in the conspiracy.25 Moreover, the fact that a particular Appellant may not have participated in every act of the conspiracy is no defense. See United States v. Cole, 755 F.2d 748, 755 (11th Cir.1985).
Even assuming arguendo that the government proved multiple conspiracies rather than the single conspiracy alleged in Count One, Appellants are unable to explain how this prejudiced them. See United States v. Coy, 19 F.3d 629, 634 *1292(11th Cir.1994) (proof of multiple conspiracies in place of a single alleged conspiracy only prejudicial if (1) there is inadequate opportunity to prepare a defense, or (2) the jury is likely to be confused about which defendants were involved in which conspiracies). Appellants make no argument that they were unable to prepare an adequate defense on account of the joinder of all four shootings into a single conspiracy charge. Moreover, given that the jury acquitted three of the original defendants and hung with respect to four others, there is no reason to believe the jury was unable to distinguish between the involvement of the various defendants. See United States v. Hernandez, 921 F.2d 1569, 1580-81 (11th Cir.1991) (stating that severance is only warranted where there is reason to believe the jury would be prejudicially confused, and that severance is not required merely because a complex case involves many defendants); Richardson v. Marsh, 481 U.S. 200, 210, 107 S.Ct. 1702, 1708, 95 L.Ed.2d 176 (1987) (explaining that judicial efficiency and fairness generally favor joint trials).
D. Gonzalez’s Other Convictions
Appellant Gonzalez argues that the evidence was insufficient to support his convictions on Counts Three and Five, which charged him with obstructing justice by giving false and misleading testimony in a civil deposition, in violation of 18 U.S.C. § 1512(b)(3) (Count Three), and perjury before a federal grand jury, in violation of 18 U.S.C. § 1623 (Count Five).26
On August 10, 2000, in the civil suit by Wiltshire’s estate in the United States District Court for the Southern District of Florida, Appellant Gonzalez gave a sworn deposition. Gonzalez’s August 10, 2000 deposition concerns the 1-395 shootings, and is the subject of his obstruction-of-justice conviction in Count Three. In this deposition, Appellant Gonzalez testified that at the time of the 1-395 shootings, he believed Wiltshire was armed and he saw what he believed was a handgun in Wilt-shire’s right hand, as follows:
Question: When you look, if you did look, what did you observe on North Miami Avenue?
Answer (Gonzalez): I observed one of the offenders [i.e., Wiltshire] running northbound on a sidewalk on the west side of Miami Avenue.
Question: One of the offenders running northbound on Miami Avenue on the west side away from your position?
Answer: Correct.
Question: Do you see any officers in hot pursuit of that offender?
Answer: No, I do not.
Question: What do you do next?
Answer: I see that, I believe him to be armed. I see what I believe is a handgun.
Question: What made you believe the offender was armed?
Answer: Well, I saw what I believe is a handgun as he ran down the street.
Question: Did he have anything in his or her hands?
Answer: Yes.
Question: Okay. What?
Answer: What I believed was a handgun.
*1293Question: And what did you do next?
Answer: At that time, I fired twice at the individual.
Question: Okay. Now, when you said you saw something in the individual’s hand, did you see something in his right hand or left hand or both?
Answer: I saw his right hand.
March 10, 2004 Trial Tr. at 87-89 (quoting Gonzalez Dep.) (emphasis added). Later, during the same sworn deposition, the following exchange occurred:
Question: You say “known to be armed.” Isn’t the most accurate way to say it at least is you felt he was armed?
Answer: Right. I felt he was armed, that’s correct.
Id. at 93. On August 16, 2001, Appellant Gonzalez testified before the federal grand jury, again about the 1-395 shootings. Gonzalez’s August 16, 2001 sworn testimony is the subject of his perjury conviction in Count Five. Before the grand jury, Appellant Gonzalez testified that he saw a weapon in Wiltshire’s right hand and believed that he was armed, as follows:
Question: Tell us what happened after you were focused on the person [i.e., Wiltshire] running on Miami Avenue?
Answer (Gonzalez): I discharged my weapon twice. It appeared that it had no effect, so I surmised maybe I missed him, and I told [sic] continued running northerly. I felt any other attempt to discharge my weapon again was going to be futile. If there was any likelihood that I was going to strike him, it would be with the first two, so I am not going to get him with the next two.
Question: You stopped after firing twice?
Answer: Yes.
Question: Subsequent to that, are you hearing gunshots?
Answer: Yes, I am still hearing gunshots.
Question: Is it 20 yards past the overpass? Is it — at that point did you see what you believed to be a weapon?
Answer: I believed that he was armed, yes.
Question: Do you recall where you saw the weapon?
Answer: As he was running, in his right hand.
Question: And you were able to see it as he is running ?
Answer: Yes.
Question: Did you ever see him turn around?
Answer: He did not turn around.
Question: Did he point the weapon?
Answer: No, he did not.
Question: You just saw him carrying the weapon as he was running, right ?
Answer: Right.
Id. at 94-96 (emphasis added). Later, during the same grand jury testimony, Appellant Gonzalez testified that although he was not certain, he believed what he saw was a handgun in Wiltshire’s hand, as follows:
Question: How certain are you that what you saw in his hand was a gun?
Ansioer: I am not certain. I believe it to be a handgun. That is factoring all the circumstances as well.
Question: One of the things that led you to believe it was the fact that you heard gunshots?
Answer: Certainly, yes.
Question: Is that something that affected your perception of whatever it was that you saw in his hand?
Answer: Certainly that factored into it, yes. I surmised it was.
Id. at 100-01 (emphasis added). Appellant Gonzalez’s sufficiency-of-the-evidence ar*1294gument focuses on his contention that his testimony in both the civil deposition and before the criminal grand jury conveyed only his uncertainty about whether Wilt-shire had a gun.
Gonzalez’s brief states: “[w]e readily acknowledge that a witness cannot avoid a perjury charge merely by inserting the words ‘I believe’ in every answer.” Gonzalez Br. at 17. Gonzalez is correct, because a witness’s false sworn statement about the witness’s knowledge and beliefs can support a perjury conviction.27 What Gonzalez argues instead is that “a full reading of the entire transcript demonstrates that is not what we have here.” Id. Gonzalez asserts that his testimony, when viewéd in context, was not that he believed Wiltshire had a gun but only that he was not sure or certain whether Wilt-shire had a gun. In effect, Appellant Gonzalez’s position is that his testimony, taken as a whole, was that “he was not certain whether Wiltshire had a gun.” Id. at 19.
We agree with Gonzalez that in perjury cases, district courts should view a witness’s testimony as a whole and his statements should not be taken out of context. See Van Liew v. United States, 321 F.2d 674, 678 (5th Cir.1963)28 (“A charge of perjury may not be sustained by the device of lifting a statement of the accused out of its immediate context and thus giving it a meaning wholly different than that which its context clearly shows.”) (quotation marks and citation omitted). We disagree, however, with Gonzalez’s reading of his testimony as a whole. We have quoted the entire relevant portion of both his deposition and grand jury testimony because it speaks for itself. In his civil deposition, Appellant Gonzalez swore repeatedly that: (1) “I believe him to be armed”; (2) “I see what I believe is a handgun”; (3) “I saw what I believe is a handgun as he ran down the street.” In response to the question, “[d]id [Wiltshire] have anything in his or her hands?,” Gonzalez replied, “[w]hat I believed was a handgun.”
Appellant Gonzalez’s testimony before the grand jury was the same. Before the grand jury, Gonzalez swore that “I believed that he was armed.” As to where Gonzalez saw the weapon, Gonzalez replied, “[a]s he was running, in his right hand.” To the question, “[w]ere you able to see it as he is running,” Gonzalez answered, “[y]es.” To the question, “[y]ou just saw him carrying the weapon as he was running right?,” Gonzalez answered, “[r]ight.” Even when Gonzalez expressed at the end that he was not certain, he still *1295repeated his belief that Wiltshire had a handgun, stating that “I am not certain. I believe it to be a handgun.”
Appellant Gonzalez’s testimony that he believed Wiltshire was armed, viewed in context, clearly conveyed not merely the possibility that the suspect was armed, but that he, as an officer on the scene, thought or believed the suspect was armed.29 The key question, then, is whether there was sufficient evidence for the jury to have concluded that Gonzalez knew in fact that Wiltshire was unarmed and never actually believed that Wiltshire had a gun. As noted at trial, there was overwhelming evidence that Young and Wiltshire were not in fact armed, and that Appellants conspired to plant guns and then lie about whether Young and Wiltshire had been armed. Indeed, Officers Mervolion and Hames testified at trial that at the time of the shooting — even when they fired at the suspects — they never believed that Young or Wiltshire was armed.30 Likewise, Officer Bell testified that he could see Young and Wiltshire’s hands in the air as they dropped from the 1-395 overpass — before any shots were fired- — and observed that they did not have guns.
Especially significant is the substantial evidence of Gonzalez’s participation in the conspiracy to mislead investigators about the 1-395 shootings. According to the testimony of Officer Mervolion, at a lunch meeting on November 8, 1995, the day after the 1-395 shootings, Appellant Gonzalez and the other officers involved in the shootings explicitly conspired to lie about the shootings. The officers agreed upon a cover story, namely to tell investigators that Wiltshire and Young had been armed and had been holding firearms in their right hands. Shortly thereafter, the officers who attended the conspiratorial meeting, including Appellant Gonzalez, each gave nearly identical sworn testimony that mirrored exactly the fabricated story they settled upon at the November 8, 1995, lunch meeting. Notably, in Appellant Gonzalez’s sworn statement on November 14, 1995 — seven days after the shooting— Gonzalez declared without reservation (and exactly in concert with the statements of his co-conspirators) that “I see that he is armed with a handgun and I recall it was in his right hand.” Gonzalez also claimed to have seen that the gun was a “semiautomatic” and was “black or blue steel.”31
*1296Gonzalez’s false testimony in his initial statement and his participation in a conspiracy to mislead investigators gave the jury ample basis to doubt the veracity of his second and third sworn statements that at the time of the 1-395 shootings he believed Wiltshire had a handgun. The jury also understood that Appellant Gonzalez’s testimony in Wiltshire’s civil suit and before the federal grand jury occurred years after the 1-395 shootings themselves, at a point when Gonzalez knew that a federal criminal investigation was pending. In light of the evidence of the shootings themselves and the officers’ orchestrated efforts to mislead, the jury reasonably concluded that even Gonzalez’s “belief’ testimony of 2000 and 2001 was untrue.
Finally, we reject Appellant Gonzalez’s claim as to Count Three that there was no possibility that his civil deposition would be communicated to federal authorities. See 18 U.S.C. § 1512(b)(3). At the time of Gonzalez’s civil deposition in 2000, the FBI already had opened its extensive civil rights investigation into the shooting incidents and had begun to review all of the post-shooting statements by the officers involved. The deposition was given in a lawsuit brought by Wiltshire’s estate for the wrongful death of Wiltshire, one of the victims in the 1-395 shootings. Given that the lawsuit and the FBI’s investigation were focused on the same shooting and the same victim, the jury could reasonably infer that Gonzalez intended to hinder and obstruct not only Wiltshire’s civil rights lawsuit but also the federal investigation into the conduct of the officers arising out of the same incidents at issue in the lawsuit.32
Accordingly, we conclude that sufficient evidence supported Appellant Gonzalez’s convictions on Counts Three and Five.
E. Extrinsic Evidence in Second Trial
During the second trial, Appellants Garcia, Gonzalez, and Quintero moved for a mistrial on the grounds that the jury was tainted by extrinsic evidence introduced by two former jurors dismissed during deliberations. After their convictions, Appellants also moved for a new trial on the same grounds. The district court denied both their motions for a mistrial and their motions for a new trial, and Appellants reassert their tainted jury claim on appeal.33
After a five-week trial followed by five days (approximately twenty-five hours) of deliberation, a series of notes from jury members in the second trial led the district court to dismiss three jurors and seat two alternates. The dismissals all occurred in succession over two days. We first review why three jurors were dismissed during the trial, and then explain why the district court did not abuse its discretion in denying Appellants’ motions.
*1297 (1) Juror # 10
Juror # 10 advised the court by note that her apartment had been burglarized and that she thought the burglary might have been related to her jury service. After receiving the note, the district court interviewed Juror #10 outside the presence of the other jurors, revealing that (1) Juror # 10 suspected that the defense might have been involved in the burglary; (2) Juror # 10 was concerned for her safety; and (3) Juror # 10 had read her note to the entire jury and expressed her concern to them.34 Juror # 10 claimed, however, that she could still perform her duties as a juror impartially.
The parties agreed that the other jurors needed to be questioned individually about Juror # 10’s note. Upon individual questioning, all eleven jurors testified that they were aware of the note and Juror # 10’s fear, as it had been discussed in the jury room.35 Nevertheless, all jurors stated that it would not affect their impartiality. However, one Juror (Juror # 4) individually admitted that Juror # 10’s story scared her, although Juror #4 stated that the fear would not affect her performance on the jury. At the conclusion of questioning each juror, the district court reminded each juror of their commitment to decide the case purely on the basis of the evidence produced during trial.
Appellants Garcia, Gonzalez, and Quinte-ro all moved for a mistrial, which the district court denied. Given that there would be no mistrial, the government and the defense agreed that Juror # 10 should be dismissed. Because the jury was already well into deliberating, the parties agreed to continue the trial with only eleven jurors. After bringing the entire jury back into the courtroom, the district court instructed the jury that it was convinced that the burglary of Juror # 10’s apartment was unrelated to the trial. The district court reiterated that the jury was to disregard the incident and decide the case based solely on the evidence presented in court.
(2) Juror # 1
At the same time that Juror #10 was being dismissed, Juror # 11 gave the court a note alleging that Juror # 1 had told the other jurors that she was watching the news and researching the case. Juror # ll’s note alleged that Juror # 1 had tried to find out the result in the first trial and was trying to tell the other jurors about what she knew. The note also alleged that Juror # 1 was refusing to deliberate.
Outside the presence of the other jurors, the district court questioned Juror # 1, confining its inquiry to the issue of whether she had become aware of extrinsic evidence. After initially denying watching the news, Juror # 1 admitted that she had heard the news once, after which she had joked with the other jurors because the news report had stated that she was fifty years old when she in fact was seventy-two. Juror # 1 also stated that she had seen the media “interviewing the defendants” but provided no details. Juror # 1 *1298claimed that she had not heard the results from the first trial, done any research, or told the jurors anything of substance.
The district court then questioned the remaining jurors individually. In direct contradiction to Juror # l’s explanation, six jurors asserted that Juror # 1 had told them that she had watched news reports about the trial and that she had conducted research to find out the results of the previous trial. Two of these jurors asserted that Juror # 1 had “made references” to her extrinsic knowledge and that they had asked Juror # 1 what information she had. However, all six jurors assured the court that ultimately Juror # 1 had not conveyed extrinsic evidence to them.
After hearing from each juror, the district court found Juror # l’s assertions that she had done no research not credible. The district court ruled that it was beyond a reasonable doubt that Juror # 1 had done independent research and that she blatantly disregarded the court’s instructions not to listen to the news. See United States v. Abbell, 271 F.3d 1286, 1302 (11th Cir.2001) (stating that a juror should be excused only when, beyond a reasonable doubt, no substantial possibility exists that the juror is basing her decision on the sufficiency of the evidence). The district court therefore dismissed Juror # 1.
Appellants moved for a mistrial, and objected to the removal of Juror # 1. Appellants made alternative arguments. On the one hand, Appellants claimed that the extrinsic evidence Juror # 1 had learned and shared with the other jurors tainted the entire jury, requiring a mistrial. On the other hand, Appellants claimed that Juror # 1 had testified truthfully, had not been exposed to any extrinsic evidence at all, and should not be dismissed. Appellants theorized that the jurors who complained about Juror # 1 had invented the story about her watching the news in order to remove her from the jury, when their real motivation was their frustration with Juror # l’s refusal to change her position during deliberation. The district court rejected both arguments and denied Appellants’ motion for mistrial.
(3) Juror # k
As the district court was finalizing its response about Juror # 1, Juror # 4 sent a note requesting to be dismissed because the trial was adversely affecting her sleep and health. Juror # 4’s note also indicated that she was “being insulted, yelled at and offended by other jurors.” The parties and the district court were concerned that Juror # 4’s request for dismissal was not health-related but rather was motivated by the deliberations themselves.
The district court questioned Juror # 4 outside the presence of the other jurors. She claimed that her request was not related to her opinion in deliberations, but rather truly was due to health concerns. Juror # 4 explained that she had felt the same way previously and had been hospitalized for three days.
The district court elected to delay deliberations so that Juror #4 could see her doctor, who might be able to determine whether her complaints were truly health-related. Juror # 4 went to her doctor on the afternoon of March 29, 2004. Juror # 4 did not report for jury duty on March 30, 2004, at which point the district court contacted Juror # 4’s doctor. Testifying by phone, the doctor explained that Juror #4 had a history of panic attacks and depression and that her mother had died recently. The doctor told the district court that “my medical opinion is that she just is not mentally capable of handling [jury service] at this point.” The district court accepted the doctor’s professional judgment and dismissed Juror #4 over the objections of Appellants.
*1299 (tí Two Alternates Empaneled
After the three jurors were dismissed, the parties agreed to add two alternates rather than three. Two alternates were empaneled, and the Appellants raised no objections to them. The district court ordered that all notes from the first five days of deliberation be removed from the jury room and instructed the jury to begin anew. The district court also gave the reconstituted jury the entire legal instructions once more. The new jury deliberated for a total of fourteen hours before reaching a verdict of guilty on all counts.
(5) Motions Properly Denied
On appeal, Appellants argue that a mistrial during the trial, or alternatively at least a new trial after their convictions, was required because the jury was tainted by extrinsic information, in particular (1) the jurors’ knowledge of Juror # 10’s burglary-related fear, and (2) the fact that Juror # 1 told the other jurors that she had read the news and researched the first trial. Although the jury was clearly exposed to this extrinsic evidence on account of Jurors # .1 and # 10, we conclude that the district court did not err in denying Appellants’ motions.
A mistrial or new trial is required only if the extrinsic evidence known by the jury posed a reasonable possibility of prejudice to the defendant. United States v. Perkins, 748 F.2d 1519, 1533 (11th Cir.1984). The defendant has the burden to show that the jury has been exposed to extrinsic evidence or extrinsic contacts. Once the defendant establishes that such exposure in fact occurred, prejudice is presumed and the burden shifts to the government to rebut the presumption. See Remmer v. United States, 347 U.S. 227, 229, 74 S.Ct. 450, 451, 98 L.Ed. 654 (1954); McNair v. Campbell, 416 F.3d 1291, 1307-08 (11th Cir.2005), cert. denied, — U.S. -, 126 S.Ct. 1828, 164 L.Ed.2d 522 (2006); Parker v. Head, 244 F.3d 831, 839 (11th Cir.2001); United States v. Martinez, 14 F.3d 543, 550 (11th Cir.1994); United States v. Spurlock, 811 F.2d 1461, 1463 (11th Cir.1987); United States v. Caporale, 806 F.2d 1487, 1503 (11th Cir.1986); Perkins, 748 F.2d at 1533.36
To rebut the presumption of prejudice, the government must show that the jurors’ consideration of extrinsic evidence was harmless to the defendant. Remmer, 347 U.S. at 229, 74 S.Ct. at 451; McNair, 416 F.3d at 1307-08; Caporale, 806 F.2d at 1503; Perkins, 748 F.2d at 1533-34. To evaluate whether the government has rebutted that presumption, we consider the totality of the circumstances surrounding the introduction of the extrinsic evidence *1300to the jury. Remmer, 347 U.S. at 229-30, 74 S.Ct. at 451; McNair, 416 F.3d at 1307-08; Parker, 244 F.3d at 839. The factors we consider include: (1) the nature of the extrinsic evidence; (2) the manner in which the information reached the jury; (3) the factual findings in the district court and the manner of the court’s inquiry into the juror issues; and (4) the strength of the government’s case. See McNair, 416 F.3d at 1307-08.
In this case, the nature of the extrinsic evidence and the district court’s thorough and careful response to that evidence convince us that the presumption of prejudice has been clearly rebutted. First and foremost, the nature of the extrinsic evidence here was relatively inconsequential and ultimately does not indicate prejudice. Not a single juror learned anything substantive about the first trial from Juror # 1,37 nor did a single juror claim to believe that the burglary at Juror # 10’s residence was in any way related to the trial. The district court also took care to instruct the jury that the burglary was in no way related to the trial, and that the jury was to reach a verdict based solely on the evidence presented at trial. Thus, the government has shown that the particular extrinsic evidence at issue in this case was not only unlikely to prejudice the jury in any meaningful fashion, but also did not in fact prejudice the jury.
Second, the record demonstrates that the district court carefully, thoroughly and correctly handled these issues at trial. The district court began by carefully investigating the juror issues. See Rowe, 906 F.2d at 656-57. In addition to interviewing Jurors # 10 and # 1 outside the presence of the other jurors, the district court interviewed every other juror individually and conferred extensively with counsel to decide upon a course of action.
After speaking with each juror, the district court wisely chose to dismiss Jurors # 10 and # 1, the two jurors who had introduced the extrinsic evidence. The district court then rehabilitated the remaining jurors by curatively instructing the jury not to consider the extrinsic evidence, by reissuing the entire legal instructions to the newly constituted jury, and by emphasizing multiple times that the reconstituted jury should begin deliberations anew. The district court concluded that (as every juror individually testified) the extrinsic evidence did not prejudice the remaining jurors nor render any juror incapable of deciding the case impartially.
In light of every juror’s assurance to the district court that the extrinsic evidence would not affect their impartiality, and especially given the district court’s careful investigation and curative measures, we agree with the district court that there was no reasonable possibility of prejudice. The manner in which the district court inquired of the jury in this case closely parallels United States v. Gabay, 923 F.2d 1536 (11th Cir.1991), where this Court affirmed a district court’s conclusion that a dismissed juror’s extrinsic discussions did not preclude a fair trial. In Gabay, we deferred to the district court’s assessment *1301of the remaining jurors’ impartiality, noting in particular the district court’s “careful and lengthy investigation,” which included voir dire and curative instructions. Id. at 1542-43. We also noted the district court’s “observation of the jurors for several weeks, and the jurors’ unequivocal pledges to deliberate fairly.” Id. at 1543.
Here, the district court’s careful and lengthy efforts to ensure that the jury remained impartial are quite similar to those taken in Gabay. As in Gabay, the district court’s reasonable conclusion that the jury was not tainted demands deference because the “ ‘determination of [the jury’s] impartiality, in which demeanor plays such an important part, is particularly within the province of the trial judge.’ ” Ristaino v. Ross, 424 U.S. 589, 595, 96 S.Ct. 1017, 1020, 47 L.Ed.2d 258 (1976) (quoting Rideau v. Louisiana, 373 U.S. 723, 733, 83 S.Ct. 1417, 1423, 10 L.Ed.2d 663 (1963) (Clark, J., dissenting)). As this Court has stated, “[t]he factual determination of whether consideration of extrinsic evidence caused the defendant prejudice is committed to the trial court’s ‘large discretion.’” BankAtlantic v. Blythe Eastman Paine Webber, Inc., 955 F.2d 1467, 1472 (11th Cir.1992); see also Martinez, 14 F.3d at 547 (a district court’s denial of a motion for a new trial based on jury’s exposure to extrinsic evidence is reviewed for abuse of discretion).
Despite unusual circumstances, the district court ensured that the extrinsic evidence did not pose a reasonable possibility of prejudice. After extensive review of the trial transcript and district court orders, we are confident that the district court did not abuse its discretion in denying Appellants’ motions. Rather, the district court carefully rehabilitated and instructed the reconstituted jury, and there is no evidence that the extrinsic evidence influenced the jury in any way. See United States v. Kopituk, 690 F.2d 1289, 1307-08 (11th Cir.1982) (affirming the district court’s decision to dismiss a juror and replace her with an alternate after five days of deliberation).
IV. APPEAL OF SENTENCES
Appellants contend that resentencing is required because their sentences violate United States v. Booker, 543 U.S. 220, 125 S.Ct. 738, 160 L.Ed.2d 621 (2005). We first review Appellants’ sentences, and then the Booker issues.
A. Sentences
Appellants Ronda, Agüero, Beguiristain and Castello — those Appellants convicted in the first trial — were sentenced at a hearing held on October 9, 2003. The Presentence Investigation Reports (“PSIs”) for all four Appellants assigned them a base offense level of 12 under U.S.S.G. § 2J1.2(a). Appellants Ronda and Castello were assigned no enhancements or reductions; their offense levels of 12 and criminal history categories of I produced a Guidelines range of 10 to 16 months’ imprisonment.
While the PSIs also assigned a base offense level of 12 to Appellants Agüero and Beguiristain, their PSIs recommended several adjustments to that base offense level. Appellant Aguero’s PSI recommended a total adjusted offense level of 19, which included: (1) a two-level enhancement for playing an organizing role in the criminal activity, see U.S.S.G. § 3Bl.l(c); (2) a two-level enhancement for abusing a position of public trust in a manner that significantly facilitated the commission or concealment of the offense, see U.S.S.G. § 3B1.3; and (3) a three-level increase pursuant to the multiple count Guidelines, see U.S.S.G. §§ 3D1.1 through 3D1.5.38 Appellant Aguero’s total adjusted offense *1302level of 19 and his criminal history category of I produced a Guidelines range of 30 to 37 months’ imprisonment.
Appellant Beguiristain’s PSI recommended a total adjusted offense level of 17. In addition to Beguiristain’s base offense level of 12, the PSI added: (1) a two-level enhancement for abusing a position of public trust in a manner that significantly facilitated the commission or concealment of the offense, see U.S.S.G. § 3B1.3; and (2) a three-level increase pursuant to the multiple count Guidelines, see U.S.S.G. §§ 3D1.1 through 3D1.5.39 Appellant Be-guiristain’s total adjusted offense level of 17 and his criminal history category of I produced a Guidelines range of 24 to 30 months’ imprisonment.
At the October 9., 2003 sentencing hearing, the district court sentenced Appellants Ronda and Castello to 13 months’ imprisonment, the middle of their Guidelines range. The district court sentenced Appellant Agüero to 37 months’ imprisonment, the high end of his Guidelines range, and sentenced Appellant Beguiristain to 27 months’ imprisonment, the middle of his Guidelines range.40 In imposing these sentences, the district court stressed, inter alia, the seriousness of the offenses, the” need to deter future criminal activity, and in particular Appellant Aguero’s role as the most culpable participant in the conspiracy, as follows:
[T]he crimes committed by these former police officers are very serious in nature and should be punished as such.
It is my view that each sentence imposed should reflect not only the seriousness of the crime but each defendant’s role and participation in the overall conspiracy.
Furthermore, the sentences imposed should serve to deter against future criminal activity by these defendants as well as by others similarly situated who may be tempted to violate the law and their oath of office to protect themselves or their fellow officers from possible state or federal criminal investigations.
Therefore, I shall impose full prison terms for Defendants Ronda and Castel-lo and Beguiristain at mid-range within their respective Sentencing Guidelines.
With respect to Defendant Agüero, I find that he is the most culpable member of the conspiracy. I also shall take into account his prior inappropriate conduct while serving as a police officer in sentencing him at the upper end of the guidelines.
Oct. 29, 2003 Sentencing Tr. at 44.
Appellants Garcia, Gonzalez and Quinte-ro — the Appellants convicted in the second *1303trial' — were sentenced at a hearing on November 16, 2004. The PSIs assigned Appellants Garcia, Gonzalez and Quintero a base offense level of 12 under U.S.S.G. § 2J1.2(a) and no enhancements or reductions. As all three Appellants were assigned to criminal history category I, their Guidelines ranges were 10 to 16 months’ imprisonment.
The district court then sentenced Appellants Garcia, Gonzalez and Quintero each to 16 months’ imprisonment, the high end of the Guidelines range, after considering the nature of the crimes and Appellants’ roles in the offenses in relation to the other Appellants.41 The district court expressed compassion for the circumstances faced by Appellants and their families, but emphasized the seriousness of the crimes as follows:
These matters are beyond the compassion I feel for the individuals associated with this case. They involve serious crimes, and this 1-395 aspect involved the death of two individuals. The concealment relating to these deaths I find even more serious, both in kind and degree than the incidents involved in Coconut Grove, which in and of themselves were serious, although the injury involved was less serious.
Defendants Garcia and Quintero were very active, involved in gun planting incidents here and the cover-up. It is a sad thing that Defendant Gonzalez, who had a leadership role at the time, did not take active steps to prevent or stop the conspiracy as it evolved based upon the matters that were presented to the jury in terms of what occurred at the scene and what occurred thereafter.
All these are serious matters that require the Court to act accordingly.
Nov. 16, 2004 Sentencing Tr. at 42-43.
B. Booker
For the first time on appeal, all Appellants challenge their sentences based on Booker.42 Because Appellants did not raise the Blakely/Booker issue in district court, we review this issue for plain error.43
*1304Under a plain-error analysis, an appellant must show “ ‘(1) error, (2) that is plain, and (3) that affects substantial rights.’ ” United States v. Rodriguez, 398 F.3d 1291, 1298 (11th Cir.) (quoting United States v. Cotton, 535 U.S. 625, 631, 122 S.Ct. 1781, 1785, 152 L.Ed.2d 860 (2002)), cert. denied, — U.S. -, 125 S.Ct. 2935, 162 L.Ed.2d 866 (2005). If the appellant is able to make a showing of all three, we then may exercise discretion to notice the error if the error “ ‘seriously affect[ed] the fairness, integrity, or public reputation of judicial proceedings.’ ” Id. (citation omitted).
In the context of Booker errors, the appellant has shown that his substantial rights were affected “ ‘when there is a reasonable probability that the district court would have imposed a different sentence if the guidelines were not mandatory.’ ” United States v. York, 428 F.3d 1325, 1336 (11th Cir.2005) (citation omitted), petition for cert. filed, No. 05-1503 (U.S. May 22, 2006). The third prong of the plain-error test “almost always requires that the error must have affected the outcome of the district court proceedings.” Rodriguez, 398 F.3d at 1299 (quotation marks and citations omitted). “The standard for showing that is the familiar reasonable probability of a different result formulation, which means a probability sufficient to undermine confidence in the outcome.” Id. (quotation marks and citations omitted). In regard to the third prong, the burden rests with the appellant to show prejudice. Id. “Where errors could have cut either way and uncertainty exists, the burden is the decisive factor in the third prong of the plain error test Mat 1300.
Under Booker, “there are two types of sentencing errors: one is constitutional and the other is statutory.” United States v. Dacus, 408 F.3d 686, 688 (11th Cir.2005). “[T]he Sixth Amendment right to trial by jury is violated where under a mandatory guidelines system a sentence is increased because of an enhancement based on facts found by the judge that were neither admitted by the defendant nor found by the jury.” Rodriguez, 398 F.3d at 1298. The statutory error occurs when the district court sentences a defendant “under a mandatory Guidelines scheme, even in the absence of a Sixth Amendment enhancement violation.” United States v. Shelton, 400 F.3d 1325, 1330-31 (11th Cir.2005).
Appellants Agüero and Beguir-istain have established Sixth Amendment violations under Booker because the district court enhanced their sentences under a mandatory Guidelines system based on judicially-determined fact findings that went beyond those facts admitted by Appellants or found by the jury. Rodriguez, 398 F.3d at 1298. The Sixth Amendment violations under Booker stemmed not from the district court’s extra-verdict enhancements but from the district court’s application of those enhancements under a mandatory Guidelines system. Id. at 1301. Further, Appellants Ronda, Castello, Garcia, Gonzalez, and Quintero have established statutory Booker error because while the district court did not assign them *1305any extra-verdict enhancements, they too were sentenced under a mandatory Guidelines system. Shelton, 400 F.3d at 1330-31.
Because plain Booker errors occurred, all seven Appellants have established the first two prongs of plain-error review. Rodriguez, 398 F.3d at 1299.44 However, none of the Appellants has shown that the Booker error affected their substantial rights. Accordingly, Appellants have failed to satisfy the third prong of plain-error review. Id. The sentencing record provides no basis for a conclusion that any Appellant has a reasonable probability of receiving a more lenient sentence under an advisory Guidelines system. The district court did not make any comments that the sentences imposed were too severe. “The record indicates no frustration on the part of the district court with the severity of the Guidelines sentence^], nor did the district court indicate a desire to impose a lesser sentence in [Appellants’] case.” United States v. Underwood, 446 F.3d 1340, 1344 (11th Cir.2006).
Rather, the district court stressed the need for either a high- or mid-range sentence because of the severity of Appellants’ crimes; the need for deterrence; Appellants’ status as police officers and society’s need for law enforcement officers to follow the law; the role and participation of each Appellant in the conspiracy; and Appellant Aguero’s heightened culpability. A sentence in the middle or high end of the range, as each Appellant received, suggests that the district court did not prefer a more lenient sentence and would not have imposed one under an advisory system.45
Accordingly, Appellants have not met their burden to show a reasonable probability that the result of their sentencing would have been different but for the Booker errors. Rodriguez, 398 F.3d at 1300-01. Because Appellants have not shown that their substantial rights were adversely affected under the third prong, we need not analyze the fourth element of plain-error review. Id. at 1301.
V. CONCLUSION
Because Appellants have shown no reversible error, we affirm Appellants’ convictions and sentences on all Counts.
AFFIRMED.