KENNEDY, Circuit Judge.
In these five consolidated appeals, defendants Loretta Walton, Charles Eddie Mitchell, Eddie Lee Johnson, Larry Noal White, and Rosemary Johnson seek relief from their convictions and sentences imposed on jury verdicts finding each guilty of conspiring to distribute cocaine in violation of 21 U.S.C. §§ 841 and 846. For the reasons stated below, we: (1) AFFIRM the convictions and sentences of defendants Walton, Mitchell, and White, (2) AFFIRM the convictions of defendants Eddie and Rosemary Johnson, but (3) REVERSE the sentences of defendants Eddie and Rosemary Johnson and REMAND their cases for resen-tencing.
I. Background
Because most of the complex facts in this case are unnecessary to the resolution of the case, only an abbreviated version will be recited here. Additional facts will be stated when needed for the resolution of the issues discussed. The defendants were convicted of participating in a conspiracy to distribute cocaine in Flint, Michigan, and Columbus, Ohio. The indictment alleged that the conspiracy began in 1980 and continued through the time of the indictment in 1988. The evidence presented at trial showed that Larry White was the leader of the conspiracy in the Flint, Michigan, area. He obtained large quantities of cocaine from suppliers in other parts of the country and arranged for its transportation, dilution, repackaging, and distribution by other members of the conspiracy.
The remaining defendants occupied various positions in White’s Flint organization. Loretta Walton was one of White’s chief lieutenants and was involved in all aspects of the conspiracy, including the pickup and delivery of cocaine, the pickup and delivery of money, the preparation and repackaging of cocaine for retail sales, and the sales themselves. Charles Mitchell was a Flint police officer during the conspiracy. He transported cocaine for the ring and provided protection for its activities, many times while in uniform. The Johnsons apparently were not as deeply involved in the conspiracy as were White, Walton, and Mitchell. They obtained small quantities of cocaine from other members of the conspiracy, diluted it, repackaged it into small lots, and sold it directly to users of cocaine.
The government conducted a lengthy investigation into the ring’s activities prior to bringing the indictment. Wiretaps placed on Larry White’s phones and a microphone hidden in his house provided a considerable amount of evidence against most of the defendants. Additionally, several members of the conspiracy testified against the defendants on behalf of the government, including Rolando Hernandez, a Miami-based supplier of cocaine for the conspiracy, and Russell Brown, who transported and repackaged cocaine for the conspiracy.
A jury found all of the defendants whose cases are on appeal here guilty of conspiracy to distribute cocaine. Following the preparation of presentence reports and hearings on the disputed issues, the District Court sentenced the defendants pursuant to the guidelines established by the United States Sentencing Commission. Loretta Walton and Charles Mitchell were each sentenced to 292 months imprisonment. Eddie and Rosemary Johnson were each sentenced to 41 months imprisonment. Larry White was sentenced to 480 months imprisonment, the maximum sentence authorized by statute.
II. Witness Brown’s Reference to a Polygraph Test
Defendants Walton, Mitchell, and White challenge a passing reference by [1292]*1292Russell Brown, a government witness, to a polygraph examination. The reference occurred during Brown’s direct examination by the Assistant United States Attorney:
Q All right. Did you tell the DEA agent Dodson something about the cocaine business in Flint?
A Yes.
Q Why did you do that?
A Well I thought it would help with my case.
Q The uttering and publishing charge was pending against you then?
A Right.
Q Okay. And you spoke with the DEA agent Dodson at that time?
A Right.
Q As a result of your conversation with Agent Dodson, were you called to a Grand Jury?
A No. I had to take a polygraph before that.
Following Brown’s mention of the polygraph, defense counsel asked for and received a sidebar conference and moved for a mistrial. The court denied the motion but agreed to accept briefs on the subject. Following the sidebar conference, the government, in an attempt to minimize any possible prejudice from the witness’ answer, asked the court to strike the witness’ answer as nonresponsive and to instruct the jury to disregard it. The court advised the jury that the answer was not responsive to the question asked by the prosecutor, ordered the response stricken, and advised the jury to disregard it. No other reference to this or any other polygraph examination was made at any time during the five-week trial.
The following day, the court heard argument on the motion for a mistrial. Brown was examined outside of the jury’s hearing. He testified that the polygraph examination he took was actually requested and administered by county prosecutors, not by any federal agent or prosecutor. He also stated that he never mentioned the polygraph examination to any federal agent or Assistant United States Attorney involved in the case and had no way of knowing whether the Assistant United States Attorney prosecuting the case knew he had taken a polygraph examination. The court again denied the motion for a mistrial and found that the government had not intended the reference to the polygraph. Although noting that the prosecutor had stated that the witness’ unresponsive statement might have been deliberate, the court found that there was no evidence that Brown had made the statement in a deliberate attempt to prejudice the defense. At this hearing, the government also offered to stipulate that no agent of the United States had requested that a polygraph test be given to Brown and no agent of the United States had been informed of the results of any polygraph test Brown may have taken. Defense counsel declined to have the stipulation read to the jury.
Brown’s testimony concerned, among other things: (1) a shipment of frozen fish containing hidden packages of cocaine that he transported for White, (2) his work for White breaking down several one kilogram shipments of cocaine into one ounce packages for resale, (8) his knowledge that Mitchell, then a police officer for the city of Flint, delivered cocaine.for White, and (4) his observation that Walton often made cocaine pickups with White. Brown, who was in prison on other charges at the time of his testimony, admitted that he only agreed to testify in hope of receiving leniency in a state prosecution and a $2,000 payment by the federal government. Thus the credibility of his testimony was disputed.
Other witnesses and evidence also linked Walton, Mitchell, and White to the conspiracy. For example, Rolando Hernandez, who supplied White with bulk quantities of cocaine, testified to his supply activities and to Walton’s actions as a courier for and confidante of White. Dennis Barker, who like Brown was only involved intermittently in White’s operations, provided additional evidence against Mitchell. He testified that he had seen Mitchell, often in uniform, receiving large lots of cocaine from White for delivery. Phone wiretaps and a microphone located in White’s house provided strong evidence against White and Walton [1293]*1293and at least some additional evidence against Mitchell.
The defendants argue that any mention of a polygraph examination by a government witness should result in a mistrial, at least where that witness’ testimony is important to the government’s case. The defendants also argue that because Brown is a convicted felon we should presume he is sophisticated in matters of criminal procedure and intentionally referred to the polygraph in order to bolster his credibility. The District Court refused to find that the government had any advance knowledge that Brown would mention the polygraph test. Although the court acknowledged the possibility that Brown may have made the reference intentionally, it concluded there was no evidence to support this proposition and refused to create a presumption in favor of the defendants.
Had the District Court found that the government intended Brown to introduce the information concerning the polygraph examination, the reference might lead to reversal under circuit precedent. In United States v. Murray, 784 F.2d 188 (6th Cir.1986), for example, we held that a defendant’s conviction must be reversed where an experienced FBI agent stated that he had asked the defendant to take a polygraph examination. That case is distinguishable for two reasons, however. First, the majority believed that the statement was deliberately made by an experienced government agent. Id. Regardless of Brown’s motive, there is no suggestion of government misconduct in the case before us. Second, though it was not explicitly mentioned by the Murray Court, a statement suggesting that a criminal defendant either took and failed a polygraph examination or refused to take an examination directly relates to guilt and implicates a defendant’s fifth amendment right not to incriminate himself. Here, the statement did not go to whether one of the defendants had taken or refused to take a polygraph test. Rather, it relates only to the credibility of a witness.
The mere fact that Brown was a convicted felon does not lead us to presume that he both knew his reference to his polygraph test to be inadmissible and made the statement with bad motive. Further, even were we to accept the proposition that Brown intended to prejudice the jury, a mistrial would not necessarily be required. Where the witness is not a government official, the question is not whether the witness intended to prejudice the jury or bolster his credibility but whether the reference was harmless.
On the facts before us, Brown’s passing reference to having taken a polygraph examination appears to have been harmless. Although references to polygraph testing must be treated with great care by a trial court, we have not mandated that all references to polygraph testing require a mistrial. See United States v. Betancourt, 838 F.2d 168, 175 (6th Cir.), cert. denied, 486 U.S. 1013, 108 S.Ct. 1748, 100 L.Ed.2d 210 (1988) (admission of information concerning a government witness’ polygraph exam at the request of four defendants did not prejudice trial of fifth defendant who objected to the admission of this evidence). Even the circuits which have held that polygraph evidence is generally inadmissible have allowed an inadvertent statement concerning the evidence to be remedied by a curative instruction. See United States v. Tedder, 801 F.2d 1437, 1444-45 (4th Cir.1986), cert. denied, 480 U.S. 938, 107 S.Ct. 1585, 94 L.Ed.2d 775 (1987); United States v. Holman, 680 F.2d 1340, 1352 (11th Cir.1982); United States v. Martino, 648 F.2d 367, 390 (5th Cir.1981), cert. denied, 456 U.S. 943, 102 S.Ct. 2006, 72 L.Ed.2d 465 (1982); see also United States v. Dietrich, 854 F.2d 1056, 1058-60 (7th Cir.1988). The standard for when such a curative instruction is effective has been stated in various ways: whether the statement “is so prejudicial as to be incurable,” Holman, 680 F.2d at 1352; whether the statement “measurably [affected] the jury’s verdicts,” Martino, 648 F.2d at 390; or, a two factor test “ ‘(1) whether an inference about the result of the test may be critical in assessing the witness’s credibility, and (2) whether the witness’s credibility is vital to the case,’ ” Tedder, 801 F.2d at 1444 (quoting [1294]*1294United States v. Brevard, 739 F.2d 180, 182 (4th Cir.1984)).
The defendants assert that Murray stands for the proposition that any admission of polygraph evidence must be harmless beyond a reasonable doubt in order to avoid a mistrial. We do not believe that Murray is so broad. Murray applies this standard, the standard for errors of constitutional magnitude, in a case where the polygraph reference is to a defendant failing or refusing to take a polygraph, a violation that implicates rights of a constitutional magnitude. Where such a polygraph reference is not to a defendant, we believe an instruction or an offer to make such an instruction to disregard the reference is sufficient if it meets the nonconsti-tutional error test adopted by the Fourth Circuit. Compare Chapman v. California, 386 U.S. 18, 24, 87 S.Ct. 824, 828, 17 L.Ed.2d 705 (1967) (constitutional errors must be “harmless beyond a reasonable doubt”) with Opper v. United States, 348 U.S. 84, 95, 75 S.Ct. 158, 165, 99 L.Ed. 101 (1954) (“Our theory of trial relies upon the ability of a jury to follow instructions”).
As did the Fourth Circuit in Tedder, we conclude that the curative instruction in this case was sufficient to cure any harm that may have occurred as a result of the passing reference to a polygraph examination. As to the first element of the test,
the jury heard no evidence as to the results of the test, and while it may have assumed that the test buttressed [Brown’s] credibility, it also observed him during lengthy direct examination and cross-examination. Thus, even if some jurors drew an inference about the results of the test, there is no special reason in this case to believe that the inference was any more critical in assessing the witness’s credibility than it was in cases in which a curative instruction was found to be sufficient.
Tedder, 801 F.2d at 1445.
As to the second element, there was an abundance of other evidence against White and Walton. There was also sufficient evidence against Mitchell. Although the defendants argue that the jury could have inferred that other witnesses took and passed polygraphs, we “decline to impute to the jury so tenuous a chain of inferences.” Id.
III. The Johnsons’ Motions for Acquittal
At the close of the government’s case-in-chief and the close of evidence, both Eddie and Rosemary Johnson moved for judgments of acquittal on the grounds that the evidence introduced was insufficient. The motions were denied. The defendants renew this argument on appeal. At the outset, we note that the defendants have a difficult burden to meet. When reviewing a denial of a motion to dismiss, we must consider all the evidence in a light most favorable to the government and then determine whether there is any evidence from which a reasonable jury could find guilt beyond a reasonable doubt. See, e.g., United States v. Adamo, 742 F.2d 927, 932 (6th Cir.1984), cert. denied, 469 U.S. 1193, 105 S.Ct. 971, 83 L.Ed.2d 975 (1985). Judged under this standard, we believe the convictions must stand.
The primary evidence against the John-sons was several tape recordings of phone conversations in which the Johnsons discussed cocaine sales. These conversations were between the Johnsons and various persons at Larry White’s home. For example, the Johnsons discussed the cutting and sale of cocaine obtained from “Loretta,” apparently Loretta Walton, a defendant in this case. A jury could reasonably conclude that the Johnsons sold cocaine they obtained from defendant Walton who they knew distributed cocaine for White. Although the defendants argue that, at most, the evidence demonstrates their participation in a conspiracy separate from the one which White led, we believe that a reasonable jury could conclude otherwise. See Exhibit 2 (4/26/86: Lynn Grant, Rosemary Johnson’s sister, and Rosemary Johnson discussing complaints about the quality of cocaine the Johnsons had obtained from “Loretta” [Loretta Walton, one of Larry White’s chief confidantes] and sold to other persons); Exhibits 50, 51, and 52 (three 8/19/86 conversations: Lynn Grant beg[1295]*1295ging the Johnsons for $800 or cocaine so that she could repay “Larry” [White] for cocaine or cocaine money of White’s that she had used; the Johnsons refusing in the belief that if they helped her and White found out then White would refuse to continue doing business with them; Eddie Johnson mentioning that he wanted to help but still owed money to Larry himself.)
IV. White’s Claim of Prejudice Concerning Erroneous Advice as to Maximum Sentence
Defendant Larry White was originally charged both with conspiring to distribute cocaine (Count I) and operating a continuing criminal enterprise (Count II). At his arraignment on these charges, White signed an “Acknowledgement of Indictment” form. The form correctly stated that he faced a maximum sentence of life imprisonment for operating a continuing criminal enterprise but erroneously stated that the maximum sentence for the conspiracy count was 20 years. The actual maximum sentence was 40 years. 21 U.S.C. §§ 841 and 846. Count II was voluntarily dismissed by the government three days after the jury was selected. After his conviction on Count I, White was sentenced to 480 months imprisonment, the maximum allowed under the statute.
White, citing United States v. Fuller, 769 F.2d 1095 (5th Cir.1985), argues that the erroneous information denied him the opportunity to knowingly and intelligently plead guilty to the conspiracy count of the indictment. He asserts that he would have given more serious consideration to a guilty plea had he known that he could be sentenced to 40 years if convicted of conspiracy. Fuller involved a post-conviction action by Fuller seeking to set aside his guilty plea and sentence on the basis that the court overstated the maximum possible sentence. Fuller argued that he pled guilty because he erroneously believed he was subject to a longer sentence. The court accepted the general theory that an attack could be made on these grounds but, after examining all of the facts that confronted the defendant, determined that he had not pled guilty as a result of the misinformation. Id. at 1098-99.
Initially, we note that this case is very different from Fuller. Here, the defendant wishes to challenge his decision not to plead guilty. Under the Federal Sentencing Guidelines, it is more difficult to understand how the defendant was harmed by his decision not to plead guilty. We need not decide whether such a challenge is available, however, since the issue was not raised below. Without any record from which we can determine whether the defendant knew the penalty he was facing in spite of the erroneous information, whether the erroneous information affected his decision not to plead guilty, and whether his decision not to plead guilty caused him cognizable harm, we cannot evaluate the claim. As with other claims that involve issues outside the scope of the record, these claims can be attacked, if at all, only in a collateral proceeding. See, e.g., United States v. Swidan, 888 F.2d 1076, 1081 (6th Cir.1989) (ineffective assistance claims cannot be raised for the first time on direct appeal since the trial record is normally insufficient to evaluate counsel performance).
V. The Trial Court’s Ex Parte Communication with a Juror
During recross-examination of defendant White concerning his financial dealings with his sister Barbara Veasley (a defense witness who had already testified), the trial judge called a sidebar conference and made the following statement:
This morning a juror told me or asked me to point — asked to see me and I saw him. He wanted to point out to me that Mrs. Veasley was in the audience and signaling answers to Mr. White. I asked this morning to have Ms. Schnay sit in the courtroom during this morning’s session before the break and observe and I observed.
During the course of the playing of tapes during the examination, there was no signalling. There is clear nodding and shaking of the head that I have observed in the last five minutes. Now [1296]*1296you figure out how you want to communicate with her that she has moved from the right side of the courtroom which she was ... out of the line of vision of Mr. White to the left side of the courtroom which is directly in the line of vision of you, Mr. Grant. I will determine at some later time whether I will do anything further about this. But I would suggest that you have somebody in the courtroom in the audience that you can communicate to that can go be seated beside her. Either that or I’ll order her removed from the courtroom.
When one of the attorneys requested that the court ask the juror if he had communicated his observations to the other jurors, the trial judge responded that he had already asked and that the juror had not. The trial judge also explained that he advised the juror not to reveal his observation to the other jurors during the trial but indicated that the juror may well choose to do so during deliberations since it was the juror’s function to make observations that went to the credibility of a witness.
In spite of the fact that the issue was not raised below, Defendant White claims that the District Court on its own motion should have held a full hearing on the record to determine whether the incident biased the juror and, if so, whether he was entitled to a new trial. White urges, citing cases such as Remmer v. United States, 347 U.S. 227, 74 S.Ct. 450, 98 L.Ed. 654 (1954), that we should presume in the absence of a hearing that the judge’s conduct biased the juror and thus violated his sixth amendment right to an impartial jury.
In Remmer an unknown person suggested to a juror that he could make some money by bringing in a verdict for the defendant. The trial court discussed the matter with the prosecution and as a result an FBI investigation was conducted. The investigation suggested the remark was a joke, but neither the incident nor the report were ever revealed to the defense during trial. The Supreme Court determined that, under the circumstances, the contact was presumptively prejudicial. Although the Court did not believe that the presumption was irrebuttable, it believed such a rebuttal could only be made after notice to the parties and a hearing on the record. Id. at 228-29, 74 S.Ct. at 450-51.
In Rushen v. Spain, 464 U.S. 114, 104 S.Ct. 453, 78 L.Ed.2d 267 (1983), the Court again determined that contacts with a juror are subject to harmless error analysis. Id. at 118-19, 104 S.Ct. at 455-56. In Rushen, the Court determined that any constitutional error that may have occurred in the murder trial of a Black Panther was harmless. A juror had contacted the trial judge in the middle of trial to inform him that the trial had triggered a childhood memory of a friend who was murdered by a Black Panther. Although the juror told the judge that the matter upset her, she assured him that she could continue to hear the case without prejudice. The judge did not inform the parties of the contact during the trial. Id. at 115-16, 104 S.Ct. at 454. On a collateral motion for a new trial, the trial judge determined after a hearing that the contact did not prejudice the defendant’s right to a fair trial. The defendant then brought a habeas corpus petition in the federal courts. On review of a lower court’s decision to grant the petition, the Supreme Court reversed, affirming his conviction. Id. at 119, 104 S.Ct. at 456.
The Court did not hold that a hearing must always be held when any contact between a juror and a judge occurs. That issue was not before it. The Court did make several observations that aid in the resolution of this case:
There is scarcely a lengthy trial in which one or more jurors do not have occasion to speak to the trial judge about something, whether it relates to a matter of personal comfort or to some aspect of the trial. The lower federal courts’ conclusion that an unrecorded ex parte communication between trial judge and juror can never be harmless error ignores these day-to-day realities of courtroom life and undermines society’s interest in the administration of criminal justice.
This is not to say that ex parte communications between judge and juror are never of serious concern or that a federal [1297]*1297court on habeas may never overturn a conviction for prejudice resulting from such communications. When an ex parte communication relates to some aspect of the trial, the trial judge generally should disclose the communication to counsel for all parties. The prejudicial effect of a failure to do so, however, can normally be determined by a post-trial hearing.
Id. at 118-19, 104 S.Ct. at 455-56 (footnotes omitted).
In this case the trial judge complied with the Supreme Court’s mandate. He informed counsel of the contact and the nature of the limited information the juror gave him. Although there was some concern expressed that the juror might be unfairly biased against defendants other than White, there was never a formal request by any of the attorneys for an evi-dentiary hearing and findings of fact. Having failed to ask for these findings or seek any other relief in the District Court, defendant White now seeks relief for the first time on appeal.
Because of the extremely limited nature of the contact, we do not believe that the District Court was required to hold a hearing on its own motion. Thus, in order to prevail, White must demonstrate from facts in the record that actual prejudice occurred. From the information that is in the record, we conclude that there has been no showing of actual juror bias as is required in order to overturn the verdict. United States v. Pennell, 737 F.2d 521, 532 (6th Cir.1984), cert. denied, 469 U.S. 1158, 105 S.Ct. 906, 83 L.Ed.2d 921 (1985). A juror is not only entitled to make observations in the courtroom during trial that go to the credibility of a witness, a juror should be encouraged to do so. The juror’s only contact with the jqdge was a brief conversation in which the juror reported his observations. The judge’s only response to the juror was to ask if he had reported the contact to other jurors. After learning that he had not, the judge instructed the juror not to do so for the remainder of the trial. There was no attempt by the judge to influence the juror; there was no indication in the juror’s remarks to the judge that the incident unduly prejudiced the defendant.
VI. The Prosecution’s Use of Peremptory Challenges
Relying on Batson v. Kentucky, 476 U.S. 79, 106 S.Ct. 1712, 90 L.Ed.2d 69 (1986), defendant White alleges that the prosecution used its peremptory challenges to strike potential black jurors from the venire on the basis of their race. In order to establish a prima facie case of such discrimination, a defendant must show that he is a member of a cognizable racial group, that the prosecution struck members of the defendant’s group from the venire, and that these and “any other relevant circumstances raise an inference that the prosecutor used [peremptory challenges] to exclude the veniremen from the petit jury on account of their race.” Id. at 96, 106 S.Ct. at 1723. Once the defendant is able to establish a prima facie case, the burden shifts to the prosecution to come forward with a neutral explanation for challenging the jurors of the defendant’s group. Id. at 96-97, 106 S.Ct. at 1723-24.
All of the defendants in this case are black. The venire was composed of 56 members, 9 of whom were black. Questioning of the members was done by the court. No other information from which a court could discern possible prosecution bias was submitted. Thus the defendants must establish a prima facie case solely by reference to the strikes actually made by the prosecution. The Assistant United States Attorney was given 15 peremptory strikes, the defendants 30 in total divided among them. The third, fifth, thirteenth, and fifteenth strikes exercised by the prosecution resulted in the removal of potential jurors who were black. The prosecution did not exercise its right to strike a potential juror with its ninth and tenth strikes. The pool remaining at the close of jury selection consisted of 15 persons, three of whom were black. The Assistant United States Attorney explained that all of the black jurors it had struck, except one, as well as a number of the white jurors, were struck because they were in a household where no adult was employed — either be[1298]*1298cause the potential juror was (1) single and unemployed, or (2) unemployed and married to someone who was unemployed.1 The prosecutor explained that he struck the final black juror because he had prosecuted her brother in federal court and because her brother-in-law was employed by the Flint Police Department. One of the defendants as well as several of the witnesses were police officers working in Flint.
Twelve of the fifteen members of the pool remaining after all strikes were exercised, including all of the three black members, were designated as jurors. The remaining three were designated alternates. Because one of the black jurors did not show up for trial, a white alternate took his place.2 After the submission of briefs, the District Court overruled defense motions to discharge the jury and the trial proceeded.
Reviewing all the facts before us, we cannot say that the defendants established a prima facie case of discrimination by the prosecution. White admits that the juror whose brother the Assistant United States Attorney had prosecuted was struck for cause. The mere fact that the government used three of its remaining fourteen strikes, 21%, to remove black jurors does not raise an inference of discrimination on the facts before us. The pool of 56 jurors was composed of almost 15% black members. Had the government used one less strike against a black member of the pool, it would have only exercised just over 14% of its strikes against black members. Where a change in one strike would result in an outcome that on its face is not prejudicial then it is rare that a defendant can prevail on a claim of discrimination absent additional evidence of discriminatory motive. Further, it is noteworthy that the government had two unused peremptory strikes, yet failed to use them to strike any of the three remaining black jurors from the panel. Had the government been intent on obtaining a non-black jury it is unlikely that these strikes would have gone unused. This is particularly true where the final composition of the pool of 15 had a higher percentage of black members than did the initial pool of 56: 20% black for the pool of 15 versus 15% black for the initial pool of 56. See United States v. Sangineto-Miranda, 859 F.2d 1501, 1520-22 (6th Cir.1988) (setting forth similar factors as useful in determining whether discrimination occurred). Because we find that defendant White has not made a prima facie showing of discrimination, we need not decide whether the prosecution’s proffered explanation for striking the remaining black jurors was credible and nondiscriminatory.
VII. Admission Of Drug Paraphernalia Found in the Johnsons’ Home
Rosemary Johnson continues to assert on appeal, as she did in the court below, that evidence seized from the basement of the Johnsons’ home at the time of their arrest on October 18, 1988 should not have been admitted at trial. The evidence, seized after a written consent search, consisted of a large triple-beam balance scale with cocaine residue on the pan, a sifter with cocaine residue, a bottle of mannitol (a cocaine dilutant), a bottle of procaine (a cocaine adulterant), and a large number of small plastic bags suitable for packaging cocaine for resale in small lots.
Rosemary Johnson argues that this evidence should not have been admitted because it does not tend to establish her involvement in the conspiracy as of the last date the conspiracy is alleged to have continued, September 30, 1988 (the date the indictment was returned). She asserts that the evidence tends to establish that she was transacting cocaine at the time of her arrest but does not support the proposition [1299]*1299that she was engaged in transactions with the other conspirators at that time. As such, the evidence is highly prejudicial, she claims, and admitting it violates Federal Rule of Evidence 404(b) (evidence of other crimes) and/or Rule 403 (more prejudicial than probative).
The government argued at trial and on appeal a different purpose for the evidence. It asserts that since the tape recorded conversations linked the Johnsons to the conspiracy and since the evidence does tend to establish that the Johnsons were continuing to distribute cocaine at the time of their arrest, the jury could infer they continued to be a part of the conspiracy at the time of their arrest.
Since the evidence was admitted to prove continued involvement in the conspiracy, rather than other similar crimes, we believe that the court correctly determined that Rule 404(b) was inapplicable. Thus Rosemary Johnson’s citation to United States v. Zelinka, 862 F.2d 92, 99 (6th Cir.1988), is inapposite. In Zelinka, the government conceded that the conspiracy had ended almost a year and a half prior to the defendant’s arrest. Id. at 98. Thus the evidence seized during the arrest could not have been evidence of the conspiracy in which the defendant was charged with participating. Since it was admitted that the evidence did not tend to establish an element of the conspiracy, the evidence had to be admissible, if at all, as evidence of another crime under Rule 404(b).
In this case, by contrast, the conspiracy continued at least until the return of the indictment, three weeks before the defendant’s arrest. Although the defense disputed Johnson’s continued involvement in the conspiracy in 1988, the fact of whether the defendant had actually withdrawn was a jury question. United States v. Hamilton, 689 F.2d 1262, 1268-69 (6th Cir.1982), cert. denied, 459 U.S. 1117, 103 S.Ct. 753, 74 L.Ed.2d 971 (1983).
The admissibility of the evidence under Rule 403, which allows evidence to be excluded if its' probative value is substantially outweighed by the danger of unfair prejudice, is a separate question. Although a strong argument can be made for excluding the evidence given the fact that there was no other evidence of continued involvement in the conspiracy after 1986, on balance, we cannot say that the trial court erred in admitting it.
VIII. Ex Post Facto Challenge to the Applicability of the Guidelines
All of the defendants except Walton argue that since the conspiracy began before the effective date of the federal sentencing guidelines, November 1, 1987, then sentencing under the guidelines violates the ex post facto clause of the Constitution. In the alternative, Eddie Johnson and Mitchell argue that the government must affirmatively establish that they performed some act in furtherance of the conspiracy after the applicable guideline date in order to sentence them under the guidelines. We reject both of these arguments.
We agree with the overwhelming weight of authority holding that a defendant who commits a continuing offense beginning before the effective date of the guidelines and ending after the effective date of the guidelines can be sentenced under the guidelines without violating the ex post facto clause of the Constitution. See, e.g., United States v. Lee, 886 F.2d 998, 1003 (8th Cir.1989), cert. denied, — U.S. -, 110 S.Ct. 748, 107 L.Ed.2d 765 (1990); United States v. Boyd, 885 F.2d 246, 248 (5th Cir.1989); United States v. White, 869 F.2d 822, 826 (5th Cir.), cert. denied, — U.S.-, 109 S.Ct. 3172, 104 L.Ed.2d 1033 (1989). It is undisputed that the conspiracy at issue in this case continued long after the November 1, 1987 effective date for the guidelines.
As to Johnson and Mitchell, we do not believe that the government must prove that they committed an act in furtherance of the conspiracy or knew of acts committed by other co-conspirators after the effective date in order to be sentenced under the guidelines. Conspirators are generally held liable for the known or foreseeable acts of their co-conspirators committed in furtherance of the' conspiracy. [1300]*1300Pinkerton v. United States, 328 U.S. 640, 647, 66 S.Ct. 1180, 1184, 90 L.Ed. 1489 (1946). Even if these defendants did not participate in the conspiracy after the effective date of the guidelines, the size of the drug distribution conspiracy in this case made it foreseeable that the conspiracy would continue after the effective date. In order to escape liability for the acts committed after the effective date of the guidelines, the defendants must prove that they affirmatively withdrew from the conspiracy before the effective date. See Hamilton, 689 F.2d at 1268-69. The defendants presented no evidence tending to show that they withdrew. Having not met this burden, it was not error for the District Court to sentence the defendants under the guidelines. Lee, 886 F.2d at 1003.
IX.Due Process Challenges to the Guidelines
Larry White claims that his sentence under the guidelines violates due process because it requires courts to make findings based on a preponderance of the evidence rather than beyond a reasonable doubt. We have already rejected due process challenges to the guidelines by defendants who have alleged that a judge must have greater sentencing discretion. United States v. Allen, 873 F.2d 963 (6th Cir.1989). The defendant’s claim here fares no better. The Supreme Court has specifically “rejected the claim that whenever a State links the ‘severity of punishment’ to ‘the presence or absence of an identified fact’ the State must prove that fact beyond a reasonable doubt,” McMillan v. Pennsylvania, 477 U.S. 79, 84, 106 S.Ct. 2411, 2415, 91 L.Ed.2d 67 (1986) (quoting Patterson v. New York, 432 U.S. 197, 214, 97 S.Ct. 2319, 2329, 53 L.Ed.2d 281 (1977)). It is sufficient that the fact be proved only by a preponderance of the evidence. Id. 477 U.S. at 91-92, 106 S.Ct. at 2418-19; see also United States v. Moreno, 899 F.2d 465 (6th Cir.1990). We see no reason to treat a challenge to the federal sentencing guidelines any differently from a challenge to a state procedure. Accord, United States v. Wilson, 900 F.2d 1350 (9th Cir.1990); United States v. Fredericks, 897 F.2d 490 (10th Cir.1990); United States v. McDowell, 888 F.2d 285, 290-91 (3d Cir.1989); United States v. Guerra, 888 F.2d 247, 250-51 (2d Cir.1989), cert. denied, — U.S. -, 110 S.Ct. 1833, 108 L.Ed.2d 961 (1990); United States v. Urrego-Linares, 879 F.2d 1234, 1237-38 (4th Cir.), cert. denied, — U.S. -, 110 S.Ct. 346, 107 L.Ed.2d 334 (1989); United States v. Wright, 873 F.2d 437, 441 (1st Cir.1989). Contra, United States v. Davis, 715 F.Supp. 1473 (C.D.Cal.1989).
X.Eighth Amendment Challenge to the Sentence
Defendant White also argues that his 40 year sentence is not proportionate to the crime he committed and therefore violates the eighth amendment. The defendant, who directed a major cocaine distribution ring, received the maximum statutory sentence. We have upheld against eighth amendment challenge sentences in drug cases far in excess of that which White received for defendants who engaged in far less serious conduct. See, e.g., Young v. Miller, 883 F.2d 1276, 1282-86 (6th Cir.1989) (first time offender convicted of one time possession with intent to deliver 1,300 grams of heroin sentenced to life in prison without parole). We do not find White’s sentence to be unusual in today’s climate of drug intolerance.
XI.Determination of the Quantity of Cocaine Involved for Purposes of Sentencing under the Guidelines
Defendants Walton, Eddie Johnson, and Rosemary Johnson argue that the District Court incorrectly determined the amount of cocaine they distributed.3 They face a difficult burden. A district court’s decision on the amount of cocaine a defendant is to be held accountable for is a [1301]*1301finding of fact which must be accepted by a court of appeals unless clearly erroneous. See, e.g., United States v. Barrett, 890 F.2d 855, 867 (6th Cir.1989).
It is more difficult to determine the actual amount of cocaine each individual defendant in the conspiracy knew or had reason to know would be handled during the conspiracy, where, as here, there was no cocaine actually seized. The Application Notes to U.S.S.G. § 2D1.4, the relevant section for sentencing these defendants, recognize this problem:
If the defendant is convicted of conspiracy, the sentence should be imposed only on the basis of the defendant’s conduct or the conduct of co-conspirators in furtherance of the conspiracy that was known to the defendant or reasonably foreseeable.
* * * * * *
Where there is no drug seizure or the amount seized does not reflect the scale of the offense, the sentencing judge shall approximate the quantity of the controlled substance. In making this determination, the judge may consider, for example, ... similar transactions in controlled substances by the defendant.
Former Application Notes 1 and 2 to U.S. S.G. § 2D1.4, reprinted in United States Sentencing Commission Guidelines Manual (October 15, 1988) (Application note as of the date of sentencing).
Defendant Walton was found to be accountable for at least 50 kilograms of cocaine. There was testimony by one of the conspirators, Rolando Hernandez, that he distributed 70-80 kilograms to co-conspirators Marzellus Wilson and Larry White. Further, the evidence showed that Larry White, the leader of the conspiracy in Flint, had several other sources of cocaine in addition to Hernandez. Walton played an important role in White’s operation, acting as a courier, seller, and trusted lieutenant in the operation. She had frequent conversations with White about deliveries and made numerous cocaine pickups and cash deliveries. We do not believe the District Court was clearly erroneous in concluding she either knew of or could have reasonably foreseen that the conspiracy would involve at least 50 kilograms of cocaine.
The sentencing of Eddie and Rosemary Johnson is more troublesome. There was a great deal less evidence about the extent of their involvement in the conspiracy. The Johnsons argued below and argue here that they cannot be sentenced for dealing any more cocaine than the amounts mentioned in the 1986 tape recordings, an amount less than 25 grams. The District Court concluded that they had dealt in at least 455 grams of cocaine. This figure was calculated based on the following evidence and assumptions: (1) there was evidence that they were dealing at least an “eight ball” (one-eighth ounce or 3.5 grams) of cocaine per week in April 1986; (2) the large quantity of cocaine paraphernalia found in their basement in October 1988 indicates they continued to cut in grade, repackage, and sell cocaine at the time of their arrest in October 1988; (3) assuming that they continued to deal cocaine every week throughout the two and one-half year period between April 1986 and October 1988 as a part of the conspiracy, then they could be found to have dealt in at least 455 grams of cocaine (3.5 grams/week times 130 weeks).
Findings of fact important to calculating a defendant’s offense level or criminal history category must generally be made by a preponderance of the evidence. See, e.g., Wilson, 900 F.2d 1350 (9th Cir.1990); Fredericks, 897 F.2d 490 (10th Cir.1990); McDowell, 888 F.2d at 290-91; Guerra, 888 F.2d at 250-51; Urrego-Linares, 879 F.2d at 1237-38; Wright, 873 F.2d at 441-42. This standard is difficult to apply when a number of plausible theories concerning the quantity of drugs are presented and the evidence is insufficient to say that any one choice is more likely than not true. Relying on relevant commentary to the guidelines which directs the sentencing judge to “approximate the quantity of the controlled substance,” the government asserts that a preponderance of the evidence standard should be relaxed when a trial judge must choose between a number of plausible esti[1302]*1302mates concerning the quantity of drugs involved in a conspiracy case. Former Application Note 2 to U.S.S.G. § 2D1.4. In the context of such approximations, the government believes that “preponderance of the evidence” should be defined as “at least as much evidence as that in support of any alternative finding.” The Johnsons assert that the amount of cocaine should be limited to the amounts discussed in the recorded phone conversations, arguing that any higher figure would necessarily be based on too many inferences. We disagree with the reasoning of both parties. Adopting the standard urged by the government could result in a defendant being held responsible for a quantity of drugs that is more likely than not in excess of the quantity for which the defendant is actually responsible. Applied literally, it could permit a finding based on no evidence. Adopting the result urged by the Johnsons would prohibit the District Court from drawing reasonable inferences that are supported by the facts.
We.believe that the guidelines do not permit the District Court to hold a defendant responsible for a specific quantity of drugs unless the court can conclude the defendant is more likely than not actually responsible for a quantity greater than or equal to the quantity for which the defendant is being held responsible. If the exact amount cannot be determined, an estimate will suffice, but here also a preponderance of the evidence must support the estimate. Thus when choosing between a number of plausible estimates of drug quantity, none of which is more likely than not the correct quantity, a court must err on the side of caution. A review of the transcript of the Johnsons’ sentencing hearing suggests that the District Court itself was struggling with the computation of amounts. It halved the amount claimed by the government. Adopting this standard allows a District Court to perform its traditional fact-finding mission, yet still protects defendants from being held responsible for drug quantities in excess of the amounts for which they more likely than not are responsible. Allowing a court to find a defendant responsible for the maximum quantity of drugs that can plausibly be found could result in defendants receiving excessive sentences based on a finding of quantity that is more likely than not excessive. Such a result would violate a defendant’s due process rights. See Townsend v. Burke, 334 U.S. 736, 741, 68 S.Ct. 1252, 1255, 92 L.Ed. 1690 (1948) (sentences may not be based on “materially false” information); McDowell, 888 F.2d 290-91; Wilson, 900 F.2d at 1353. While this may result in an underestimation of the quantity of drugs involved in some few cases, we believe it is nonetheless constitutionally required to prevent excessive sentences.
We do not decide whether every fact that enters into a sentencing decision must be proven by a preponderance of the evidence. Not only is that question not before us, but it is quite possible that all facts do not have to be proven by a preponderance of the evidence. See McMillan, 477 U.S. at 91, 106 S.Ct. at 2419 (“Sentencing courts have traditionally heard evidence and found facts without any prescribed burden of proof at all”). We only hold that where a fact is crucial to the determination of a defendant’s guidelines base offense level or criminal history score then it must be proven to such a degree.
The problem with approximating the quantity of cocaine for which the John-sons are responsible, the basis on which the District Court sentenced the Johnsons, is compounded in that first one must determine a weekly quantity and then select a time period over which it is more likely than not that they were dealing in that quantity. Here, the weekly amount selected .is the lesser of the two problems. The quantity selected, 3.5 grams a week, is modest, and there is reliable evidence through the wiretaps that such quantities were being distributed in April and August 1986. These dates, we believe, are close enough in time to infer the Johnsons were dealing on a regular basis in the interim. However, the only direct evidence that they continued to deal during the next two years and two months is the drug paraphernalia with cocaine traces, which was found in [1303]*1303their home when they were arrested in October 1988. There is no circumstantial evidence to support the continuous distribution. Indeed it points the other way. The Johnsons’ only asset was a 1975 automobile. Their rent is $250.00 a month. Rosemary Johnson was receiving Aid to Dependent Children, thus providing a source of subsistence aside from drugs. Eddie Johnson acknowledged he had a cocaine habit. However, if the Johnsons were regularly dealing in drugs during the two years after August 1986 and their arrest in October 1988, one would expect to find some evidence of additional income. The evidence is simply insufficient to support a finding of continuous drug dealing during this period. We express no opinion as to whether on resentencing the District Court may wish to consider the conduct of the co-conspirators that was known to one or both of the Johnsons, or that was reasonably foreseeable, the other standards permitted by the guidelines.
XII. Failure to Grant the Johnsons a Reduction as “Minor” or “Minimal” Participants
The Johnsons also assert that the District Court incorrectly applied the facts in this case by failing to grant them a two or four point reduction as minor or minimal participants under U.S.S.G. § 3B1.2. The evidence does establish that they were minor participants if one compares their activities to the scope of the conspiracy as a whole and the activities of the other defendants. We do not believe this is always the proper inquiry, however. In this case, the Johnsons have only been held responsible for cocaine that they were actively involved in distributing — not the additional amounts involved in the entire conspiracy. As to the distribution of this amount, they were not minor participants. If they are not held responsible in resentencing for the remaining cocaine distributed by the conspiracy, their minor or minimal role in the remaining portion cannot benefit them. Since we do not know what the District Court may do on resentencing, we do not decide whether they were minor participants if the sentence is based on the amounts they knew were distributed by others or reasonably foreseen.
XIII. Conclusion
Accordingly, we: (1) AFFIRM the conviction and sentences of defendants Walton, Mitchell, and White, (2) AFFIRM the convictions of defendants Eddie and Rosemary Johnson, but (3) REVERSE the sentences of defendants Eddie and Rosemary Johnson and REMAND their cases for resen-tencing.