
    TRUSTEES OF the CHICAGO PAINTERS AND DECORATORS PENSION, HEALTH AND WELFARE, AND DEFERRED SAVINGS PLAN TRUST FUNDS, Plaintiffs-Appellees, v. ROYAL INTERNATIONAL DRYWALL AND DECORATING, INC., an Illinois corporation, Defendant-Appellant.
    No. 06-2367.
    United States Court of Appeals, Seventh Circuit.
    Argued Jan. 18, 2007.
    Decided July 3, 2007.
    
      Steven F. McDowell (argued), Donald D. Schwartz, Arnold & Kadjan, Chicago, IL, for Plaintiffs-Appellees.
    Joseph P. Berglund (argued), Berglund & Niew, Oak Brook, IL, for Defendant-Appellant.
    Before BAUER, MANION, and ROVNER, Circuit Judges.
   MANION, Circuit Judge.

The Trustees of the Chicago Painters and Decorators Pension, Health and Welfare and Deferred Savings Plan Trust Funds (“Trustees”) sued an employer of its union members, Royal International Drywall & Decorating, Incorporated (“Royal”). The Trustees claimed that Royal violated collective bargaining agreements by failing to compensate the Funds for each hour the covered employees worked. Following a bench trial, the district court found that Royal owed contributions to the Funds and, after further proceedings, entered an award for damages. Royal appeals, contesting the admission of expert testimony at trial, the district court’s factual findings, and the amount of liquidated damages awarded. We affirm.

I.

The district court fully recounted the background of this case in its findings issued after the bench trial, so we recite the facts only as necessary. See Trustees of the Chicago Painters v. Darwan, No. 01-C-2458, 2004 WL 1459553 (N.D.Ill. June 29, 2004). Royal employed individuals, known as “tapers,” to install drywall in residential tract housing. Tract housing is a manner of constructing subdivisions with a limited number of model designs and options. Historically, tapers received compensation by the number of pieces of drywall they installed, not by the hours they worked. However, a collective bargaining agreement between Royal and its employees required Royal to compensate the employees and contribute to the Funds based on each hour worked, not by the piece. The Trustees filed suit against Royal, alleging that Royal violated the collective bargaining agreement by compensating its employees by the piece, and not by the hour, and claiming that the contributions made to the Funds were insufficient for the actual number of hours worked.

At trial, the Trustees claimed that Royal’s time sheets inaccurately recorded the hours worked, and were instead rigged to reflect the hours budgeted for the project. To prove the actual number of hours worked, the Trustees offered evidence of a “materials audit” that the Trustees conducted on Royal. The audit used the amount of raw materials Royal purchased for installation to calculate the amount of drywall installed. Then, the number of hours worked was calculated based on the average rate of drywall installation for a taper. The parties vigorously contested the average rate at which a taper works. Royal also contended that its time sheets were accurate and sufficient to calculate the contributions.

After hearing the evidence, the district court determined that Royal’s time sheets for its employees did “not accurately reflect hours actually worked.” Accordingly, the court had to establish an alternative means of ascertaining the hours worked in order to determine the proper amount of contributions due to the Funds. The district court, after considering testimony from experts, employees, and various published sources, followed the materials audit and settled on a rate of 2.86 boards of drywall per hour. The determination of this rate is the crux of this appeal. Based on this rate, and following further briefing from the parties, the district court awarded the Trustees $30,508.13 in unpaid contributions and $68,266.16 in liquidated damages for the period of October 1, 1999, through March 31, 2000. The district court also awarded the Trustees $17,955.82 in liquidated damages for late contributions and $712.08 in owed disbursements for the period of December 1, 1998, through March 31, 2000. Royal appeals.

II.

In an appeal from a bench trial, “[w]e review a district court’s conclusions of law de novo, and we review its findings of fact, as well as applications of law to those findings of fact, for clear error.” Reach v. U.S. Trust Co., 419 F.3d 626, 634 (7th Cir.2005) (citation omitted).

On appeal, Royal first objects to the district court’s finding that the time sheets Royal maintained for its employees were inaccurate. . An employer must “maintain records with respect to each of his employees sufficient to determine the benefits due or which may become due to such employees.” 29 U.S.C. § 1059(a)(1). The collective bargaining agreement in this case required Royal to contribute benefits based on the hours worked. Therefore Royal had an obligation to maintain records of the hours worked that would be “sufficient to determine the benefits due.” Id. Royal did maintain time sheets; simply maintaining time sheets, however, is not enough to comply with the statute unless the time sheets are sufficient to determine the actual hours worked, and thus the benefits due. Previous cases in this circuit have characterized sufficient records as “reliable, contemporaneous records,” Chicago Dist. Council of Carpenters Pension Fund v. Reinke Insulation Co., 347 F.3d 262, 264 (7th Cir.2003), that demonstrate “accuracy” in recording the work performed, Laborers’ Pension Fund v. RES Envtl. Servs., 377 F.3d 735, 739 (7th Cir. 2004) (citation omitted).

Royal claims that it properly main-' tained reliable, contémporaneous time sheets reflecting the hours worked. The district court, however, made extensive findings about the insufficient quality of Royal’s time sheets and records. Specifically, the district court noted that “only five of the 263- time sheets for the period [at issue] record daily hours worked; the remainder list only weekly totals.” While weekly, instead of daily, totals of hours are not necessarily insufficient under the agreement, other evidence raised suspicion that the weekly totals were inaccurate. For example, the district court noted that the hours budgeted and the hours worked on the time sheets matched on a substantial majority of the time sheets. The district court also remarked on various notations made on the time sheets, such as “a hand written calculation dividing the number of hours budgeted (156.25) by the number of crew members' (4); the sheet assigns 39.25 hours to [the crew chief], while each remaining crew member is reported to have worked only 39 hours.” Furthermore, employees testified that they regularly worked more hours than were reflected on the time sheets, and the district court found their testimony credible. Although Royal presents alternative explanations for the quality of the time sheets, the district court’s conclusion reflects a reasonable examination of the evidence and testimony. We are not “left with the definite and firm conviction that a mistake has been committed.” Gaffney v. Riverboat Servs. of Indiana, 451 F.3d 424, 447-48 (7th Cir.2006) (citation and internal quotation omitted). Accordingly, based on the exhibits and testimony, the district court did not clearly err in concluding that Royal’s records were not credible and did not satisfy the requirement to maintain sufficient records to determine the hours actually worked by the employees. United States v. Biggs, 491 F.3d 616 (7th Cir. 2007) (“Although we review for clear error, determinations of witness credibility can virtually never be clear error.” (internal quotation, citations, and footnote omitted)).

Because Royal’s records were insufficient, the district court proceeded to seek another method for determining the number of hours the employees worked in order to calculate the proper benefits. It did so by finding an hourly rate at which drywall tapers installed drywall boards, a rate that the parties vigorously contested before the district court. On appeal, Royal objects to the district court’s admission of the testimony of two experts, John Hull and Ian Parr, who offered their opinions on the rate of drywall taping. We review de novo the district court’s application of the principles articulated in Daubert v. Merrell Dow Pharmaceuticals, Inc., 509 U.S. 579, 113 S.Ct. 2786, 125 L.Ed.2d 469 (1993), which guides the admission of expert testimony under Federal Rule of Evidence 702. That is, we review de novo “whether (1) the proposed witness would testify to valid scientific, technical, or other specialized knowledge and (2) his testimony will assist the trier of fact.” Ammons v. Aramark Unif. Sens., 368 F.3d 809, 816 (7th Cir.2004) (citations and internal quotation omitted). If we determine that the district court applied the proper framework, the district court’s decision to admit an expert’s testimony is then reviewed for abuse of discretion. NutraSweet Co. v. XL Eng’g Co., 227 F.3d 776, 788 (7th Cir. 2000) (citations omitted).

Following this standard, we first examine whether the district court applied the proper framework. This step “evaluates the reliability of the testimony” and should “reject any subjective belief or speculation.” Ammons, 368 F.3d at 816 (citations and internal quotation omitted). The court “must ensure that the expert testimony at issue ‘both rests on a reliable foundation and is relevant to the task at hand.’ ” United States v. Cruz-Velasco, 224 F.3d 654, 660 (7th Cir.2000) (quoting Daubert, 509 U.S. at 597, 113 S.Ct. 2786). During the bench trial, both proposed experts testified about their experience, indicating “specialized knowledge” related to the rate of installation of drywall. Fed. R.Evid. 702. Hull testified to his extensive experience in taping drywall and in training the apprentice tapers. Parr testified to his extensive experience in cost estimating in the construction industry. Royal argues that the witnesses’ methodology was insufficiently scientific or reliable. This court has previously noted, however, that “[although the Daubert Court identified a number of factors to be considered when evaluating the admissibility of expert testimony — including testing, peer review, error rates, and acceptability within the relevant professional community — these factors do not establish a definitive checklist.” Cruz-Velasco, 224 F.3d at 660 (citations omitted). The Daubert Court also recognized that expert testimony was not limited to scientific testimony by acknowledging that “Rule 702 also applies to ‘technical, or other specialized knowledge[,]’” even though the Daubert opinion was “limited to the scientific context because that is the nature of the expertise offered” in that case. Daubert, 509 U.S. at 589-90 n. 8, 113 S.Ct. 2786. This court has “recognized that while extensive academic and practical expertise in an area is certainly sufficient to qualify a potential witness as an expert, Rule 702 specifically contemplates the admission of testimony by experts whose knowledge is based on experience.” Unit ed States v. Parra, 402 F.3d 752, 758 (7th Cir.2005) (internal quotation and citation omitted). Accordingly, we “consider a proposed expert’s full range of practical experience as well as academic or technical training when determining whether that expert is qualified to render an opinion in a given area.” Smith v. Ford Motor Co., 215 F.3d 713, 718 (7th Cir.2000). Given the qualifications of Hull and Parr, their testimony appears to be sufficiently reliable. Because it is based on specialized knowledge as opposed to subjective beliefs or speculations, the witnesses meet the first prong of the two-part test under Rule 702. See Ammons, 368 F.3d at 816.

Next we examine whether the experts’ testimony would assist the trier of fact. The district court referred to Daubert in admitting the experts and noted that, in the context of a bench trial, the judge would be able to consider any shortcomings of their expertise drawn out through cross-examination. See In re Salem, 465 F.3d 767, 777 (7th Cir.2006) (noting that if “the gatekeeper and the factfinder are one and the same — that is, the judge — the need to make such decisions prior to hearing the testimony is lessened” (citation omitted)). The district court judge, the finder of facts in this case, remained open to any criticism of their testimony on cross-examination, but it can be inferred from her statements that she considered their testimony helpful in determining the facts. We conclude that their testimony was relevant and helpful to the issues at trial. Accordingly, under Daubert, Hull and Parr were qualified to offer expert testimony regarding the rate of drywall taping.

Since the district court applied the proper framework, we next consider whether the district court abused its discretion in admitting the testimony of the experts. The district court aptly observed in overruling Royal’s objection to Hull’s testimony as an expert that he

worked for many years in the industry, directly, himself, taping drywall, and now training for more than 10 years, all the apprentices in the area, including those employed by the defendant himself. If this gentleman is not an expert on the issue of how much can be reasonably expected [of] a drywall taper to do in a day, I don’t know [if] there is such a person.

Regarding Parr, the district court observed that “[t]his is a gentleman who does cost estimating on a regular basis, he uses a source that’s widely recognized and agreed to in the construction industry .... [and] has done [cost estimating] for 30 years.” The district court exercised discretion in admitting the experts, and we find no abuse in admitting the testimony of these two individuals as experts at trial. Royal’s objections are therefore without merit.

Royal next contends that the district court clearly erred in determining the productivity rate to be 2.86 boards per hour. Royal pointed out a number of inconsistencies and inadequacies in the Trustees’ witnesses’ testimony and in the documentary evidence. Contrary to Royal’s assertion, the determination of the productivity rate is a factual question separate from a legal interpretation of a collective bargaining agreement, and is therefore reviewed for clear error. Reach, 419 F.3d at 634 (citation omitted). Royal submits a dozen alleged inconsistencies and errors in the district court’s assessment of witnesses’ testimony and documentary evidence. For example, Royal argues that the court should have considered higher productivity rates contained in several exhibits, and it should have ignored the materials audit because it was flawed. Royal also alleges that the district court mischar-acterized the testimony of Royal’s owner Salma Darwan and made factual errors, such as finding that Royal hired more apprentices than other contractors. Our review of these objections and alleged flaws discloses no clear error in the district court’s ultimate findings. Individually or cumulatively, these objections do not overcome the rationale of the district court’s opinion; again, we are not “left with the definite and firm conviction that a mistake has been committed.” Gaffney, 451 F.3d at 447-48 (citation and internal quotation omitted). To the extent that the parties contest the burden of proof, we note that such burden shifting is inapplicable once the question reaches the fact finder. See Reinke, 347 F.3d at 265 (“This case was tried .... [t]he only question that matters thus is whether the district judge’s decision is clearly erroneous.”). Royal’s objections to the 2.86 boards per hour productivity rate are therefore without merit.

Finally, in a cursory, two-paragraph argument in its opening brief, without citations to the record or authority, Royal asserts that the district court’s award of liquidated damages for late payments in the amount of $17,955.82 was without support in the record. We could consider such “unsupported and undeveloped arguments” to be waived. United States v. Turcotte, 405 F.3d 515, 536 (7th Cir.2005) (citations omitted). We will, however, address Royal’s argument since proving a negative proposition is inherently difficult and since we can discern the crux of the argument. United States v. Newman, 144 F.3d 531, 543 n. 13 (7th Cir.1998) (noting “the difficulty in proving a negative proposition”). Royal’s argument fails regardless because evidence in the record supports the district court’s award of damages.

We review the district court’s award of damages for clear error. Wheel Masters, Inc. v. Jiffy Metal Prods. Co., 955 F.2d 1126, 1131 (7th Cir.1992) (“Our standard of review of [the] damage award is the same as our standard of review of any factual finding — the damage award cannot be overturned unless the factual basis for the award is clearly erroneous.”). The collective bargaining agreements provided for liquidated damages of ten percent of the amount owed for tardy contributions that remained unpaid for twenty days after the month in which the hours were worked. The district court’s damage award relied on a document admitted at trial as Plaintiffs Exhibit 15, in which an auditor calculated the liquidated damages for late payments at $23,875.50. At trial, Royal’s owner was asked if he had any basis for disputing the amount of liquidated damages revealed by that audit, and he responded that he had none. The district court based the liquidated damages on this amount, reducing the amount by $5,044.18 (the sum of two months’ worth of penalties) because a previous settlement and release between the parties encompassed those two months, arriving at the $17,955.82. Royal points to no authority in its opening brief indicating that such un-contradicted evidence is an insufficient basis for the award. Because this sum was reasonably based on evidence that was admitted at trial and contained in supplemental briefing, the district court did not commit clear error in assessing $17,955.82 in liquidated damages.

Royal similarly objects to the Trustees’ reliance on the final audit report, arguing that the Trustees were required to show each tardy payment, not just the total amounts. As discussed above, however, the Trustees did produce evidence of the total amount of liquidated damages through the final audit and the requested supplemental briefing, and further elicited from Royal’s owner that he had no basis to dispute the totals. While Royal remained free to produce evidence to contradict the Trustees’ evidence, sufficient evidence was present in the record at trial and in the supplemental briefing for the district court to discern an amount of liquidated damages without committing clear error. Royal’s argument is therefore without merit.

Royal also asserts that it cannot be held liable for liquidated damages because the Trustees’ complaint did not properly plead a claim for such damages. This is a distinct argument against the award of liquidated damages and Royal did not present this argument in its opening brief. This argument is therefore forfeited. NLRB v. IBEW, Local Union 16, 425 F.3d 1035, 1041 (7th Cir.2005) (“An argument raised for the first time in a reply brief is forfeited.”) (citations omitted).

III.

The district court did not commit clear error in finding that Royal’s time sheets were insufficient or in determining the productivity rate from which the contributions to the Funds could be calculated. Additionally, the district court did not abuse its discretion in admitting expert testimony to assist the trier of fact in determining the rate of drywall installation to be used. Furthermore, the district court did not clearly err in awarding the Trustees liquidated damages. Accordingly, we AffiRM the judgment of the district court. 
      
      . Royal argues that the district court’s determination that the time sheets were sufficient should be reviewed de novo. The sufficiency of time sheets, however, is a factual question requiring the district court to determine whether the time sheets are, in fact, accurate. Further determining whether those time sheets are sufficient to ascertain benefits requires an application of the law to the facts. Both findings are reviewed for clear error. Reach, 419 F.3d at 634 (citation omitted).
     
      
      . Royal also argues that a third individual, Steven Klomfar, should not have been admitted as an expert. Klomfar, however, was not offered as an expert and testified as a factual witness based on his personal knowledge. Royal also mentions a fourth individual in this section of its brief, Howard Levinson, an accountant who conducted the materials audit. Levinson similarly was not offered as an expert at trial, and he did not propose a productivity rate of his own. Instead, Levin-son used a rate provided by others and testified as to his own knowledge of the audit process. We therefore need not conduct a Daubert analysis of the testimony of Klomfar or Levinson.
     
      
      . Royal complains that the district court's liquidated damages award was based on a document and affidavit tendered to the court in response to the district court’s request for additional briefing on damages, but which was not admitted at trial. Royal contends that only documents admitted at trial could be considered in the supplemental briefing. Regardless, the critical document listing the amount of liquidated damages was admitted at trial as Plaintiff's Exhibit 15. Supplementation of the trial record before the district court, which Royal argues the Trustees should have done, was arguably unnecessary. Furthermore, if the district court desired to consider such additional evidence (as it apparently did by citing to it in its decision), such "decisions regarding the admission and exclusion of evidence are peculiarly within the competence of the district court.” Adams v. City of Chicago, 469 F.3d 609, 612 (7th Cir. 2006) (citation and internal quotation omitted). Royal had the opportunity to respond to the Trustees’ briefing and exhibits with its own proposed calculations, and the district court did not abuse its discretion by admitting or relying on the additional materials. Id.
      
     