In the
 United States Court of Appeals
               For the Seventh Circuit
                          ____________

No. 06-2367
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.
                          ____________
             Appeal from the United States District Court
        for the Northern District of Illinois, Eastern Division.
             No. 01 C 2458—Rebecca R. Pallmeyer, Judge.
                          ____________
      ARGUED JANUARY 18, 2007—DECIDED JULY 3, 2007
                          ____________


 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
2                                              No. 06-2367

hour the covered employees worked. Following a bench
trial, the district court found that Royal owed contribu-
tions 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. 01-2458, 

(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 sub-
divisions 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 com-
pensating 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
No. 06-2367                                              3

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 accu-
rately reflect hours actually worked.” Accordingly, the
court had to establish an alternative means of ascertain-
ing 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
4                                                 No. 06-2367

findings of fact, as well as applications of law to those
findings of fact, for clear error.” Keach v. U.S. Trust Co., 

, 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 bene-
fits due.” 

maintain time sheets; simply
maintaining time sheets, however, is not enough to
comply with the statute unless the time sheets are suf-
ficient to determine the actual hours worked, and thus the
benefits due. Previous cases in this circuit have character-
ized sufficient records as “reliable, contemporaneous
records,” Chicago Dist. Council of Carpenters Pension Fund v.
Reinke Insulation Co., 

, 264 (7th Cir. 2003), that
demonstrate “accuracy” in recording the work performed,
Laborers’ Pension Fund v. RES Envtl. Servs., 

,
739 (7th Cir. 2004) (citation omitted).
  Royal claims that it properly maintained reliable, contem-
poraneous 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
No. 06-2367                                                     5

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 re-
ported 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 Indus., 

, 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.1 United States v. Biggs, ___ F.3d ___, ___, No. 05-
6413, 

(7th Cir. June 14, 2007) (“Although


1
  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 re-
viewed for clear error. 

(citation omitted).
6                                                 No. 06-2367

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 dis-
trict court. On appeal, Royal objects to the district
court’s admission of the testimony of two experts,2 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., 

(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. Servs., 

, 816 (7th Cir. 2004)
(citations and internal quotation omitted). If we determine


2
  Royal also argues that a third individual, Steven Klomfar,
should not have been admitted as an expert. Klomfar, how-
ever, 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 Levin-
son, 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, Levinson used
a rate provided by others and testified as to his own knowl-
edge of the audit process. We therefore need not conduct a
Daubert analysis of the testimony of Klomfar or Levinson.
No. 06-2367                                                 7

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. X-L
Eng’g Co., 

, 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.” 

(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, 

, 660 (7th
Cir. 2000) (quoting 

). 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 construc-
tion industry. Royal argues that the witnesses’ methodol-
ogy was insufficiently scientific or reliable. This court has
previously noted, however, that “[a]lthough 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.” 

(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 knowl-
edge[,]’ ” even though the Daubert opinion was “limited to
8                                                No. 06-2367

the scientific context because that is the nature of the
expertise offered” in that case. 

n.8. 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.” United
States v. Parra, 

, 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., 

, 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 

.
  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, 

, 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
No. 06-2367                                                9

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 objec-
tion 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, includ-
    ing 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 col-
lective bargaining agreement, and is therefore reviewed
10                                               No. 06-2367

for clear error. 

(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 mischaracterized 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 ratio-
nale of the district court’s opinion; again, we are not “left
with the definite and firm conviction that a mistake has
been committed.” 

(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 

(“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, 

, 536 (7th
Cir. 2005) (citations omitted). We will, however, address
No. 06-2367                                                    11

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, 

, 543 n.13
(7th Cir. 1998) (noting “the difficulty in proving a nega-
tive 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., 

, 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 Plaintiff’s Exhibit 15,3 in


3
  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 exclu-
                                                    (continued...)
12                                                 No. 06-2367

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 re-
sponded 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 uncontradicted 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



3
  (...continued)
sion of evidence are peculiarly within the competence of the
district court.” Adams v. City of Chicago, 

, 612 (7th
Cir. 2006) (citation and internal quotation omitted). Royal had
the opportunity to respond to the Trustees’ briefing and ex-
hibits with its own proposed calculations, and the district
court did not abuse its discretion by admitting or relying on
the additional materials. 

                                              13

free to produce evidence to contradict the Trustees’ evi-
dence, 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 commit-
ting clear error. Royal’s argument is therefore without
merit.
  Royal also asserts that it cannot be held liable for liqui-
dated 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, 

, 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 find-
ing that Royal’s time sheets were insufficient or in deter-
mining the productivity rate from which the contribu-
tions 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 judg-
ment of the district court.
14                                         No. 06-2367

A true Copy:
      Teste:

                     _____________________________
                     Clerk of the United States Court of
                       Appeals for the Seventh Circuit




               USCA-02-C-0072—7-3-07