Disney Enterprises, Inc. et al v. Hotfile Corp. et al

Filing 448

NOTICE by Hotfile Corp., Anton Titov Defendants' Notice of Filing the Declaration of Dr. Daniel S. Levy In Opposition to Plaintiffs' Motion to Strike Portion of Testimony of Professor James Boyle (Attachments: # 1 Exhibit A)(Munn, Janet)

EXHIBIT A

PUBLIC VERSION
UNITED STATES DISTRICT COURT
SOUTHERN DISTRICT OF FLORIDA
CASE NO.: 11-CIV-20427-WILLIAMS/TURNOFF

DISNEY ENTERPRISES, INC.,
TWENTIETH CENTURY FOX FILM CORPORATION,a
UNIVERSAL CITY STUDIOS PRODUCTIONS LLLP,
COLUMBIA PICTURES INDUSTRIES, INC., and
WARNER BROS. ENTERTAINMENT INC.,
Plaintiffs,
v.
HOTFILE CORP., ANTON TITOV, and
DOES 1-10.
Defendants.
/
HOTFILE CORP.,
Counterclaimant,
v.
WARNER BROS. ENTERTAINMENT INC.,
Counter-Defendant.

/

DECLARATION OF DR. DANIEL S. LEVY IN
OPPOSITION TO PLAINTIFFS’ MOTION TO
STRIKE PORTION OF TESTIMONY OF
PROFESSOR JAMES BOYLE

I, Daniel S. Levy, declare as follows:
1.

I am National Managing Director for Advanced Analytical Consulting Group.

This declaration is based on personal knowledge and all statements contained in this declaration
are true and correct to the best of my knowledge. If called as a witness, I could and would testify
to the facts set forth in this declaration.
2.

I have a Ph.D. in Economics from The University of Chicago. I have designed

and implemented statistical sampling protocols for business analysis and litigations over the
course of more than 25 years. I have provided testimony involving surveys, sampling, statistics,
econometrics, economics and business, among other topics, before state and Federal courts. I
have served as an expert for the US Department of Justice, the US Securities and Exchange
Commission, the New York State Attorney General and served as an Expert Arbitrator for the
Internal Revenue Service. I have testified in a range of matters over a number of years. My
curriculum vitae is attached to my original report in this matter dated January 6, 2011 in support
of Hotfile’s opposition to Plaintiffs’ motion for summary judgment.
3.

It is my understanding that Plaintiffs have moved to exclude Prof.Boyle’s analysis

of conversion rates, presented in his Rebuttal Report, on the grounds that it is based on statistical
and sampling expertise which Plaintiffs claim Professor Boyle does not have. Plaintiff motion is
baseless because Professor Boyle’s analysis stands as a matter of straight forward calculation of
percentages and does not require the support of statistics or sampling science. So while
ProfessorBoyle’s analysis is based on precisely the same set of download records that Dr.
Waterman used in his analysis, which left Dr. Waterman’s analysis invalid and outside the
bounds of statistical science, Professor Boyle’s analysis, even though it rests on precisely the
same set of records, is valid for the purposes it is put to in Professor Boyle’s report, which is a
calculation of the characteristics of the population of 1750 downloads selected by Dr. Waterman.
4.

The reason that Professor Boyle’s use of this set of downloads is valid, when Dr.

Waterman’s analysis is incurably errant, is that unlike Dr. Waterman, Professor Boyle’s report
does not attempt to extrapolate his findings about the 1750 downloads to a broader population.
Dr. Waterman, in significant contrast, repeatedly stated in his report and at his deposition, that
his purpose in selecting the 1750 download records was to extrapolate his findings to the broader
population of downloads in January 2011. Although Dr. Waterman’s sampling process was
invalid and unscientific for the use of extrapolating to the full population of downloads even in

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CASE NO.: 11-CIV-20427-WILLIAMS/TURNOFF

January 2011, let alone to the broader population of downloads from Hotfile during the course of
its operations, it did produce a set of 1750 download records that Professor Boyle can validly
comment about. Professor Boyle’s report is simply reporting what one finds by looking within
these 1750 downloads.
5.

In contrast to Dr. Waterman’s report, Professor Boyle’s report does not make any

statements about the broader population of downloads in his analysis. Since Professor Boyle’s
analysis is about the population of the 1750 records and the proportion of conversions within that
population and does not attempt to extrapolate to some broader population, his statements stand
on the basis of straightforward calculations of percentages within the population of 1750 records
selected by Dr. Waterman. Professor Boyle is not appealing to the properties of sampling and
statistics, and does not need to, in order to make statements are about the conversion percentages
found within the 1750 download records. Dr. Waterman’s, albeit invalid, attempt to extrapolate
his findings to the broader population of downloads beyond the 1750 downloads does require the
properties of sampling and statistical science because he is attempting to extrapolate from a
sample to a population.
6.

To clarify this point, take the example of four class rooms of 35 sixth graders

each. Based on a selection of 20 taken from one sixth grade class, the average age of those 20
sixth graders can be calculated. There is no variance on that average. The average age of those
20 sixth graders is known for certainty. In fact, it is not even necessary that the 20 sixth graders
be selected randomly to calculate the average age of those 20 sixth graders as long as there is no
use of that average to estimate the average age of sixth graders in the broader population of sixth
graders such as in the four classes. Furthermore, the fact that the 20 selected students came from
just one class and that the selection process was clearly not random would not impact the
calculation of the average age of those 20 sixth graders. As long as the statements about the
average age of those sixth graders is meant to apply only to the 20 selected sixth graders, the
selection or sampling process does not impact the calculation, and the average age of the 20 sixth
graders can be calculated with complete certainty; there is no variance, and the calculation does
not rest on the process of, or knowledge of, the science of statistics or sampling.
7.

However, if one wants to extrapolate the findings based on the 20 selected sixth

graders to make some statements about the entire sixth grade then the fact that the sample was
not scientifically drawn becomes critical. In this example, since the average age of the 20

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selected sixth graders was not a valid scientific probability sample of the broader population, it
cannot be used as a valid scientific probability sample from which to draw an estimate of the
overall sixth grade. 1 However, the average age of the 20 selected students, if sampled with a
valid scientific probability sample, would estimate the average age of the overall sixth grade with
some degree of variance. After all, even if the 20 students were selected based on a valid
scientific probability sample, it would still only be a sample. The actual average age of the
whole sixth grade would likely differ from the estimate by some amount, and the calculated
variance of the estimated mean would be used by a scientist to reflect the degree of precision
with which the average age of the entire sixth grade could be estimated based on the average age
of the 20 selected students. But if statements about age are limited only to the selected records,
and no statements are made about the broader population, then there is no variance to be
calculated; it simply does not exist. In this example, the average age of the selected students is a
specific number that is calculated without variance. The variance only comes into play when the
selected records are to be used as an estimated measure of the age in the general population of
sixth graders. But there is no variance when simply reporting the average age of the 20 sixth
graders, and there is no reliance or required knowledge of the science of sampling or statistics to
make such a calculation about the average age of the 20 students selected.
8.

This is analogous to what Professor Boyle did in reporting the conversion rate in

the 1750 records for each of the sub-populations he discussed. He presented no variance, because
there is no statistical variance in the calculated average of the conversion rates when that average
is simply a description of the characteristics of the population of 1750 download record. As long
as Professor Boyle’s statements and conclusions in his report are about the 1750 download
records and do not require some extrapolation to some broader population, he does not require a
background in the science of statistics or sampling. Further, there is no variance that is required
to be presented for that purpose.
9.

It is important to point out that the analysis of Professor Boyle, since it is based

on precisely the same 1750 downloads chosen by Dr. Waterman, would have the same sampling
properties as found in Dr. Waterman’s analysis when addressing questions in which the relevant
unit of observation is a download. Likewise, Professor Boyle’s analysis would have the same
sampling properties as Dr. Waterman’s analysis if Professor Boyle attempted to extrapolate his
1

William G. Cochran, Sampling Techniques, third edition, John Wiley & Sons, New York, 1977, P. 5.

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results to the broader population of downloads. However, Professor Boyle did not attempt to
make such an extrapolation and instead chose to present his findings in his report as they apply
to the population of 1750 downloads. Not some broader population. As mentioned above, in
performing his calculations and making his statements only about the 1750 downloads, Professor
Boyle’s analysis requires only the properties of basic math, not the properties of sampling or
statistical science.
10.

As a further point, Dr. Waterman claims that if Professor Boyle had altered his

analysis to conform to what Dr. Waterman believes would be the right analysis, Professor Boyle
would have reweighted the selection of 1750 downloads, which Dr. Waterman chose, and then
extrapolated those results to a broader population, which would have required Professor Boyle to
provide variances of these estimated conversion rates. Further, Dr. Waterman says that if
Professor Boyle had done what Dr. Waterman had wanted, Professor Boyle would have found
that the conversion rates between the “Highly Likely” (Infringing in Dr. Waterman’s Exhibits)
category and the “Noninfringing” category would not be statistically significant. Dr.
Waterman’s proposed analysis answers a different question than the one Professor Boyle
apparently intended to address, so there is no reason why Professor Boyle should have performed
the analysis that Dr. Waterman proposes. Professor Boyle’s analysis details the average
historical conversion rate based on a selection of downloads which were subsequently grouped
into four categories. It is a valid calculation.
11.

Dr. Waterman’s analysis in his Reply Declaration attempts to provide estimates of

the conversion rates in some broader population based on his sample of 1750 selected
downloads. His analysis is again invalid because his sample is not a valid scientific sample of the
broader population even for January 2011. However, putting that aside, it is simply a different
question than Professor Boyle answered. In his Reply Declaration, Dr. Waterman states that
“[p]erforming that calculation produces conversion rates for the infringing and non-infringing
categories that are different from those reported by Prof. Boyle but not statistically different from
each other.” Dr. Waterman does not list these new conversion rates in his report, but fortunately
they are listed in the Exhibits to his Reply Declaration. I have copied those results into Table 1
below.

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Table 1

Illegal
Infringing
Noninfringing
Unknowable
12

Estimate
0.00012
0.00060
0.00102
0.00028

Std.err

WEIGHTED
MoE
95%CI LL

95%CI UL

0.00007
0.00029
0.00024

0.00014
0.00057
0.00047

0.00074
0.00160
0.00075

0.00046
0.00045
‐0.00020

cv
11.93%
28.51%
87.17%

I have added the labels of the categories, at the left, which were simply missing

from Dr. Waterman’s results on Page 44 of his Exhibits. But otherwise this is a replica of Dr.
Waterman’s findings presented on Page 44 of his Exhibits to his Reply Declaration. Dr.
Waterman produced these results based on his selection of records and statistical calculations.
The “Estimate,” which is Dr. Waterman’s estimate of the average conversion rate is listed for
four categories. Of interest are the estimated average conversion rates for Infringing and
Noninfringing. Dr. Waterman estimated a conversion rate for the Noninfringing in his broader
population of .00102, with a Std. err. of .00029. Professor Boyle found the average conversion
rate of .001074 (written .1074% in Professor Boyles Report) within the population of 1750
downloads. For the Infringing category, Dr. Waterman finds an estimated conversion rate for his
broader population of .00060 with a Std. err or .00007. Professor Boyle found the average
conversion rate of .000586 (written .0586% in Professor Boyles Report, combined .000551 for
the combined Highly Likely and Confirmed) within the population of 1750 downloads. Simply
to facilitate comparison of the averages calculated by Professor Boyle and Dr. Watermen, I
isolate them in Table 2 below.

Table 2

Infringing
NonInfringing

Waterman
Estimates of
Averaged
Conversion Rate
for Broader
Population
0.00060
0.00102

Boyle Calculation of
Average Conversion Rate
within Population of 1750
Downloads
0.000551
0.001074

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CASE NO.: 11-CIV-20427-WILLIAMS/TURNOFF

Sources : Rebuttal Expert Report Professor James Boyle, Jan 6, 2012, P. 23, Repl y
Decla ra tion Exhibi ts Dr. Ri cha rd Wa terman. P. 44. Boyle presented

13

Reading across Table 2, where Professor Boyle found .000551, Dr. Waterman

found .00060. Where Professor Boyle found .001074, Dr. Waterman found .00102. In think it is
fair to say that Dr. Waterman’s finding for the estimated average conversion rates validate, not
contradict, Professor Boyles calculations of the average conversion rates he found within the
population of 1750 downloads. Table 2 shows that Dr. Waterman’s point estimates of the
conversion rates in his broader population exhibit higher point estimates for the Noninfringing
category than the Infringing category as did Professor Boyle’s calculations for population of
1750 downloads. Reading the text of Dr. Waterman’s Rebuttal Declaration one might expect
that his reweighting and reanalysis had produced results that were very different from Professor
Boyle’s. However, side-by-side comparison of the results in Table 2 show that Dr. Waterman’s
results about the point estimates of conversion rates confirm Professor Boyle’s finding, even
under alternative weightings of the underlying records, which Dr. Waterman advocates and
presumably finds acceptable.
14

Dr. Waterman also says that in his own reweighted result, the difference in his

point estimates between Noninfringing and Infringing categories are not statistically significantly
different. Below I shed further light on Dr. Waterman’s calculation and claims about statistical
significance which helps to clarify precisely what Dr. Waterman has found.
15

Dr. Waterman does not list within his Rebuttal Declaration the standard errors or

the specific test of significance he used in his Rebuttal Declaration. However, he does list the
“Std.err” of his point estimates in the related Exhibits. Based on these Std.errs, the point
estimates and the numbers of observations in Dr. Waterman’s selection of downloads as he
reweights them on P.44 of his Exhibit, a standard test of statistical significance can be
performed. Because Dr. Waterman has not provided the calculations he performed to support his
opinion about his test of statistical significance, it is not possible to determine the exact
calculation that Dr. Waterman used as the basis of his opinion that the conversion rates of the
Infringing and Noninfringing categories were not statistically significant. It is certainly
consistent with Dr. Waterman’ s statements that Dr. Waterman used what is known as a t-test for
the difference between two means to perform his test of statistical significance. We do know
form footnote 1 of Dr. Waterman’s Rebuttal Declaration that his statement about the lack of

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statistical significance was “…determined using a 0.05 level of significance, which is equivalent
to 95% confidence.” The standard interpretation of this confidence interval is that if the same
sampling process were used repeatedly in a population the true average conversion rate would
tend to be within confidence bounds 95% of the times. 2 The same statement would hold for a
statement about a 99% confidence interval; in repeated samplings the true population average
would tend to be within the alternative 99% confidence intervals 99% of the time. Similarly for
a 90% confidence interval, the true population average would tend to be within the boundaries of
the 90% confidence interval in 90% of the samples. I have performed calculations of the 95%
and 90% confidence intervals based on a t-test of the difference between two means. I found that
the difference in the conversion rates between Infringing and Noninfringing would be
statistically significant at the 90% confidence level. So while Dr. Waterman says that “from a
statistical point of view the rates are indistinguishable, ”3 a fuller exposition of what Dr.
Waterman’s own data shows is that, in common parlance, the conversion rate in the
Noninfringing files is likely to be greater than the conversion rate in the Infringing files. That
difference does may not reach the 95% confidence level standard set by Dr. Waterman, but it
would more than reach a 90% confidence level standard.
16

Contrary to the tone and language of Dr. Waterman’s Rebuttal Declaration, Dr.

Waterman’s actual data and results, when featured side-by-side with Professor Boyle’s, are very
useful in confirm and validating Professor Boyle’s findings and statements, even under the
alternative methods of calculation and reweighting proposed and implemented by Dr. Waterman.
Specifically, like Professor Boyle’s findings for the population of 1750 downloads, Dr
Waterman’s finding under his chosen methods, setting aside the invalid sampling it is based on,
demonstrate that the point estimates of the conversion rate for Noninfringing are greater than
that for Infringing. And further, that while this difference may not be statistically significant at a
95% confidence level, it would be statistically significant if the standard were set at the 90%
confidence level.
17

Therefore, even though Professor Boyle’s report does not discuss any

extrapolation to a broader population outside the 1750 downloads selected by Dr. Waterman, Dr.
Waterman has performed such an extrapolation based on methods that he presumably finds

2
3

William G. Cochran, Sampling Techniques, third edition, John Wiley & Sons, New York, 1977, P. 12.
Rebuttal Declarat ion of Dr. Richard Waterman, March 19, 2012, P. 4.

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acceptable, and that analysis buttresses Professor Boyles’ conclusion that conversion rates
among Noninfringing files are at least as great as Infringing files, and statistically significantly
greater at the 90% confidence level.
18

It is also important to point out that on P. 43 of Dr. Waterman’s Exhibits to his

Rebuttal Declaration, Dr. Waterman provides calculations of the average convers ion rates and
the Std.err of those estimated conversion rates, without the reweighting of Dr. Waterman’s
selection of the 1750 download; that is, based on the same weighting that Dr. Waterman used for
his original report. Those unweighted average conversion rates and standard errors (Std.err)
show that the conversion rates for the Noninfringing downloads are statistically significantly
greater than the conversion rates for the Infringing files at the 95% confidence level chosen by
Dr. Waterman. I provide these calculations in Appendix 1. Dr. Waterman does not discuss these
results in his report, but the average conversion rates and Std.err needed to perform these
calculations are available on P. 43 of the Exhibits to Dr. Waterman’s Rebuttal Declaration.
While Dr. Waterman asserts that such a calculation on the unweighted downloads, which Dr.
Waterman himself selected, is not appropriate, the calculation of the conversion rate based on
downloads appears to be what Professor Boyle was addressing in his original report. If the
conversion rate based on each of these categories based on downloads is the point of interest for
Professor Boyle, then Dr. Waterman has provided that statistical know-how and the calculations
needed to extend Professor Boyle’s calculation, which was limited to the 1750 download
records, to the broader population of downloads, to the extent Dr. Waterman’s unscientific
sample can be employed for any extrapolation. And to the extent that any of Dr. Waterman’s
samples can be used, the results in Dr. Waterman’s Exhibits to his Rebuttal Report show that
Noninfringing files based on the sample of downloads have a statistically significantly greater
conversion rate than Infringing files based on the sample of downloads.
19

In summary, Professor Boyle’s analysis requires only the basis of standard

mathematical calculation because he makes no statements about any broader population outside
the population of 1750 download records chosen by Dr. Waterman. Professor Boyle’s analysis
therefore is unassailable on any statistical or sampling science grounds since these disciplines are
not required to validate Professor Boyle’s analysis. Dr. Waterman’s sample itself is woefully
invalid and unscientific for use in extrapolating to the broader population of January 2011 or any
broader population. However, Dr. Waterman’s Rebuttal Declaration analysis supports, rather

8

SAS Code:
libname hotfile "E:\Main\Clients\Hotfile\Team A\Report Analysis\Data";
options compress=yes;
proc import datafile="E:\Main\Clients\Hotfile\Team A\Report Analysis\Data\boyle_conv_sum.csv"
out=boyle
dbms=csv
replace;
run;
proc import datafile="E:\Main\Clients\Hotfile\Team A\Report
Analysis\Data\wm_conv_sum_weighted.csv"
out=weighted
dbms=csv
replace;
run;
proc print data=boyle;
run;
proc print data=weighted;
run;

title "Test for Equality of Means Conversion Rate Infringing-Conversion Rate NonInfringing:
Boyle";
proc ttest data=boyle sides=L;
class CATEGORY;
var CONVERSION;
run;
title "Test for Equality of Means Conversion Rate Infringing-Conversion Rate NonInfringing:
Waterman-Weighted";
proc ttest data=weighted sides=L;
class CATEGORY;
var CONVERSION;
run;
boyle_conv_sum.csv:

Category
Infringing
Infringing
Infringing
NonInfringing
NonInfringing
NonInfringing

_STAT_
N
MEAN
STD
N
MEAN
STD

Conversion
1579
0.00053
0.004768 0.00012
87
0.00116
0.001772 0.00019

wm_conv_sum_weighted.csv

Category
Infringing
Infringing
Infringing
NonInfringing
NonInfringing
NonInfringing

_STAT_
N
MEAN
STD
N
MEAN
STD

Conversion
1579
0.0006
0.00278156
87
0.00102
0.00270494

0.00007

0.00029

Note: Sample standard deviations were estimated using the following formula:
sqrt(std(mean)^2*n).
SAS Output
The SAS System
2012

13:44 Monday, April 2,

1
Obs
1
2
3
4
5
6

Category

_STAT_

Infringing
Infringing
Infringing
NonInfringing
NonInfringing
NonInfringing

N
MEAN
STD
N
MEAN
STD

Conversion

VAR4

1579
0.00053
0.0047684
87
0.00116
0.0017722

.
.
0.00012
.
.
0.00019

The SAS System
2012

13:44 Monday, April 2,

2
Obs
1
2
3
4
5
6

Category

_STAT_

Conversion

VAR4

Infringing
Infringing
Infringing
NonInfringing
NonInfringing
NonInfringing

N
MEAN
STD
N
MEAN
STD

1579
0.0006
0.00278156
87
0.00102
0.00270494

.
.
0.00007
.
.
0.00029

Test for Equality of Means Conversion Rate Infringing-Conversion Rate NonInfringing:

Boyle

3
13:44 Monday, April 2,
2012
The TTEST Procedure
Variable:
Category

N

Mean

Std Dev

Std Err

Minimum

Maximum

1579
87

Infringing
NonInfringing
Diff (1-2)
Category
Dev

Conversion

0.000530
0.00116
-0.00063

0.00477
0.00177
0.00466

0.000120
0.000190
0.000513

.
.

.
.

Method

Mean

Infringing

95% CL Mean

95% CL Std

0.000295 0.000765

0.00477

.

0.000782

0.000530

Std Dev

0.00154

0.00177

.

0.00466

0.00451

.
NonInfringing

0.00116

.
Diff (1-2)
0.00482
Diff (1-2)

Pooled

-0.00063

-Infty

0.000215

Satterthwaite

-0.00063

-Infty

-0.00026

Method

Variances

Pooled
Satterthwaite

Equal
Unequal

DF

t Value

Pr < t

1664
166.85

-1.23
-2.80

0.1099
0.0028

Equality of Variances
Method
Folded F

Num DF

Den DF

F Value

Pr > F

1578

86

7.24

<.0001

Test for Equality of Means Conversion Rate Infringing-Conversion Rate NonInfringing: WatermanWeigh 4
13:44 Monday, April 2,
2012
The TTEST Procedure
Variable:
Category

N

Mean

Std Dev

Std Err

Minimum

Maximum

1579
87

Infringing
NonInfringing
Diff (1-2)
Category
Dev

Conversion

0.000600
0.00102
-0.00042

0.00278
0.00270
0.00278

0.000070
0.000290
0.000306

.
.

.
.

Method

Mean

Infringing

95% CL Mean

95% CL Std

0.000463 0.000737

0.00278

.

0.000443

0.000600

Std Dev

0.00160

0.00270

.

0.00278

.
NonInfringing

0.00102

.
Diff (1-2)
0.00288
Diff (1-2)

Pooled

-0.00042

-Infty

0.000083

Satterthwaite

-0.00042

-Infty

0.000075

Method

Variances

Pooled
Satterthwaite

Equal
Unequal

DF

t Value

Pr < t

1664
96.295

-1.37
-1.41

0.0850
0.0812

Equality of Variances
Method
Folded F

Num DF

Den DF

F Value

Pr > F

1578

86

1.06

0.7591

0.0026

Illegal
Infringing
Noninfringing
Unknowable

Estimate
0.00014
0.00053
0.00116
0.00036

Std.err
0.00035
0.00012
0.00019
0.00020

MoE
0.00068
0.00023
0.00038
0.00039

UNWEIGHTED
95%CI LL

95%CI UL

cv
247.73%
21.72%
16.64%
54.50%

var
1.225E-07
1.44E-08
3.61E-08
0.00000004

Estimate_Noninfringing-Estimate_Infringing
Var(Estimate_Noninfringing-Estimate_Infringing)
Std Err(Estimate_Noninfringing-Estimate_Infringing)

0.00063
0.000000051
0.000225

Lower Bound of One-sided 90% Confidence Interval
Lower Bound of One-sided 95% Confidence Interval

0.0003420
0.0002604

Source: William G Cochran, Sampling Techniques, 3rd Edition, Wiley & Sons, New York, 1977, P. 180.

Illegal
Infringing
Noninfringing
Unknowable

Estimate
0.00012
0.0006
0.00102
0.00028

WEIGHTED
95%CI LL

Std.err

MoE

0.00007
0.00029
0.00024

0.00014
0.00057
0.00047

0.00046
0.00045
‐0.00020

95%CI UL
0.00074
0.0016
0.00075

cv

var

11.93%
28.51%
87.17%

4.9E-09
8.41E-08
5.76E-08

Estimate_Noninfringing-Estimate_Infringing
Var(Estimate_Noninfringing-Estimate_Infringing)
Std Err(Estimate_Noninfringing-Estimate_Infringing)

0.000420
0.000000089
0.000298

Lower Bound of One-sided 90% Confidence Interval

0.000038

Source: William G Cochran, Sampling Techniques, 3rd Edition, Wiley & Sons, New York, 1977, P. 180.