Population Estimates of *Hyla cinerea* (Schneider) (Green Tree Frog) in an Urban Environment

Lanminh Pham, Seth Boudreaux, Sam Karhbet, Becky Price, Azmy S. Ackleh, Jacoby Carter, and Nabendu Pal

*Southeastern Naturalist,* Volume 6, Number 2 (2007): 203–216

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2007 SOUTHEASTERN NATURALIST 6(2):203–216
Population Estimates of Hyla cinerea (Schneider) (Green
Tree Frog) in an Urban Environment
Lanminh Pham1, Seth Boudreaux2, Sam Karhbet2, Becky Price2,
Azmy S. Ackleh2, Jacoby Carter3,*, and Nabendu Pal2
Abstract - Hyla cinerea (Green Treefrog) is a common wetlands species in the
southeastern US. To better understand its population dynamics, we followed a
relatively isolated population of Green Treefrogs from June 2004 through October
2004 at a federal office complex in Lafayette, LA. Weekly, Green Treefrogs were
caught, measured, marked with VIE tags, and released. The data were used to
estimate population size. The time frame was split into two periods: before and after
August 17, 2004. Before August 17, 2004, the average estimated population size was
143, and after August 24, 2005, this value jumped to 446, an increase possibly due to
tadpoles metamorphosing into adults.
Introduction
Recently, declines in some amphibian populations around the world have
been reported (Young et al. 2001). There has been much discussion in the
literature about causes and general nature of the reported declines (Sala et al.
2000). There is now growing recognition of the need for long-term monitoring
of amphibian populations. The University of Louisiana at Lafayette (UL
Lafayette) and the United States Geological Survey National Wetlands Research
Center (USGS NWRC) have initiated a project in partnership to
monitor and model frog populations at the National Wetlands Research
Center/Estuarine Habitat and Coastal Fisheries Center research complex,
with an initial focus on Hyla cinerea (Schneider) (Green Treefrog).
With the increase in urbanization and the spread of suburbs, wildlife
populations at the urban/suburban interface may become isolated. By monitoring
a local, relatively isolated population of frogs, we hope to gain a greater
understanding of the population dynamics of this species. We followed the
population over a breeding season, and used the field data to develop a
population dynamics model and make weekly estimates of population size.
The study site was located at the UL Lafayette campus on land leased
to the NWRC and the National Marine Fisheries Service (NMFS) (henceforth
the NWRC/EHCFC complex) in Lafayette, LA. The site was chosen
because of previous observations showing an abundance of Green
Treefrogs and suitable breeding habitat for these frogs. Over a five-year
period preceding this study, several anuran species had been seen or
1Department of Biology, University of Louisiana at Lafayette, Lafayette, LA 70506.
2Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA
70506. 3USGS National Wetlands Research Center, 700 Cajundome Boulevard,
Lafayette, LA 70506. *Corresponding author - jacoby_carter@usgs.gov.
204 Southeastern Naturalist Vol. 6, No. 2
heard at the NWRC/EHCFC complex. These include Bufo fowleri
(Hinckley) (Fowler's Toad), Bufo nebulifer (Gulf Coast Toad),
Elutherodactylus planirostris (Cope) Greenhouse Frog, Gastrophryne
carolinensis(Holbrook) (Eastern Narrow-mouthed Toad), Rana
catesbeiana (Shaw) (American Bullfrog), Rana utricularia (Cope)
(Southern Leopard Frog), Hyla squirella (Bosc) (Squirrel Treefrog),
Rana clamitans clamitans (Latreille) (Bronze Frog), and Pseudacris crucifer
(Wied-Neuwied) (Spring Peepers) (J. Carter, pers. observ.).
The Green Treefrog is a common anuran in the southeastern United
States, ranging from Delaware southward to southern Florida and the Florida
Keys, and westward through Mississippi River Valley into southern Illinois,
and extends further westward into eastern and southern Texas (Winston
1997). They are often found in floodplains, large lakes, smaller ponds, and
swamps (Gunzburger and Travis 2004); marshy areas that have an abundance
of emergent and floating vegetation, grasses, and cattails along the
banks (Winston 1997); and in temporary aquatic habitats (Wright 1932).
Adult Green Treefrogs feed on small arthropods and live on plant stems in
trees and shrubs near water, while the tadpoles feed on algae and prefer
shallow water in dense vegetation. Colors of the Green Treefrog range from
green to reddish-brown, and its size ranges from 3.2 cm to 6.4 cm. Green
Treefrogs have been documented to mate from April to the end of August
(Dundee and Rossman 1989).
Methods
Field-site description
The NWRC/EHCFC complex contains a network of artificial ponds and
reflecting pools (Fig. 1). The landscape fringing the ponds simulate different
wetland types, including emergent marshland and swamp. Buildings limit
access to the field surrounding the NWRC complex. The eastern border of
the pond complex is made of sidewalks, four lane roads, and is across the
street from a medium-density housing complex.
In the fields adjacent to the ponds are seasonal wetlands, with the closest
being approximately 60 m away. The ponds in front of the NWRC were
created in 1992 at the time the NWRC was built. The ponds and reflecting
Figure 1. NWRC/EHCFC Pond complex. Pond outlines are highlighted in white.
Arrows show likely migration routes between ponds.
2007 L. Pham, S. Boudreaux, S. Karhbet, B. Price, A.S. Ackleh, J. Carter, and N. Pal 205
pools adjacent to the EHCFC were created in 1999 when that complex was
built. Before construction of the two building complexes, the area was an open
field with some shallow seasonal wetlands. The region is at the eastern edge of
a historical coastal prairie. More recently, the parcel of land was used as a pig
farm and then later converted to open field. Land management in the adjacent
fields has varied over the years, but at different times, different parts of the
area have been leveled or contoured to prepare the land for construction, to
remove wetlands, or to encourage meadow-nesting birds.
Mark-recapture technique
Capturing was conducted at least one hour after sunset, and lasted between
2100 and 2300 hours central daylight time. Each pond was divided into
approximately four quadrants. For each quadrant, a group of three people
searched the area for 15 minutes while listening for Green Treefrog calls.
Individuals were visually sighted and caught by hand. Captured individuals
were placed in plastic bags that were labeled to indicate capture site and the
individual who captured it. Only one frog was placed in each bag, and bags
were not reused. We measured the snout to tail length, and if not previously
marked, the frogs were marked with Visible Implantable Fluorescent
Elastomere (VIE) tags (Gillette 2003). A combination of colors and locations
on the body were used to indicate week of first capture. A UV-light and
specialized UV glasses were used during the examination to detect any
fluorescent marks that may have been present. Placement locations on the
body (Fig. 2) were chosen after we conducted a pilot study to determine which
locations retained tags (J. Carter, unpubl. data). After marking, frogs were
released in approximately the same locations where they were caught. For
previously caught frogs, we noted their length and the week of initial capture.
All Green Treefrogs that were captured were marked, and we made no
distinction between adult, juvenile, and recent metamorphs, except to note the
length of the animal and an incompletely absorbed tail, if present. We did not
use a “termination” mark with recaptured frogs.
The survey was conducted once a week for 18 weeks, from June to
October of 2004. For each night of surveying, we recorded weather conditions,
time spent in the field, and catch efforts. In addition, these methods
Figure 2. Letter codes indicate locations
where elastomere markers were
placed on the frog's body.
206 Southeastern Naturalist Vol. 6, No. 2
Table 1. Adult frog capture summary over 18-week period. Total captures (TC) = new captures + recaptures. NC = new captures. R = recaptures.
Recaptures by week
WeekNCTCR123456789101112131415161718
1 18 18 0 ~
2 14 16 2 2 ~
3 24 26 2 2 0 ~
4 15 25 10 5 1 4 ~
5 18 30 12 3 2 2 5 ~
6 9 22 13 9 1 0 3 0 ~
7 297411100~
8 624188325000~
9 310720121100~
10 187201010012~
11 121310000000100~
12 912310000000020~
13* 000000000000000~
14 6820010010000000~
15 25300000000000102~
16 462001000000000010~
17 5940000110010010000~
18 710300100100000000100~
Total 164 9 8 14 16 3 4 0 2 3 2 0 2 0 3 1 0 0
*No captures during sampling on week 13.
2007 L. Pham, S. Boudreaux, S. Karhbet, B. Price, A.S. Ackleh, J. Carter, and N. Pal 207
were supplemented by calling surveys to evaluate which species were
present, if they were breeding, and what their abundance was.
Frequent visits to the complex were made as early as March; however, no
mating calls were heard until early April. Similar to Gerhardt (1987), we
observed that most captured individuals resumed performing mating calls
within a few minutes.
Statistical estimation of the weekly population size and further inferences
Our capture-recapture protocols followed the Unknown Capture History
Protocol as outlined in Burnham et al. (1987). This sampling scheme involves
drawing a random sample, marking the individuals, and releasing them. Data
of previously marked individuals in the sample allows estimation of the
population. A heuristic idea of this sampling process is outlined in Figure 3.
Frog population in week-i was estimated using the sample information of
marked frogs that had been captured in the previous weeks. The probability
distribution of the number of marked frogs in a particular week appears to be a
generalization of the well-known hypergeometric distribution.
The probability (conditional) distribution of the observed data (number
of marked and unmarked frogs) is:
P(Xki = xki, 1 i k - 1 | Xlj = xlj; l = 1, 2, ... ,[k - 1], j = 1, 2, ... ,[l - 1]) =
=
= =
=
=
k
k
k
i
k ki
k
l
l
j
k k l lj
l
kl
l
j
l lj
M
N
M x
N M x
x
M x
1
1
1
1 1
1
1
1
, (1)
where: Nk = population size in week-k, unknown; Mk = size of the sample
drawn in week-k; and Xk = number of frogs in the sample (in week-k) of size
Mk that were caught and marked earlier (Fig. 4).
Figure 3. Statistical model used to make population estimates.
208 Southeastern Naturalist Vol. 6, No. 2
Note that Xk = Xk1 + Xk2 + ... + Xk(k – 1), where Xki = number of frogs in
week-k sample that were marked in week-i, i = 1, 2,… (k-1); for convenience,
Xij = 0 for i = j (since a frog cannot be recaptured the same week it is
originally marked). The model development is described in detail in Appendix
1. Maximizing (1) with respect to Nk gives Ñk = the estimated population
size in week-k.
Results
Weekly captures and recapture numbers are summarized in Table 1.
The last week we heard frogs during the monitoring run was week 10 (10
August 2004).
The lowest weekly population estimate (Table 2) was 125 frogs in week
6, and the highest weekly estimate was 1429 frogs in week 11 (17 August
2004). By using week 11 as a dividing line, the data can be separated into
two periods. Before week 11, the average population estimate was 173 with
a standard deviation of 92.19. After week 11, the average population estimate
was 445.7 with a standard deviation of 114.42.
Average length data for Green Treefrogs caught are in Table 3. For the
first 10 weeks, average frog length was greater than 30 mm. From weeks 11–
18, the average length varied from week to week between 24 and 38 mm.
Discussion
An important premise of this study is that the population we are studying
is in fact a year-round population, not simply a breeding population.
Figure 4. A visual respresentation of the sampling scheme that gives the probability
(conditional) expression in week 1.
2007 L. Pham, S. Boudreaux, S. Karhbet, B. Price, A.S. Ackleh, J. Carter, and N. Pal 209
While it is probable that Green Treefrogs migrate to the pond complex
from the surrounding areas, we feel this migration is limited and consider
the area a habitat island within the large landscape. There are several
lines of evidence to support this view. First, we regularly surveyed areas
around the NWRC/EHCFC for frogs away from the ponds, but rarely
found them. Frogs were only occasionally found away from the pond
Table 3. Weekly summary data on green treefrog body lengths in millimeters.
Week # N Min Max Mean Median STD
2 14 20 50 37.5 40 10.0
3 24 16 50 32.5 35 10.9
4 15 30 40 34.2 35 4.2
5 18 20 45 32.1 30 6.7
6 9 20 50 37.0 40 7.6
7 2 20 50 32.7 30 9.0
8 6 15 50 30.2 30 8.7
9 3 25 50 34.5 30 7.6
10 1 20 50 35.0 35 11.6
11 12 15 55 27.6 20 13.7
12 9 15 55 27.5 20 11.9
13* 0
14 6 20 40 27.5 25 5.9
15 2 20 25 24.0 25 2.2
16 4 30 55 38.3 37.5 9.3
17 5 15 40 25.1 22 9.5
18 7 20 55 31.5 27.5 12.9
*No captures during sampling on week 13.
Table 2. Population estimates per week.
Lower 95% Upper 95%
Week CI bound Estimated population size CI bound
2 57 143 519
3 143 415 1520
4 99 140 236
5 128 177 285
6 119 150 228
7 106 125 218
8 116 133 179
9 120 151 269
10 113 123 207
11 331 1429 8249
12 231 487 1395
13*
14 222 523 1873
15 156 228 605
16 199 416 1469
17 196 341 769
18 246 523 1398
*No capture data for sampling week 13.
210 Southeastern Naturalist Vol. 6, No. 2
complex. Second, we have set out PVC pipes to act as artificial habitat.
While we have found Green Treefrogs in our pipes around areas immediately
adjacent to the pond complex, we have rarely found them away
from the area. One of the authors (J. Carter) has inspected the adjacent
roadway after rains for evidence of frog movement from a housing complex
across the street to the study site and has not found any dead frogs
on the road. Finally, call monitoring was conducted in conjunction with
the mark-recapture work. We have never heard Green Treefrogs calling
from areas adjacent to our study site. So while it is likely that some frogs
migrate from and to the study site, we feel that the site, because of
geography, remains relatively isolated from other centers of population.
We caught frogs from 17 June 2004 to 22 October 2004. The data can be
separated into two periods, before and after week 11 (August 17th). Before
this date, mating calls for Green Treefrogs were noted, but not afterwards. If
we treat week 3 as an outlier, the first period’s estimated population average
is 142.75 ± 17.33. Week 11 appears to be an outlier with an estimated
population of 1429. This value is a one-week ten-fold increase. There are
two possible explanations for that increase.
One possibility is that our estimations were affected by differences in
catchability. Only male frogs call, and when they stop calling, they are
harder to catch. Before week 11, our sample sizes averaged about 25
frogs per night, and after week 11, the average sample size decreased to
approximately 10 frogs each night. Because our catching methods relied,
in part, on listening for frog calls, our method is biased towards males,
and our estimate for this period will be based on mainly the male population.
If we assume that there is an equal ratio of males to females in the
population, we may have underestimated the population by a half during
the first half of our study. After the breeding season, males no longer call.
Thus, our catch efforts are a result of unbiased sampling of males and
females. Under this scenario our estimate is more representative of the
population in the second half of the capture period. In that case, the population
didn’t change, instead our estimates became better.
A second explanation may be that our estimates reflected an actual
increase in population size during week 11 due to an influx of recent
metamorphs into our frog population. This could be due to immigration, but
is more likely due to the flush of recent metamorphs into the population that
our simulation modeling efforts predicted. An increase due to immigration
of breeding adults is unlikely because calling activity decreased after the
increase in population. One would expect calling activity to increase under
these circumstances. Furthermore, adult frogs are larger than juveniles and
recent metamorphs. If the increase was due largely to a influx of adults, we
would have expected a significant increase in the average size. Instead, the
median size of frogs caught decreased.
2007 L. Pham, S. Boudreaux, S. Karhbet, B. Price, A.S. Ackleh, J. Carter, and N. Pal 211
In the above estimation procedure, it is assumed that in week-k
(k = 1, 2, … 18) the population size Nk was constant at the time of sampling
(in other words, population size should not vary significantly for any reason).
The estimated population sizes for week 3 and week 11 had unexpected
variance. Reexamining the original samples from week 3 and week 11
revealed that there were a small percentage of frogs captured with marks
from previous weeks (7.7% and 8% respectively). Compared to the rest of
the data set, these particular weeks’ samples seem to be outliers. The timeseries
plot supports this hypothesis (Fig. 5). The time-series plot separates
into two periods, before and after week 10. After week 10, the population
size appears to have abruptly increased, which may have been due to
metamorphs entering the frog population.
We also considered the measured length of frogs caught. Before week
11, the average length of the frogs caught was greater than 30 mm. In
week 11, the average dropped to 27.7 mm and did not rise above the first
10-week average for 5 weeks. However, the variance associated with
these lengths overlapped, and therefore, the significant difference in
lengths cannot be justified. Nevertheless, the median length did decline
Figure 5. Time-series plot of the estimated population size by the maximum-likelihood
estimator model (for each week, the vertical line provides an approximate 95%
confidence interval of the population size, with the dot indicating the point estimate).
212 Southeastern Naturalist Vol. 6, No. 2
by 10 mm or 1/3 for the two weeks after Week 11 and did not rise to pre-
Week 11 levels for 6 weeks. During this same period, our estimated
population size increased significantly. Both of these observations could
be explained by metamorphs entering the frog stage. We created an age
structured metapopulation dynamics model for the ponds in the NWRC/
EHCFC to help us better understand how our population may change over
the season (J. Carter et al., unpubl. data). The results from this model also
support the idea that a population increase might be expected midseason
as a result of the influx of metamorphs.
Wright and Wright (1995) report that adult male and adult female
Green Treefrogs range in size from 37–59 mm and 41–63 mm, respectively.
The sizes of frogs caught in the second half of the study would lend
support to the idea that more subadults were being caught, and not significantly
more adult females.
It is also interesting to note that calling activity significantly declined
after week 10. The Green Treefrog is noted for calling through the end of
August if the weather is suitable (Dundee and Rossman 1989). In order to
effectively attract females, male frogs expend large amounts of energy in
producing calls (Bosch and de la Riva 2004). The surplus energy needed to
make these calls is not available if the frog experiences stress due to a
decrease in food supply. Therefore, we speculate that if a sudden increase in
population size did occur, this might cause increased competition for food,
and males would stop calling. If this is the case, we may be able to use this
relationship as a method for estimating the local carrying capacity for adults
in the system.
In addition to estimating the populations of adult Green Treefrogs, we
attempted to estimate the tadpole populations for that species using the
techniques outlined in Jung et al (2002). However, initial capture and recapture
rates of tadpoles were too low to make reliable population estimates.
Conclusions
Our population estimates varied during the study; there were approximately
140 individuals during the first 10 weeks, then an increase in
population size around week 10, and finally a comparative decline for the
rest of the sampling period. These results are in agreement with population-
dynamics modeling work (A.S. Ackleh, S. Boudreaux, S. Karhbet, L.
Pham, and B. Price, unpubl. data) which predicts that the adult population
in the pond complex should increase as tadpole cohorts that hatched out
earlier in the summer metamorphose to adult frogs. The sudden decrease in
Week 11 of median size of frog length supports this conclusion. After the
abrupt population increase in week 10, the frog count steadily declines for
the rest of the summer. This larger population estimation could either
reflect (1) an actual decline (caused by recent metamorphs moving out of
2007 L. Pham, S. Boudreaux, S. Karhbet, B. Price, A.S. Ackleh, J. Carter, and N. Pal 213
the ponds), (2) an artifact of our catch method (males calling less reduces
our catch success rate), or (3) a combination of these two factors.
Although the use of VIE tags was helpful in identifying Green Treefrogs
for weekly mark-recapture analysis, this technique did not allow for individual
identification. This method limited our ability to look at capture
history, develop accurate estimations of survival probability, or estimate
growth rates. In the future, we are plan to mark frogs as individuals using
alpha-numeric tags.
Acknowledgments
This project was funded through the National Science Foundation under grants
#DMS-0311969 and #DUE-0531915. We would also like to thank the NWRC for
providing training, equipment, and access to and use of their facilities. Jim
Delahoussaye provided frogs for our pilot study. Kathleen Roberts provided technical
assistance with VIE tagging. The order of authorship was randomly chosen and
does not reflect relative contribution to this work.
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2007 L. Pham, S. Boudreaux, S. Karhbet, B. Price, A.S. Ackleh, J. Carter, and N. Pal 215
Appendix 1: Statistical model.
First we consider the case of week 2. Draw a sample of size M2 (known) from the
population of size N2 (unknown). Let X2 denote the number of marked frogs in this
sample that were caught earlier (in week 1). Note that X2 is a random variable and
hence has a probability distribution. The probability that X2 = x2 (some observed
value) is:
( )
= =
2
2
2 2
2 1
2
1
2 2
M
N
M x
N M
x
M
P X x , (A1)
where N2 - M1 > 0, (N2 - M1) - (M1 - x2) > 0, and N2 > M2. According to our notations in
(1), x2 = x21. Note that N1 = population size in week 1, and M1 is the first week's
sample size (caught, marked, and released).
The only unknown element in the probability expression (A1) is the population
size N2 which is the unknown parameter in the probability model, known as the
hypergeometric probability model. The restrictions given after (A1) can be summarized
as N2 > max { M1, M2, (M1 + M2 - x2)}. When the probability expression (A1) is
viewed as a function of x2 (the observed value of the random variable associated with
the experiment of mark-recapture sampling scheme for week-2), it is called a (discrete)
probability distribution. The same expression (A1), when viewed as a function of the
unknown parameter N2, is called a likelihood function, and it is denoted by L(N2| x2, M1,
M2) (i.e. a function of N2 given that x2 is the observed value of X2, and the sequential
sample sizes up to week-2 are M1 and M2, respectively). We follow the method of
Maximum Likelihood Estimation (MLE) to estimate N2. The MLE of N2, denoted by Ñ2
is obtained by maximizing L2(Ñ2| x2, M1, M2) = P(X2 = x2) with respect to N2; i.e.,
Max L2(N2| x2, M1, M2) = L2(Ñ2| x2, M1, M2). (A2)
N2
The MLE is a preferred estimation technique because the estimate of the parameter
is known to have nice theoretical asymptotic properties. The estimate Ñ2 is the
value of the parameter N2, which makes X2 = x2 most probable since we have already
observed it.
Next we consider the case of week 3. Draw a sample of size M3 (known) from the
population of size N3 (unknown). Let X3 denote the number of marked frogs in this
sample that were caught earlier. X3 is a random variable and has two components X31
and X32 such that
X3 = X31 + X32, (A3)
whereX31 = number of frogs in the week 3 sample marked in week 1, and X32 = number
of frogs in the week 3 sample marked in week 2.
The probability distribution of observing (X31 = x31, X32 = x32) is (given that we
had observed X2 = x2 in week 2):
( )
{ ( )}
+
= = = =
3
3
3 31 32
3 1 2 2
32
2 2
31
1
31 31, 32 32 2 2
M
N
M x x
N M M x
x
M x
x
M
P X x X x X x , (A4)
Again, note that the above expression has the only unknown element N3.
216 Southeastern Naturalist Vol. 6, No. 2
Maximum Likelihood Estimation yields Ñ3 such that
Max L3(N3| x2: x31, x32; M1, M2, M3) = L3(Ñ3| x2: x31, x32; M1, M2, M3).
N
3
Notice that the estimate Ñ3 is dependent not only on (x31, x32) (i.e., what we
observe in week 3), but also on x2 (i.e., what we had observed in week 2). This is
because the probability distribution (A4) is the conditional distribution of the data
observed in week 3, given what has been observed in week 2. Continuing in this
fashion, we obtain the probability model (1) for week k.
Interestingly, we have noted that if we use slightly less precise information, then
still our population estimates remain the same with the observed frog data. The
probability model (1) is a generalization of the standard hypergeometric distribution,
which uses the precise, past weekly recapture data on a conditional basis. If we
combine the past weekly data as a simple “past data” and use the standard hypergeometric
distribution, then
= = = k k l l P(X x X x , 1 l k 1)
+ + + +
+ + + +
=
k
k
k k
k k k
k
k k
M
N
M x
N M M x M x M x
x
M (M x ) (M x ) ... (M x ) [ ( ) ( ) ... ( )] 1 2 2 3 3 1 1 1 2 2 3 3 1 1
, (A5)
where M(k – 1) = M1 + (M2 – x2) + ... + (M(k – 1) – xk – 1) is the total number of marked frogs
in the population in week-k. (It is assumed that all the frogs marked earlier but which
didn't get caught in week k are still alive.)
One can maximize the above (A5) with respect to Nk to get another estimate, say,
N*
k , of Nk, the population size in week-k. Our numerical computations have shown
that N*
k Ñk.
The slightly less precise model (A5) (which combines all the week-wise recapture
data into a single “recapture observation”) has one advantage. The moment
expressions for Xk are known for (A5), whereas no such expressions are available for
the model (1). Therefore, we use (A5) to get approximate confidence bounds for the
estimates of Nk.
Using the probability model (A5), mean of Xk = k =
=
k
k
k N
M
M ( 1) and standard
deviation of Xk = k
=
1
1
1 1 ( 1) ( 1)
k
k
k
k
k
k
k N
M
N
M
N
M
M . It is expected that Xk
should fall within 2k of k with probability 0.95 (roughly), i.e.,
0.95 P(k - 2k Xk k + 2kk).
The two inequalities inside the last probability expression can be inverted to get
upper and lower confidence bounds, which have been calculated in Table 2.
Weekly population estimates and the corresponding approximate 95% confidence
bounds are plotted in Figure 5.
Notice that the upper confidence bound for the population size is further away from
the population estimate compared to that of the lower confidence bound. This indicates
that the population estimates are highly positively skewed, and this is expected
considering the fact that each population size Nk (2 k 18) takes values over the space
{1,2,3, … }, which has a finite lower bound, but infinite upper bound.