A STOCHASTIC HODEL FOR PLANT SPACING
by
Nasser Ordoukhani and L. A. Nelson
Institute of Statistics
Mimeograph Series No. 780
Raleigh - 1971
iv
TABLE OF CONTENTS
Page
,.
1.
INTRODUCTION....
1
2.
REVIEW OF LITERATURE ••
3
2.1
2.2
2.3
2.4
3.
Cotton Plant Spacing Related to Yield • •
Gap Filling and Effect of Skip or Missing Row
in Case of Cotton Plant. • • • • • ••
• ••••
Mathematical Model for Plant Space Distribution •
Distribution of Plant Spacing Used for
Mechanical Thinning Devices ••
5.
RESULTS AND DISCUSSION
6
7
12
14
MATER IALS AND METRO DS • •
14
Field Sampling. • • • • • •
Modeling the Plant Distances. •
4.
3
20
.......
23
4.1 Distribution of Cotton Plant Spacing. . . . • .
4.2 Estimation of the Parameter of the Negative
23
Exponential Distribution • • • • • • • • • •
27
CONSTRUCTION OF A CONFIDENCE INTERVAL FOR THE PARAMETER OF
THE PLANT SPACING DISTRIBUTION. • • • • • • • • •
5.1
5.2
5.3
Confidence Interval for the Parameter of the
Negative Exponential when Sample Size, n,
is Less than 100
Shortest Confidence Interval for the Parameter
of the Negative Exponential Distribution
when n < 100
Construction of a Confidence Interval for the
Parameter of the Negative Exponential
Distribution When n > 100.
.....······...····
29
29
.......······
····
36
······
.····
38
v
TABLE OF CONTENTS (Continued)
Page
•
6.
DEVEIDPMENT OF A PLANT STAND INDICATOR AND ITS APPLICATIONS. •
6.1 Plant Stand Indicator • • • • • • • • • • • • • • •
6.2 Use of Plant Stand Indicator in Regression Model. •
41
41
43
6.2.1 Plant Stand Indicator as an Independent Variable 43
6.2.2 Use of Plant Stand Indicator for Replanting. • • 44
6.2.3 Plant Stand Indicator as a Dependent Variable •• 52
7.
SUMMARY AND CONCLUSIONS.
53
8.
LIST OF REFERENCES
56
9. APPENDICES
9.3
2
The X Goodness-of-fit Test
Estimation of the Parameter of the Negative
Exponential Distribution • • • • • • • • • • • • • •
The Method of Construction of Confidence Limits
59
59
64
68
1.
INTRODUCTION
In order to maximize cotton yields, a properly spaced uniform
stand is essential.
•
A uniform plant spacing distribution insures
optimum utilization of solar radiation, CO , moisture, and nutrients
2
for the growth of the crop.
Non-uniform stands result in row segments
with high plant densities and other segments with relatively low plant
densities.
plant basis.
Under high plant density, less bolls are produced on a per
Under low plant densities, additional bolls are produced
later; thus, resulting in different maturity stages.
When both high
and low plant densities occur in the same field, additional costs due
to multiple harvests and loss of cotton due to adverse weather conditions
late in the season may occur.
Non-uniformity in plant stands may result
from planter inaccuracies, and/or random seedling failure caused by low
quality of some seeds or adverse environmental conditions.
Planter
inaccuracies are either non-random failures caused by some mechanical
problem or they too can be due to some random phenomenon.
This study
is concerned with measuring the effects of a combination of random
seed placement problems and random environmental effects.
From the standpoint of optimum utilization of growth factors and
harvesting procedures, it is important to have a plant stand indicator,
which indicates both the number of plants per acre and the uniformity
within the row.
Prior to this time, plant stand indicators have not
been available for cotton.
If a suitable stand indicator were available for cotton, it would
assist producers in making management decisions relative to replanting
2
entire fields or reseeding the low plant density segments.
The plant
stand indicator may be used in combination with such factors as soil
properties, climatic factors and management practices to obtain a
mathematical model for yield prediction, and can be useful in evaluating
and improving the performance of mechanical planters.
The plant stand
indicator can also be used in the development of mechanical thinning
devices.
The major objective of this study was to develop a plant stand
indicator for cotton which would be based on a sound theoretical
statistical model and which would be used in making decisions about
replanting, yield estimation and several other purposes.
This
development involved the determination of a reasonable probability
density function of cotton plant spacings in the field and techniques
of estimation of the parameters of the sampling distribution and the
construction of a confidence interval for the parameter.
There is good reason to believe that suitable stand indicators
may be developed for other crops using the techniques described herein.
The density functions may vary from crop to crop but the underlying
methodology would be similar.
Stand may be much more critical in some
crops than it is in cotton because these crops may not have the tendency
to compensate for unevenness of stand to the extent cotton does.
•
3
2.
2.1
REVIEW OF LITERATURE
Cotton Plant Spacing Related to Yield
The literature on cotton spacing is extensive and recommendations
have varied from wide to very close spacing between plants in the row.
These contradictory recommendations are concrete evidence of the wide
adaptation of the cotton plant and of the varied conditions under which
it may be grown.
Spacing influences earliness in the opening of bolls.
In the warmer part of the cotton belt, thick stands seem to promote
earliness.
Usually the boll size increases slightly as the space
between plants increases from 3 to 12 inches.
Close spacing reduces
both the number of negative links on the plant and the number of bolls
per fruiting link, and raises node-wise the position of the first square.
Dugger (1886) obtained nearly constant yield with plants spaced 1,
2 and 3 feet apart.
Moores (1928) summarized the data from the various
experimental stations over the cotton belt and found that the "best"
average spacing for the belt as a whole was between 11 and 16 inches.
Ware (1930) made comprehensive spacing studies in Arkansas and concluded
that the adaptability of the cotton plant to spacing is such that under
favorable growing conditions there would be no significant difference
attributable to the spacing effect within the population range of
20,000 to 50,000 plants per acre.
Further evidence of adaptability of
the cotton plant to spacing is shown by Neely (1941).
Simpson and
Duncan (1942) conducted experiments with cotton and concluded that
narrow rows yield more per acre but wider rows yield more per row, and
4
they suggested that row widths should be adjusted to the point at
which the cost of increased production will result in the lowest cost
per pound of cotton produced or the greatest production return for labor
and equipment.
Lane (1956) found that close spacing decreases dry
weight of a single plant but increases the dry weight per acre of the
parts of plants above grou.i1.d.
He fitted a
qt-~adratic
model by the
method of least squares for the data which were obtained for several
years, resulting in the following equation:
A
Y
= 90.0
2
+ .356 X - .005 X
A
where Y is the predicted relative yield (percent of maximum yield) and
X is the number of plants per acre (thousands) for the range of X from
5,000 to 70,000 plants per acre.
He also discovered that the relation
between bolls per plant and stands is negative exponential.
Duncan (1958) proposed that under corn belt conditions, the
logarithm of the grain (corn) yield per plant was a linear function of
plant density.
Since yield per acre is the product of yield per plant
and plant density, he multiplied the yield per plant equation by plant
density to obtain an equation for predicting yield per acre.
Duncan
pointed out that use of such equations permitted estimation of the
maximum yield of a variety and the plant density producing that yield
from trials with as few as two plant densities per variety.
Warren
(1962) used an equation similar to Duncanfs to describe the effect of
plant density on yield of fresh market sweet corn.
However, his
approach differed from Duncan's in that the basic equation used is a
..
5
linear relation between plant density and yield per plant, rather than
the logarithm yield per plant.
.
Corr8equently~
a parabolic function was
obtained for yield per acre instead of an exponential function.
These
particular yield equations also can be obtained from as few as two
plant populations per variety and are simpler to compute than exponential
equations.
In further studies~ Warren (1963) obtained additional
evidence indicating that such parabolic equations adequately describe
yield responses of fresh market sweet corn within the limits of plant
density and environment that had been examined.
Carmer and Jakobs (1965) represented the relationship between corn
yield per unit area and plant density by the exponential regression
model~
A
where Y is the yield per unit area, N is the number of plants per unit
area, A and K are unknown parameters having biological significance.
These parameters may be expected to vary with variation in geneticenvironmental conditions.
When there is only one plant per unit area,
the product of AK represents the maximum yield per plant under the
particular set of genetic-environmental conditions.
The proportionality
constant K, is indicative of the plants' competitive abilities.
It is
apparent that K has a positive value less than unity, since yield per
plant decreases with increases in plant density.
They pointed out that
use of this model implies the assumption that the plant density is great
enough to insure effective competition among plants growing in that area.
6
Ray (1959) showed that within the range of 20,000 to 50,000 cotton
plants per acre the yield was nearly constant.
Cowley (1960) studied
several spacings with cotton and concluded that the 4-inch spacing is
associated with higher yields.
Toomey* (1962) showed that no great
differences in yield were obtained in stand sampling ranging from 4-6
plants per foot of row.
He also pointed out that for no less than two
and no more than eight plants per foot, replanting or thinning would
not be justified in most instances.
2.2
Gap Filling and Effect of Skip or Missing Row
in Case of Cotton Plant
Replanting is seldom carried out if the stand is uniformly thin,
but if the field contains many rather long skips, it is customary to
replant the entire row or field.
In this operation most or all of the
earlier plants are destroyed, frequently it is difficult for the grower
to determine whether replanting is likely to be profitable, and little
experimental evidence is available as a guide in deciding on the
advisability of replanting.
Extensive data summarized by Brown (1923)
and Reynolds (1926) showed that essentially equal yields of cotton may
be expected within relatively wide spacing limits at individual
locations representing the major sections of the cotton belt.
Pope (1947)
studied 27 skip correction tests, based on perfect-stand checks and skip
distances of 2, 3, 4, 5, 6, 7, 8, 9 and 10 feet in the middle row of
* Toomey, W. G. 1963. Report of cotton stand studies.
Report, North Carolina State University at Raleigh.
Unpublished
7
3-row 25-feet plots during the four-year period and concluded that the
skips occurring on interior rows of multiple row plots were largely
compensated for by increasing production on end plants in the row
containing the skips and by lateral compensation on the adjacent rows.
Also in
single~row
plots there were definite reductions in yield due to
skips, and these reductions appeared to be substantially linear for
distances greater than three feet.
MacDonald (1947) studied the use-
fulness of filling gaps in bad stands of cotton.
He found that any
reduction in stand, though reducing yield per acre, had to produce
large and significant increases in yield per plant, even when the
spacing was 3 by 3 feet, wider than that normally used on the station
and replanting the gaps in bad stands did not lead to any increase in
yield per acre.
He concluded that no useful purpose is served by
refilling the gaps in poor stands of cotton having up to
2.3
40% of gaps.
Mathematical Model for Plant Space Distribution
Chittey et ale (1967) first pointed to the importance of the
distribution of plant spacing.
They developed a system for recording
plant distribution data by a perforated paper strip.
then automatically compiled.
The data are
They did not work with the mathematical
distribution of plant spacing but merely compiled frequency data for
field spacings.
Rohrbach et aL (1969) at Ohio State University studied the
distribution of plant stands.
They attempted to provide an analytical
basis for evaluating accuracy and precision of horizontal cell plate
planters.
They proceeded as
follows~
8
Let p. equal to probability that i seeds occupy a seed cell at the
l
time of discharge.
Therefore
t
p.
i:::l
~.
1
l
The theoretical drop R of planter is defined in terms of p. as
l
tip.
R --
i=O
l
The number of seeds per cell at the jth choice, N., is a random
J
variable with a discrete probability density, P(N
k
= 0,
1, •••
j
= k) = Pk'
Let g equal the probability that a given seed becomes a
seedling plant, and hence:1 1 - g equal the probability that any given
seed will not survive to seedling.
The random plant development
variable G~ is defined to take on values of
J
g and (l - g), respectively.
Thus,
P (G~
r
° and +1 with probability
J
= 0) = g
and
k
=
J
P (G.
r
1)
=1 -
g.
The desired uniform plant locations would be given on the numbers
by jSa' j
p
ro
= 0,
1, 2, ••• when sa is the intended uniform spacing.
Let
be the probability that two plants defining s., a plant spacing,
l
came from the same seed drop.
- (l/R)
t
Pk[(k
k=2
The expression for p
ro
k
is
1) - «1 - g) - (1 - g) )/g J.
9
Let p 1 be the probability of observing s. from a pdf, that is,
r
1
2
2
N(s , 20 ), where N(s , 20 ) refers to a normally distributed population
a
a
2
with mean, sa' and variance, 20.
Let p
rn
The general expression forPrl
is~
be the probability that the second plant is from the nth
successive seed drop after the seed drop that established the first
plant, then
p
rn
= wn-lz n/R[Pl
+ [1 + (1
~ g) Jp2 + ••• + [1 + (1 - g) +
(2.1)
where z
is defined as the probability that at least one plant from
n
the nth successive seed drop survives, and w as the probability that no
plants survived from any of the (n - 1) intervening seed drops.
On
the other hand, w is given by
00
w = t P (1 _ g)m •
m=O m
Finally, they introduced a probability density of spacing by a planter
which drops i seeds with probability p.,
at each desired plant
point,
1
,
every sa feet down the
00
f(s.)
1
row~
k
p [(k - 1) - (1 - g) - (1 - g) IN(O, 202 )+ ~
R k=2 k
g
Rg
=! t
00
t
n=l
j
10
2
where N(ns
, a , 20 ) refers to a normally distributed random variable with
mean nS
2
a
2
and variance 20 , and N(O~ 20 ) refers to a normally distributed
random variable
Wl'th
'
2 o.
2
mean zero an d
varlance
Suppose this probability
density function is an appropriate one, then one must find the sampling
distribution and try to estimate the parameters by means of statistical
methods such as maximum likelihood estimation.
find the sampling distribution.
But they neglected to
Instead they considered four definition
equations:
t p.
. 0
1=
:=.
1
1
c:o
R
= . t·
01p.
1=
1
(1 _ g) _ (1 _ g)k ]
g
c:o
W
=
t P (1 _ g)m
m=O m
The equations can also be expressed in matrix product notation:
1
1
1
1
(I-g)
(I-g)
2
0
PO
0
PI
0
0
g
-Pro
0
1
2
-1
1
W
=
P2
0
R
0
(2.3)
Numerical values for PO' PI' P2 and R can be obtained by solution
of (2.3), and numerical values for p rn ,n:=. 1, 2, ••• , in equation (2.1)
from the histogram for field data.
This procedure is not based on any
11
statistical methods.
Furthermore, the solution does not give any
estimate for s , or for the estimate of the variance of the density.
a
•
Rohrbach et al. (1969) also used computer simulation procedures
for predicting field plant spacings.
Specifically, corn, cucumber and
sugar beet populations were predicted and tested, against field
observations.
Testing consisted of observing field populations,
deducing model parameters using equation (2.2) and predicting with a
computer a comparable field population based on these model parameters.
The chi-square test of goodness of fit was used for testing deviation
of simulated models from actual field results.
freedom in testing goodness of fit should be:
But, the degrees of
d.f.
=k
- c - 1,
where d.f. are the degrees of freedom, k is the number of class
intervals and c is the number of parameters which were estimated.
They neglected to consider c, so their computed degrees of freedom
are incorrect.
For corn, the d.f. should be four instead of nine
since PO' Pl' P2' g and cra were estimated.
By the same reasoning,
the d.f. for cucumbers should be five instead of ten and for sugar
beets the d.f. should be 11 instead of 15.
Consequently, from the
implications of these three tests with the correct d.f., rejection
of the hypothesis that the field data follow the proposed distribution
is the end result.
This is just the opposite from the conclusions
drawn by these authors.
Rohrbach et al. (1970) used this proposed distribution (2.2) and
analyzed another set of data through a Monte Carlo approach.
2
Again
the computed d.f. for the X test goodness of fit was incorrect as
12
before.
They also used the Hasselblad
parameters of the distribution.
a mixture of normal
(1966) procedure to estimate the
However J the Hasselblad program is for
distributions~
co
Q.1
=
E q .. p.
i~l 1.J
1
where
2
=
q."
lJ
2
2 -(x.-~.) /2cr. ,
(1/ ./c'ncr.)e
1.
J
J
J
and
co
E p. -- 1
i=l
1
whereas the density proposed by Rohrbach et al. is for the sum of two
mixtures of normal di.stributions which involved additional parameters,
Rand G.
Therefore the Hasselblad program does not apply to this
problem and should not be used for this purpose.
2.4
Distribution of Plant Spacing Used for Mechanical
Thinning Devices
Mechanical thinning devices have been designed by engineers to
produce precision spaced rows.
The operations of these devices have
been based upon theoretical knowledge of planter spacings and the
combined effects of planting and thinning.
Palmer
(1958) did some of
the early theoretical work on the harmonics of random thinning treatments
for sugar beets.
By assuming a uniform plant distribution, he showed
that the harmonic characteristics of the planter and the thinner must
13
be made to coincide in order to obtain a uniformly thinned distribution.
He also studied the effects of thinning the same planted row more than
once and found no advantage in so doing.
Ririe and Hill (1957) analyzed the problems of California sugar
beet growers in establishing plant stands.
During the period in which
their study was conducted, the practice of complete mechanical thinning
of sugar beets was increasing greatly because seed quality had improved
to a level that growers were planting to stand.
They conducted several
studies to determine sugar beet tolerance to close spacing in the row.
They also emphasized the inadequacy of using the number of beets per
hundred feet of row as an index for the expected yield.
Garret (1966) made a significant contribution to the art of
mechanical down-the-row plant thinning with his development of a
selective thinning principle.
His research machine was designed to sense a plant in the row and
remove anything in a specific interval ahead of it.
Since its initial
conception, this principle has been developed into commercial machines
by manufacturers whose field resul.ts have generally been much more
acceptable than those obtained with random thinning machines.
14
3.
MA.TERIALS AND METIDDS
3.1 Field Sampling
This study, conducted to develop a suitable plant stand indicator,
was based on two sets of data.
1
data set 1, was collected
One set of data, herein referred to as
from the Southern National Bank Test Farm,
Red Spring, North Carolina, in 1968.
This farm was chosen because the
experimental fields were located on several different soil series and
topography situations and the seeding had been carried out by skilled
operators using mechanical planters.
The cotton production area consisted of four fields with a total
of 80 acres.
In each field, several plots were randomly selected.
Each plot was four rows wide (the distance between adjacent rows was
40 inches) by 25 feet long.
soil variability.
The plots were arranged to cover maximum
The distances between plants in the row in each plot
were measured and two harvests of cotton bolls were made.
was carried out by hand.
The harvesting
The numbers of plants per plot and yields per
acre are given in Table 3.1.
The second set of data, herein referred to as data set 2, on
plant spacing of cotton was obtained from two farms near Laurinburg in
Scotland County, North Carolina.
1
Scotland County is an important cotton
These data were supplied to the author by Dr. C. D. Sopher, Assistant
Professor, Department of Soil Science, North Carolina State University
at Raleigh.
15
Table 3.1
Plot No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
Numbers of plants per plot and yields for 64 plots studied
in 1968 ~data set 1), Red Spring, North Carolina
No. of plants
per plot
245
261
260
281
235
260
247
244
198
274
215
277
24·9
275
244
251
249
265
249
231
210
233
304
235
236
239
222
238
274
253
165
135
135
280
243
201
249
206
204
236
.181
Plot yield of
cotton Pounds/acre
1954
1981
1596
1293
1514
881
1569
1514
1459
1871
632
1981
1844
1789
2009
1514
1183
1706
1651
1679
1486
1734
1761
1211
1128
1514
1514
1348
1101
1706
771
908
1238
1156
1156
1376
716
605
660
688
653
16
Table 3.1
Plot No.
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
(Continued)
No. of' plants
per plot
226
189
220
206
147
119
142
136
111
125
170
206
168
175
155
163
210
158
132
182
192
188
232
Plot yield of
cotton Pounds/acre
743
550
771
495
1651
1596
1899
1926
1871
2064
1954
1706
2284
1596
2119
2394
2532
2009
1651
1596
1018
1761
1320
17
producing area and the two farms were selected on the basis of their
long experience in using seed planting machinery.
Ten fields were selected from one farm (location 1)2 and four fields
from the other farm (location 2)3.
Within each selected row in each
field, 25-foot sections were marked
off~
approximately 100 yards apart.
The distances were measured in location 1 and in location 2 and are
summarized in Tables 3.2 and 3.3, respectively.
The analyses of variance for data combined over these two locations
for the mean distances and the logarithms of the variances of plant
distances were performed and are given in Tables 3.4 and 3.5.
The purpose of these analyses of variance was to establish the
optimality of the sampling procedures which had been used in the field.
Knowledge of the relative magnitudes of the several sources of variation
was considered essential in deciding upon the adequacy of the procedures
which had been used.
Individual fields were rather homogeneous with
respect to soil but there were differences in soil series from field to
field and also, as might be expected, even greater differences from
location to location.
These were reflected in a large variance component
for locations with considerably smaller variance components for fields
in location and for samples in fields.
The samples-in-field component
was somewhat smaller than the fields-in-location component for the
logarithm of the variances but the reverse was true for the mean plant
2 J. T. Johns Company Farm, managed by Henry McLeod, Laurinburg, N. C.
3 McNair Farm, Laurinburg, North Carolina
18
Table 3.2
Number of plants in all samples from each field,
Johns Farm (location 1)
Field
number
No. of 100 ft.
plots sampled
1
3
5
2
3
4
5
6
7
8
9
10
Total
Table 3.3
Field
number
Number of plants in all
samples from fields
10
724
1,303
834
890
1,246
1,914
433
448
2,361
2,612
51
12,765
4
4
5
7
2
2
9
Number of plants in all samples from each field,
McNair Farm (location 2)
No. of 100 ft.
plots sampled
1
6
2
3
4
27
35
31
Total
99
Number of plants in all
samples from fields
1,941
6,652
8,451
7,740
24,784
19
Table 3.4 Analysis of variance for mean plant distances
Source
d.f.
Sum of
squares
Mean
squares
Total
Corrected
149
1.6938
Among
locations
1
1.3309
1.3309
Among fields
12
within
locations
.0908
.0076
2
2
a +9.470f
Among plots
within
136
locations
.02721
.0020
a
1\2
.00059, ap
Table 3.5
..
=
Expected
mean squares
%of
variance
component a
2
2
a +14.39Of+67.32op
2.64
2
9.01
.0196
Analysis of variance for logarithm of variances
%of
Mean
squares
Expected
mean squares
62.3744
62.3743
222
a +14.390f+67.3201
Among fields
12
within
locations
3.4763
.2897
2
2
a +9.470f
Among plots
within
136
locations
13.0887
.0962
a
Source
d.L
Sum of
squares
Total
Corrected
149
78.9394
Among
locations
1
1\2
cr
=
88.34
1\2
.0962, of
=
2
variance
component a
88.76
1.97
9.28
1\2
.0204, 0 = .9207
1
a Each variance component estimate is expressed as a Rercentage of
the sum of the three variance component estimates (l·~·, ~ +
+
01 cri).
20
distances.
In general, the relative sizes of the variance components
for mean distances and the logarithm of variances were very similar.
This is related to the observation which will be discussed later that
for the density function chosen to represent cotton stands the mean
and variance are highly correlated.
The observed frequencies of the plar"t distances are reported in
Section 9, Tables 9.1 and 9.2.
The data were compiled in 11 classes.
The means and variances were calculated directly without the use of
group formulae.
Since the most commonly occurring distance was .1
feet, the classes were chosen as .05-.15, .16-.25, .26-.35, .36-.45,
.46-.55, .56-.65, .66-.76, .76-.85, .86-.95, .96-l.05 and the last
class was provided for the distances occurring at greater than 1.05
feet.
3.2
Modeling the Plant Distances
In order to determine a suitable distribution for the cotton plant
spacing, several distributions were given consideration and the procedure
was as
fo11ows~
In Section 3.1, we discussed the division of the range of the cotton
plant spacing into 11 intervals.
If F(Y) is the true distribution
function of the spacing, we can compute p., the probability that Y will
1
fall in the i
th
interval, from:
Pl· - Pr (a.1-1 <Y < a.)
1
21
a.
.- S
1.
f{y')dY
ai~l
- F(a.)
- F(a.1.- 1)
1.
where a. and a. 1 are the limits of the i
1.
1.-
th
class interval.
The expected number of observations falling into the i
is obtained by e.
1.
~
th
interval
n • p. where n is the number of observations in
1.
the sample.
The p. 's representing individual classes may vary depending upon
1.
which distribution function is used.
Because researchers in the past
have reported use of the negative binomial distribution for fitting
plant distribution data, this function initially was used for obtaining
the p. 's.
1.
Using this method for the negative binomial distribution, we have:
p {y
r
= y) = (r+y-l)pr qy
y
It equals the probability that Y falls in specific intervals where
1
r is positive integer.
For example, ~(o ~ Y ~ 1) =
I: f(y) and so on.
y=o
The chi-square goodness of fit test was used to test at a
significance level of a the hypothesis that the negative binomial
approximated the true distribution function of cotton plant spacing.
For a better understanding of this chi-square test and its
theoretical basis, a discussion is given in Section 9.1.
Since the values of chi-square for the goodness of fit tests for
all plots were very large, (ranging from 126 to 825) the hypothesis
22
that the negative binomial distribution approximated the true distribution
function was rejected.
Next the Gamma Distribution was fitted.
This distribution is a
2-parameter family distribution with a density defined as:
f(x;
Cl',
e) =
1
eCX+l
Cl'.
x
e
xCl'e
a <x<ao
I
a < e<ao
=a
elsewhere,
-1 < Cl' < ao
The parameters are Cl' and e.
Since changing
e merely
changes the scale on the two axes, an
extensive search was made to examine the distribution for different
values of
Cl'.
From results of the chi-square goodness of fit test, the
most suitable distribution was the negative exponential which is a
special case of the Gamma when the value of Cl' is equal to zero.
This
distribution will be singled out for extensive description and discussion
throughout the remainder of the thesis.
.
23
4.
4.1
RESULTS AND
DISCUSSION
Distribution of Cotton Plant Spacing
From the chi-square goodness of fit, the negative exponential was
the only distribution studied that provided a good fit to these data.
2
For tests involving this distribution, the values of X for 207 of 214
plots were not significant at the 5 percent level of probability.
A
summary of the tests of goodness of fit for the negative exponential is
given in Table 4.1.
Since the plant spacing data follow the negative exponential
distribution quite well, some properties of this distribution are
discussed as
follows~
a random variable X is said to have the negative
exponential distribution if its probability density is given by:
x
fEx; e)
=
e1 e
e
O<X<CD
o<e<CD
o
where
e is
elsewhere
the parameter of the distribution.
is the mean) of this distribution is~
CD
E(X) =
So !e e
::: e
x
exdx
The first moment (which
Table 4.1
Summary of the result of the X2 goodness of fit using the negative
exponential distribution for data obtained in 1968 and 1970
Sample
Designation
1
2
3
4
5
6
7
8
e
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23.
•
d.f.
2
X
Value
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
11.79
5.67
6.56
5.94
7.24
5.15
6.04
3.92
7.99
3.94
5.80
6.64
5.54
7.16
5.34
9.09
7.70
6.37
9.02
8.90
15.40
11.72
15.73
Sample
Designation
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
d.f.
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
e
2
X
Value
Sample
Designation
d.f.
4.27
5.95
6.68
6.69
5.14
6.93
7.81
10.15
12.30
11.53
6.60
5.27
5.13
6.10
9.41
5.22
5.51
5.56
6.70
6.76
10.05
7.26
7.15
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
Sl,Fl,Fl
S2,Fl,Ll
S3,Fl,Ll
Sl,F2,Ll
S2,F2,Ll
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9,
2
X
Value
12.30
5.27
9.45
78.97*
9.84
5.06
12.68
9.79
8.73
12.77
9.98
6.58
11.50
22.20*
8.39
11.16
13.61
9.55
5.84
8.30
7.37
7.92
~.96
e+
[\)
e
e
•
Table 4.1
Sample
Designation
S3,F2,Ll
S4,F2,Ll
S5,F2,Ll
Sl,F3,Ll
S2,F3,Ll
S3,F3,Ll
S4,F3,Ll
Sl,F4,Ll
S2,F4,Ll
S3,F4,Ll
S4,F4,Ll
S2,F5,Ll
S2,F3,Ll
e
(Continued)
2
d.f.
9
9
9
9
9
9
9
9
9
9
S3,F5,Ll
S4,F5,Ll
S5,F5,Ll
Sl,F6,Ll
9
9
9
9
9
9
9
S2,F6,Ll
S3,F6,Ll
S4,F6,Ll
S5,F6,Ll
9
9
9
9
S6,F6,Ll
9
9
9
S7,F6,Ll
Sl,F7,Ll
!
'X.
Value
11.55
8.04
Sample
Designation
S2,F'7,Ll
Sl,F8,Ll
9.86
S2,F8,Ll
7.03
7.89
8.66
6.48
Sl,F9,Ll
S2,F9,Ll
S3,F9,Ll
S4,F9,Ll
S5,F9,Ll
S6,F9,Ll
12.99
7.69
9.56
5.27
8.84
7.06
6.04
6.83
8.91
15.27
10.86
8.71
9.10
10.51
11.56
8.01
13.86
S7,F9,Ll
S8,F9,Ll
S9,F9,Ll
Sl,Fl0,Ll
S2,Fl0,Ll
S3,Fl0,Ll
S4,Fl0,Ll
S5,Fl0,Ll
S6,Fl0,Ll
S7,Fl0,Ll
S8,Fl0,Ll
S9,Fl0,Ll
S10,Fl0,Ll
Sl,Fl,L2
S2,Fl,L2
2
d.f.
9
9
9
'X.
Value
10.47
13.68
12.23
10.46
Sample
Designation
S3,Fl,L2
S4,Fl,L2
2
d.f.
7
7
9.31
14.08
S5,Fl,L2
S6,Fl,L2
Sl,F2,L2
S2,F2,L3
7
7
7
7
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9.01
8.88
16.41
10.86
S3,F2,L2
S4,F2,L2
S5,F2,L2
S6,F2,L2
7
8
8
10.93
8.65
14.70
10.88
8.61
9·99
10.41
12.11
S7,F2,L2
S8,F2,L2
S9,F2,L2
S10,F2,L2
Sll,F2,L2
S12,F2,L2
S13,F2,L2
s14,F2,L2
S15,F2,L2
s16,F2,L2
14.74
9
7
7
9.48
9
9
9
9.19
10.43
9.95
11.33
9
7
7
7
'X.
Value
8.61
6.73
4.36
11.26
12.38
11.29
12.50
11.63
10.76
12.18
12.23
8.59
11.24
11.64
S17,F2,L2
9
9
9
9
9
7
7
9
s18,F2,L2
S19,F2,L2
S20,F2,L2
7
7
7
12.63
11.10
10.50
13.82
10.42
11.83
8.32
11.27
16.20*
12.71
I\)
\.Jl
e
e
e
Table 4.1
Sample
Designation
S21,F2,L2
S22,F2,L2
S23,F2,L2
s24,F2,L2
S25,F2,L2
s26,F2,L2
S27,F2,L2
Sl,F3,L2
S2,F3,L2
(Continued)
d.f.
2
X
Value
7
7
7
7
7
7
7
9
8
8
8
18.43*
10.53
13.57
11.74
13.80
12.72
9.61
6.83
12.98
S6,F3,L2
8
8
6.59
12.83
S7,F3,L2
S8,F3,L2
7
7
S9,F3,L2
S10,F3,L2
Sll,F3,L2
S12,F3,L2
7
7
7
7
9·30
12.54
10.68
6.01
11.68
6.08
S13,F3,L2
s14,F3,L2
S15,F3,L2
s16,F3,L2
7
7
9
7
S3,F3,L2
S4,F3,L2
S5,F3,L2
13.55
10.05
7.27
10.10
8.03
10.96
11.21
Sample
Designation
s18,F3,L2
S19,F3,L2
S20,F3,L2
S21,F3,L2
S22,F3,L2
S23,F3,L2
s24,F3,L2
S25,F3,L2
s26,F3,L2
S27,F3,L2
s28,F3,L2
S29,F3,L2
S30,F3,L2
S31,F3,L2
S32,F3,L2
d. f.
9
9
9
7
7
7
8
9
9
9
9
9
8
8
8
S33,F3,L2
S34,F3,L2
S35,F3,L2
Sl,F4,L2
9
8
S2,F4,L2
S3,F4,L2
S4,F4,L2
9
9
8
S5, F4,L2
S6,F4,L2
9
7
7
8
2
X
Value
12.06
5.80
11.60
14.33*
12.30
13.84
12.70
10.58
16.15*
9.55
12.80
9.03
10.02
9.09
6.31
7.32
9.19
5.07
11.21
11.68
5.80
8.13
7.16
12.55
Sample
Designation
S7,F4,L2
S8,F4,L2
S9,F4,L2
S10,F4,L2
Sll,F4,L2
S12,F4,L2
S13,F4,L2
s14,F4,L2
S15,F4,L2
s16,F4,L2
S17,F4,L2
S18,F4,L2
S19,F4,L2
S20,F4,L2
S21,F4,L2
S22,F4,L2
S23,F4,L2
s24,F4,L2
S25,F4,L2
s26,F4,L2
S27,F4,L2
s28,F4,L2
S29,F4,L2
d.f.
2
X
Value
7
7
7
8.83
6.71
14.05
9
9
7
8
8
6.38
14.10*
4.51
9
7
7
7
8
'7
'7
8
8
8
8
8
8
8
7
7.89
16.08*
6.45
6.05
14.35*
8.27
6.71
4.50
11.52
10.11
8.55
10.15
13.31
8.66
8.04
10.76
12.56
S30,F4,L2
6.78
I
S31,F4,L2
7.74
7
2
2
* The computed X value is greater than the tabulated X for corresponding d.f. and 5% level
of probability
S17,F3,L2
7
'7
(\)
0'\
27
The second moment is:
co
=
_
So !e e
= 2e2
•
The variance is
2
(J
x
2
2
= E(X ) _ E (X)
2
2
= 2e _ e
2
= e
.
A property of this distribution which is useful from a practical point
of view is that the mean and standard deviation are equal.
4.2
Estimation of the Parameter of the Negative
Exponential Distribution
The negative exponential distribution involves the parameter
When applied to plant spacing data,
deviation of the plant spacing.
e refers
e.
to the mean and standard
Thus, if the value of
e is
known
exactly, our state of knowledge concerning the outcome of cotton plant
spacing is complete.
In general, the parameter
but we can make an estimate of
taken from a cotton field.
involved,.
e from
e is
not known exactly,
spacing data obtained from samples
Thus, the general problem of estimation is
28
From a series of observations or measurements, it is required to
make some sort of a statement about the parameter of the distribution
from which the measurements were taken.
The inferences or estimates
that may be made from a set of observations, or sample, as it is usually
called, are open to error.
However, it usually is possible to assess
the extent of this error and thus have a measure of the preciseness of
the inference.
By the method of estimation given in Section 9.2, the minimum
variance unbiased estimator of
A
e is
n
e = t x.1
n=l
l
n
found to be:
29
5.
CONSTRUCTION OF A CONFIDENCE INTERVAL FUR THE PARAMETER
OF THE PLANT SPACING DISTRIBUTION
5.1
Confidence Interval for the Parameter of the Negative Exponential
when Sample Size, n, is Less than 100
In addition to estimation of the parameter of the sampling
distribution, it is also useful to know how accurately we have determined
the estimate;
!.~.,
within what interval we are fairly certain that it
lies.
The generally accepted method of approaching this aspect is to
construct what are known as confidence limits.
lower 100 (1 -
a)
We derive upper and
percent limits such that, when we say that the true
a)
value of the parameter lies in that interval, 100 (1 -
percent of
all such statements will be correct and 100a will be incorrect.
Obviously, we choose a small, for example, .05 or .01.
The general method of construction of confidence limits is given
in Section 9.3.
For our case
X
fAX, e)
1
= - e
e
=0
The density function
follows:
"e,
e
X~O,
6>0
elsewhere
EK/n, can be obtained for our case as
The moment generating function of f(X;
function of variable t, is defined as follows:
e),
M (t) which is a
x
30
~ (t) = E[e tx ]
t
CD
=S
f(x)e x dx
_CD
which is for our case
CD
_
1
=
So -a e
=
(1- at)
x
a e tx dx
1
The term "moment generating function" arises from the fact that
~
(t) does generate all the moments.
The moment generating function
(if it exists) is unique and completely determines the distribution of
the random variable.
We can use one of the property of moment
generating function:
Let
Xl,
X , ••• ,
2
~
be N independent random
variable having a distribution function F(X), and let C , C2 , ••• , C
l
N
be any constant, then we have
Sor for n
random samples taken from the density (5.1), the random variable
n
Z = t X. has a moment generating function
i=l J.
n
However, it is not
n
rather in
t
. 1
J.=
X.I
J. n
t X. with which our primary interest lies but
i=l J.
which =
"a.
Therefore we let
n
tx
Z i=l
W=-=--=x.
n
n
31
Then the moment generating function of
~(t)
=
x = a is
A
(1 _ ~t)-n •
We know that the above moment generating function corresponds to the
Gamma Distribution, with a
density~
_
A
g( a; a) -, g{x)
~-le-W/(e/n)
= .;.;..---:;.---
r(n) (~)n
n
where r{n) is the Gamma function.
In the exponential density function, if we take n samples the
joint distribution of this new random variable, Z = DC will be
= -1-
g(z;a)
r~n) an
z
n-l e -z/ a •
we make another transformation as follows:
Let
The Jacobian of the transformation which is the determinant of the
first partial derivative of the inverse function
Then the density of q can be expressed
w(q)
9 >
as~
(qe)n-l e -,(q9)/a.
=
I,J I = e since
a
r(n) an
= q
n-l -q
e
r~n)
The resulting distribution is free of the parameter, 9.
fund two numbers t
l
and t
2
such that
Now we may
o.
32
=Pr [DC s:
at }
_Pr[DC
t,
s:e}
1
..L
and
One rational basis for particularizing the choice would be to
make the interval (~,e) as short as possible, in some sense.
In practice,
for symmetrical distributions it is standard practice to place equal
probability in each tail, !.~., to put PI
= a/ 2 = 1
- P2' so that
P2 = 1 - a/2.
For any common asymmetrical distrubution, the possible gain in
shortening the interval by adjusting the two tail areas is outweighed
by practical convenience of equal areas.
Of course, if the consequences
of being in error in one direction are more serious than the consequences of being in error in the qther direction, then we may construct
asymmetrical limits.
In our case, the distribution function
~(q)
is not symmetric,
therefore putting equal probability in the tails does not give the
~
33
shortest confidence interval.
However, the consequences of error are
no more serious in one direction than in the other.
Thus, for the
purpo se of convenience we will put equal probability in the tails.
In putting equal probability in the tails, we have
Pr[~ <
e<
2
so we need to find t
l
and t
~x]
(5.10)
= 1 - O!
1
2
in the integral, ~(q), lo~.,
Since the value of this integral is not accessible, we proceed to
evaluate it as follows:
t
Pr[Q~t]
=S
°
q
n-l-q
e
dq
fen)
o<q<co
This integral must be evaluated by numerical methods unless n is a
positive integer, in which case the function can be found by successive
integration by parts to be
2
F(t)
where (n-l)~
=
t
1 - [l+t+ 2~ +
= (n-l)(n-2) ••• 1.
t n- l
-t
+ (n_l)]e
We computed the values of this integral
for values of n from 10 to 100, for probabilities .050, .100, .250, .500,
.750, .900, .950, .975, .990 and .995 and these are tabulated in
Table 5.1.
To illustrate the use of the table (501) for computing a confidence
interval for
e,
we compute as an example for n = 30 (the number of plant
spacing measurements) and
O!
= .1 as follows:
Table
Value
of n
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
5.1 Pr (Q so t)
t
=::
J
rhT
o r
n
34
qn-.le·~qdq
pdT :s:: t)
.950
5.5
6.3
6.9
7.8
8.6
9.4
10.2
10.9
11.5
12.5
13.3
14.3
14.9
15.8
16.5
17.5
18.2
19.5
19.8
20.8
21.9
22.8
23.8
24.5
24.5
25.8
26.5
27.8
28.5
29.5
30.5
31.4
31.9
32.9
33.9
34.8
35.5
36.5
37.5
38.5
38.9
.900
6.2
7.1
7.9
8.9
9.8
10.5
11.3
12.0
12.9
13.8
14.8
15.4
16.3
17.2
18.0
18.9
19.5
20.9
21.5
22.5
23.5
24.3
25.0
26.0
27.0
27.9
28.5
29.5
30.5
31.5
32.6
33.5
34.0
35.0
36.0
36.8
37.9
38.5
39.5
40.5
41.5
.750
'7.8
8.3
9.5
10.5
11.5
12.4
13.5
14.2
15.0
15.9
16.7
17.9
18.7
19.6
21.0
21.5
22.5
23.5
24.5
25.5
26.2
27.2
28.2
28.9
29.9
30.5
31.9
32 .9
33.9
34.9
35.8
36.5
37.5
38.6
39.5
40.6
41.8
42.5
43.5
44.5
45.5
.520
9.9
10.9
11.9
12.8
13.6
14.8
15.9
16.9
17.5
18.5
19.9
20.8
21.7
22.7
23.9
24.9
25.6
26.6
27.8
28.8
29.5
30.8
31.8
32.9
33.8
34.5
35.5
36.8
37.9
38.8
39.9
40.5
41.5
42.9
43.8
44.7
45.9
46.8
47.9
48.9
49.8
.250
11.9
13.1
14.6
15.3
16.5
17.4
18.4
19.8
20.4
21.6
22.9
23.8
25.0
26.1
27.1
28.1
29.7
30.5
31.5
32.5
33.5
34.5
35.9
36.9
37.9
38.9
39.9
40.8
41.9
43.1
44.1
45.3
46.2
47.8
48.5
49.8
50.6
51.5
52.5
53.9
54.9
.100
14.5
15.9
16.9
18.0
19.0
20.5
21.5
22.9
23.9
24.9
26.0
27.2
28.2
29.4
30.9
31.9
33.0
34.0
35.0
36.3
37.2
38.2
39.9
40.8
42.0
43.0
44.0
45.0
46.5
47.9
48.5
49.2
50.9
52.0
53.0
54.0
55.0
56.0
57.2
58.9
59.9
.05
15.9
17.0
18.2
19.8
20.9
22.0
23.8
24.3
25.9
26.9
2'7.9
29.3
30.3
31. ')
32.9
33.9
34.9
36.5
37.5
38.9
39.9
40.5
41.9
43.0
44.5
45.8
46.8
47.6
48.8
49.9
51.0
52.1
53.9
54.5
55.9
56.8
57.5
58.9
60.0
61.8
62.9
.025
17.0
18.2
19.9
20.9
22.2
23.8
24.9
26.0
27.8
28.5
29.9
30.9
32.1
33.4
34.6
35.8
37.0
38.1
39.5
40.5
41.9
43.0
44.0
45.4
46.5
47.9
48.9
49.5
51.0
52.1
53.5
54.3
55.9
56.8
58.0
59.8
60.5
61.5
62.9
63.8
65.0
.01
18.9
20.9
21.8
23.0
24.8
25.9
26.9
28.4
29.9
31.0
32.0
33.9
34.4
35.6
37.0
38.8
39.9
41.0
42.0
43.0
44.9
46.0
47.0
48.0
49.5
50.5
51.9
52.3
54.0
55.0
56.5
57.2
58.9
60.0
61.0
62.5
63.5
64.2
65.9
67.0
68.0
.005
20.0
21.9
22.9
24.9
25.8
27.0
28.2
29.9
31.0
32.2
33.8
35.8
36.0
37.3
38.9
40.0
41.2
42.3
43.9
45.0
46.0
47.1
48.1
49.9
50.5
52.8
53.9
54.9
55.9
57.0
58.5
59.2
61.0
62.0
63.0
64.6
65.5
67.0
67.9
69.0
70.5
35
Table 5.1
Value
of n
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
.950
39.5
40.9
41.5
42.5
43.5
44.5
45.5
46.6
47.0
48.0
48.9
49.9
50.7
51.9
52.5
53.8
54.8
55.2
55.9
56.6
57.3
58.1
59.0
61.1
61.8
62.5
63.4
64.2
65.0
65.8
66.9
67.8
68.5
69.6
70.6
71.5
72.6
73.3
74.3
75.0
75.8
76.9
77.8
78.7
79.7
80.8
81.6
82.5
83.6
84.5
(Continued)
.900
42.8
43.0
44.0
45.0
46.0
47.0
47.8
48.5
49.9
50.5
51.9
52.5
53.2
54.0
55.0
55.9
56.8
58.0
58.9
59.7
60.8
61.7
62.9
63.2
64.5
65.4
66.0
67.0
68.0
69.0
70.0
71.0
71.9
72.5
73.5
74.5
75.5
76.3
77.4
78.3
79.1
80.0
81.0
82.0
83.1
84.0
84.9
85.3
86.7
87,8
.750
46.0
47.0
48.0
48.9
49.5
50.8
51.9
52.9
53.8
54.8
55.8
56.8
57.5
58.5
59.5
60.7
61.8
62.6
63.3
63.2
64.4
65.2
66.5
68.0
69.0
69.9
71.0
71.9
72.9
73.9
74.9
75.8
76.9
77.8
78.8
79.8
80.8
81.8
82.8
83.5
84.4
85.5
86.7
87.6
88.3
89.3
90.6
91.4
92.5
93.4
.520
50.9
51.9
52.5
53.9
54.5
55.9
56.9
57.9
58.9
60.0
60.9
61.9
62.9
63.8
65.0
65.9
67.0
68.0
68.9
70.0
71.2
72.1
73.2
73.4
74.9
75.9
76.9
78.0
78.9
79.9
80.9
81.9
82.8
83.8
84.9
85.9
86.8
87.8
88.8
90.0
90.5
91.8
92.8
93.8
94.8
95.9
96.7
97.8
98.8
99.7
.250
55.9
56.8
57.8
58.9
60.0
60.8
61.9
62.9
64.0
65.0
66.3
67.3
68.5
69.2
70.5
71.5
72.5
73.9
74.5
75.6
76.9
77.8
79.9
80.8
80.8
81.8
82.8
83.8
84.9
85.8
86.8
87.9
88.9
90.0
91.0
92.2
93.3
94.3
95.3
96.4
97.3
98.5
99.5
100.4
101.6
102.8
103.5
104.9
105.6
106.5
Pr(T ~t)
.100
60.8
61.2
62.8
64.0
65.0
66.0
67.0
68.0
69.5
70.9
71.5
72.9
74.3
74.8
76.0
77.0
78.0
79.0
80.0
81.1
83.3
84.2
85.1
85.9
86.8
87.9
89.0
90.0
91.0
92.0
93.0
94.0
95.0
96.0
97.5
98.2
99.6
100.2
101.5
102.5
104.0
105.0
106.0
107.0
108.0
109.1
110.0
111.0
112.1
113.0
.05
63.9
64.8
66.0
66.0
67.9
69.0
70.0
71.0
72.5
74.0
74.9
76.0
76.7
78.0
78.9
80.0
81.0
82.9
83.9
85.0
85.9
87.1
88.2
89.1
89.8
90.5
92.0
93.0
94.8
95.5
96.9
97.5
98.5
99.5
100.5
101.4
102.5
104.0
105.0
106.5
107.5
108.5
109.5
110.5
111.5
112.6
113.6
114.8
116.0
117.0
.025
66.0
67.0
68.5
69.5
71.3
71.9
73.0
74.0
75.0
76.0
77.5
78.5
79.6
80.5
81.9
82.9
84.0
85.2
86.6
87.8
88.9
89.1
90.3
91.8
93.0
94.0
95.5
96.2
97.5
98.5
99.5
101.0
101.8
103.0
104.0
105.3
106.4
107.5
108.4
109.5
110.5
111.8
112.8
114.0
115.0
116.4
117.5
118.3
119.4
120.8
.01
69.9
71.0
71.9
72.5
74.0
75.0
76.5
77.9
79.0
80.0
80.9
82.0
83.0
84.5
85.9
86.9
87.9
89.0
90.0
90.9
91.8
93.0
93.9
95.8
97.0
98.0
99.0
100.0
101.0
103.0
104.0
104.5
106.0
107.0
108.0
109.0
110.0
111.0
112.8
113.8
115.0
116.0
117.0
118.9
119.5
120.8
121.3
122.9
124.1
125.2
.005
72.0
73.0
74.0
75.0
76.0
77.5
79.0
80.0
81.0
81.9
83.0
84.5
85.9
87.0
88.0
89.0
90.0
91.2
92.8
93.5
94.5
95.7
96.9
98.1
99.9
101.0
101.8
102.5
103.9
105.0
106.0
107.6
108.5
109.5
111.0
112.0
113.0
114.0
115.5
116.0
117.5
119.0
119.5
121.0
122.0
123.3
125.0
125.9
126.8
128.0
36
From equation (5.10) we have
<
e<
nx
-]
ex
t ,l.l
.~
•. ,
1
c~
ex
.
'2
The value of t , ex/2 and t , 1-0'/2 may be obtained from
2
l
Table 5.1.
= .05
The value of t 2 , a/2 for n ~ 30 and 0'/2
from
Table 5.1 is equal to 39.9 and the value of t , 1 - ex/2 for n =, 30,
1
and 1 - 0'/2 :; .95 is equal to 21. 9.
By substituti.ng these values in
the above probability statement, we have
30x
PI' [39. 9
We may say that
e lies
<
e<
30x
21. 9 J-
. 90.
in the interval with probability of .90.
5.2 Shortest Confidence Interval for the Parameter of
the Negative Exponential Distribution when ni < 100
Since the probability density function ~(q) i.s not symmetric, we
proceed to find the shortest confidence interval as follows:
From equation (5.9) on Section (5.1) we had
n-l -q
,I'(q'
...
'I'(~)
_
q,
e
,.
I
,.
e - f(n)
The integration needed to be performed is:
t
S2
t
~(q)dq ~ 1 - ex
l
First the value of (I-ex) is chosen, say .95, then t
found for the confidence interval
2
and t
1
will be
37
Pr [t
1.
< nx < t ]
T
1. _
::=
2
ct
or
The length of confidence
interva1.~
L, is
using the Lagrange multiplier to mi.nimize the length of confidence
interval subject to the restriction (5.12) we have:
L*
-
1.
= nx(--.
.t
1.
- --) -
2
t
l
1
~1q)dq - (l-ct)),
1
where A is the Lagrange multiplier.
-
t
A<St 2
Equation (5.13) is equivalent to:
1
L* '" nx(t - t ) - M(t 2 ) + M(tl )
2
l
where ~(t2) and ~(tl) are the cumulative distribution function of ~
evaluated at t
2
and t , respectively.
l
Now differentiating L* with respect to t
and t
l
2
and equating to
zero, we obtain:
oL* = n-x{- -1:.)
2
~t
o
t
1
'I
=
0
1
oL*
- 1
--ot = nX(--2)
t
2
t1t(t)
1
+~,
- A~£t2)
=0
2
where ~(t2) and ~(tl) are the points at which the probability density
function of q are evaluated at t
and (5.15) for A we get
2
and
tl~
respectively.
Solving (5.14)
38
me
which can be written
nx
as~
(5.16)
where t
l
and t
2
the sample size.
t
l
can be obtained for desired probability level and n,
The ~(tl) and ~(t2) are the ordinates of~(q) at
and t , respectively.
2
To obtain the shortest confidence interval for
(5.16) must be satisfied.
To do
this~
e,
the equality
one must find the ordinates
~(tl) and ~(t2) which is time-consuming.
Another procedure for deriving
the confidence interval for n > 100 is given in the next section.
5.3
Construction of a Confidence Interval for the Parameter
of the Negative Exponential Distribution When n > 100
It is desireable to make the confidence interval as short as
possible and this can be achieved by increasing the sample size.
Since
obtaining a sample size of cotton plant spacing of about 200 to 300 is
simple and inexpensive, the distribution of ~(q) when n becomes large
was given consideration.
39
The distribution of ~(q) tends toward the normal distribution as n
becomes large.
Although as n becomes
the normal
the distribution of
large~
we found an appropriate
distribution~
~
tends toward
transformation~
such
that the transformed variable tends toward the normal distribution more
readily.
Consider the density of
~{q) =
e
~~
-qq n-l
O<q<~
r(n)
_. 0
elsewhere
We make the transformation
~
=
v
[~ +
n
jn -
1/2J2
where n is the sample size and y is normal deviate.
distributed approximately normal if n > 100.
Then
~
is
As n increases, the
approximation is closer to normal.
One of the advantages of using the normal approximation is that
the values of the normal distribution are tabulated and accessible.
Since the normal distribution is symmetric, we may put equal probability
in each tail and achieve the shortest confidence interval.
To illustrate the use of the given transformation to normal for
construction of a confidence
a
interval~
we compute the confidence limits
for the parameters (mean) of the negative exponential distribution for
case where n = 200 and ~ = .05.
mean is .3.
Also suppose that
The probability statement is
x estimated
sample
40
The values of ~ and ~ are obtained by using (5.17) as follows:
The value of y is found in a normal table for probability .025 to
be -1.96 for the left tail, and the valQe of y is equal to 1.96 for the
right tail.
Substituting these values in the
=
(5.18)
we have:
22801.6.
For
By substitution of values
~
and ~, x in to
Pr [200 ( •3) <
228.16
Pr[.263 <
(5.18)
we have
e < 200 ( •3) J = 95
172.79'
e < .347J
==
.95
These limits provi.de the cotton producer and farm manager with a
guide to the plant stand status and also provide information the
precision upon which plant stand estimates are based.
•
41
6.
DEVELOPMENT OF A PIANT STAND INDICATOR
AND ITS APPLICATIONS
6.1
Plant Stand Indicator
In Section 4.2 we concluded that the cotton plant spacing
distribution follows the negative exponential and, since the mean and
the standard deviation of the distribution are the same, it was pointed
out that one can make an estimate of the uniformity of plant stand by
measuring only the mean of cotton plant spacing.
The degree of
uniformity and accuracy would be based on the closeness of the estimate
of the mean to the intended spacing,
~.!.,
the planter setting.
Since cotton planters may have different settings, for example,
.1, .2 feet, etc., it is desireable to have a measurement of plant
stands which is not dependent on the cotton planter setting.
In order
to develop such an indicator, we insert the planter setting as another
parameter, 6, in the negative exponential distribution.
Then we have
_ (x- 0)
e
1
f(x; 0, e) = - e
e
x
~
0
~
0,
e>0
(6.1)
elsewhere.
= 0
To find the mean and the variance of this distribution, we proceed as
follows:
Let y
=x
- 0, then
f(y, e)
Y..
= -e e e
y ~ 0, 0 ~
=0
elsewhere.
1
°
42
Since we found the first and second moment forf,(x, e) in Section
2.2.1, by substitution for E(X-6) for (4.1) we can make a similar
substitution forE(Y) to find the first and second movement for f(x;e,6).
E(Y)
= E(x
- 6),
E(X)
= E(Y)
+ 6
or
= e + o.
The variance of X
by substituting for X we have
=
222
E[Y + 0 + 2Yo] - (e + 6)
= e2 .
So the mean increased by 0, but the variance remained the same.
Now, by subtraction of the planter setting from the estimate of the
mean distance, we obtain a quantity herein referred to as stand indicator
which is not dependent on the planter setting.
c.
Estimation of the parameter
Although the parameter C, the planter setting is known, one should
be able to estimate this parameter from the data.
We used the maximum
likelihood method for estimation.
The likelihood function of n independent random samples from the
density (6.1) is
n
-
(x. - c)
1::
1
L = - e i=l
1.
e
x
en
;;=:
c ;;=:
0
e >0
l
This function is maximized at xl the first order statistic, !.~.,
the minimum of X. is the maximum likelihood estimate of
1.
c.
However, in
practice due to some factors such as planter vibration, soil physical
condition, etc., a single minimum of x., would not give a good estimate
1.
of
c.
Since we usually obtain a frequency of plant distances, and as
our data has shown, the most frequent plant distances is the reliable
estimate of the
6.2
6.2.1
c.
Use of Plant Stand Indicator in Regression Model
Plant. Stand Indicator as an Independent Variable
The plant stand indicator may be used as an independent variable
...
in combination with some environmental factors and harvesting procedures
1
See Sarahan and Greenberg, 1962.
44
to develop a regression equation for predicting the yield of cotton.
The yield obtained (one harvest) from field number 2 in Red Springs,
North Carolina was regressed upon plant stand indicator alone for
purposes of evaluating the need for replanting.
The regression
equation was
Y
" = 1830.2 - 1392.2 SI
(6.2)
"
where Y is the predicted yield of cotton and SI is the value of stand
indicator.
The regression coefficient was significant at the one
percent probability level.
It should be noticed that even for SI = 0
(perfect stand) in 6.2 the maximum yield is equal to 1830.2.
not necessarily the maximum yield that can be obtained.
This is
For instance
Toomey (1962) reported that in some cases 2747 pounds cotton was
obtained in the Piedmont Research Station.
as a more conservative figure.
But we used 1830.2 pounds
It is recognized that by entering other
variables into the regression, the plant stand indicator coefficient
probably would change somewhat, but for our purposes this simple
regression was thought to be adequate.
6.2.2 Use of Plant Stand Indicator for Replanting
To assess the need for replanting, a cotton producer needs to
measure the uniformity of cotton stand after the seedlings have emerged.
The results of this study showed that he can achieve this by taking
samples from his field, estimating the mean of the plant spacing and
subtracting the planter setting value from the sample mean to obtain
the stand indicator.
He also can find a confidence interval at the
desired probability level for the parameter.
The question arises as to what constitutes a satisfactory value of
the plant stand indicator.
In the ideal situation this value is zero
(perfect stand), but usually in actual field practice it is greater
than zero.
By using the prediction equation obtained in Section 6.2.1,
the cotton producer can predict the yield of cotton for the particular
value of stand.
The product of the yield and the price of cotton (a
weighted price including government payment and market list price and
the price of seeds) which now is about .156 cents per pound,
1
gives
the value of cotton per acre.
To make any decision as to replanting, two things should be
considered.
First, the chance of obtaining a better stand is usually
increasing from April 15 to May 20 in North Carolina.
Second, although
the stand may improve over time as planting is delayed in the spring,
the chances of a high yield are decreased as planting is delayed.
The mean of the stand indicator for Red Springs and Laurinburg
was .27 feet.
Since this was derived for the early planting, it is
reasonable to assume that the later planting would result in a stand
indicator at least as low as .27 feet.
IThe $.156 was derived as follows: Usually 33 percent of cotton
yield consists of lint which was priced at $.22 per pound in January 1971.
The 67 percent remainder is cotton seeds which was priced at $.02 per
pound. The government payment currently is $.195 per pound of lint.
Therefore the value of yield of cotton would be $.156 per pound. These
values and costs for production were obtained from Crops for Profit, 1971,
The North Carolina Agricultural Extension Service, Circular No. 519.
46
By using equation 6.2 for SI
will be estimated.
:=:
.27, the expected yield of cotton
However, this expected yield is not realistic,
since a reduction in yield due to late planting was not taken into
consideration.
The percent of maximum yield of time of planting decreases
steadily from April 20 to May 20.
Table 6.1
2
This is shown in Table 6.1.
Percent of maximum yield of cotton for different dates
of planting
Percent of maximum yield
Date of Planting
April 10-20
100
April 25
98.7
April 30
96.8
May 5
May 10
93.7
May 15
83.7
May 20
75.0
90.0
So the expected yield obtained by using SI
=
.27 in equation 6.2 should
be multiplied by percent of maximum yield for the specific date of
replanting.
There would be an additional cost for replanting which is
currently about $5.50 per acre. 3
Then the total cost of production
2Dr • J. Phillips, Associate Professor, Department of Soil Science,
North Carolina State University (personal communication).
3Crops for Profit,
Ope
cit.
47
would be $177.32 per acre.
The decision rule therefore
is~
if the
expected net revenue obtained after replanting is higher than one by
expected from the existing stand, the cotton producer should replant.
Cotton producers can follow the procedure summarized above by
carrying out the following steps:
1.
Estimate the stand indicator in the field.
2.
Compute the expected yield using equation 6.2 for the value
of stand indicator obtained in Step 1.
3.
Obtain the value of the yield using the current cotton price
and government payment to cotton producers.
4.
Subtract from the value of y"ield obtained in Step 3 the total
cost of production (currently $171.82 per acre) which gives
the net revenue to land and management for the existing plant
stand.
5.
Estimate the uncorrected expected yield using equation 6.2 for
the specific stand indicator.
6.
Obtain the corrected expected yield by multiplying the uncorrected expected yield by the percent. of maximum yield for
the specific date of replanting Jusing Table 6.1).
7. Obtain the value of the corrected yield using the current
cotton prices and government payments to cotton producers.
•
These v"alues for the current cost of production including the
additional assumed cost of $5.50 per acre for replanting and
net revenue assuming $.156 per pound of cotton are given in
Table 6.2.
48
Table 6.2
The net revenue to land and management for replanting
(per acre)
%of
Uncorrected
expected
yield/A
for SI:.::.27
Corrected
expected
yield/A
Gross
revenue
per
A
Total cost
inc. assumed
cost of replanting
Net revenue
to land and
management
Date of
planting
max.
yield
April 25
98.7
1454
1435
223.90
177.32
46.58
April 30
96.8
1454
1407
219 •.50
177.32
42.18
May 5
93.7
1454
1362
212.50
ITT .32
35.18
May 10
90.0
1454
1309
204.20
177.32
26.88
May 15
83.7
1454
1217
189.80
177.32
12.48
May 20
75.0
1454
1090
170.00
177.32
-7.32
8.
If the net revenue to land and management obtained in step 8
is greater than the one estimated for the existing stand
indicator in step 4, replanting would be recommended.
As an example, suppose a cotton producer found that the stand
indicator in his field is .5.
would be 1134 pounds per acre.
Using equation 6.2, the expected yield
By multiplying this yield by the
weighted price per pound of cotton which currently is about $.156, a
value of $176.90 is obtained.
acre.
The total cost is about $171.82 per
Then the net revenue to land and management would be $5.08 per
acre.
Suppose the cotton producer can replant on May 5.
Then, using
the average stand indicator, .27, in equation 6.2, the expected yield
would be 1454 pounds per acre.
The corrected expected yield and net
revenue to land and management can readily be obtained for May 5 from
Table 6.2 ($35.18 per acre).
Thus comparing this net revenue with the
$5.08 figure, one would conclude that replanting would be profitable.
The reader should remember that these results are highly dependent upon
cost and price assumptions.
The most important point to remember here
is that the cost of replanting currently is very small compared to the
total cost of planting management and harvesting.
This would imply
that under a wide variety of circumstances, replanting would be
profitable.
As another illustration, suppose the value of stand indicator in
the field is
...;
.4 feet.
Using equation 6.2, the expected yield for
~
S1= .418 1273.32 pounds per acre.
So the value of the yield would
be $198.63 (using $.156 per pound as before).
land and management is $20.81 per acre.
as late as May 20.
Then the net revenue to
Suppose the replanting date is
The net revenue to land and management by replanting
in May 20 from Table 6.2 is $-7.32 (loss of 7.32 per acre).
Thus
comparing this loss and net revenue with the $20.81 per acre figure
estimated for non-replanting situations, the replanting for this case
produces losses of $28.13 per acre.
Tpis, of course, is figured for the most conservative set of
.circumstarlCes (replanting on May 20).
It appears ~hat under current
replanting cost figures and the current price of. cotton that a critical
stand indicator would need to be reached before replanting would be
. prof~table.
50
Since replanting involves addit.iol2al cost" the corrected expected
yield for different dates of plar:ting should be increased by an amount
of cotton (value) equivalent to this cost.
l
Then a stand indicator
value corresponding to this yield may be calculated.
price of cotton and production
costs~
For the current
the critical stand indicators
were obtained and are reported in Table 6.3.
Table 6.3
The critical values of :3tand Indicat.or where replanting
costs are offset (given by dat;e of planting)
Critical value of
81
Date of
Planting
Expected
Yield/A
April 25
1470
1442
April 30
May 5
May 10
May 15
May 20
1397
1344
1252
1125
where replant.ing
costs are offset
.259
.279
.311
.349
.415
.506
Percent of
desireable stand
assuming desireable has
four plants per foot
69.6
65.9
60.8
55.6
48.9
41.2
So replanting would be profitable if the existing stand indicator for
specific date of replanting is significantly greater than the critical
.
I
values of S1 given in Table 6.3.
IThe amount of cotton required to offset the cost of replanting
is about 35 pounds per acre.
51
Toomeyl in 1963 reported that the stand of 4 to 6 plants per foot
produced the maximum yield, and if the stand is less than 65 percent
of the desireable stand (4 plants per foot) then replanting would be
profitable.
Comparing this decision rule to that using the method
described above the lower bound of maximum yield corresponds to a
stand indicator of 81
of
4
=
.15, and the stand indicator for 65 percent
plants per foot of row is .28.
2
In fact, the critical value of
stand indicator for replanting by Toomey's procedure, 81 - .28, is
very close to the mean stand indicator which was used in construction
of Table 6.2,
~.~.,
SI = .27.
The percent of stand of 4 plants per
foot of row corresponding to critical values of stand indicator are
given in the last column of Table 6.3.
It appears that Toomey's
critical value would be correctly applied if replanting were to be
done during the period from April 25 to 30.
However, it is an over-
estimate of the percentage which should be used if replanting is to
be done at a later date.
In other words if replanting has to be
done in May, and the initial stand indicator was only .28 it is
almost necessary to have a 50 percent reduction in desireable stand
before replanting can be justified economically.
1
Toomey, W. G. 1963. Report of Cotton Plant Stand Studies.
North Carolina State University. Unpublished report.
2
Toomey, W. G. Extension Cotton Specialist, North Carolina
State University at Raleigh (personal communication).
52
6.2.3 Plant Stand Indicator as a Dependent Variable
The variability of stand indicator values may be influenced by
the planter and by environmental factors, such as moisture, temperature,
physical and chemical properties of the soil and management practices.
Thus, these environmental and soil variables can be used to obtain a
regression equation to p.:::'edict stand indicator values.
As an
illustration, stand indicator was regressed on exchangeable calcium
for the
64 plots in Red Springs, North Carolina.
The F value for this
regression was significant at the I percent level and about 15 percent
of the variability in the stand indicator was explained by only this
one soil chemical factor.
.
j
53
7.
SUMMARY AND CONCLUSIONS
This thesis reports the results of an attempt to develop a cotton
stand indicator.
To do this, the following steps were undertaken.
From an 80-acre farm in Red Springs, North Carolina, 64 plots were
selected and the distances between 13,688 plants were measured in 1968.
Each plot was four rows wide by 25 feet long.
Another set of cotton
plant distance measurements were obtained from two different locations
in 1970; the Johns and McNair farms near Laurinburg, North Carolina.
From the Johns farm, 10 fields were selected for sampling and the
distances between 12,765 plants were measured.
From the McNair farm
four fields were chosen for sampling and the distances between 24,784
plants were measured.
To determine a suitable probability density function of cotton
plant spacing, several possible stochastic models were given
consideration.
The chi-square goodness of fit was used to test the
hypothesis that the various distribution functions uPfler consideration
represented the true cotton plant spacing distributions.
The chi-square
values for the negative binomial were very large and therefore the
hypothesis of good fit was rejected.
For most forms of the Gamma
distribution, the chi-square value was very high except for a special
case of the Gamma which is a one-parameter distribution called the
negative exponential distribution.
The chi,-square value for testing
the appropriateness of the negative exponential for 207 out of a total
54
of 214 plots was not significant at the five percent level of probability
and therefore we conclude that this model provided a satisfactory fit.
Estimation of the parameter of the negative exponential distribution was discussed.
~ihe
maximum likelihood estimation of the mean of
the distribution turned out to be the minimum variance unbiased
estimator.
In order to construct a confidence interval for the mean distance,
the theoretical distribution of the mean distance was derived.
Behavior
of the distribution was examined and it was shown that it was aSYmmetrical
for sample sizes less than 100.
Since when the distribution is not
SYmmetrical, putting equal probability in each tail does not provide the
shortest confidence interval, a method for computing the shortest
confidence interval was developed.
Since the percentile probability
table for the distribution does not exist, a table for sample sizes
from 10 to 100 for different levels of probability was constructed
which can be used for construction of the confidence intervals for the
mean distances when the sample size is less than or equal to 100.
It
is desireable to make the confidence interval as short as possible and
this can be achieved by increasing the sample size.
Because obtaining a sample size of about 200 to 300 is simple and
inexpensive, the properties of the distribution for large samples were
considered and it was found that with sample size greater than 100, it
is possible to make a transformation to the normal distribution.
This
expedites matters because probability tables for the normal distribution
are accessible.
were made.
Several examples of the application of the transformation
55
The plant spacing model which included planter setting was
developed.
It was found that the planter setting distance moves the
mean of the plant spacing to the right by this same distance.
The
cotton stand indicator was defined as the mean plant spacing minus the
planter setting distance.
Since the mean and standard deviation of the
plant stand indicator are the same, the stand indicator thus may be
used as a measurement of uniformity of cotton stand.
Replanting when poor stand occurs was considered and a decision
rule for replanting was given.
Feasibility of developing predicti.on equations for the stand
indicator using some soil factors as independent variables was
indicated and as an illustration, a relationship between stand indicator and exchangeable calcium was obtained.
2
The value of R for
this relationship was 15 percent.
The stand indicator may be used as an independent variable along
with soil properties, climatic factors, and management practices in a
regression equation for predicting the yield of cotton.
The stand indicator may also be used for comparison of performance
of different planters.
Development of plant stand indicator may be done for other crops
using simi.lar techniques.
The form of density function may vary for
each crop and seed planter types but the underlying methodology would
be simi.lar.
Since most crops do not have a wide adaptation to spacing,
as does cotton, the stand indicator might be a more informative index.
56
8.
Brown, H. B. 1923.
pp. 16-25.
LIST OF REFERENCES
Cotton Spacing.
Miss. Agr. Expt. StaG Bul. 212.
Brownlee, K. A. 1960. Statistical Theory and Methodology in Science
and Engineering. John Wiley and Sons, Inc. New York.
Carmer, S. G. 1964. Exponential regression and a computer program of
the estimation of parameters. Agron •.J. 56:515-518.
Carmer, S. G. and J. A. Jackobs. 1965. An exponential model for
predicting optimum plant density and maximum corn yield. Agron. J.
57:241-244.
Chittey, E. T. and D. J. Perki.ns. 196'7. A method of recording and
evaluating seedling distributions. J. Agr. Eng. Res. 12:133-141.
Cochran, W. G. 1952. The chi·-square test of goodness of fit.
Math. Stat. 23:315-321.
Ann.
Cochran, W. G. 1954. Some methods for strengthening the common chisquare tests. Biometrics 10:417-435.
Dick, J. B. and H. H. Remey. 1957. Cotton yield increased by skiprow
planting. Agr. Exp. Farm. Res. 30:1-10.
Dugger, J. P. 1886. Preparation and cultivation of the soil for
cotton in Alabama. Ala. Agr. Exp. StaG Bul. 107:215-224.
Duncan, W. G. 1958. The relationship between corn population and
yield. Agron. J. 50:82-84.
Ehrenfeld, S. and S. B. Littaure. 1964. Introduction to Statistical
Method. McGraw-Hill Book Company. New York, New York.
Ferguson, T. S.
New York.
1967.
Mathematical Statistics.
Academic Press,
Fisher, R. A. 1922. On the mathematical foundations of theoretical
Statistics. Philosophical 'Transactions of the Royal Society,
London Series A. 222:349-368.
R. E. 1966. Device for synchronous thinning of plants.
Journal of the Amer. Soc. of Ag. Eng. 47(12):652-653.
Garret~
The
57
Gowley, W. R. and L. Hoverson. 1960. Growth and yield of cotton in
lower spacing and topping at different stages of maturity. Texas
Agr. Exp. Frog. Rep. 2133. pp. 1-5.
Graybill, F. A. 1961. An Introduction to Linear Statistical Models.
Vol. 1. McGraw-Hill Book Company, Inc. New York.
Hasselblad, V. 1966. Estimation of parameters for a mixture of
normal distributions. Technometrics 8(3):431-444.
Hawkins, B. S. 1963. Skip-row pla~ting increases cotton yields.
Cal. Ag. Exp. Stag Sere 172, pp. 1-,12.
Kendall, M. G. and A. Stuart. 1967. The Advanced Theory of Statistics.
VoL II. Hafner Publishing Company. New York.
Lane, H. C. 1956. Cotton Spacing, A Review and Discussion.
Agr. Exp. Stag Miscel. Publ. 170, pp. 1-5.
Texas
MacDonald, D., W. L. Fieldings, and D. F. Ruston. 1947. A study of
the effect of gap filling on the development and yield of cotton
plants in poor stands of hand planted cotton. J. Agr. Sci. 37:297-300.
McCollum, R. E. 1962. Skip row planting boosts cotton yield.
Agr. Exp. Res. and Farming 20:7.
N. C.
Mood, A. M. and F. A. Graybill. 1963. Introduction to the Theory of
Statistics. McGraw-Hill Book Co., Inc. New York.
Moores, C. A. 1928. The effect of spacing on the yield of cotton.
Agron. J. 20:211-230.
Neely, J. W. 1941. The effect of irregular stand on the yield of
cotton at the Delta Branch Station. Miss. Agr. Exp. Stag Sere
Sheet 311.
Neyman, J. W. 1937. Outline of a theory of statistical estimation
based on the classical theory of probability. Phil. Trans. A:
236-333.
Palmer, J. D. 1958. An examination of the interaction of harmonic
treatment. Jour. of Agr. Eng. Res. 3(1):47-55.
Pearson, K. 1900. On the criterion that a given system of deviations
from the probable in the case of a correlated system of variables
is such that it can be reasonably supposed to have arisen from
sampling. PhiL Mag. Series 5, 50:157-172.
58
Pope, O. A. 1947. Effect of skip or missing row segments on yield of
seed cotton in field experiments. J. Agr. Res. 74~1-13.
Rao, C. R. 1965. Linear Statistical Inference and Its Applications.
John Wiley and Sons, Inc. New York.
Ray, L. L~, E. B. Hudspath and E. R. Holekamp. 1959. Cotton Planting
Rate Studies on the High Plains. Texas Agr. Exp. Misc. Publ. 358,
pp. 1-8.
Reynolds 9 E. B. 1926. The effect of spacing on the yield of cotton.
Texas Agr. Expt. StaG Bul. 340, pp. 77-85.
Rich, P. A. 1964. Skip row cotton production.
Rep. 2306, pp. 1-2.
Texas Agr. Expt. Frog.
Ririe, David and E. J. Hill. 1957. Effect of in the row spacing of
single, double and multiple plant hills on sugar beet. Sugar
Research Foundation. Technological Report Series IX(4):360-366.
Rohrbach, R. P., R. D. Brazee and H. J. Barre. 1969. On spacing
statistics of plant populations produced by uniform seed-placement
devices. J. Agr. Eng. Res. 14(3):210-225.
Rohrbach, R. P., R. D. Brazee and H. J. Barre. 1970. Evaluating
precision planting based on a Monte Carlo planter model. Paper
No. 70-100, Jour. of Agr. Eng.
Sarhan, E. A. and G. D. Greenberg. 1962. Contributions to Order
Statistics. John Wiley and Sons, Inc. New York.
Simpson, D. M. and E. N. Duncan. 1942.
production. Agron. J. 34:544-552.
Row width and cotton
Steel, R. G. D. and J. H. Torrie. 1960. Principles and Procedures of
Statistics. McGraw-Hill Book Co., Inc. New York.
Strukle, D. G. and G. K. Boseck. 1962. Skip-row cotton produce highest
yield. Alabama Agr. Exp. Res. Bull. 9,/:pp • 4-8.
Ware, J.
o.
1930.
Cotton Spacing.
Ark. Agr. Exp. StaG 253.
Warren, J. A. 1963. Use of empirical equations to describe the effect
of plant density on the yield of corn and application of such
equations to variety evaluation. Crop Sci. 3:197-201.
59
90
9.1
APPENDICES
?
The X Goodness,-of-fit Test
Suppose we are making a test of the hypothesis H
a
(i
= 1,
2, ••• , k) against ~:HO is false.
space 0 all points (Pl' P2 ,
Pl + P2 + ••• + Pk = 1.
0
•• ,
~p.
1
= P'O'
1
We have as the parameter
Pk ) s~ch that 0 ~ Pi ~ 1,
For k, any positive integer, 0 is a (k-l)
dimensional hyperplane and the null hypothesis H refers to a point
O
(PIO' P20' ••• , PkO) on O.
This point is designated by W, which, in
general, is some subset of O.
In order to derive an efficient test,
we shall follow the likelihood-ratio procedure, assuming N observations
and k categories.
occurrence in the
{Xl' x 2 '
It
.th
1
follows directly, then, if the probability of
category is denoted by p., that the sample
1
X ), where xl. + x +
k
2
o •• ,
L(x , x 2 '
l
... ,
xk )
=
...
+ x
k
= N, has likelihood:
xl x 2
1
Y
v
x l ·x2 • ••• x k • Pl P2
N~
x
k
Pk
.
(9.1)
Finding the maximum of L with respect to the probabilities can
be done simply with the aid of LaGrange multipliers, but we shall try
direct elementary methods.
It is apparent, however, from the multi-
plicative form of the variables of differentiation in (9.1) that some
restrictions on the p. must be introduced in order to avoid the
1
meaningless result p.
1
=
00.
The p. are, fortunately, restricted by the
1
relation Pl + P2 + ••• + Pk ::: 1, so that we can let Pk ::: 1 - PI - P2 ••• - Pk-l.
Even so, the manipulation is clumsy, and we shall resort
to the maximum of .tn L, which occurs at the same value of p. as does
1
the maximum of L.
60
Taking the natural logaritrilll of' L, we have
k
+ i-n
N~
-
I: i-n x.l
i=l
~
and
0 IDL
op.
l
x.
=
l
Pi
x
k
Pk
n_
i
...
1, 2,
k
~
- l-
Setting each partial first derivative of' i-n L equal to zero, we
i = 1, 2, ••. , k.
But the solutions given in (9.2) imply that the frequencies with
which the various categories occur in the sample are proportional to
their probabilities of occurrence as determined by the population from
which the random sample was drawn.
This result for maximum L on
n is
intuitively reasonable.
We can reduce this result to an elegant form by making use of the
theorem:
the sum of the numerators divided by the sum of the
denominators of any number of equal fractions is equal to the value of
the original fraction.
Hence, given
x.
l
-=
p.
constant (i = 1, 2, ••• , k)
l
.,
x +x2+
l
Pl+P2+
or Pi -- x.jN
l
i = 1, 2,
...
')
k.
+ x
+Pk
k
=:
x.
l
N
=1
Pi
61
Making this substitutio:J. in (9.1), we obtain
.
The maximum of Lover W, which consists only of (PIO' P20' """' PkO ) ,
is
obtained by substituting .P.;O for .Pi •
..L
~
Thus the likelihood ratio is
Max L
A
=
W
Max L
=
(1
The null hypothesis is rejected when A is too small or
equivalently when -2 .tn A is too large.
We require, then, the
restriction of
x.
k
-2 .tn A == 2
I: x . .tn
i==l l
l
~
PiO
"
Since, however, the exact distribution of -2 £n A is very difficult
to compute, we resort to the fact that under H , -2 .tn A has
o
2
asymptotically a X distribution with degrees of freedom equal in
number to the difference between the dimensions of
(1
and W, which in
the present case is k - I.
We use the 1 -
~
2
percentile of X with k - 1 degrees of freedom,
then,
Xi
2
I: x . .tn - - ~ Xl
k I"
i=l l
NPiO'~~'k
-2 .tn A == 2
62
The probability of rejecting H when it is true is ex percent or less.
o
As stated before, it seems intuitively I'E:8,sonatle" when H is true,
o
that x.jN is a IVgood" estimate of
1
p~O'
~
Hence, when H is true, .en(x.jNp.O) is small and a lower bound
011
rejection region on -2 .en f.. is reasonable.
. t'lon 1S
. ra th e:c
X2 approx1ma
k
good~
It might be added that the
. Slze
.
even for moderate
N, so long as
> 2.
Karl Pearson (1900) su-ggested meastu:'ing the discrepancy between
the sample results Xi and the consequences of "true" probability PiO'
in a rather intuitively direct marmer, by taking the differences
x.1 - Np.o'
1.
In order to obtain a measure whose distribution was not to be
influenced by the size of N, that is to say, in order that N not be a
parameter of the distribution which was to reflect departure from H ,
o
he chose as the element of the measure (x.-Np.o)/~o'
forming the
1
1.
1
function~
Z .-
~
(i-
NP
iO)2 __
(x.1. -Np 1'0)
2
i=l /NPiO
Intuitively, it seems proper to reject H when Z is too large.
o
2
Fortunately, when H is true, Z is distributed approximately as X
o
with k - 1 degrees of freedom.
In fact, testing the H type of hypothesis-goodness of fit- with
o
function Z is equivalent to testing with -2 .en f... for large N.
In general, the following procedure is used in testing goodness
of fit:
63
There is assumed a distribution; there are k events or categories
of occurrence; the expected number' of occ·u.:rrenc:e E. in category i is
l
derived from knowledge of the distribution and the total number of
occurrences N.
The ac·tual number o.f' occurrences x., resul. ting from
1
experiment, in each category i is observed.
k
Z =
1:
i=l
(x.-E.)
Z has the
1
l
1
2.
IE.
is formed.
'Then the random variable
Under the null hypothesis for large N,
1.
distribution with k - 1 degrees of freedom.
The H is
o
~
rejected when Z is greater than x-:
-, •
1", ex, K~.l.
The equivalence of -2
~n
A and Z can be readily established by
making use of expansion of the natural logaritrilll about unity.
The
general term x. J.,n (x .jNp, 0)' can be modified by the device of adding
1
1
I
yielding
and subtracting the numerator, Np·O'
1.
x. J.,n
x.+Np·O-Np·o
I
I
I
1.
-,
x. J.,n(1+0. )
I
NPiO
1
where
o.I .~.
whence by substitution in
x.-Np·o
1
1
NPiO
(9.5)
k
-2 J.,n A -. 2 ~~ Xi J.,i".(l+O)
..1---1
But, x.
1
k
-2 in A
:=
1: Np'o(l+O,) J.,n(ltO.).
. 1
1""
1
1
1
64
Since under H we may accept O. to be small for large N, we may
o
1
develop the sum in (9.10) on the
asslli~ption
that O. is small.
1
But for
small 0., in (1 + 0.) .- O. and therefore
1 1 1
k
-2
in A ==
k
k
2
1: NP'O(l+O. )0. + 1: Np.O 0.+ 1: Np.O O. •
i=l 1
1
1
i=l 1
1 i==l
1
1
Examining each sum in (9.11), we find that the second sum is
k
k
positive but that the first sum is 1: Np.O. = 1: (x.-Np.o) == O. Thus
i::::l 1 1
i=l 1
1
(9.11) reduces to
k
-2 in A;"
1: Np.o
i=l 1
O~
(9.12)
1
It was desired to establish this equivalence.
9.2
Estimation of the Parameter of the Negative
Exponential Distribution
We assume that X is a random variable with distribution function
Ftx;
e)
where
e is
the ur~nown parameter.
... , xn of variables has been observed and
the true value of
e.
A random sample xl' x2 '
it is required to estimate
We compute the value of a
f~~ction
of the n
observations,
called a statistic or an estimate of
"e(xl ,
x ' ••• , x ) does not depend on
2
n
e.
It is assumed that the function
e.
be derived from its sampling distribution.
"
The properties of -6 may then
The main problem within the
theory of estimation is to formulate the desired properties of estimates
and to derive methods for finding estimates having these properties.
65
We used a method of estimation given by R. A. Fisher, (1922) the
method of maximum likelihood for estimation of the parameter of the
negative Exponential Distribution.
Using the method of maximum likeli-
hood, first we define the likelihood function L of a random sample as
We then choose as an estimate of
a the
value which maximizes
L for the given values of (xl' x2 '
As it is more convenient to work with log
the required value of
a is
e
L than with L itself,
usually found by solving the likelihood
equation:
o .tn
L
08
In other words, the maximum likelihood estimate
the value of
a that
"e of
the parameter is
makes the probability function of the sample that
was actually observed a maximum.
The likelihood function of a random sample from the negative
exponential distribution
x
e
1
f(x; a) = - e
e
x
~
0,
e>0
x
~
0,
a > o.
n
I:: x.
i=l 1.
can be written as:
L = -
1
an
e
e
The log of the likelihood function is:
n
fu L = -n tn
a-
I:: x.
i=l 1
e
66
Differentiating with respect to 6 and setting equal to zero, we have
n
a
tn
_.
L
216
solving for
e,
we get
-n
t x.
i=l 1.
2
e+ e
= 0;
n
t x.
1\
e ..-
. 1 1
1=
n
which is the arithmetic mean and standard deviation.
The method of maximum likelihood was a useful approach for
estimating the paramet.er of the negative exponential distribution
because it has the property of being a minimum variance unbiased
estimator.
In terms of the plant spacing this means that in the long
run, sample means will average to the true plant spacing for the
population of plants.
It also implies that sample variances {which
are essentially indices of unevenness of stand and measures of the
reliability of the average stand estimate) are a minimum.
These are
both very desireable properties.
Proof that the maximum likel.ihood estimator for the negative
exponential is unbiased is as follows:
1\
We wish to show that EJ( 6)
="
6.
n
1\
Since E(6) = E( t x./n)
i=l 1
1
= n
n
t E(x.).
i=l
1
67
However,
x
a:>
E(X)
By substitution of
= S :h
oe
e
ex
e.
dx
A
e for
E(X) in (9~9) for E(a) we have
To prove that the maximum likelihood estimator of the negative
exponential has the property of min.imum variance, in addition to
unbiasedness, we need to show that
1)
A
The a is a sufficient statistic for
e (an
estimator is said
to be sufficient if it uses efficiently all the information
in the samples regarding the parameter).
2)
A
The density of a is complete.
Neyman (1933) has developed a criterion for examining a statistic
e for
sufficiency as follows:
If the joint density of random samples
of size n from the density f(x; e) can be factored as g(x ,
l
x
2
' ..• ,
A
x n ; e) = h(a; a) q(x l , x 2 ' ••• , x n ) where q(x1 , x 2 ' • DO' x n ) does not
depend upon the
e,
A
then a is a sufficient statistic.
theorem for maximum likelihood estimates of
n
exponential, we have:
t x.
i=l
a)
= -1
an
e
e
1
e for
Using this
the negative
68
If we let
"
1
"a
-n(-)
h( a; a) ""-e
an
a
and
"
then by the above theorem, the estimator a is a sufficient statistic
for
a.
"
The density of a is also complete, since i.t is a member of the
exponential family of distributions; therefore the maximum likelihood
estimate of the parameter of the negative exponential distribution is
n
a minimum-variance unbiased estimator.
_
2
(x) = lin
Var(tx.) ~ lin
For the variance of x =
2
n
t x./n,
.:I..1 :I.
t [Var(x.)J. But the variance
i=l
:I.
2
of x. was equal to a which was obtained in Section 4.1 and by sub-
we have var
1
:I.
2
2
stitution we get Var(x) = l/n (ne )= e2/n.
9.3 The Method of Construction of Confidence Limits
The general method of construction of confidence limits is as
follows:
We suppose that we have a family of populations each with a known
density function p(x; e}, x being the random variable and
e the
parameter
"
in question.
We suppose that we have an estimator a to estimate e,
"
that e is a function of observed x, and that we can define the density
"
"
function of e, g(a; a}.
value, say
eo ,
If we assume that e equals some particular
we can insert this value and get the density function
g(T; ao) of the distribution of T under this assumption, where T is a
function of the observed X.
69
Under the assumption
e = eo ,
there will be a p, point for the
~
distribution of T, say T which will. be determined by the equation
l
'r1
Pr[T ~ Tl , a
ao}
=
J
(9. 21)
g(T:ao)dT -- PI
-CD
likewise, under the same assumption
e=
ao' there will be a P2 point
for the distribution of T, say T , determined by the equation:
2
co
Pr ['r ~ T2 : a =
eo } =
S
g [T: a }dT = 1 - P2'
0
'11
-'-2
These points are indicated in Figure 9.1.
The area to the left of
T is equal to P2 and the area between T and T is equal to (P2 - Pl)'
l
2
2
Now, in (9.21) and (9.22), if we change the value of
corresponding values of T and T •
2
l
eo ,
we change the
We therefore can regard T and T
l
2
as functions of a, say Tl(a) and T2 (a), respectively.
can plot these functions Tl(a) and T (e) against
2
e.
In principle, we
In idealized form,
these are plotted in Figure 9.2.
Now assume that the true value of a is actually
e'.
Then Tl(e)
and T (e) take the values Tl(a') and T (e'), respectively, and
2
2
Pr IT ~ T ( e' )}
l
= PI' PrlT
~ T2 (e' )}
= 1 - P2' which imply:
Now suppose that we have taken a sample observation and have computed
the numerical value of the estimate, say T.
o
We draw a horizontal line
parallel to e axis through the point T on the T axis.
o
This line will
intersect the two curves T (a) and Tlte) at points A and B.
2
The points
70
P [T: e}
peT: e
/0
}
T
Figure 9.1
Illustration of confidence interval for Gamma distribution
including probabilities in the tails ,(asswnption e = e )
o
T
e
Figure 9.2
Plots of the functions Tl(e) and T (e) against e showing
the relationship between the confi~ence limits and the
parameter
71
A and B are dropped on to the 6 axis to give 6 and 6.
a(P2- Pl) confidence interval for
e is
We assert that
(~,e), ~.~.,
The justification for this assertation is as follows.
Enter the
true value of 6 1 on the e axis; erect the perpendicular at this point
to cut the curves T (6) at C and T (e) at D.
2
l
At both these points e has the value e'; so, ate'r=T(6')
,
1
'
and at D, T = T (e').
2
Now draw the horizontal lines through C and D;
these will intersect the T axis at Tl(e ' ) and T (e'), respectively.
2
Now examine Figure 8 .2 6 I may be anywhere on the 6 axis, but, if
AB intersects CD, then To must lie in the interval (Tl(e l
and simultaneously the interval (~,e) must include e'.
)),
T (6 1 ) )
2
In other words,
the two statements
and
(ii) The interval (~,e) includes e l
,
are always true simultaneously or not true simultaneously.
But by
(9.23) the event (i) has probability (P2- Pl); so the event (ii) must
also have probability (P2- Pl) if these statements are true simultaneously.
Hence, we can write
and this completes the justification of (9.24).
72
At the point A, the function T (e) has e
2
To' ~.~., T2(~)
=
e and takes the value
= To·
Now T (e) was defined as the solution of (9. 22 ),
2
Cl)
J'I' g[T;e
2
}dT
=1
- P2;
0
so we can use this equation to find
.~;~
is obtained by solving
co
J
T
g
[T ~ e}d'I' ..
IT ~
1. .- .p 2. .- Pf'
,·-t·
ni.
..l000
e ::: e}
_
0
o
Similarly, at the point B, the function Tl(e) has e:::: e and takes the
value T ; so
o
T1Ce)
.
and
e can
c::
T ,
0
be found as the solution
T
J
of~
o
g[T:e}dT _.. Pl- Pr [T :s: T ;
_Cl)
o
e : : e}.
73
Table 9.1 Observed and Expected Frequenci.es for Cotton Plant Spacing
and Results of Goodness of Fit Tests for 64 Initial Samples
(Data Set 1)
Distance
FeetY
Sample 1
Sample 2
Sample 3
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
.1
.2
.3
.4
.5
.6
·7
.8
.9
1.0
>1.0
70
38
34
21
22
11
12
5
9
6
16
53.23
41.67
32.61
25.52
19.98
15.64
12.24
9.58
7.49
5.87
21.14
Swn of
Frequencies
244
Mean Distance .4082
2
11.79
X
Distance
Feet
39
36
30
23
14
16
10
7
7
13
59.52
45.89
35.39
27.29
21.39
16.22
12.50
9.64
7.43
5.73
19.31
260
.3846
5.67
67
42
39
34
23
17
16
12
6
9
16
68.83
51.97
39.24
29.63
22.37
16.89
12.75
9.63
7.27
5.49
16.92
281
.3559
6.56
Sample 4
Sample 6
Sample 5
Expected
Observed
Expected
Observed
Expected
Observed
Frequency Frequency Frequency Frequency Frequency Frequency
.1
.2
.3
.4
.5
.6
.7
.8
·9
1.0
>1.0
Swn of
Frequencies
63
41
35
29
23
17
12
11
8
9
12
260
Mean Distance .3846
2
5.94
X
Y
6~
j
59.53
45.89
35.38
27.28
21.03
16.22
12.50
9.64
7.43
5.73
19.31
55
36
33
27
21
12
15
8
10
3
15
235
.4184
7.24
50.80
40.01
31.50
24.80
19.53
15.37
12.11
9.53
7.50
5.91
21.89
64
40
36
29
23
16
11
14
7
3
17
59.53
45.89
35.39
27.28
21.04
16.22
12.50
9.64
7.43
5.73
19.31
260
.3846
5.15
In Tables 1 and 2, classes are denoted by midpoints only. To obtain
the intervals corresponding to these midpoints, see Section 3.1.
74
Table 9.1 (Continued)
Distance
Feet
Sample 7
Sample 8
Sample 9
Observed Expected Observed Expected Observed Expected
Frequency Freq'J.ency Frequency FreqJ.ency Frequency Frequency
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
53
40
34
28
20
18
13
10
4
10
17
54.05
42.22
32.98
25.76
20.12
15.72
12.28
9.59
7.49
5.85
20.89
Sum of
Frequencies
52.82
41039
32.42
25.40
19.90
15.59
12.21
9.:;7
'7.50
5 .0 1
AI?
21.26
244
.4098
3.92
247
Mean Distance .4049
2
6.04
X
Distance
Feet
56
36
34
28
22
15
13
8
10
6
16
36
32
21
28
18
10
r ,)
9
6
4
21
35.57
29.17
23.9 4
19.63
16.11
13.21
10.84
8.89
7.29
5.98
27.33
198
.5051
7.99
Sample 10
Sample 11
Sample 12
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency' Frequency Frequency Frequency
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
Sum of
Frequencies
68
41
38
31
24
17
14
11
9
3
18
274
Mean Distance .3650
2
3.94
X
65.66
49.93
37.96
28.86
21.94
16.68
12.68
9.46
7.33
5.57
17.69
47
32
30
24
19
11
13
7
9
6
1'7
215
.4651
5.80
41.59
33.54
27.06
21.82
17.60
14.19
11.44
9.23
7.45
6.01
27.04
68
41
38
31
25
15
17
9
11
7
15
277
.3610
6.64
67.01
50.80
38.51
29.19
22.13
16.77
12.71
9.64
7.30
5.53
17.35
75
Table 9.1 (Continued)
=
Distance
Feet
Sample 13
Sample 14
Sample 15
Observed Expected Observed Expected Observed Expected
Frequency Frequency Freg,uency' .Fres...uency Frequency Frequency
.4
58
37
35
28
.5
22
.6
16
10
13
.1
.2
.3
.7
.8
.9
5
1.0
>1.0
9
16
54.88
42.'78
33.35
26.01
20.2'7
15,80
12.32
9.60
7.48
5.83
20.64
Sum of
Frequencies
249
Mean Distance .4016
2
5.54
X
Distance
Feet
67
41
3'7
3~J
21
19
12
14
'7
8
14
66.12
:.,0.22
38.15
38.97
22.01
16.72
12.69
9.64
'7.32
5 .~i6
1'7.58
57
36
34
28
22
18
10
12
6
'7
15
275
.3636
245
.4098
7.16
5.34
52.83
41.39
32.43
25.41
19.91
15.59
12.22
9.57
7.50
5.87
21.26
Sample 16
Sample 17
Sample 18
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
Sum of
Frequencies
60
38
30
30
16
22
11
13
10
3
18
251
Mean Distance . .3984
2
9.09
X
52.72
43.34
33.73
26.23
20.41
15.11
12.35
9.61
7.48
5.81
20.39
56
37
38
24
1~-)
22
13
8
10
8
18
54.88
42.78
33.35
26.01
20.27
15.80
12.32
9.60
7.49
5.83
20.64
65
40
37
30
23
14
17
8
11
5
15
249
.401.6
265
.3774
7.70
6.37
61.69
47.33
36.31
27.85
21.37
16.39
12.58
9.65
7.40
5.68
18.72
76
Table 9.1 '(Continued)
Distance
Feet
Sample 19
Sample 21
SamJ21e 20
Expected
Observed Expected Observed
Observed Expected
Frequency Frequency F'requenc;f Frequency Frequency Frequency
.1
.2
.3
.4
61
37'
34
28
16
22
14
9
10
5
13
.5
.6
.7
.8
.9
1.0
>1.0
54.88
42.78
33.35
26.01
20.27
15080
12.32
9.60
7.48
5.84
20.64
Sum of
Frequencies
2
9.02
Distance
Feet
:<C)
J.
26
32
20
16
9
12
6
...,
I
It')
47.64
37.81
30.01
23.82
18.91
1~i.01
11.91
9. l +5
7.50
5.95
22.92
32
45
30
23
13
19
8
11
)
...,
I
1'7
39.77
32.24
26.13
21.18
17.17
13.91
11.28
9.14
7.41
6.01
25.71
210
04762
15.40
231
.4329
8.90
249
Mean Distance .4016
X
::<3
Sample 22
Sample 23
Sample 24
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequencl Frequencl Frequencl
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
Sum of
Frequencies
54
32
35
26
22
13
10
12
3
10
16
233
Mean Distance .4292
2
11.72
X
48.00
38036
30.38
24.07
19.06
15.10
11.96
9048
7.50
5.94
22.66
80
42
4'7
31
30
16
19
9
13
7
15
309
.3289
15073
79.69
58.80
43038
32.01
23.62
17.43
12.86
9.48
7.00
5.16
14.54
51
35
33
37
21
11
15
11
8
5
18
235
.4253
4.27
49.21
38.90
30.76
34.32
19.22
15.19
12.01
9.49
7.50
5.93
22 041
77
Table 9.1 (Continu.ed)
Distance
Feet
Sample 2.5
Sample 26
Sample 2'7
Observed Expected Observed Expected Observed Expected
Frequency Frequer:.c::y Frequency Freg,uency Freguency Frequency
.1
.2
.3
.4
.5
06
.7
.8
.9
1.0
>1.0
54
35
32
27
20
12
17
10
5
6
18
49.61
39.18
30.94
24.49
19.30
15.24
12.03
9.:,0
7G51
5.93
22028
Sum of
Frequencies
.36
33
27
21
15
13
9
11
3
15
40.28
31.68
24.92
19060
15.42.
12 013
9. ~i4
7050
5.90
21.77
240
.416'7
6.68
236
Mean Distance .4237
2
5.95
X
Distance
Feet
~)1.21
38
43
31
25
20
12
14
7
9
5
18
44.19
35.39
28.35
22.70
18.18
14.56
11.66
9.34
7.48
5.99
24.11
222
.4505
6069
Sample 29
Sample 30
Sa~le 28
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
r;:~
.1
/)
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
36
36
27
18
12
15
9
8
3
19
Sum of
Frequencies
238
Mean Distance .4202
2
5.14
X
50.41
39.73
31.32
24.68
19.45
15.33
12.• 08
9.52
7.50
5092
22.07
6'7
)/
41
46
34
22
14
12
10
9
3
16
2'74
.3650
6.93
65.67
49.93
37.96
28.78
21094
16.68
12.68
9.60
7033
5.57
17069
61
37
35
28
22
16
13
11
r::
./
9
13
250
.4
7.81
55.29
43.06
33.54
26.12
20.34
15.84
12.33
9.68
7.48
5.82
20.52
78
Table 9.1 (Continued)
Distance
Feet
.1
.2
2'-;.39
1~~,1
17'00,1:;
L::-:
... , ......
.3
H).22
15.43
13.07
11.07
9.37
'7.94
6.72
5.69
31.56
n
14
14.39
13.01
11.3'7
9.93
8.68
7.58
6.62
)078
5.08
34.99
~)";
_ ...... u
.4
.5
.6
.7
.8
.9
1.0
>1.0
Sum of
Frequencies 166
Mean Distance .6024
2
X
Distance
Feet
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
13
12
12
11
9
9
7'
23
136
.7407
12.30
10.15
135
.7407
11.53
Sample 34
Sample 36
SaJEE1e 35
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequen~Fre9.2?~er;cy Frequency Frequency
70
51
41
33
27
16
11
8
4
2
17
Sum of
Frequencies 280
Mean Distance .3484
2
6.60
X
71.60
53.73
40.33
30.26
22.71
17.04
12.79
9.60
7.20
58
36
34
28
18
18
13
8
10
5.40
4
16.27
16
243
.411:)
5.21
52.42
41.11
32.24
25.28
19.83
15. ~?5
12.19
9.56
7.50
5.88
21.39
39
2:)
2'"
23
15
12
13
12
8
8
21
r-
./
201
.4975
5.13
36.59
29.93
24.48
20.02
16.37
13.39
10.95
8.96
7.33
5.99
26.93
79
Table 9.1 (Continued)
Distance
Feet
Sample 38
Sample 39
Sample 37
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
60
37
34
28
22
13
16
12
5
3
19
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
54.88
42.78
33.35
26.01
20.27
15.80
12 •.32
9.60
7.48
5.83
20.64
Sum of
Frequencies
38.35
31.21
25.40
20.67
16.82
13.69
11.14
9.06
7.37
6.00
26.25
206
249
Mean Distance .4016
2
6.10
X.
Distance
Feet
47
25
24
23
19
10
12
10
9
9
18
38
28
27
22
14
11
12
12
11
7
22
37.64
30.69
25.03
20.41
16.64
13.57
11.07
9.02
7.36
6.01
26.52
204
.4854
.4902
9.41
5.22
Sample 41
Sample 42
Sample 40
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
Sum of
Frequencies
54
36
33
27
21
17
10
11
3
6
18
236
Mean Distance .4237
2
5.51
x.
49.61
39.18
30.94
24.44
19.30
15.24
12.03
9.50
7.50
5.93
22.28
30
23
21
17
16
10
11
10
9
10
24
181
29.96
25.01
20.86
17.41
14.52
12.12
10.11
8.44
7.04
5,87
29.62
52
30
27
20
26
26
12
8
8
7
20
226
.5525
.4425
5.56
6.70
45.71
36.46
29.09
23.20
18.51
14.76
11.78
9.39
7.49
5.98
23.58
80
Table 9.1 (Continued)
Distance
Feet
Sample 43
Sample 44
Sample 45
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
.1
.2
30
25
21
23
19
10
13
11
8
8
21
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
Sum of
Frequencies
32.54
26.94
22.30
18.46
15.28
12.65
10.47
8.66
7.17
5.94
28.55
43.44
3 4 .86
27.98
22.45
18.02
14.46
11.60
9.31
7.47
5.99
24.37
221
189
Mean Distance .5291
2
6.76
X
Distance
Feet
40
32
30
25
]·9
15
13
12
9
11
15
36
27
26
19
23
15
13
10
10
8
19
~8.35
31.21
25.40
20.67
16.82
13.69
11.14
9.06
7.37
6.01
26.25
206
.4545
.4854
10.05
7.26
Sample 46
Sample 47
Sample 48
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Fresuency
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
Sum of
Frequencies
23
19
16
14
12
9
10
8
8
7
21
147
Mean Distance .6803
2
7.15
X
20.09
17.34
14.97
12.92
11.16
9.63
8.23
'7.18
6.11
5.35
33.79
16
14
13
13
12
9
8
4
5
4
21
119
13035
11.85
10.52
9.34
8.29
7.36
6.53
5.80
5.15
4.57
36.20
20
17
17
14
11
10
9
8
7
6
23
142
.8403
.7042
12.30
5.27
18.79
16.30
14.15
12.27
10.65
9.24
8.01
6.95
6.03
5.23
34.32
81
Table 9.1 (Continued)
Distance
Feet
Sample 49
Sample 'jO
Sample 51
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequens; Frequency Frequency Frequency
21
16
11
10
13
8
12
8
7
7
24
.1
.2
.3
.4
.5
06
.7
.8
.9
1.0
>1.0
1'7.29
15.09
13.17
11.49
10003
8.76
7064
6.67
5.82
5008
34.90
Sum of
Frequencies
.L
0
23
11.66
10.43
9.34
8.35
'7.48
6.69
'5.99
5.36
4.79
)+.29
36.58
15
13
14
12
11
11
10
'7
5
6
22
14.68
12.96
11.43
10.09
8.90
7.86
6.93
6.12
5040
4.76
35.81
126
111
.9009
78.9'7
137
Mean Distance 07353
2
9.45
x.
Distance
Feet
31
1'7
14
12
9
1
2
1,
793
9.84
Sample 52
Sample 53
Sample 54
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
.1
02
03
04
05
06
.7
.8
.9
1.0
>1.0
Sum of
Frequencies
24
22
23
14
10
11
10
11
9
8
29
171
Mean Distance .5882
2
5.06
X
26.57
22.42
18.91
15.95
13.46
11.35
9.58
8.08
6.82
5.75
31005
32
27
27
23
15
10
12
12
13
11
25
207
.4854
12.68
38035
31.21
25.40
20.67
16.82
13.69
11.14
9.08
7.38
6.01
26.25
24
21
21
18
14
13
10
10
12
3
22
168
.5952
9.79
25.98
21.96
18.56
15.69
13.26
11.21
9.48
8.01
6.77
5.72
31.31
82
Table 9.1 (Continued)
Distance
Feet
Sample 56
Sample 55
Sample 57
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
32
19
18
15
17
11
11
10
12
7
23
Sum of
Frequencies
28.09
23.58
19.79
16.61
13.95
11.71
9.83
8.25
6.92
5.81
30.41
175
Distance
Feet
10
8.'78
13
7.52
6.44
5.51
32.89
9
12
6
9
22
11
8
24
13
24.51
20.82
17.69
15.03
12.77
10.85
9.21
7.83
6.65
5.65
31.93
163
.6135
12.77
8.73
X
20
23
16
18
15
12
22.25
19.06
16.32
13.97
11.97
10.25
155
.6452
Mean Distance .5714
2
20
16
13
15
13
9.98
Sample 58
Sample 60
Sample 59
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
Sum of
Frequencies
41
30
29
24
19
16
9
11
7
8
16
210
Mean Distance .4762
2
6.58
X
39.77
32.24
26.13
21.18
17.17
13.91
11.28
9.44
7.41
6.01
25.71
25
16
18
17
10
13
12
10
8
9
20
158
.6329
11.50
23.09
19.71
16.83
14.37
12.27
10.48
8.94
7.64
6.52
5.57
32.54
19
16
14
12
7
10
10
9
12
7
16
132
.7576
22.20
16.32
14.30
12.53
10.98
9.62
8.43
7.39
6.47
5.67
4.97
35. 26
Table 9.1 (Continued)
Distance
Feet
Sample 62
Sample 61
Sample 63
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequencz-Frequency Frequency
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
23
27
18
20
11
16
8
10
11
8
29
Sum of
Frequencies
29.96
25.00
20.86
17.41
14.52
12.12
10.11
8.44
7.04
5.87
29.62
181
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
Sum of
Frequencies 232
Mean Distance .4310
2
'X
9.55
--"1 ~
..
8
12
11
10
20
11.16
Sample 64
Observed Expected
Frequency Frequen~
50
30
32
20
26
16
12
13
9
8
16
21
18
192
.5208
Mean Distance .5525
2
8.39
'X
Distance
Feet
30
23
26
48.03
38.09
30.20
23.94
18.99
15.05
11.94
9.46
7.51
5.95
22.79
33.54
27.68
22.84
18.85
15.56
12.84
10.56
8.74
7.21
5.95
28.14
28
23
28
22
17
10
13
5
12
9
21
188
.5319
13.61
32.28
26.69
22.12
18.33
15.18
12.58
10.42
8.64
7.16
5.93
28.62
84
Table 9.2 Observed and Expected Frequencies for Cotton Plant Spacing
and Results of Goodness of Fit Tests for 64 Initial Samples
(Data Set 2)
Distance
Feet
82, Fl, Ll
Sl, Fl, Ll
S3, Fl, Ll
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
Sum of
Frequencies
55
42
28
27
23
13
10
10
8
7
14
237
Mean Distance .421
2
5.84
X
50.01
39.45
31.13
24.56
19.37
12.29
12.06
9.52
7.50
5.92
22.15
63
40
25
30
25
13
10
9
8
9
23
255
.3922
8.30
57.39
44.47
34.46
26.70
20.69
16.03
12.42
9.63
7.46
5.78
19.91
40
43
28
27
15
12
14
10
8
10
25
232
.4310
7.37
48.03
38.09
30.20
23.94
18.99
15.05
11.94
9.47
7.50
5.95
22.79
85
Table 9.2 (Continued)
Distance
Feet
84, F2, Ll
85, F2, Ll
81, F3, Ll
Observed Expected Observed Expected Observed Expected
Frequency Frequency Fre~uency Frequency Frequency Frequency
60
43
28
27
25
13
10
8
5
2
25
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
Sum of
Frequencies
53.65
41095
32.79
25065
20.05
15.68
12.26
9.59
7050
5.86
21.02
246
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
Sum of
Frequencies
27
28
25
14
17
12
10
7
5
1
33
179
Mean Distance .5587
X
32
33
23
21
12
14
12
10
8
2
34
36.60
23093
24.48
20.03
16.38
13.40
10.96
8.95
7.33
5.99
26.93
201
.4000
.9975
9.86
7.03
S2, F3, Ll
83, F3, Ll
84, F3, Ll
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
.1
.2
2
55.30
43.07
33.54
26.12
20.34
15.84
12 034
9.61
7.48
5.83
20.52
250
Mean Distance .4065
2
8.04
X
Distance
Feet
58
48
27
30
25
10
11
10
5
2
24
7.89
29.34
24.52
20.51
17.15
14.3 4
11.90
10.02
8.38
7001
5.86
29088
64
47
30
30
23
12
14
6
3
4
24
257
58 024
45004
34.83
26.94
20.83
16.11
12.46
9.34
7.45
5.76
19.67
30
31
26
17
16
10
11
9
4
2
30
196
.3891
.5102
8.66
6048
34.88
28 068
23.57
19.38
15.93
13.09
10.76
8.85
7.27
5.98
27.61
86
Table 9.2 (Continued)
Distance
Feet
81, F4, L1
82, F4, Ll
83, F4, Ll
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency' Frequency Frequency
36
21
22
18
16
10
8
3
2
6
39
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
Sum of
Frequencies
29.97
25.01
20.86
17.41
14.53
12.12
10.11
8.44
7.04
5.87
29.62
181
43
41
34
20
22
16
14
5
8
3
28
48.82
28.63
30.57
24.19
19.15
15.15
11.99
9.49
7.51
5.94
22.54
234
58
37
35
26
22
12
10
13
5
10
15
243
Mean Distance .5525
.4274
.4115
l
7.69
9.56
12.99
Distance
Feet
52.42
41.11
32.24
25.29
19.83
15.55
12.20
9.57
7.50
5.88
21.39
s4, F4, Ll
Sl, F5, Ll
82, F5, Ll
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Fre~ency Frequency
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
Sum of
Frequencies
43
42
31
27
21
13
13
10
4
3
25
232
Mean Distance .4310
2
5.27
X
48.04
38.09
30.20
23.95
18.99
15.06
11.94
9.47
7.51
5.95
22.80
58
40
31
27
21
14
13
5
8
3
30
250
55.30
43.07
33.54
26.12
20.34
15.84 '
12.34
9.61
7.48
5.83
20.52
38
37
26
24
14
10
12
10
8
3
33
215
.4000
.4651
8.84
7.06
41.59
33.57
27.06
21.82
17.60
14.19
11.44
9.24
7.45
6.01
25.04
87
Table 9.2 (Continued)
Distance
Feet
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
83, F5, Ll
s4~ F'"
,); Ll
85, F5 2 Ll
Observed Expected Observ'ed Expected Observed Expected
Frequency Freque2cy Frequency Frequency Frequency Frequency
5,4
52.02
68.83
72
58.67
57
40
48
40.83
48
51097
45.33
32.06
41
27
39.24
35.02
33
26
25.16
23
29.63
27.06
30
22
24
18
19.76
22 37
20.90
21
17
15.51
13
16.89
16.,15
12
12.18
12.48
12.75
17
7
11
13
9.56
9.63
9.64
7
10
4
7.50
7 027
7045
3
8
5.89
5.49
7
3
5.75
21
20
24
21.52
18.91
19.55
-
Q
Sum of
Frequencies
242
281
Mean Distance .4132
2
6.04
X
258
.38'76
.3559
6.83
8.91
;;;
Distance
Feet
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
S1, F6. Ll
S2:... F6;) Ll
S3, F6, Ll
Observed Expected Observed Expected Observed Expected
Frequency Frequensz Frequency Frequency Frequency Frequency
79
53
43
28
25
12
17
7
4
2
26
Sum of
Frequencies 296
Mean Distance .3378
2
15.27
X
75.84
56.40
41.95
31.21
23.21
17.26
12.85
9.56
7.10
5.28
15.34
62
43
40
22
22
13
15
13
4
1
2.4
259
.3861
10.86
59.10
45.61
35.20
27.17
20.97
16.19
12.49
9.64
7.44
5.74
19.43
77
57
48
37
17
20
9
10
9
9
12
305
.3279
8.71
80.18
59.10
43.56
32.11
23.67
-L7 .44
12.86
9.48
6.98
5.15
14.45
88
Table 9.2 (Continued)
Distance
Feet
84, F6, Ll
85, F6, Ll
86, F6, Ll
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Fre~uensy Frequency Frequency
75
50
42
27
24
12
17
7
10
2
21
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
Sum of
Frequencies
71.60
53.73
40.33
30.26
22.71
17.05
12.79
9.60
7.21
5.41
16.27
287
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
Sum of
Frequencies
61
52
32
33
19
19
9
13
10
9
17
274
Mean Distance .3650
X
55
37
36
19
21
12
15
13
4
1
25
50.41
39.73
31.]2
24.68
19.46
15.33
12.09
9.53
7.51
5.92
22.03
238
.3922
.4202
10.51
11.56
S7, F6, Ll
Sl, F7, Ll
S2, F7, Ll
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequencl
.1
.2
2
7
12
3
8
25
57040
44048
34.47
26.71
20.70
16.03
12.43
9.63
7.46
5.78
19.91
255
Mean Distance .3484
2
9.10
X
Distance
Feet
53
47
32
29
18
21
8.01
65.67
49.93
37.96
28.86
21.95
16.69
12.69
9.65
7.33
5.58
17.69
46
31
25
18
19
9
16
7
10
2
35
218
42.70
34.34
27.61
22.20
17.85
14.36
11.54
9.28
7.46
6.01
24.64
46
31
32
16
19
11
14
13
4
2
27
215
.4587
.4651
13.86
10047
41.59
33.55
27.06
21.82
17.60
14.19
11.45
9.23
7.45
6.01
25.04
Table 9.2 (Continued)
Distance
Feet
m'~2U
.1
.5
.6
37
34
21
26
15
17
.7
8
.8
.9
9
4
3
38
.2
.3
.4
1.0
>1.0
Sum of
Frequencies
40.50
32.76
26.50
21.44
17.34
14.03
11.35
9018
7043
6.01
25.45
212
53
39
28
21
21
10
1'7
6
10
3
31
76
62
41
35
21
22
7
11
49.61
39.18
30.94
24.44
19.30
15 024
12004
9051
7.51
5.93
22.28
3
8
19
80.17
59.10
43.56
32.11
23.67
17.45
12 086
9.48
6.99
5.15
14.44
305
.4237
.3279
12.23
10.46
S2, F9, U
s4, F9, Ll
S3, F92 Ll
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
•
ffiJ~2U
236
Mean Distance .4717
2
X
13.68
Distance
Feet
~,~~ll
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequer.cy Frequency Frequency
Sum of
Frequencies
76
51
45
25
24
14
15
13
4
1
18
286
Mean Distance .3497
2
9031
X
71.13
53.44
40.15
30.16
22.66
17.02
12.79
9.61
7.22
5.42
16.38
68
51
35
35
20
20
9
5
10
9
25
287
71.60
53.74
40.33
30.27
22.72
17005
12079
9.60
7 021
5.40
16 027
78
52
43
28
25
12
17
7
10
2
20
294
.34-84
.3401
14.08
9001
74.89
55.81
41.59
31.01
23.10
17.22
12.83
9.56
7.13
5.31
15.54
Table 9.2 (Continued)
==
::)7IJ F9, Ll
s6, F9 2 Ll
S5, F9, Ll
Distance Observed Expected Observed Expected Otserved Expected
Frequency FreqQenSl Frequency Fre~uency Frequency Frequency
Feet
61
60.38
64.77
.1
64.33
56
59
46.46
.2
29.05
49.34
47
51
49
42
32
37.41
25.75
37.59
.3
33
28.64
.4
23
28.53
27 •.51
30
33
21.82
21.17
23
21.75
19
.5
19
16.62
21
.6
16.29
16.59
13
19
12.66
12.65
12.53
15
.7
9
7
9.64
.8
11
9.64
13
9.65
5
4
10
4
7.42
7.35
7.35
.9
2
5.60
1.0
8
5.61
5.71
9
24
24
18.03
>1.0
17.91
19.07
30
Sum of
Frequencies 262
272
271
Mean Distance .3817
.3690
.3676
2
16.41
10.86
8.88
X
Sl, FlO, Ll
s8, F9, Ll
S9, F9, Ll
Distance Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
Feet
.1
52.82
82.83
51.20
56
47
73
40.28
.2
41.39
43
38
49
51.97
32.42
39.24
31.68
34
34
29
.3
.4
25.40
24.92
29.62
27
25
32
24
21
19.60
22.37
19.90
17
.5
20
.6
11
15.42
16.89
15
15.59
12.21
12.13
12.75
15
.7
17
7
11
9.62
.8
9.54
13
9.57
7
4
4
10
7.26
7.50
7.50
.9
2
2
1.0
5.48
"I
5.87
5.90
21.26
28
28
16
16.91
>1.0
21.77
Sum of
240
281
Frequencies 244
Mean Distance .4098
.4167
.3359
2
14.70
8.65
10.93
X
•
91
Table 9.2 (Continued)
Distance
Feet
s4 , FlO, L1
S2, FlO, Ll
S3, FlO, L1
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
Sum of
65
53
34
34
20
19
9
5
10
9
23
68.83
51.97
39.24
29.62
22.37
16.89
12.75
9.62
7.26
5.48
16.91
76
62
41
35
21
22
7
11
3
8
16
80.17
59.10
43.56
32.11
23.67
17.44
12.86
9.48
6.98
5.15
14.44
305
.3279
9.19
s6, FlO, Ll
S7, FlO, Ll
85, FlO, Ll
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
.1
..
72.06
54.03
40.51
30.37
22.77
17.07
12.80
9.59
7.19
5.39
16.16
288
.3472
8.61
281
Mean Distance .3559
2
10.88
X
Frequencies
Distance
Feet
76
51
42
27
24
12
17
7
10
2
20
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
Sum of
72
49
43
24
24
13
15
13
4
1
21
279
Mean Distance .3584
2
10.43
X
Frequencies
67.92
51.38
38.87
29.40
22.25
16.83
12.73
9.63
7.28
5.51
17.13
59
49
31
32
19
19
9
5
10
9
24
266
.3759
9.99
62.12
47.61
36.49
27.97
21.43
16.43
12.59
9.65
7.39
5.66
18.60
68
46
39
25
23
11
17
7
10
2
24
272
.3676
10.41
64.77
49.34
37.59
28.64
21.82
16.62
12.66
9.64
7.35
5.60
17.91
92
Table 9.2 (Continued)
Distance
Feet
88, FlO, Ll
89, FlO, Ll
810, FlO, Ll
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
39.77
32.24
26.13
21.18
17.17
13.91
11.28
9.14
7.41
6.01
25.71
35
35
24
24
15
18
6
11
4
3
35
8um of
Frequencies
210
155
Mean Distance .4762
2
12.11
X
Distance
Feet
22.25
19.06
16.32
13.97
11.97
10.25
8.78
7.52
6.44
5.51
32.89
27
17
21
8
11
7
11
11
3
1
38
66.11
50.22
28.14
28.97
22.01
16.71
12.69
9.64
7.32
5.56
17.58
63
52
33
33
20
19
9
5
10
9
22
275
.6452
.3636
14.74
9.48
81, Fl, 12
82, Fl, 12
83, Fl; 12
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequenc~ Frequency Frequency Frequency
.1
.2
.3
.4
.5
.6
.7
.8
.9]
1.0
>1.0
8um of
Frequencies
142
60
45
16
15
4
4
1
J
129.92
72.31
40.24
22.39
12.46
6.93
3.86
2.14
5
293
Mean Distance .1706
2
9.95
X
8.69
111
60
46
29
10
11
3
4
1
6
280
120.06
68.58
39.17
22.37
12.78
7.30
4.17
2.38
3.17
138
64
47
27
9
3
3
1
J
129.93
72.31
40.24
22.39
12.46
6.93
3.86
2.14
1
293
.1786
.1706
11.33
8.61
2.69
93
Table 9.2 (Continued)
86 , F ..... , L2
84, Fl, L2
859 F1 2 L2
Observed Expected Observed Expected Observed Expected
Frequency Frequency' Frequency Frequency Frequency Frequency
t
",
Distance
Feet
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
110
7'7
35
25
15
6
5
1
J
120.06
68.58
39.17
22.38
12.78
7.30
4.17
2.38
f6
8wn of
Frequencies
3.17
280
.1
.2
.3
.4
.5
.6
.7
.8
.
.9
1.0
>1.0
5
L
218.45
98.55
44.46
20.06
9.05
4.08
3.35
398
.1256
Mean Distance .1786
2
6.73
X
Distance
Feet
203
108
49
17
11
217.54
98.34
44.45
20.09
9.08
4.10
239
81
39
26
5
2
~
5
3.39
397
.1259
11.26
4.36
82 2 F2, L3
81, F2, L2
~3, F2, L2
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
i
8wn of
Frequencies
96.28
58.75
35.85
21.87
13.35
8.14
4.95
3.03
87
68
40
17
20
10
2
1
)
2
247
Mean Distance .2024
2
12.38
X
4.75
59
53
37
23
15
5
8
2
1
0
2
205
68.95
45.75
30.37
20.15
13.37
8.88
5.89
3.90
2.59
1.72
3.39
75
36
34
17
11
10
3
7
1
2
7
203
.2439
.2463
11.29
12.50
67.74
45.13
30.07
20.04
13.35
8.89
5.92
3.95 .
2.63
1.75
3.50
94
Table 9.2 (Continued)
Distance
Feet
s4, F2, L2
86, F2, L2
85. F2, L2
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
.1
.2
.3
.4
98
80
30
25
15
4
5
1
1
.5
.6
.7
.8
.9
1.0
>1.0
}4
Sum of
Frequencies 263
Mean Distance .1901
2
11.63
X
r
Distance
Feet
107.57
63.57
37.56
22.20
13.12
7.75
4.58
2.70
1.60
2.31
122
52
42
16
16
5
6
1
1
J4
109.01
64.16
37.77
22.23
13.08
7.70
4.53
2.66
1.57
2.24
265
73
60
26
30
11
12
4
5
1
1
4
82.83
52.60
33.41
21.21
13.47
8.55
5.43
3.45
2.19
1.39
2.42
227
.1887
.2203
10.76
12.18
S8, F2, L2
S7, F2, L2
S9, F2, L2
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
.1
.2
.3
.4
.5
.6
.7
.8
.91
1.0
>1.0
Sum of
Frequencies
112
53
41
15
18
5
3
4
}8
259
Mean Distance .1931
2
12.23
X
104.71
62.37
37.15
22.13
13.18
7.85
4.67
2.78
4.10
112
4'7
41
16
16
6
5
2
J6
251
99.06
59.96
36.29
21.97
13.30
8.05
4.87
2.94
4.52
29
34
13
18
9
12
5
7
2
2
11
142
.1992
.3521
8.59
11.24
35.10
26.42
19.89
14.97
11.27
8.48
6.38
4.80
3.61
2.72
8.29
95
Table 9.2 (Continued)
Distance
Feet
.
810, F2, L2
811 1 F2, L2
812, F2, 12
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
44
22
23
11
13
6
9
7
2
1
4
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
8um of
Frequencies
35.10
26.42
19.89
14.97
11.27
8.48
6.38
4.80
3.61
2.72
8.29
142
32.83
24.96
18.98
14.43
10.97
8.34
6.34
4.82
3.66
2.78
8.84
137
Mean Distance .3521
2
X
11.64
Distance
Feet
24
33
14
17
8
11
4
6
2
4
14
54
29
27
10
15
6
7
3
3
1
4
43.31
31.51
22.92
16.68
12.13
8.83
6.42
4.67
3.40
2.47
6.61
159
.3650
.3145
13.82
10.42
814, F2, L2
815,' F2, 12
813, F2, L2
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
.1
.2
.3
.4
.5
.6
.7
.8
.9
LO
>1.0
8um of
Frequencies
54
50
22
26
11
12
5
6
1
3
6
196
Mean Distance .2251
2
11.83
X
63.56
42.94
29.02
19.60
13.25
8.95
5.04
4.08
2.76
1.86
3.88
93
45
37
19
15
5
7
1
4
1
2
229
84.14
53.22
33.66
21.29
13.47
8.52
5.39
3.40
2.15
1.36
2.34
121
81
36
25
10
9
1
4
!
129.93
72.31
40.24
22.39
12.46
6.94
3.86
2.15
6
293
.2183
.1706
8.32
11.27
2.69
96
Table 9.2 (Continued)
Distance
Feet
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
J
818, F2,L2
816, F2, L2
8172 F2 2 L2
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
148
127
84.15
139.27
115.59
75
66
68
66.83
54
53.23
75.67
26
44
41.11
38.63
33.67
45
22
22.34
21.30
22.33
17
19
14
12.14
16
11
12.91
13.47
12
1
6.60
8.52
7.46
5
4
4.31
3.58
5.39
3
7
0
0
3.41
2.49
1.95
5
1
2.16
2
3.42
2.31
1.36
)3
2.35
5
\8
8um of
Frequencies 274
Mean Distance .1825
2
16.20
X
Distance
Feet
229
.2183
305
.1639
12.63
12.71
819, F2,L2
820, F2, L2
821, F2, L2
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
.1
.2
.3
.4
.5
.6
.7
.8
.9}
1.0
>1.0
8um of
Frequencies
135
59
45
16
15
5
4
}6
289
126.86
71.17
39.93
22.40
15.57
7.05
3.95
2·:~83
157
70
45
15
17'
2
5
L
148.03
78.68
41.82
22.23
11.82
6.30
3.34
3.79
134
91
39
14
10
10
2
16
Mean Distance .1730
316
.1582
310
.1613
11.10
10.50
18.43
i
143.23
77.05
41.45
22.29
11.99
6.45
3.47
1.:
17
...
97
Table 9.2 (Continued)
S22, F2, L2
824 , F2 L2
S23 2 F2. L2
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Freqaency Frequency Frequency
j
Distance
Feet
121
81
34
27
8
11
2
4
.1
.2
.3
.4
.5
.6
.7
.8
09
l.0
>1.0
129.16
72.02
70.16
22.40
12.49
6.96
3.88
2.16
$4
1
Sum of
Frequencies
2.73
.1
.2
.3
.4
.5
.6
.7
.8
09
1.0
>1.0
54.41
37.96
26.48
18.48
12.89
8.99
6.27
4.3'7
3.05
2.13
4.91
180
292
Mean Distance .1712
2
X
10.53
Distance
Feet
59
35
31
13
14
6
10
8
1
0
3
144
61
47
16
15
133.79
'73.72
40.62
22.38
12.33
6.79
3.74
2.06
5
"i
.-'
0
J
5
2.53
298
.2778
.1678
13.57
11.74
s26, F2, L2
S27, F2, L2
S25, F2, L2
Observ"ed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
149.64
79.22
41.94
22.20
11.75
6.22
3.29
1.74
139
8'7
35
29
9
10
2
3
1
Sum of
Frequencies
J
5
318
Mean Distance .1572
2
13.80
X
1.96
159
67
46
16
14
4
2
0
J5
317
.1577
12.72
148.84
78.95
41.88
22.21
11.78
6.25
3.31
1.75
1.98
151
64
45
19
14
3
5
0
J4
306
.1634
9.61
140.06
75.95
41.18
22.33
12.11
6.56
3.56
1.93
2.28
98
Table 9.2 (Continued)
Distance
Feet
Sl, F3, L2
82. F3. L2
83, F3, L2
Observed Expected Observed Expected Observed Expected
Frequency Frequency Fre~uency Frequency Frequency Frequency
66
50
27
23
18
11
4
6
1
2
2
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
Sum of
Frequencies
72.02
47.32
31.09
20.42
13.42
8.81
5.79
3.80
2.50
1.64
3.14
210
91.49
56.61
35.03
21.67
13.41
8.29
5.13
3.17
1.96
?'7
-t
16
6
2
1
0
!
2
3.19
240
Mean Distance .2381
2
6.83
X
Distance
Feet
102
44
40
85
69
28
27
11
7
7
2
4
95.59
58.44
35.73
21.84
13.35
8.16
4.99
3.05
1.86
1
6
2.93
246
.2083
12.98
.2033
13.35
84, F3, L2
s6, F3, L2
85, F3. L2
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0}
>1.0
Sum of
Frequencies
94
73
32
25
11
11
2
4
0
}4
256
Mean Distance .1953
2
10.05
X
102.58
61.47
36.84
22.07
13.23
7.92
4.75
2.84
1.70
2.55
118
56
41
19
15
4
6
4
S2
265
.1887
6.59
109.01
64.16
37.77
22.23
13.08
7.70
4.53
2.66
3.81
114
50
44
16
18
5
3
1
~7
258
.1938
12.83
103.99
62.07
37.05
22.11
13.20
7.88
4.70
2.80
4.157
.,
99
Table 9.2 (Continued)
Distance
Feet
s8, F3, L2
89, F3, L2
S7~ F3, L2
Observed Expected Observed Expected Observed Expected
Frequency Frequency Fre~uency Frequency Frequency Frequency
.1
.2
.3
.4
.5
.6
.7
.8
99
71
30
29
11
11
3
4
.9j
1.0
>1.0
J6
Sum of
Frequencies
108.29
63.87
37.67
22.21
13.10
7.72
4.55
2.68
3.86
33
92.85
57.22
35.26
21.73
13.39
8.25
5.08
3.13
5.03
242
264
Mean Distance .1894
2
9.30
X
Distance
Feet
111
45
39
16
15
5
7
1
75
64
31
18
11
12
3
5
1
J7
82.18
52.29
33.28
21.17
13.47
8.57
5.45
3.47
2.20
3.86
226
.2066
.2212
12.54
10.68
S12, F3, L2
811, F3, L2
S10, F3, L2
Observed
Expected
Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
.1
.2
.3
.4
.5
.6
.7
.8
.9)
1.0
>1.0
Sum of
Frequencies
108
52
40
19
15
5
6
1
J6
252
Mean Distance .1984
2
6.01
X
99.75
60.26
36.40
21.99
13.28
8.02
4.84
2.92
4.47
77
60
26
28
10
12
4
5
1
0
4
82.83
52.60
33.41
21.21
13.47
8.55
5.43
3.45
2.19
1.39
2.42
83
64
30
24
10
11
7
5
J5
227
.2203
239
.2092
11.68
6.08
90.81
56.30
34.91
21.64
13.42
8.32
5.15
3.19
5.21
100
Table 9.2
(~ontinued)
Distance
Feet
813, F3, L2
814" F.?, L2
S15. F3, 12
Observed Expected Observed Expected Observed Expected
Frequency Fre~enc'y Freg,uencx Fre3,:...i.er:,cy Frequency Frequency
108
58
41
15
16
6
5
1
.1
.2
.3
.4
.5
.6
07
.8
.9
1.0
>1.0
Sum of
Frequencies
99076
60.26
36.40
21099
13.28
8.02
4.84
2.92
1
2
4047
252
~
7
1
3
0
4
~5;;007
34.42
21.51
13.44
8.40
5.25
3.28
2.05
1.28
2.13
87
52
29
24
11
11
3
5
0
0
2
80088
51.67
33.01
21.09
13.47
8.61
5.50
3.51
2.24
1.43
2.53
224
02232
10.10
7.27
Distance
Feet
i8
15
88012
235
.2128
Mean Distance .1984
2
X
97
4'7
38
8.03
816 2 F33 12
s18, F3, L2
817" F'~- 9 12
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency FreqQency Frequency Frequency
.1
.2
.3
04
.5
.6
.7
.8
.9
1.0
>1.0
Sum of
Frequencies
101
72
30
27
11
13
3
4
!
110.46
64.'76
37.96
22.25
13.04
7.65
4.48
2.62
6
267
Mean Distance .1873
2
10.96
X
3.'72
83
64
30
24
10
12
3
5
0
2
4
237
.2110
11.21
89046
55.69
34.66
21.58
13.43
8.36
5.20
3.24
2 001
1.25
2 007
76
36
34
15
17
10
3
6
1
3
1
202
.2475
12.06
67013
44082
29092
19.97
13033
8.90
5094
3.96
2.65
1.76
3055
101
Table 9.2 (Continued)
Distance
Feet
•
jt
820, F3,L2
821, F3, L2
819, F3, L2
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
74
57
28
25
12
9
6
6
3
1.
1.
8um of
Frequencies
79.59
51.05
32.75
21.01
13.47
8.64
5.54
3.55
2.28
1.48
2.61
222
145.63
77.87
41.64
22.26
11.90
6.36
3.40
1.82
159
65
46
16
14
3
.--,
0
J
5
2.09
313
.2119
11.60
.1597
14.33
822, F3, L2
824, F3, 1.2
823, F3, L2
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
•
88.79
55.38
34.54
21.54
13.44
8.38
5.22
3.26
2.03
1.26
2.10
236
Mean Distance .2252
2
5.80
X
Distance
Feet
93
58
29
17
9
12
7
6
1
0
4
8um of
Frequencies
133
57
33
27
15
5
5
1
57
283
Mean Distance .1767
2
12.30
X
122.31
69.44
39.43
22.38
12.71
7.21
4.09
2.32
3.05
119
79
33
26
10
12
2
4
I6
292
129.16
72.02
40.15
22.40
12.49
6.96
3.88
2.16
2.73
95
40
30
25
19
12
3
1
2
}
5
232
.1712
.2155
13.84
12.70
86.126
54.15
34.04
21.40
13.46
8.46
5.32
3.34
2.10
3.56
102
Table 9.2 (Continued)
Distance
Feet
s26 , F3, L2
825, F3, L2
827, F3, 12
Observed Expected Observed Ex:pected Observed Ex:pected
Frequency Frequency Frequency Frequency Frequency Frequency
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
83
42
37
17
15
5
9
2
1
0
4
8um of
Frequencies
75.1)+
48.87
31.79
20.68
13.45
8.75
5.69
3.70
2.40
1.56
2.91
215
2
1
1
1
3
84.80
53. ~-j3
33.79
21.33
13.46
8.50
5.36
3.38
2.13
1.35
2.31
230
Mean Distance .2326
2
10.58
X
Distance
Feet
73
65
42
28
10
4
91
60
27
19
16
6
3
1
1
0
3
82.83
52.60
33.41
21.21
13.47
8.55
5.43
3.45
2.19
1.39
2.42
"
227
.2174
.2203
16.15
9.55
828, F3, L2
829, F3, 12
830, F3,12
Observed Ex:pected Observed Ex:pected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
Sum of
Frequencies
53
50
21
23
11
12
5
6
2
3
8
194
Mean Distance .2577
2
12.80
X
62.38
42.32
28.71
19.47
13.21
8.96
6.08
4.12
2.79
1.89
4.01
91
47
30
25
9
12
6
2
1
0
4
227
.2203
9.03
82.83
52.60
33.41
21.21
13.47
8.55
5.43
3.45
2.19
1.39
2.42
90
67
29
26
11
12
3
5
2
J5
250
.2000
10.02
98.36
59.66
36.18
21.94
13.31
8.07
4.89
2.97
1.80
2.77
•
103
Table 9.2 {Continued)
Distance
Feet
S31 2 F32 L2
832, F3 2 L2
833 2 F3 2 L2
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
102
52
40
30
10
4
3
2
1
.1
.2
.3
.4
.5
.6
.7
.8
.9
13
1.0
>1.0
Sum of
Frequencies
96.28
58.75
35.84
21. 87
13.34
8.14
4.96
3.03
1.85
2.89
247
65
35
24
22
10
12
8
3
1
1
6
58.34
40.14
27.61
18. 99
13007
8.99
6.18
4.25
2.92
2.01
4.44
187
.1992
.2674
6.31
7.32
81, F4, L2
S34, F3, L2
835, ]'3, L2
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
.1
.2
.3
.4
.5
.6
.7
08
.9
1.0
>1.0
Sum of
Frequencies
..
15
99.06
59.96
36029
21. 97
13.30
8.05
4.87
2.94
1.78
2.73
251
Mean Distance .2024
2
9.09
X
Distance
Feet
94
64
39
18
17
6
3
4
1
103
50
39
24
15
5
7
1
0
94.90
58.14
35.61
21.82
13.36
8.18
5.01
3007
1
1
245
Mean Distance 02041
2
9.19
X
1.88
2.97
110
56
34
25
9
10
103.99
62.07
37.05
22.• 11
13.20
7.88
4.70
2 080
5
2
J
7
258
77
63
26
25
10
13
4
5
1
84.80
53.53
33.79
21.33
13.46
8.50
5036
3.38
2.13
4.15
] 6
230
.1938
.2174
5.07
11.21
3066
104
Table 9.2 (Continued)
Distance
Feet
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
S2, F4 , L2
83, F4, L2
S4..L..F4 , L2
Observed Expected Observed Expected Observed Ex.pected
Frequency Frequency Frequency Freq~ency Frequency Frequency
110
101.168
82.83
70
75.14
90
52.60
48.87
60.87
47
54
.52
40
36.62
33.41
30
35
31.79
1'7
26
20068
21 021
22 003
19
·1
12
13.26
13045
13047
15
15
6
8055
8075
7097
7
5
2
6
4.80
5.43
5.69
3
1
2.88
3.45
30'70
5
5
2
2040
1
1.
2.19
1. 73
2
1.56
1.39
3
2.62
2 042
2 091
5
5
J5
Sum of
Frequencies
227
215
Mean Distance .2203
11.68
x.2
Distance
Feet
254
.2326
.1969
8.13
5.80
s6, F4, L2
S5, F4, L2
S7, F4, L2
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
Sum of
Frequencies
94
60
38
17
15
6
4
1
0
1
1
237
Mean Distance 02110
2
7016
x.
89.46
55.69
34.66
12.58
13.43
8.36
5.20
3.24
2 001
1.25
2.07
88
67
29
18
15
9
8
7
17
248
96097
59005
35.96
21.89
13033
8.12
4.94
3001
4068
117
56
41
19
15
4
6
0
] 6
264
.2016
.1894
12.55
8083
108.29
63.87
37.67
22.12
13.10
7.72
4.55
2.68
3.86
105
Table 9.2 (Continued)
•
Distance
Feet
s8, F4, 1.2
810, F4, 1.2
89, F4~ L2
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
105
65
28
26
10
9
3
4
J
Sum of
Frequencies
99.76
60.26
36.40
21.99
13.28
8.02
4.84
2.92
2
252
Mean Distance .1984
2
6.71
X
Distance
Feet
4.47
108.29
63.87
37.67
22.21
13.10
7.72
4.55
2.68
99
71
30
26
10
12
3
6
1
7
3.86
81
59
37
18
16
6
4
3
4
1
4
264
86.79
54.46
34.17
21.44
13.45
8.44
5.29
3.32
2.08
1.30
2.20
233
.1894
.2146
14.05
6.38
811, F4, L2
812, F4 2 L2
813, F4, 1.2
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
8um of
Frequencies
96
46
38
17
15
5
7
1
4
o
5
234
87.45
116
54.77
59
34.29
34
21.48
25
13.45
11
8.42
9
5.27
5
3.30
1
2.06}
1.29
6
2.17
266
109.74
64.46
37.86
22.24
13.06
7.67
4.50
2.64
3.77
112
50
40
19
14
6
5
6
2
l 3
J .
257
Mean Distance .2137
.1880
.1946
i
4.51
7.89
14.10
103.28
61.77
36.94
22.09
13.21
7.90
4.72
2.82
1.69
2 51
106
Table 9.2 (Continued)
Distance
Feet
81 5 2 F4-l 2 L2
816, F4, L2
s14, F4? 12
Observed Expected Observed Expected Observed EKpected
Frequency Frequency Fre~ency Frequencz Frequenc~ Frequency
.1
.2
03
.4
.5
06
.7
.8
.9
LO
75
62
26
28
9
12
3
1
82 083
52 060
33.41
21 021
13.47
8.55
5.43
3.45
2019
17
>1.0
3.81
Sum of
Frequencies
44.82
29092
19.97
13.33
r
8090
5094
3.96
2065
1.76
3055
202
227
Mean Distance .2203
2
16.08
X
Distance
Feet
67013
62
49
33
17
l ')
7
5
4
4
0
6
123.06
69.73
39.51
22.39
132
61
43
19
15
4
6
1
12 068
7.19
4007
2030
1
3
3.01
284
02475
.1761
6.45
6.05
s18, F4, L2
S17, F4, L2
S19, F4, L2
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Fre9"uency Frequenc;y: Frequenc;y:
01
.2
.3
.4
.5
.6
.7
.8
.9
LO
>1.0
Sum of
Frequencies
120
80
33
26
10
10
2
4
57
292
123.82
70.02
39.60
22.39
129.16
72.02
40.16
22.40
132
68
43
12 049
6096
3088
14
4
12066
6
4005
2029
2.16
2.'73
17
2.16
0
1
1
285
Mean Distance .1712
01'754
2
X
8.2'7
14.35
2.98
98
66
43
18
15
6
3
1
3
J4
257
.1946
6.71
103.28
61.77
36.94
22.09
13.21
7.90
4.72
2.82
L69
2.51
•
107
Table 9.2 (Continued)
•
Distance
Feet
820, F4, L2
821, F4, L2
822 2 F4, L2
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
126.86
71.17
39.92
22.40
12.56
7.05
3.95
2.21
131
66
44
18
10
7
5
4
I
8um of
Frequencies
4
2.83
289
2
2
S6
3.54
103
52
32
27
11
6
3
2
0
I6
92.85
57.22
35.26
21.73
13.39
8.25
5.08
3.13
1.93
3.10
242
.1845
.2066
11.52
10.11
824, F4, L2
823, F4, L2
825, F4, L2
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency Frequency Frequency Frequency
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
Sum of
Frequencies
•
c:;
,'/
113.39
65.94
38.35
22.30
12.97
7.54
4.38
2.55
271
Mean Distance .1730
2
4.50
X
Distance
Feet
104
73
31
30
18
93
45
37
16
15
5
8
4
4
13
230
Mean Distance .2174
2
8.55
X
84.80
53.53
33.79
21.33
13.46
8.50
5.36
3.38
2.13
3.66
86
46
29
24
8
9
6
2
5
17
222
79.59
51.05
32.75
21.01
13.47
8.64
5.54
3.55
2.28
4.08
86
72
39
16
9
5
7
3
3
16
246
.2252
.2033
10.15
13.31
95.59
58.44
35.73
21.84
13.35
8.16
4.99
3.05
1.86
2.93
108
~
Table 9.2 (Continued)
S?'7
"282. F4, L2
826 9 F4, L2
F'4 o 1.2
.,
Observed Expected Observed Ex:pected. Observed Expected
Frequency Freq'~ency Frequ.ency Freg,(;;.ency Freg,uency Frequency
~ (
Distance
Feet
..
100
52
32
25
10
6
8
6
3
.1
.2
.3
.4
.5
.6
07
.8
.9
1.0
>1.0
J2
8um of
Frequencies
94.21
:"5'7.83
35050
21.-;'9
13.37
R
.?~
___ .' 0 ,-' .......
5 00 4
3 009
1.89
3.01
244
43
30
10
9
6
4
100.46
600'57
36051
22 001
13 027
8.01
4.82
2 090
2
10'7:3
92
53
f4
2.66
105
66
27
15
10
9
8
3
3
J1
•
96.28
58.75
36.84
21087
13.34
8014
4096
3.03
1.85
2.89
21.7
253
Mean Distance .2049
2
8.66
X
Distance
Feet
0
.2024
.1976
8.04
10.76
829, F4, L2
831, F4, L2
S30~ F)4, L2
Observed Expected Observed Expected Observed Expected
Frequency Frequency Frequency .Frequency Frequency Frequency
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
>1.0
8um of
Frequencie s
101
75
31
18
22.
8
5
3
J6
269
Mean Distance .1859
2
12.56
X
111.92
65.35
38.16
22 028
13001
7.59
4.43
2.59
3.63
118
56
42
15
16
9
3
4
J3
266
109.74
64.46
37086
22.24
13006
7.6'7
40.50
2.64
] 3077
80
61
38
18
17
10
:3
1.
2
0
3
86.79
54.46
34.17
21.44
13.45
8.44
5.29
3.32
2.08
1.30
2.20
"
.1880
233
02146
6.78
7.74
"