BIBLIOGRAPHY ALOS, EMILY JUNE C., BAT-OY,...
BIBLIOGRAPHY
ALOS, EMILY JUNE C., BAT-OY, ELEONOR B. April 2008. Efficiency of
Ratio and Regression Estimators in Estimating Strawberry Production. Benguet, State
University, La Trinidad, Benguet.
Adviser: Marycel H. Toyhacao
ABSTRACT
The study was conducted to determine the efficiency of the estimates on
strawberry production using ratio and regression estimators.
A total of 195 plots were selected as samples from the total population of 899
plots from the Sariling Sikap area at strawberry farm. The data were summarized,
tabulated and analyzed. The total production and total number of plants per area were
used as the variables in the estimation.
Being an unbiased estimator, the Simple Random Sampling has the least mean
and total production of strawberry. Among the bias estimator, classical ratio estimator is
the most precise among the three estimators used in estimating strawberry production
with its least coefficient of variability.
Regression estimator is the most consistent estimator and it is considered the most
efficient estimator among the estimator used in estimating the strawberry production.
TABLE OF CONTENTS
Page
Bibliography ……………………………………………………. i
Abstract
……………………………………………………………. i
Table of Contents …………………………………………..………. ii
INTRODUCTION ………………………………………………….. 1
Background of the Study
…………………………………… 1
Objective of the Study
…………………………………… 3
Importance of the Study
…………………………………… 3
Scope and Delimitation
…………………………………… 4
REVIEW OF RELATED LITERATURE
………………………….. 5
THEORETICAL FRAMEWORK
………………………………….. 11
Simple Random Sampling
…………………………………... 11
Classical Ratio Estimation
…………………………………… 13
Hartley-Ross ratio-type Estimation …………………………… 16
Regression Estimation …………………………………………… 20
Magnitude Efficiency
…………………………………… 24
Definition of Terms …………………………………………… 25
METHODOLOGY …………………………………………………… 26
Respondents of the Study
……………………………………... 26
Data Gathering
……………………………………………... 26
Data Analysis ……………………………………………………… 26
ii
RESULTS AND DISCUSSION
……………………………………… 28
Estimates of the Strawberry Production
………………………. 28
Consistency of the Estimates
……………………………… 29
Precision of the Estimates
……………………………………… 30
Relative Efficiency of the Different Estimates
……………… 31
SUMMARY, CONCLUSION AND RECOMMENDATION
…….. 33
Summary
……………………………………………………… 33
Conclusion
……………………………………………………… 34
Recommendation
……………………………………………… 34
LITERATURE CITED
……………………………………………… 35
APPENDICES
……………………………………………………… 37
Appendix A. Application for Oral Defense ………………………… 37
Appendix B.The raw data of the strawberry production ………….. 38
Appendix C. Sample Computations ……………………………….. 43
iii
1
INTRODUCTION
Background of the Study
A sample survey may be considered as an absolute (enumeration)
experiment whose main objective are to obtain estimators of parameters and to
derive measures of precision of these estimators. In analytical surveys one of the
objectives is to test hypothesis with the use of appropriate estimation procedures.
Thus, it is apparent that simple and sound estimation techniques must be
developed to provide precise and accurate statistics whether such endeavors are of
the absolute or analytical type or both. (Burton T. Onate & Julia Mercedes O.
Bader: 1990)
Estimation of the population mean and total sometimes based on a sample
of response measurements y1,y2,….yn are obtained by simple random sampling
and stratified random sampling. William Mendenhall (1990) stated that measuring
Y and one or more subsidiary variables are used to estimate the mean of the
response Y. It is basic to the correlation and provides means for development of a
prediction equation relating Y and X by the method of least square.
The basic concept in estimation procedure is to determine the unknown
values of the parameters using the sample values. By doing so, many estimators
can be utilized. Among of those are using ratio estimators and regression
estimators. However, the correct approach can lead to better estimates, that is, less
bias is obtained. Ratio estimators and regression estimators, as pointed by Onate
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
2
and Bader (1990), increased its precision or usually attained by observing an
additional variable Zi, in addition to the original characteristics.
It is most effective when the relationship between the response y and the
subsidiary variable x is linear through the origin and the variance of y is
proportional to x.
In Regression estimation the technique is for incorporating information on
a subsidiary variable, this is better if the relationship between the y’s and x’s is a
straight line not through the origin. It works well when the plot of y and x reveals
points by lying uniformly close to a straight line with unit slope.
Ratio estimation usually leads to biased estimators. Thus, a consideration
for the magnitude of the bias should be observed. However, for a large sample
size (n>30) and he (σx/μx) ≥10, the bias is negligible. Ratio estimators are
unbiased when the relationship between y and x is linear through the origin. As
mentioned, we can assume to draw sample from the population, and possibly
estimate its average (strawberry production) from y, sample mean. Since there are
underlying associations between the mentioned variables, the number of plants
per plot can be multiplied from the production of strawberries per plot.
Objective of the Study
Specifically, the study was conducted to determine the efficiency of the
estimates on the production of strawberries in Strawberry Farm, La Trinidad,
Benguet using the ratio estimates and using regression estimates.
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
3
Importance of the Study
The researchers find the study important; it gives information to the
concern individuals specially in estimating the production of strawberry in La
Trinidad, Benguet using the ratio and regression estimators. Through this study,
students would be aware that strawberry production can be estimated or it can be
measured using statistical tools by studies and researchers.
The useful and relevant information acquired from the study will
encourage future researchers to estimate strawberry production applying the
choice of their analysis. This can help them to boost their ability in analyzing
given data’s from the specific tools they will conduct.
Result of the study will provide the strawberry management knowledge if
the ratio and regression estimators can be appropriately applied to the production
of strawberries. This will make them aware on the specific materials needed for
strawberry production.
This study will serve as a valuable reference to farmers that production of
strawberry can be estimated through different estimation procedures. This will
also be a basis on how many strawberries will be produced in one plot.
Finally, this study will be an instrument or a benchmark for other
researchers who are planning to study on different estimation procedures.
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
4
Scope and Delimitation
The main focus of this study was to determine the efficiency of ratio and
regression estimators in the production of strawberries in Strawberry Farm at La
Trinidad, Benguet.
The data was gathered from the Strawberry Farm of La Trinidad, Benguet.
Specifically at the Sariling Sikap Area.
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
5
REVIEW OF LITERATURE
Ratio Estimation
The ratio estimation is a different way of estimating population total or
mean that is useful in many problems. It involves the use of known population
totals for auxiliary variables to improve the weighting from sample values to
population estimates. It operates by comparing the survey sample estimate for an
auxiliary variable with the known population total for the same variable on the
frame. The ratio of the sample estimate of the auxiliary variable to its population
total on the frame is used to adjust the sample estimate for the variable of interest.
Ratio method of estimation is frequently used in sample surveys to
estimate the population mean of the variable under investigation. Several ratio-
type estimates can be formed. All these estimates are unsatisfactory in the sense
that they are biased. Hartley and Ross, who proposed an unbiased ratio-type
estimator for uni-stage sampling designs, overcame this difficulty. In practice,
however, we are generally faced with multi-stage sampling designs. This
communication gives a generalized form of Hartley-Ross unbiased ratio-type
estimator for multi-stage designs.
This method aims to obtain an increased precision by taking advantage of
the correlation between Yi and Xi .It applies population knowledge of adjustment
variable Xi to improve estimation of the population total should be known.
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
6
The ratio estimates can be used when Xi is done some other kind of
supplementary variables. However, for successful application, the ratio Yi / Xi
should be relatively constant over the population and the population total should
be known.
Lohr (1999) enumerated the uses of ratio estimation which include: ratio
estimators is simply used to estimate a ratio; to estimate a population total but the
population size N is unknown and to adjust estimates from the sample so as to
reflect demographic totals.Often,it is used to adjust for no response.
Hsu and Kuo (2000) used the ratio estimation to estimate the recycled and
the production values and employment opportunities induced by household waste
recycling. Ratio estimation was used in their study because the true population
size of recycling plant is unknown; and the audited amount of household waste
recycling is known; and the audited amount and recycling amount are highly
correlated.
Angel et.Al (2004) evaluated the impact of using ratio estimation in their
study, as the estimate census of night dwelling was much closer to the actual
value.
M.Gossop, J.Strang, P.Griffiths, B.Powis and C.Taylor presents an
approach in estimating the prevalence of cocaine use, based upon a new ratio
estimation technique. This method can be applied to random samples of
overlapping populations for which no sampling frames exist. When the ratio
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
7
estimation method is applied to the two study samples (drawn from populations of
people using cocaine and people using heroin) the ratio of cocaine users to heroin
users (C/H) was 1.55, with a 95% confidence interval of +/- 0.48. Such estimates
should be applied with caution. However, if used with reference to national
estimates of about 75,000 heroin users, application of the present estimate
suggests that there may be about 116,000 cocaine users in the UK.
Myers and Thompson (1989) pioneered the concept of a generalized
approach to estimating hedge ratios, pointing out that the model specification
could have a large impact on the hedge ratio estimated. While a huge empirical
literature exists on estimating hedge ratios, the literature lacks a formal treatment
of model specification uncertainty. These researches accomplish the task by
taking a Bayesian approach to hedge ratio estimation, where specification
uncertainty is explicitly modeled. Specifically, a Bayesian approach to hedge ratio
estimation that integrates over model specification is uncertain; it yields an
optimal hedge ratio estimator that is robust to possible model specification
because it is an average across a set of hedge ratios conditional on different
models. Model specifications vary by exogenous variables (such as exports,
stocks, and interest rates) and lag lengths. The methodology is applied to data on
corn and soybeans and results showed potential benefits and insights from such
approach.
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
8
Damoslog and Tomin (2007) considered the Classical Ratio Estimation as
a precise and efficient estimator because it has the lowest coefficient of variation
and 15% higher relative efficiency than simple random sampling. They also cited
that classical ratio estimator has the least estimated bias and is a consistent
estimator with its narrow 95 % confidence interval
.
Regression Estimation
Regression estimation aims to find a relationship between a dependent
variable and set of independent variable (Kweon and Kocelman, 2004).
In a study conducted by Wei Chen et.Al (2005), linear regression was used
to evaluate the parameters with the least squares estimation to estimate an original
model for daily shopping with the consideration of individual influence the
estimation was verified to be straight forward and efficient.
Myall (2000) noted a known feature of regression estimation, that is, some
data will generate negative and weights, which leads to having a download effect
on aggregate estimates.
Valliant (2002) used the general regression estimator to construct variance
estimators that are approximately model-unbiased in single estimates.
Barrios (1995) explored the used of various small area estimators for
various socio-economic indicators. Regression estimation was noted to be
reasonable use.
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
9
Watson (1937) from Cochran’s Citation (1977) illustrated an example of a
general situation in which regression estimates are helpful. Watson used a
regression on the leaf area and leaf weight to estimate the average area of the
leaves on a plant. The procedure was to weight all the leaves of the plant. The
area and the weight of each leaf were determined from the small sample of leaves.
The sample area was then adjusted by means of the regression on leaf weight
means of the regression on leaf weight.
Usha Rani (2001) used the regression model Y=k (LxP) where Y = fruit
surface area in Chili (capsicum annuum L.)Fruit, D= diameters of the fruit and k
is the regression coefficient. She found the values of k=1.6827.Thus, one can
estimate the fruit diameter x 1.6827.
Palaniswamy (1990) reported that the linear regression model can be
employed in the prediction of grain yield per plot in rice and consequently for
crop estimation.
Lih-Yuan Deng and RAJ S. Chhikara (1990) cited that in a classroom
setting students often find the concept of super population and the assumption of a
model somewhat artificial when one needs to estimate the mean of a finite
population. They introduce the concept of a finite population decomposition
based on a regression fit to the population values and then discuss the bias and
variance of each estimator from the sampling design viewpoint. It shows by
fitting a regression line of y and x to the finite population, that the leading term of
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
10
the bias of yˆ is accounted for it on terms of the intercept of the regression line.
R
On the other hand, using quadratic regression fit, describe the leading term of the
bias of yˆ in terms of the coefficient of the quadratic term. Moreover, the sign of
ir
intercept in the fitted regression fit would be indicative of an over-or
underestimation of Y by yˆ or yˆ ,respectively. They hope that this will add to
R
ir
the understanding of the sampling properties of yˆ and yˆ without having to
R
ir
assume the super population structure.
Dorfman, J.H and Sanders, D.R (2004) cited that regression and
regression-related procedures have become common in survey estimation. It
reviews the basic properties of regression estimators, discuss implementation of
regression estimation, and investigate variance estimation for regression
estimators. The role of models in constructing regression estimators and the use of
regression in non-response adjustment are also explored.
A class of ratio and regression type estimators is given such that the
estimators are unbiased for random sampling, without replacement, from a finite
population. Nonnegative, unbiased estimators of estimator variance are provided
for a subclass. Similar results are given for the case of generalized procedures of
sampling without replacement. Efficiency is compared with comparable
estimation sample selection methods for this case (Rand Corp Santa Monica
Calif).
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
11
THEORETICAL FRAMEWORK
Simple Random Sampling (SRS)
Simple random sampling is a method of selecting n units out of the N
⎛ N ⎞
population elements such that every ⎜
⎟ distinct sample has an equal chance of
⎝ n ⎠
being selected. (Beligan et al., 2004)
Y n is the unbiased estimator of Y N and it is given by:
n
∑Yi
Y
i=1
n =
(sample mean)
(1.1)
n
and its sample variance is given by:
∑ Y −Y
2
(
n
i
)2
s =
(1.2)
n −1
An estimated estimator of the variance of Y n is
N − n ⎛ s2 ⎞
V (Y n ) (
)
=
⎜
⎜
⎟
⎟
(1.3)
N
⎝ n ⎠
where:
⎛ N − n ⎞
⎜
⎟ is the finite population factor(fpcf).The correction factor is used
⎝ N ⎠
because with small populations, the greater the sampling fraction n/N, the more in
formation there is about the population and the smaller the variance to be derived.
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
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The estimated coefficient of variation (C.V) are usually computed using
the estimated standard error (S.E)
SE(Y n )
CV (Y n ) =
(1.4)
Y n
where:
N − n
SE
(Y
2
n ) =
s = V (Y n )
(1.5)
Nn
n Y i
∑
N
1
Since Y
i=1
N =
=
∑Y and to estimate population mean is given by
N
i
N i=1
Y N = N Y n (1.6)
The variance, standard error (SE) and the coefficient of variation (C.V) of the
population total are
N (N − n)
Vˆ (N Y n ) =
2
s
n
proof:
Vˆ (N Y n ) = N2(V(Y n ))
⎛ N − n ⎞ ⎛ S ⎞
= N2 ⎜
⎟
⎝ N ⎠ ⎜⎜
⎟
⎟
⎝ n ⎠
N(N − n) 2
=
s
n
If S 2 is not available, therefore the estimates of V(N Y n ) is
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
13
N (N − n) 2
Vˆ (N Y
=
s
n )
n
(1.7)
N(N − n)
SE
(N
Y
2
n ) =
s = V (NY n )
(1.8)
n
Confidence interval is a range of possible values for the unknown
parameter with some measure for the degree of certainty, (1-α ) called the level of
confidence coefficient. Hence, the confidence interval for the mean (Y n ) and total
(NY n ) are as follows:
Confidence Interval of the mean estimate:
⎡
⎤
P⎢Y n − t
SE Y n ≤ Y n ≤ Y n + t
SE Y n ⎥ = 1−
α
( )
( )
α
α
(1.9)
⎣
,n−1
,n−1
2
2
⎦
Confidence interval of the total estimate:
⎡
⎤
P⎢NY n − t
SE NY n ≤ NY n ≤ NY n + t
SE NY n ⎥ = 1−
α
( )
( )
α
α
(1.1)
⎣
,n−1
,n−1
2
2
⎦
Classical Ratio Estimation
In ratio estimation, the ratio estimate of Y N using SRS is
Y
Y n
R =
X N
(2.1)
X n
where
X N (Total area of the strawberry)
n
∑ yi
Y
i=
n =
1
(Sample mean for the production of strawberry) (2.2)
n
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
14
n
∑xi
X
i=
n =
1
(Sample mean of the number of strawberry plant) (2.3)
n
According to Hartley and Ross, Rn is biased for R N ,and the bias is
⎛
⎞
N − n
bias ( R
⎜
⎟
2
n ) = ⎜
{Rns − s (2.4)
x
xy }
2 ⎟
⎝ Nnxn ⎠
where:
Y n
Rn =
(sample ratio of the mean)
(2.5)
X n
∑(x − xn)2
s 2 =
i
(sample variance of x
x
n −1
i )
(2.6)
∑(x − xn
i
)(y − y
i
n )
s 2
xy =
(sample covariance of x
n −1
i and yi ) (2.7)
Hence,
Bias (Y R ) = X N bias ( Rn )
(2.8)
The variance of classical ratio estimate and its standard error are
N − n
2
V (Y R) =
∑[y − Rnx (2.9)
i
i ]
Nn(n − )
1
SE (Y R ) = V (Y R ) (2.10)
Estimate coefficient of variation of the estimate is estimated using the S.E. but the
square root of the MSE will be used to estimate the CV where the variance is
adjusted for its bias:
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
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MSE(Y R )
CV(Y R ) =
100
(2.11)
YR
where:
MSE (Y R ) = V (Y R) + bias ( (Y ) ) 2
(2.12)
R
To estimate the population mean
Y N = NY R is used
(2.13)
The variance, standard error (S.E.) and bias of NY R are as follows:
V (NY R ) = N2 V (Y R)
(2.14)
SE (NY R ) = V (NY R )
(2.15)
and bias of population total is given by
bias (NY R ) = N(bias (Y R)) = N[ X N bias ( Rn ) ]
(2.16)
The confidence interval for the mean and total are shown below:
Confidence Interval of the mean estimate:
⎡
⎤
P⎢Y n − t
SE Y n ≤ Y n ≤ Y n + t
SE Y n ⎥ = 1−
α
( )
( )
α
α
(2.17)
⎣
,n−1
,n−1
2
2
⎦
Confidence Interval of the total estimate:
⎡
⎤
P⎢NY n − t
SE NY n ≤ NY n ≤ NY n + t
SE NY n ⎥ = 1−
α
( )
( )
α
α
(2.18)
⎣
,n−1
,n−1
2
2
⎦
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
16
Hartley-Ross Ratio-Type Estimation
Y
Instead of using
n
Rn =
,we use
X n
n
1
r = ∑ri (sample mean of ratio)
(3.1)
n
n i=1
Where:
y
r = i ;( individual sample ratio)
(3.2)
i
xi
y
r = 2
1
x2
y
r = 2
2
x2
:
:
y
r = n
n
xn
Note that E( r n ) ≠ Rn ⇒ r n is inconsistent, therefore Y r = r n X N ⇒ Y r is biased
and consistent. Bias estimator of Y r is given as
N −1
Bias(Y r ) =
(s )
(3.3)
rx
N
Proof:
∑ y
Bias(Y
i
r ) = ( X N r N ) − Y N = ( X N r N ) −
N
1
=
[∑y − NX NrN
i
]
N
1
=
[∑r x − NX NrN
i i
]
N
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
17
⎡
1
X
r ⎤
∑
=
⎢∑
i ∑
r x − N
i
i i
⎥
N ⎢⎣
N
N ⎥⎦
⎡
1
X
r ⎤
∑
=
⎢∑
∑
r x − N
i
i
i i
⎥
N ⎢⎣
N
⎥
⎦
Multiplying by N-1
N −1 ⎡
X
r ⎤
= -
⎢∑
∑ i∑
r x −
i
i i
⎥
N ⎢⎣
N
⎥
⎦
N −1
= -
(S
rx )
N
N −1
Bias(Y r ) = -
(s
rx )
N
Wherein the unbiased estimator of srx
1
S
[y −rn xn (3.4)
n
]
rx = n −1
Therefore,
N −1
Bias(Y r ) = -
(y −rn xn (3.5)
n
)
N
To get the unbiased estimator which make use of r n ,subtract the bias from
y = rx , therefore,
r
N
(N − )1n
y = r X
(y −rn xn
(3.6)
n
)
N +
r
N (n − )
1
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
18
As cited by Hartley and Ross, the formula of the variances of the ratio
type estimation for large sample is given by
2
1 (N − )
1
1
N −1
V ( y ) =
S − r S +
S + S S
r
( 2
2
2 N
y
x )
(
) ( 2 2 2
rx
r
x )
2
n
N
n(n − )
1
N
The unbiased estimator is given by
2
1 (N − )
1
1
N −1
Vˆ ( y ) =
S − r S +
S + S S (3.7)
r
( 2
2
2 N
y
x )
(
) ( 2 2 2
rx
r
x )
2
n
N
n(n − )
1
N
Where
∑(x − xn
i
)2
2
S =
(3.8)
x
n −1
∑(y − y
i
n )2
2
S =
(3.9)
y
n −1
∑(r − rn
i
)2
2
S =
(3.10)
r
n −1
n
S =
y − r x (3.11)
rx
[
n
n
n
]
n −1
∑ x − xn y − y
2
( i
)( i n)
S =
(3.12)
xy
n −1
Estimated standard error is,
SE( y ) = V y (3.13)
r
( r)
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
19
This is the square root of the variance. The estimated coefficient of
variation of the estimate is shown below,
MSE(yr )
CV ( y ) =
100 (3.14)
r
y r
Where,
MSE( y
= V y + Bias y
(3.15)
r
( r) ( ( r)2
)
All these result apply to the estimation of the population total Y N .The
unbiased estimator of Y N = NYn and the variance of standard error(SE) and bias
of NYn is defined by
V(NY r =N2V(Y r ) (3.16)
SE(NY r )= V (N y (3.17)
r )
N −1
Bias(NY r )=N(Bias(Y r ))=-N
(s ) (3.18)
rx
N
The confidence interval for the mean and total are shown below confidence
interval for the mean ( Y r )
Confidence Interval of the Mean
⎡
⎤
P⎢Y r − t
SE Y r ≤ Y r ≤ Y r + t
SE Y r ⎥ = 1−
α
( )
( )
α
α
(3.19)
⎣
,n−1
,n−1
2
2
⎦
Confidence Interval of the Total
⎡
⎤
P⎢NY r − t
SE NY r ≤ NY r ≤ NY r + t
SE NY r ⎥ = 1−
α
( )
( )
α
α
(3.20)
⎣
,n−1
,n−1
2
2
⎦
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
20
Regression Estimation
The linear regression estimator is designed to increase precision by the use
of auxiliary variant xi that is correlated with yi. Suppose yi and xi are obtained for
every unit in the sample that the population mean X N of xi is known then the
regression estimator of Y N is
Y reg = y + β X − x (4.1)
n
( N n)
Where the subscript reg denote as regression and β is the estimate of the
change in y and x when x is increased by unity. The model is shown,
∑(x − xn y y
i
)( −
i
n )
β =
(4.2)
∑n(x − x 2
i
n )
i=1
Where,
n Y i
∑
Y
i=1
n =
(Sample mean for the production of strawberry) (4.3)
n
n
∑xi
X
i=
n =
1
(Sample mean of the number of strawberry plant ) (4.4)
n
Variance of regression estimation is estimated using the variance of residual as
⎛
n ⎞ se2
V Y
( reg ) = ⎜1−
⎟
(4.5)
⎝
N ⎠ n
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
21
Where
e
ˆ
i = yi - Y n + β
(residual sample) (4.6)
1 (x − x n
i
)
where,
ˆ
n∑ x − y −
x y
i
i
∑( i )( i )
β =
(slope) (4.7)
1
n∑ −
x
i
(∑ i)2
2
1
en = ∑e (Mean of the residual) (4.8)
i
n
∑(e − en
i
)2
2
s =
(Variance of the residual) (4.9)
e
n −1
Standard error is defined by
SE(Y reg ) = V (Y reg ) (4.10)
And the bias of (Y reg ) is estimated as
1− n
2
N e x
x
N
i (
− n
i
)
bias (Y reg ) =
(4.11)
2
∑
n
1
N 1
sx
i=
−
The coefficient of variation of the estimator (Y reg ) which measures the variability
of the estimate is,
CV (
MSE Y
Y reg )
( reg )
=
100 (4.12)
Y reg
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
22
where
MSE(Y reg ) = V (Y reg )+ (Bias(Y ) 2
reg
(4.13)
To estimate the population mean, Y N , we used the unbiased estimator
Y N =N (Y reg ) (4.14)
And estimate variance of the total N (Y reg ) is
V (NY reg ) = N2 V (Y reg ) (4.15)
SE(NY reg ) = V (NY reg ) (4.16)
And its bias is given by
1− n
2
N e x
x
N
i (
− n
i
)
bias(NY reg ) = N (bias(Y reg )) = -N
(4.17)
2
∑
n
1
N 1
sx
i=
−
At the regression estimation in the measure of accuracy of the estimate is
approximately 95% confidence interval(C.I) where half width of the C.I is the
margin error of an estimate that is,
Confidence Interval of the mean:
⎡
⎤
P⎢Y reg − t
SE Y reg ≤ Y reg ≤ Y reg + t
SE Y reg ⎥ = 1−
α
( )
( )
α
α
(4.18)
⎢
⎣
2,n−1
2,n−1
⎥
⎦
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
23
Confidence Interval of the total
⎡
⎤
P ⎢N Y reg − t
SE N Y reg ≤ N Y reg ≤ N Y reg + t
SE N Y reg ⎥ = 1 −
α
(
)
(
)
α
α
⎢
⎣
2,n−1
2,n −1
⎥
⎦
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
24
Magnitude of Efficiency
This is used to determine the efficiency of each estimator with respect to
the y-only estimator. The magnitude of efficiency can be computed using the
formula bellow:
V(Y n )
RE(Classical ratio over “Y-only”, in %) =
(5.1)
V (Y )
)
100
(
R
V(Y n )
RE (Hartley-Ross over “Y-only”, in %) =
(5.2)
V (Y )
)
100
(
r
V(Y n )
RE (Regression over “Y-only”, in %) =
(5.3)
V (Y )
)
100
(
reg
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
25
Definition of Terms
Strawberry. Pulpy red fruit with a seeded-studded surface plant with
runners and white flower bearing this.
Production. Something produced.
Estimate. Approximate judgment, especially of cost, value, size,
ect.statement of approximate charge for work to be undertaken
Estimation. The process of providing a numerical value for population
parameter based on information collected from a sample.
Estimator. A statistic used to provide an estimate for a parameter. A
sample mean for example is of the population mean.
Ratio Estimation. The use of the ratio estimator when the relationship
between the response y and the subsidiary variable x which is proportional to y.
Regression Estimation: It is the relationship between the mean value of a
random variable and the corresponding values of one or more independent
variables.
Variable .Used to estimate the relationship between the y’s and the x’s not
through the origin but is a straight line.
Subsidiary variable. Used to estimate the mean of the response y.
Dependent variable. It is the variable to be determined or explained by one
or more explanatory variable.
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
26
METHODOLOGY
Sampling Design
Simple random sampling was employed in the selection of sample plots
using random numbers. The population consisted of 899 plots from 11,488 square
meters of the Sariling Sikap area and 22 farmers. The plots were planted with
strawberries.195 plots were drawn as samples. Each plot has different number of
strawberry plants and sizes of plots.
Data Gathering
The data was gathered through harvesting strawberries. The researchers
asked permission from the farmers to harvest the strawberry from the selected
plots and weighted it. The researchers gathered data until the sixth harvesting
season of the strawberry. Data gathering started on the 29th day of December,
2007 and ended on the 3rd day of February, 2008.Harvest is done twice a week
since farmers must spray and exposed the plants for two days before the
harvesting.
Data Analysis
The data gathered were summarized using the Microsoft excel program.
The mean, variance, mean square error, and bias of ratio, regression and
simple random sampling estimates were computed and compared from each other
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
27
through their relative efficiency with the strawberry production and number of
plants as auxiliary variable used.
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
28
RESULTS AND DISCUSSION
The Estimates of Strawberry Production/grams
in Strawberry Farm, La Trinidad, Benguet
The estimates of the mean and total production of strawberry using the
four estimators are presented in Table 1. It showed that simple random sampling
had the least estimated mean that is about 7,208.718 grams and a total of
6,480,637.482 grams in estimating strawberry production. Hartley-Ross
estimation had the largest mean and total estimation that is about 9,888.170 grams
and 8,889,464.170 grams. Classical estimation had 9,844.239 grams estimated
mean and 8,849,970.861 grams estimated total production while regression
estimation had the largest mean and total estimation that is about 9,888.170 grams
and 8,889,464.170 grams. Classical estimation had 9,844.239 grams estimated
mean and 8,849,970.861 grams estimated total production while regression
estimation has 8,613.167 grams and 7, 743.133 grams estimated mean and total
production of strawberry per plot in Strawberry Farm.
According to Cochran (1977) simple random sampling is a method of
selecting n units out of the N such that every one of the C distinct samples has
N
n
an equal chance of being drawn. He stated also that Ratio estimate is consistent. It
is biased except for some special types of population, although the bias is
negligible for large samples.
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
29
Table1. Estimates of mean and total of Strawberry Production using four methods
of estimation.
ESTIMATION ESTIMATOR ESTIMATE
± STANDARD BIAS
PROCEDURE
ERROR
for average
SRS
Y n
7,208.72
± 209.331 0
Classical Ratio Y
207.501 7.218
R
9,844.239
±
Hartley-Ross
Y
239.867 121.99
r
9,888.170
±
Regression
Y
8,613.167
± 201.426 0.0223
ir
for total
SRS NY
188,188.469 0
n
6,480,637.482 ±
Classical Ratio NY
186,548.308 6,488.982
R
8,849,970.861 ±
Hartley-Ross
NY
516,943.809 109,669.01
r
8,889,464.170 ±
Regression
NY
7,743,237.133 ± 181,082.317
20.048
ir
Consistency of the Estimates
Table 2 shows the Confidence Interval of the true mean and total of the
strawberry production using the different estimation procedures. Result shows
that Hartley-Ross ratio-type has the largest 95% Confidence Interval for the true
mean with a value of 9,598.167 grams to 10,357.957 grams and true total with a
value indicated to be 8,467,126.872 grams to 9,311,802.788. Simple random
sampling, a biased estimator has the least confidence interval for true mean with a
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
30
values of 6,798.429 to 7,619.007 grams and total production of 6,111,788.083
grams to 6,849,486.881 grams. Moreover, for the bias estimators, regression
Table 2. Confidence Interval of the true mean and total strawberry production
using three methods of estimation
ESTIMATION
LOWER LIMIT
UPPER LIMIT
PROCEDURE
for average
SRS 6,798.429
7,619.007
Classical Ratio
9,473.537
10,250.941
Hartley-Ross 9,598.167
10,421.065
Regression 8,218.372
9,007.962
for total
SRS
6,111,788.083
6,849,486.881
Classical Ratio
8,484,345.977
9,215,595.743
Hartley-Ross 8,628,750.649
10,011,902.3
Regression 7,388,315.792
8,098,158.474
estimator has the least confidence interval with 8,218.372 to 9,007.962 for the
true mean and 7,388,315.880 to 8,098,158.386 for the true total production.
Precision of the Estimates
The study of Damoslog and Tomin (2007) showed that classical ratio
estimator with the least variability is the more precise estimator. Table 3 showed
that classical ratio estimation had the least variability with 2.108 is the more
precise estimation but regression estimator also had a close value of coefficient of
variation which is 2.339. Specifically, simple random sampling has the highest
coefficient of variation with a value of 2.904 compared to the other estimators. It
was also found that the different estimators have small variability with Hartley
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
31
Ross ranked second from the highest variability obtained followed by regression
with 2.339 coefficient of variation.
Table3. Coefficient of Variability of the four methods of estimation.
ESTIMATION
COEFFICIENT OF
PROCEDURE VARIATION
SRS 2.904
Classical Ratio
2.108
Hartley-Ross 2.720
Regression 2.339
Efficiency of the Estimates
This is supported by Cochran (1977) as cited by Beligan (2004). If ˆo and
1
ˆo are unbiased estimators for the parametersθ , that is E( ˆo ) = E( ˆo ) = θ , then
2
1
2
ˆo is more efficient than ˆo if V( ˆo ) < V( ˆo ).
1
2
1
2
In the study of Damoslog and Tomin (2007), an estimator with higher
relative efficiency is considered an efficient Thus, in the said study classical ratio
was efficient compared to simple random sampling.
Based from the mentioned criterion for efficient estimators, regression is
the most efficient estimators in estimating strawberry production with a variance
of 40,572.592 and higher relative efficiency as shown in Table 4. Hartley-Ross
ratio-type having a value of 57,449.827 has the largest variance in estimating
strawberry production and classical ratio is next to the regression estimation with
a variance of 43,189.421.
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
32
And this result is the same with the study of Barrios (1995) where
regression estimator was considered efficient in estimating socio-economic
indicators.
Table 4. Relative Efficiency and Variance of the different estimates compared to
simple random sampling.
RELATIVE
VARIANCE
PROCEDURE EFFICIENCY
SRS
Classical Ratio
101.77
43,189.421
Hartley-Ross 76.274
57,449.827
Regression 108
40,572.592
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
33
SUMMARY, CONCLUSION AND RECOMMENDATION
Summary
The study was conducted to determine the efficiency and consistency of
ratio and regression estimation in estimating the strawberry production in
Strawberry farm at La Trinidad, Benguet over the simple random sampling
method.
The data was gathered from the 29th day of December 2007 until 3rd day of
February 2008 wherein the researchers ask permission from the owner of the
strawberry that was selected as one of the study site. The sample plots were
determined using simple random sampling and there were about 195 plots
considered. The data gathering ended until the sixth harvesting of strawberries.
The production was estimated using simple random sampling classical
ratio estimation, Hartley-Ross ratio-type and regression estimation. It was found
that each estimated values obtained due to different estimation schemes utilized.
The classical ratio, Hartley–Ross ratio-type and regression estimators have
greater estimated mean and total than the simple random sampling. The regression
estimator was said to have the least bias while the classical ratio was considered
to have the most bias. Though simple random sampling estimator is not bias, it is
considered as a consistent estimator since its 95 % confidence interval for the true
mean and true total of the strawberry production is narrow. Among the considered
bias estimator, regression was more consistent with its 95 % confidence interval
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
34
which is limited or having the least confidence interval. Classical ratio estimate is
also considered as the most precise among the estimates since it has the lowest
coefficient of variation. Result of the relative efficiency shows that regression
estimator is the more efficient compared to simple random sampling.
Conclusion
It is therefore concluded that classical ratio estimator is the most precise
among the four different estimators used in estimating strawberry production with
its least coefficient of variability. Moreover, simple random sampling was a
consistent and unbiased estimator.
However, regression estimator is the most efficient and most consistent
among the bias estimators as applied to strawberry production.
Recommendation
It is recommended that some study has to be conducted using the
independent variable that is the number of plants per plots to determine the
efficiency of estimator for more meaningful estimate .To maintain a high degree
of precision, the variability among sampling units within a plot should be kept
small. It is also recommended that in estimating strawberry production it is
efficient to use the regression estimator over simple random sampling.
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
35
LITERATURE CITED
BARRIOS, E. B. 1998. Small area Estimator of Employment Rate.7th National
Convention on Statistics. Manila, Philippines.
BELIGAN, S.Z. 2004. Survey and Sampling Techniques Syllabus
COCHRAN, G. W. 1997.Sampling Techniques.3rd Edition. John Wiley and Sons,
United States of America.
DAMOSLOG, F.T. AND TOMIN, J.T. 2007. Efficiency of Ratio and Regression
Estimator s in Estimating Farmers’ Income in La Trinidad Benguet.
DENG, LI-YUAN AND CHHIKARA.R.S (1990). On the Ratio and Regression
Estimation in Finite Population Sampling
DORFMAN, J.H. AND SANDERS, D.R. 2004. Generalized Hedge Ratio
Estimation with an unknown model
EVERITT, B.S. 1998. Dictionary of Statistics.
FAEDI W., MOURGUES F AND ROSATI. Strawberry Breeding and Varieties:
Situation and Perspective
HSU AND KUO. 2000. Household Solid Waste Recycling Included Production.
LOHR,S.L.1999. Sampling:Design and Analysis.Brooks/Cole
PublishingCompany
M.GOSSOP, J.STRANG, P.GRIFFITHS, B.POWIS AND C.TAYLOR. Drug
Transitions Study, National Addiction Centre, Maudsley Hospital, London
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/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
36
ONATE, B.T.1967. Ratio Estimators in Field Research. The Philippine
Statistician
Vol.VXI, No.3-4:62-74
(Oxford Quick Reference)Dictionary
PALANISWAMY, U.R AND PALANISWAMY, K.M.(2006). Handbook of
Statistics for teaching and research in plant and crop sciences.
RAND CORP SANTA MONICA CALIF. Some Finite Population Unbiased
Ratio and Regression Estimators.
SUKHATME, B.V. Hartley-Ross Unbiased Ratio-Type Estimator
TOTEMIC N. All season Strawberry Growing with day-neutral Cultivars.
WALPOLE, R.E.ET.AL.1998. Probability and Statistics for Engineers and
Scientists.Prentice-Hall, Inc.6th edition.
WEI CHEN, YOSHINAO OOEDA AND TOMONORI SUMI. 2005. A study on
a shopping Frequency Model with the Consideration of Individual
Difference. Kyushu University, Japan.
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
37
APPENDICES
Appendix A. Application for Oral Defense
Benguet State University
College of Arts and Sciences
DEPARTMENT OF MATHEMATICS-PHYSICS-STATISTICS
APPLICATION FOR ORAL DEFENSE
Date: March 17, 2008
Group Members:
Emily June C. Alos Major: Statistics
Eleonor B. Bat-oy Minor: Information Technology
Degree: BACHELOR OF SCIENCE AND APPLIED STATISTICS
Title of Thesis: “Efficiency of Ratio and Regression Estimation in Estimating the
Strawberry Production.”
ENDORSED BY : MARYCEL H. TOYHACAO
Adviser
Date and time of Defense: March 18, 2008 @ 5 PM
Place of Defense: CAS Annex 210
NOTED BY : MARIA AZUCENA B.LUBRICA
Department Chairman
REPORT OF RESULT ON ORAL DEFENSE
Name and signature *Remarks
Marycel H. Toyhacao
_______________________
Adviser
Salvacion Z. Beligan
_______________________
Member
Maria Azucena B.Lubrica
_______________________
Member
Cristina B.Ocden
_______________________
Member
*passed or failed
38
Appendix B. The raw data of strawberry production
Harvesting
plots
1st
2nd
3rd
4th
5th
6th
Production(g)
# of plants
1 1000 700 700 1000 1700 1100
6200
120
2 1000 600 800 500 1700 1100
5700
116
3 1000 900 500 600 1100 1200
5300
114
4 1000 500 1300 900 1900 1800
7400
122
5 700 700 800 500 900 1000
4600
110
6 800 700 900 600 1600
1250
5850
110
7 1000 500 1100 900 1700 1600
6800
122
8 1000 300 1000
1000 1300 1400
6000
118
9 1000 600 1100
1500 1200 1100
6500
116
10 700 500 600 1500 1200 1500
6000
116
11 700 400 700 1000 1100 1600
5500
118
12 700 600 800 1200 1300 1300
5900
120
13 600 400 1000
1000 1300 1250
5550
136
14 600 600 600 500 1400 1100
4800
110
15 1200 600 600 900 1200 1600
6100
112
16 600 600 700 500 1400 1750
5550
112
17 600 600 600 1100 1250 1900
6050
101
18 500 500 900 1200 1700 1100
5900
112
19 700 200 900 1500 2300 1150
6750
118
20 500 600 800 800 1700 1900
6300
120
21 600 400 800 700 1300 1800
5600
118
22 700 400 900 900 1400 1400
5700
112
23 600 400 900 1700 1450 1450
6500
110
24 700 400 900 1700 1950 1000
6650
108
25 600 300 800 1400 1850 1100
6050
110
26 600 500 900 1800 1500 1150
6450
112
27 500 300 900 1400 1900 1250
6250
106
28 600 500 600 1700 1500 1000
5900
130
29 700 500 900 1700 1850 900
6550
114
30 600 600 600 1600 1700 1800
6900
112
31 500 500 800 1600 1700 1800
6900
114
32 500 500 500 1600 1450 1900
6450
110
33 700 300 1200
1300 1700 1100
6300
114
34 500 300 700 1350 1450 1200
5500
106
35 600 300 800 1250 1500 1000
5450
110
36 400 300 700 1000 1700 1700
5800
130
37 400 500 800 1600 1200 1600
6100
108
38 900 500 1100 600 2250 1850
7200
110
39 600 400 1000
1400 1900 1550
6850
104
40 300 800 900 600 1300 1250
5150
106
41 500 200 700 600 900 1900
4800
124
42 300 100 400 800 1100 1800
4500
106
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
39
43 600 400 800 400 300 1000
3500
118
44 400 600 600 400 500 1150
3650
120
45 400 500 900 500 500 1100
3900
106
46 500 400 400 300 500 1000
3100
108
47 600 300 700 300 400 1500
3800
106
48 700 800 100 400 400 1000
3400
108
49 300 600 1000 400 500 800
3600
106
50 500 300 1200 500 500 1100
4100
94
51 400 400 700 400 300 1250
3450
108
52 300 500 900 300 300 1200
3500
104
53 400 700 900 500 500 800
3800
108
54 700 200 600 200 300 900
2900
110
55 500 600 500 200 300 1150
3250
106
56 400 400 700 300 400 800
3000
108
57 400 400 600 400 400 850
3050
118
58 400 600 700 200 400 1050
3350
116
59 600 700 600 300 300 850
3350
110
60 500 300 700 400 500 700
3100
108
61 800 900 800 500 500 1350
4850
108
62 400 400 600 400 400 800
3000
106
63 400 200 400 400 500 1250
3150
110
64 400 300 400 200 300 1000
2600
112
65 200 200 300 200 200 1000
2100
120
66 400 500 300 300 300 1100
2900
130
67 150 200 200 300 400 1150
2400
118
68 250 250 300 100 200 800
1900
114
69 200 300 400 200 200 1150
2450
116
70 400 300 200 100 300 1000
2300
122
71 200 300 400 300 400 1250
2850
120
72 500 300 300 300 500 800
2700
120
73 400 300 300 300 400 800
2500
124
74 300 400 500 200 100 900
2400
126
75 200 500 200 400 300 950
2550
130
76 300 500 200 100 100 850
2050
126
77 300 300 200 300 400 800
2300
128
78 200 400 500 400 300 1000
2800
129
79 300 300 200
200 400
1100
2500
132
80 600 500 700 700 500 1150
4150
130
81 500 900 900 900 900 800
4900
128
82 300 900 900 900 800 700
4500
122
83 1300 17000 900 1100 1200 2000
23500
115
84 1100 8000 900 1400 1400 2100
14900
96
85 1600 1000 500 1000 1600 2000
7700
116
86 1200 900 300 1000 1000 1500
5900
114
87 11100 1000 1200 1500 1900 2200
18900
123
88 1200 1000 600 1400 1600 2000
7800
120
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
40
89 1700 11000 1100 1500 1800 2300
19400
115
90 2200 1500 1700 2100 1900 2350
11750
117
91 1200 1000 1000 1600 2100 2500
9400
116
92 1700 1500 1200 1350 2100 2100
9950
107
93 1600 1500 1600 1700 1800 1500
9700
103
94 800 1300 1600
1900 1900 2200
9700
112
95 11000 1500 1500 2000 1300 2000
19300
105
96 1100 1500 800 1500 1600 1800
8300
106
97 1100 1100 500 1600 1800 1800
7900
112
98 1300 900 800 1400 1700 2000
8100
120
99 1000 900 1000 1500 1600 1800
7800
93
100 1400 900 900 1800 1700 2100
8800
112
101 12000 1200 1000 1900 2000 1200
19300
117
102 1100 1000 900 2200 2300 1250
8750
110
103 1300 1200 1000 1300 1500 2000
8300
119
104 1100 1300 1100 1600 1800 1400
8300
115
105 1400 1200 1000 1200 1600 1400
7800
114
106 1700 1800 1500 1400 1500 1450
9350
117
107 1200 1300 1500 1500 1300 1000
7800
115
108 1600 1200 1100 1600 1800 1100
8400
114
109 1300 1000 900 1800 1900 1000
7900
117
110 1500 1500 1300 1900 1600 1600
9400
115
111 1000 1500 1400 1800 1700 1900
9300
115
112 1500 1000 800 1300 1400 2000
8000
120
113 1300 1300 1100 1400 1600 2200
8900
120
114 1500 1600 1500 1700 1000 2250
9550
113
115 1600 1500 1200 1600 2000 2000
9900
118
116 1800 1500 1500 1800 2100 2200
10900
120
117 2200 2000 1800 2200 2200 2400
12800
134
118 1300 1500 1100 2100 2300 2000
10300
124
119 1200 1100 1000 1400 2100 1500
8300
123
120 1300 1400 1300 1500 1900 1600
9000
116
121 1000 1000 800 1600 1700 1800
7900
118
122 1000 1200 1500 1800 2000 1600
9100
101
123 1500 1400 1300 2100 2100 2000
10400
121
124 1500 1600 1500 2400 2100 1800
10900
112
125 1400 1200 1300 1400 1500 2200
9000
124
126 1500 1300 1000 1600 1900 2300
9600
105
127 1000 1000 800 1800 2000 2000
8600
100
128 1300 1300 1000 1000 1200 2250
8050
102
129 1400 2500 2400 2600 2300 2000
13200
150
130 1300 1100 1100 1300 1400 1250
7450
145
131 1600 1000 1600 1700 1400 1400
8700
142
132 1200 1100 1500 1800 1900 1800
9300
132
133 1300 1300 1600 1400 1500 1200
8300
131
134 1600 1200 1400 1200 1100 1000
7500
142
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
41
135 1700 700 1900 2000 1800 1500
9600
144
136 1400 700 1500 1300 1400 1900
8200
145
137 1200 750 1600 1350 1400 1400
7700
138
138 1300 1000 1400 1500 1300 1100
7600
138
139 1000 750 1000 750 800 1300
5600
138
140 700 800 600 600 800 1100
4600
120
141 1000 1100 900 1200 1300 1150
6650
140
142 1000 1100 1200 1100 1100 1300
6800
142
143 1200 2000 1800 1600 1900 1600
10100
152
144 800 500 700 900 1000 1800
5700
120
145 700 800 800 700 600 1900
5500
122
146 1000 700 1000 1800 1300 2000
7800
120
147 900 750 800 1300 1600 1400
6750
140
148 400 750 1400 1300 1500 1800
7150
150
149 300 750 400 900 1000 1200
4550
140
150 600 800 900 800 1000 1000
5100
152
151 400 1000 1000 500 600 900
4400
120
152 500 700 800 1500 1600 1100
6200
124
153 700 1000 1800 800 1900 1800
8000
130
154 1000 1100 1600 1700 1800 1200
8400
150
155 1300 1200 1400 1800 1800 1000
8500
131
156 1200 850 1200 1400 1400 1200
7250
142
157 700 850 900 1400 1500 1300
6650
145
158 500 800 1600 1000 1100 1900
6900
138
159 750 1600 1000 3500 2100 2250
11200
201
160 750 1550 1500 1400 2400 2500
10100
192
161 750 700 1600 1250 2200 2250
8750
189
162 500 800 700 2700 2000 2750
9450
189
163 1500 1000 800 1200 2200 2250
8950
201
164 1750 1000 800 1250 2250 2250
9300
219
165 1500 1100 1000 1250 2000 2200
9050
207
166 1500 700 1600 1400 2100 2250
9550
222
167 1300 800 800 1500 2600 2500
9500
210
168 1250 850 700 2500 2000 2300
9600
228
169 1000 850 750 2250 2100 2400
9350
207
170 650 900 800 2500 2000 2800
9650
180
171 650 650 850 1500 2600 2250
8500
240
172 600 600 800 2250 2100 2500
8850
189
173 500 550 800 3050 3000 2200
10100
168
174 1000 700 750 1500 2000 2250
8200
207
175 800 1000 1000 2000 2100 2600
9500
204
176 1000 800 1100 1200 2200 2250
8550
201
177 1100 1100 1150 1750 2150 2200
9450
177
178 1000 1150 1100 1750 2200 2400
9600
168
179 700 1600 1000 1500 1600 2250
8650
171
180 700 900 1150 1100 2200 2600
8650
168
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
42
181 750 850 1500 1500 2100 2500
9200
168
182 750 700 1600 2000 1900 3000
9950
174
183 800 750 1800 1900 1800 2500
9550
189
184 850 900 1600 2000 1950 2750
10050
192
185 600 1000 1650 2000 1100 2750
9100
189
186 600 950 1000 2200 2000 2000
8750
183
187 1000 750 1100 2000 1800 2000
8650
186
188 750 1100 1500 2000 1900 2100
9350
192
189 850 1000 1000 2500 1950 2000
9300
186
190 800 950 950 2250 2000 2800
9750
180
191 800 850 950 1200 2200 2800
8800
180
192 800 1000 950 1700 1500 2250
8200
177
193 750 1500 850 1500 2600 2500
9700
186
194 500 450 500 500 900 1300
4150
100
195 700 350 400 500 700 1250
3700
120
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
43
Appendix C. Sample Computation
Simple Random Sampling (SRS)(for average)
N = 899
n = 195
n
∑Yi 1405700
Y
i=1
n =
=
n
195
= 7208.718
∑(Y −Y n
i
)2
2
s =
=
596
,
911
,
10
n −1
2
V (Y n ) (N − n)⎛ s ⎞
=
⎜
⎜
⎟
⎟
N
⎝ n ⎠
899 −195 ⎛10911596 ⎞
=
⎜
⎟
899
⎝
195
⎠
= 43,819.4209
SE (Y n ) = V (Y n ) = 209 331
.
SE(Y n )
CV (Y n ) =
Y n
331
.
209
=
(100)
718
.
7208
= 2.904
Confidence Interval of the mean estimate:
⎡
⎤
P⎢Y n − t
SE Y n ≤ Y n ≤ Y n + t
SE Y n ⎥ = 1−
α
( )
( )
α
α
⎣
,n−1
,n−1
2
2
⎦
= 7,208.718 ± 1.96(209.331)
= 6,798.429 ≤ Y n ≤ 7,619.007
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
44
Simple Random Sampling(for total)
NY n = 899(7208.718)
= 6,480,637.482
N (N − n)
Vˆ (N Y n ) =
2
s
n
= 35,414,899,810
SE (NY n ) = V (Y n )
= 188,188.469
Confidence interval of the total estimate:
⎡
⎤
P⎢NY n − t
SE NY n ≤ NY n ≤ NY n + t
SE NY n ⎥ = 1−
α
( )
( )
α
α
⎣
,n−1
,n−1
2
2
⎦
= 6,480,637.482 ± 1.960 (188188.469)
= 6,111,788.082 ≤ NY n ≤ 6849486.881
Classical Ratio Estimation
Y
Y n
718
.
7208
R =
X N =
)
180
(
X n
81
.
131
= 9844.239
⎛
⎞
N − n
bias ( R
⎜
⎟
2
n ) = ⎜
{Rns − s
x
xy }
2 ⎟
⎝ Nnxn ⎠
⎛
899 −195
⎞
=
[54 69
. ( 186
,
3
596
.
)− 411
]
⎜
⎜
⎝ 899(195)(131 )
161
.
81
.
2 ⎟⎟⎠
= 0.0401
Y n
718
.
7208
Rn =
=
X n
81
.
131
= 54.69
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
45
∑(x − xn)2
s 2 =
i
= 3,186.596
x
n −1
∑(x − xn
i
)(y − y
i
n )
s 2
xy =
= 169,053.579
n −1
Bias (Y R ) = X N bias ( Rn )
= 180 (.0401)
= 7.218
N − n
2
V (Y R) =
∑[y − Rnx
i
i ]
Nn(n − )
1
899 −195
=
( 08
.
2
E09)
)
194
)(
195
(
899
= 43,056.623
SE(Y R ) = V (Y R ) = 207.501
MSE (Y R ) = V (Y R) + bias ( (Y ) ) 2
R
= 43,056.623 + 7.2182
= 43,108.723
MSE(Y R )
43061 605
.
CV(Y R ) =
100 =
100
(
)
Y
9844 239
.
R
= 2.109
Classical Ratio Estimation (Total)
NY R = 899 (9844.239)
= 8,484,346.662
V (NY R ) = N2 V (Y R) = (899)2 (43056.623)
= 34,798,405,770
SE (NY R ) = V (NY R ) = 186548.308
bias (NY R ) = N(bias (Y R))
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
46
= 899(7.218)
= 6,488.982
Confidence Interval of the mean estimate:
⎡
⎤
P⎢Y n − t
SE Y n ≤ Y n ≤ Y n + t
SE Y n ⎥ = 1−
α
( )
( )
α
α
⎣
,n−1
,n−1
2
2
⎦
= 9,844.239 ± 1.960(207.501)
= 9,437.537 ≤ Y n ≤ 10250.941
Confidence Interval of the total estimate:
⎡
⎤
P⎢NY n − t
SE NY n ≤ NY n ≤ NY n + t
SE NY n ⎥ = 1−
α
( )
( )
α
α
⎣
,n−1
,n−1
2
2
⎦
= 8,484,346.662 ± 1.960(186,543.308)
= 8,118,721.778 ≤ NY n ≤ 8,849,971.546
Hartley-Ross Ratio-Type Estimation(average)
n
1
1
r = ∑ri =
(
.
10844 39)
n
n i=1
195
= 55.612
N −1
899 −1
Bias(Y r ) =
(s ) =
(−
.
122
)
126
rx
N
899
= 121.99
1
S
[y −rnxn
n
]
rx = n −1
1
=
[7208 718
.
− 55 612
.
131
(
)
81
.
]
195 −1
=-122.126
(N − )1n
y = r X
(y −rn xn
n
)
N +
r
N (n − )
1
(899 − )1
= 55.612(180)+
(7208 718
.
− 55 62
.
131
(
)
81
.
)
195
(
899
− )
1
= 9,888.170
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
47
2
1 (N − )
1
1
N −1
Vˆ ( y ) =
S − r S +
S + S S
r
( 2
2
2 N
y
x )
(
) ( 2 2 2
rx
r
x )
2
n
N
n(n − )
1
N
1 899
(
− )
1
2
=
(
−
,
11
,
249
.
454 99 −
.
55
(
2
)(
612
.
411
)
161 )
1
(899 )1
+
195
899
195
(
195
− )
1
8992
(
102
.
1220
+
.
689
.
186
,
3
(
582
596)
=57,449.827
∑(x − xn
i
)2
2
S =
= 3,186.596
x
n −1
∑(y − y
i
n )2
2
S =
= 11,249,454.99
y
n −1
∑ r − r
2
( n
i
)2
S =
= 1220.102
r
n −1
∑ x − xn y − y
2
( i
)( i n)
S =
=169,053.580
xy
n −1
SE( y ) = V y = 209.950
r
( r)
MSE( y = V y + Bias y
= 44070.356 + (121.99)2
r
( r) ( ( r)2
)
= 72,331.387
MSE(y
.
72331 387
r )
CV ( y ) =
100 =
)
100
(
r
y
.
9888 170
r
= 2.720
Confidence Interval of the Mean
⎡
⎤
P⎢Y r − t
SE Y r ≤ Y r ≤ Y r + t
SE Y r ⎥ = 1−
α
( )
( )
α
α
⎣
,n−1
,n−1
2
2
⎦
= 9,888.170 ± 1.96( 239.687)
= 9,418.383 ≤ Y r ≤ 10,357.957
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
48
Hartley-Ross Ratio-Type Estimation(total)
NY r = 899(9,88.170)
V(NY r ) = N2 V(Y r ) = 8992 ( 9888.170)
= 4.643100763 E 10
SE(NY r ) = V (N y = 215,478.555
r )
Bias (NY r ) = N (Bias (Y r )) = 899(121.99)
=109669.01
Confidence Interval of the Total
⎡
⎤
P⎢NY r − t
SE NY r ≤ NY r ≤ NY r + t
SE NY r ⎥ = 1−
α
( )
( )
α
α
⎣
,n−1
,n−1
2
2
⎦
= 8,889,464.83 ± 1.96( 215,478.555)
= 8,467,126.872 ≤ NY r ≤ 9,311,802.788
Regression Estimation(average)
Y reg = y + β X − x = 7208.718 + 29.144 (180-131.81)
n
( N n)
= 8613.167
∑(x − xn y y
i
)( −
i
n )
β =
5570391
=
∑n(
1
.
191136
x − x 2
i
n )
i=1
= 29.144
⎛
n ⎞ se2
⎛
195 ⎞10103092.782
V Y
( reg ) = ⎜1−
⎟
= ⎜1−
⎟
⎝
N ⎠ n
⎝
899 ⎠
195
= 40572.587
ˆ
n∑ x − y −
x y
(
195 .
1 92E09) − (
(
25703 1405700)
i
i
∑( i )( i )
β =
=
1
n∑ −
x
(
195 3596429)− (
)2
25703
i
(∑ i)2
2
=32.201
e
ˆ
i = yi - Y n + β
= 2962.6841
1 (x − x n
i
)
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
49
1
1
en = ∑e =
(
.
2862 684)
i
n
195
= 15.193
∑(e − en
i
)2 96
.
1
E09
2
s =
=
e
n −1
194
= 10103092.78
SE(Y reg ) = V (Y reg ) = 40572.587
1− n
2
N e x
x
N
i (
− n
i
)
bias (Y reg ) =
2
∑
n
1
N 1
sx
i=
−
195
1899 ⎛12591 27
. (
)
6941
.
2962
−
86
.
131
⎞
=
⎜
⎟
)
238
.
985
(
195
⎝
898
⎠
= 0.0223
MSE(Y reg ) = V (Y reg )+ (Bias(Y ) 2
reg
= 40572.587 +( .0718)2
= 40572.568
CV (
MSE Y
Y reg )
( reg )
=
100
Y reg
= 40,572.568
Confidence Interval of the mean:
⎡
⎤
P⎢Y reg − t
SE Y reg ≤ Y reg ≤ Y reg + t
SE Y reg ⎥ = 1−
α
( )
( )
α
α
⎢
⎣
2,n−1
2,n−1
⎥
⎦
= 8,613.167 ± 1.960(201.426)
= 8,218.372 ≤ Y reg ≤ 9,007.962
Regression Estimation(total)
Y N =N (Y reg ) = 899(8613.167 )
= 7,743,237.133
V (NY reg ) = N2 V (Y reg ) =8992 ( 40572.587 )
= 3.27908922 E 10
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
50
SE(NY reg ) = V (NY reg ) = 181,082.272
bias(NY reg ) = N (bias(Y reg )) = 899(0.0223)
= 20.048
Confidence Interval of the total
⎡
⎤
P ⎢N Y reg − t
SE N Y reg ≤ N Y reg ≤ N Y reg + t
SE N Y reg ⎥ = 1 −
α
(
)
(
)
α
α
⎢
⎣
2,n−1
2,n −1
⎥
⎦
= 7,743,237.133 ± 1.960 (181,082.317)
= 7,388,315.88 ≤ NY reg ≤ 8,098,158.386
Magnitude of Efficiency
V(Y n )
RE(Classical ratio over “Y-only”, in %) =
V (Y )
)
100
(
R
421
.
43819
=
(100)
623
.
43056
= 101.77
V(Y n )
RE (Hartley-Ross over “Y-only”, in %) =
V (Y )
)
100
(
r
421
.
43819
=
)
100
(
827
.
57449
= 76.274
V(Y n )
RE (Regression over “Y-only”, in %) =
V (Y )
)
100
(
reg
= 108.00
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
ALOS, EMILY JUNE C., BAT-OY, ELEONOR B. April 2008. Efficiency of
Ratio and Regression Estimators in Estimating Strawberry Production. Benguet, State
University, La Trinidad, Benguet.
Adviser: Marycel H. Toyhacao
ABSTRACT
The study was conducted to determine the efficiency of the estimates on
strawberry production using ratio and regression estimators.
A total of 195 plots were selected as samples from the total population of 899
plots from the Sariling Sikap area at strawberry farm. The data were summarized,
tabulated and analyzed. The total production and total number of plants per area were
used as the variables in the estimation.
Being an unbiased estimator, the Simple Random Sampling has the least mean
and total production of strawberry. Among the bias estimator, classical ratio estimator is
the most precise among the three estimators used in estimating strawberry production
with its least coefficient of variability.
Regression estimator is the most consistent estimator and it is considered the most
efficient estimator among the estimator used in estimating the strawberry production.
TABLE OF CONTENTS
Page
Bibliography ……………………………………………………. i
Abstract
……………………………………………………………. i
Table of Contents …………………………………………..………. ii
INTRODUCTION ………………………………………………….. 1
Background of the Study
…………………………………… 1
Objective of the Study
…………………………………… 3
Importance of the Study
…………………………………… 3
Scope and Delimitation
…………………………………… 4
REVIEW OF RELATED LITERATURE
………………………….. 5
THEORETICAL FRAMEWORK
………………………………….. 11
Simple Random Sampling
…………………………………... 11
Classical Ratio Estimation
…………………………………… 13
Hartley-Ross ratio-type Estimation …………………………… 16
Regression Estimation …………………………………………… 20
Magnitude Efficiency
…………………………………… 24
Definition of Terms …………………………………………… 25
METHODOLOGY …………………………………………………… 26
Respondents of the Study
……………………………………... 26
Data Gathering
……………………………………………... 26
Data Analysis ……………………………………………………… 26
ii
RESULTS AND DISCUSSION
……………………………………… 28
Estimates of the Strawberry Production
………………………. 28
Consistency of the Estimates
……………………………… 29
Precision of the Estimates
……………………………………… 30
Relative Efficiency of the Different Estimates
……………… 31
SUMMARY, CONCLUSION AND RECOMMENDATION
…….. 33
Summary
……………………………………………………… 33
Conclusion
……………………………………………………… 34
Recommendation
……………………………………………… 34
LITERATURE CITED
……………………………………………… 35
APPENDICES
……………………………………………………… 37
Appendix A. Application for Oral Defense ………………………… 37
Appendix B.The raw data of the strawberry production ………….. 38
Appendix C. Sample Computations ……………………………….. 43
iii
1
INTRODUCTION
Background of the Study
A sample survey may be considered as an absolute (enumeration)
experiment whose main objective are to obtain estimators of parameters and to
derive measures of precision of these estimators. In analytical surveys one of the
objectives is to test hypothesis with the use of appropriate estimation procedures.
Thus, it is apparent that simple and sound estimation techniques must be
developed to provide precise and accurate statistics whether such endeavors are of
the absolute or analytical type or both. (Burton T. Onate & Julia Mercedes O.
Bader: 1990)
Estimation of the population mean and total sometimes based on a sample
of response measurements y1,y2,….yn are obtained by simple random sampling
and stratified random sampling. William Mendenhall (1990) stated that measuring
Y and one or more subsidiary variables are used to estimate the mean of the
response Y. It is basic to the correlation and provides means for development of a
prediction equation relating Y and X by the method of least square.
The basic concept in estimation procedure is to determine the unknown
values of the parameters using the sample values. By doing so, many estimators
can be utilized. Among of those are using ratio estimators and regression
estimators. However, the correct approach can lead to better estimates, that is, less
bias is obtained. Ratio estimators and regression estimators, as pointed by Onate
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
2
and Bader (1990), increased its precision or usually attained by observing an
additional variable Zi, in addition to the original characteristics.
It is most effective when the relationship between the response y and the
subsidiary variable x is linear through the origin and the variance of y is
proportional to x.
In Regression estimation the technique is for incorporating information on
a subsidiary variable, this is better if the relationship between the y’s and x’s is a
straight line not through the origin. It works well when the plot of y and x reveals
points by lying uniformly close to a straight line with unit slope.
Ratio estimation usually leads to biased estimators. Thus, a consideration
for the magnitude of the bias should be observed. However, for a large sample
size (n>30) and he (σx/μx) ≥10, the bias is negligible. Ratio estimators are
unbiased when the relationship between y and x is linear through the origin. As
mentioned, we can assume to draw sample from the population, and possibly
estimate its average (strawberry production) from y, sample mean. Since there are
underlying associations between the mentioned variables, the number of plants
per plot can be multiplied from the production of strawberries per plot.
Objective of the Study
Specifically, the study was conducted to determine the efficiency of the
estimates on the production of strawberries in Strawberry Farm, La Trinidad,
Benguet using the ratio estimates and using regression estimates.
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
3
Importance of the Study
The researchers find the study important; it gives information to the
concern individuals specially in estimating the production of strawberry in La
Trinidad, Benguet using the ratio and regression estimators. Through this study,
students would be aware that strawberry production can be estimated or it can be
measured using statistical tools by studies and researchers.
The useful and relevant information acquired from the study will
encourage future researchers to estimate strawberry production applying the
choice of their analysis. This can help them to boost their ability in analyzing
given data’s from the specific tools they will conduct.
Result of the study will provide the strawberry management knowledge if
the ratio and regression estimators can be appropriately applied to the production
of strawberries. This will make them aware on the specific materials needed for
strawberry production.
This study will serve as a valuable reference to farmers that production of
strawberry can be estimated through different estimation procedures. This will
also be a basis on how many strawberries will be produced in one plot.
Finally, this study will be an instrument or a benchmark for other
researchers who are planning to study on different estimation procedures.
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
4
Scope and Delimitation
The main focus of this study was to determine the efficiency of ratio and
regression estimators in the production of strawberries in Strawberry Farm at La
Trinidad, Benguet.
The data was gathered from the Strawberry Farm of La Trinidad, Benguet.
Specifically at the Sariling Sikap Area.
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
5
REVIEW OF LITERATURE
Ratio Estimation
The ratio estimation is a different way of estimating population total or
mean that is useful in many problems. It involves the use of known population
totals for auxiliary variables to improve the weighting from sample values to
population estimates. It operates by comparing the survey sample estimate for an
auxiliary variable with the known population total for the same variable on the
frame. The ratio of the sample estimate of the auxiliary variable to its population
total on the frame is used to adjust the sample estimate for the variable of interest.
Ratio method of estimation is frequently used in sample surveys to
estimate the population mean of the variable under investigation. Several ratio-
type estimates can be formed. All these estimates are unsatisfactory in the sense
that they are biased. Hartley and Ross, who proposed an unbiased ratio-type
estimator for uni-stage sampling designs, overcame this difficulty. In practice,
however, we are generally faced with multi-stage sampling designs. This
communication gives a generalized form of Hartley-Ross unbiased ratio-type
estimator for multi-stage designs.
This method aims to obtain an increased precision by taking advantage of
the correlation between Yi and Xi .It applies population knowledge of adjustment
variable Xi to improve estimation of the population total should be known.
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The ratio estimates can be used when Xi is done some other kind of
supplementary variables. However, for successful application, the ratio Yi / Xi
should be relatively constant over the population and the population total should
be known.
Lohr (1999) enumerated the uses of ratio estimation which include: ratio
estimators is simply used to estimate a ratio; to estimate a population total but the
population size N is unknown and to adjust estimates from the sample so as to
reflect demographic totals.Often,it is used to adjust for no response.
Hsu and Kuo (2000) used the ratio estimation to estimate the recycled and
the production values and employment opportunities induced by household waste
recycling. Ratio estimation was used in their study because the true population
size of recycling plant is unknown; and the audited amount of household waste
recycling is known; and the audited amount and recycling amount are highly
correlated.
Angel et.Al (2004) evaluated the impact of using ratio estimation in their
study, as the estimate census of night dwelling was much closer to the actual
value.
M.Gossop, J.Strang, P.Griffiths, B.Powis and C.Taylor presents an
approach in estimating the prevalence of cocaine use, based upon a new ratio
estimation technique. This method can be applied to random samples of
overlapping populations for which no sampling frames exist. When the ratio
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estimation method is applied to the two study samples (drawn from populations of
people using cocaine and people using heroin) the ratio of cocaine users to heroin
users (C/H) was 1.55, with a 95% confidence interval of +/- 0.48. Such estimates
should be applied with caution. However, if used with reference to national
estimates of about 75,000 heroin users, application of the present estimate
suggests that there may be about 116,000 cocaine users in the UK.
Myers and Thompson (1989) pioneered the concept of a generalized
approach to estimating hedge ratios, pointing out that the model specification
could have a large impact on the hedge ratio estimated. While a huge empirical
literature exists on estimating hedge ratios, the literature lacks a formal treatment
of model specification uncertainty. These researches accomplish the task by
taking a Bayesian approach to hedge ratio estimation, where specification
uncertainty is explicitly modeled. Specifically, a Bayesian approach to hedge ratio
estimation that integrates over model specification is uncertain; it yields an
optimal hedge ratio estimator that is robust to possible model specification
because it is an average across a set of hedge ratios conditional on different
models. Model specifications vary by exogenous variables (such as exports,
stocks, and interest rates) and lag lengths. The methodology is applied to data on
corn and soybeans and results showed potential benefits and insights from such
approach.
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Damoslog and Tomin (2007) considered the Classical Ratio Estimation as
a precise and efficient estimator because it has the lowest coefficient of variation
and 15% higher relative efficiency than simple random sampling. They also cited
that classical ratio estimator has the least estimated bias and is a consistent
estimator with its narrow 95 % confidence interval
.
Regression Estimation
Regression estimation aims to find a relationship between a dependent
variable and set of independent variable (Kweon and Kocelman, 2004).
In a study conducted by Wei Chen et.Al (2005), linear regression was used
to evaluate the parameters with the least squares estimation to estimate an original
model for daily shopping with the consideration of individual influence the
estimation was verified to be straight forward and efficient.
Myall (2000) noted a known feature of regression estimation, that is, some
data will generate negative and weights, which leads to having a download effect
on aggregate estimates.
Valliant (2002) used the general regression estimator to construct variance
estimators that are approximately model-unbiased in single estimates.
Barrios (1995) explored the used of various small area estimators for
various socio-economic indicators. Regression estimation was noted to be
reasonable use.
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Watson (1937) from Cochran’s Citation (1977) illustrated an example of a
general situation in which regression estimates are helpful. Watson used a
regression on the leaf area and leaf weight to estimate the average area of the
leaves on a plant. The procedure was to weight all the leaves of the plant. The
area and the weight of each leaf were determined from the small sample of leaves.
The sample area was then adjusted by means of the regression on leaf weight
means of the regression on leaf weight.
Usha Rani (2001) used the regression model Y=k (LxP) where Y = fruit
surface area in Chili (capsicum annuum L.)Fruit, D= diameters of the fruit and k
is the regression coefficient. She found the values of k=1.6827.Thus, one can
estimate the fruit diameter x 1.6827.
Palaniswamy (1990) reported that the linear regression model can be
employed in the prediction of grain yield per plot in rice and consequently for
crop estimation.
Lih-Yuan Deng and RAJ S. Chhikara (1990) cited that in a classroom
setting students often find the concept of super population and the assumption of a
model somewhat artificial when one needs to estimate the mean of a finite
population. They introduce the concept of a finite population decomposition
based on a regression fit to the population values and then discuss the bias and
variance of each estimator from the sampling design viewpoint. It shows by
fitting a regression line of y and x to the finite population, that the leading term of
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the bias of yˆ is accounted for it on terms of the intercept of the regression line.
R
On the other hand, using quadratic regression fit, describe the leading term of the
bias of yˆ in terms of the coefficient of the quadratic term. Moreover, the sign of
ir
intercept in the fitted regression fit would be indicative of an over-or
underestimation of Y by yˆ or yˆ ,respectively. They hope that this will add to
R
ir
the understanding of the sampling properties of yˆ and yˆ without having to
R
ir
assume the super population structure.
Dorfman, J.H and Sanders, D.R (2004) cited that regression and
regression-related procedures have become common in survey estimation. It
reviews the basic properties of regression estimators, discuss implementation of
regression estimation, and investigate variance estimation for regression
estimators. The role of models in constructing regression estimators and the use of
regression in non-response adjustment are also explored.
A class of ratio and regression type estimators is given such that the
estimators are unbiased for random sampling, without replacement, from a finite
population. Nonnegative, unbiased estimators of estimator variance are provided
for a subclass. Similar results are given for the case of generalized procedures of
sampling without replacement. Efficiency is compared with comparable
estimation sample selection methods for this case (Rand Corp Santa Monica
Calif).
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THEORETICAL FRAMEWORK
Simple Random Sampling (SRS)
Simple random sampling is a method of selecting n units out of the N
⎛ N ⎞
population elements such that every ⎜
⎟ distinct sample has an equal chance of
⎝ n ⎠
being selected. (Beligan et al., 2004)
Y n is the unbiased estimator of Y N and it is given by:
n
∑Yi
Y
i=1
n =
(sample mean)
(1.1)
n
and its sample variance is given by:
∑ Y −Y
2
(
n
i
)2
s =
(1.2)
n −1
An estimated estimator of the variance of Y n is
N − n ⎛ s2 ⎞
V (Y n ) (
)
=
⎜
⎜
⎟
⎟
(1.3)
N
⎝ n ⎠
where:
⎛ N − n ⎞
⎜
⎟ is the finite population factor(fpcf).The correction factor is used
⎝ N ⎠
because with small populations, the greater the sampling fraction n/N, the more in
formation there is about the population and the smaller the variance to be derived.
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The estimated coefficient of variation (C.V) are usually computed using
the estimated standard error (S.E)
SE(Y n )
CV (Y n ) =
(1.4)
Y n
where:
N − n
SE
(Y
2
n ) =
s = V (Y n )
(1.5)
Nn
n Y i
∑
N
1
Since Y
i=1
N =
=
∑Y and to estimate population mean is given by
N
i
N i=1
Y N = N Y n (1.6)
The variance, standard error (SE) and the coefficient of variation (C.V) of the
population total are
N (N − n)
Vˆ (N Y n ) =
2
s
n
proof:
Vˆ (N Y n ) = N2(V(Y n ))
⎛ N − n ⎞ ⎛ S ⎞
= N2 ⎜
⎟
⎝ N ⎠ ⎜⎜
⎟
⎟
⎝ n ⎠
N(N − n) 2
=
s
n
If S 2 is not available, therefore the estimates of V(N Y n ) is
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N (N − n) 2
Vˆ (N Y
=
s
n )
n
(1.7)
N(N − n)
SE
(N
Y
2
n ) =
s = V (NY n )
(1.8)
n
Confidence interval is a range of possible values for the unknown
parameter with some measure for the degree of certainty, (1-α ) called the level of
confidence coefficient. Hence, the confidence interval for the mean (Y n ) and total
(NY n ) are as follows:
Confidence Interval of the mean estimate:
⎡
⎤
P⎢Y n − t
SE Y n ≤ Y n ≤ Y n + t
SE Y n ⎥ = 1−
α
( )
( )
α
α
(1.9)
⎣
,n−1
,n−1
2
2
⎦
Confidence interval of the total estimate:
⎡
⎤
P⎢NY n − t
SE NY n ≤ NY n ≤ NY n + t
SE NY n ⎥ = 1−
α
( )
( )
α
α
(1.1)
⎣
,n−1
,n−1
2
2
⎦
Classical Ratio Estimation
In ratio estimation, the ratio estimate of Y N using SRS is
Y
Y n
R =
X N
(2.1)
X n
where
X N (Total area of the strawberry)
n
∑ yi
Y
i=
n =
1
(Sample mean for the production of strawberry) (2.2)
n
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n
∑xi
X
i=
n =
1
(Sample mean of the number of strawberry plant) (2.3)
n
According to Hartley and Ross, Rn is biased for R N ,and the bias is
⎛
⎞
N − n
bias ( R
⎜
⎟
2
n ) = ⎜
{Rns − s (2.4)
x
xy }
2 ⎟
⎝ Nnxn ⎠
where:
Y n
Rn =
(sample ratio of the mean)
(2.5)
X n
∑(x − xn)2
s 2 =
i
(sample variance of x
x
n −1
i )
(2.6)
∑(x − xn
i
)(y − y
i
n )
s 2
xy =
(sample covariance of x
n −1
i and yi ) (2.7)
Hence,
Bias (Y R ) = X N bias ( Rn )
(2.8)
The variance of classical ratio estimate and its standard error are
N − n
2
V (Y R) =
∑[y − Rnx (2.9)
i
i ]
Nn(n − )
1
SE (Y R ) = V (Y R ) (2.10)
Estimate coefficient of variation of the estimate is estimated using the S.E. but the
square root of the MSE will be used to estimate the CV where the variance is
adjusted for its bias:
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MSE(Y R )
CV(Y R ) =
100
(2.11)
YR
where:
MSE (Y R ) = V (Y R) + bias ( (Y ) ) 2
(2.12)
R
To estimate the population mean
Y N = NY R is used
(2.13)
The variance, standard error (S.E.) and bias of NY R are as follows:
V (NY R ) = N2 V (Y R)
(2.14)
SE (NY R ) = V (NY R )
(2.15)
and bias of population total is given by
bias (NY R ) = N(bias (Y R)) = N[ X N bias ( Rn ) ]
(2.16)
The confidence interval for the mean and total are shown below:
Confidence Interval of the mean estimate:
⎡
⎤
P⎢Y n − t
SE Y n ≤ Y n ≤ Y n + t
SE Y n ⎥ = 1−
α
( )
( )
α
α
(2.17)
⎣
,n−1
,n−1
2
2
⎦
Confidence Interval of the total estimate:
⎡
⎤
P⎢NY n − t
SE NY n ≤ NY n ≤ NY n + t
SE NY n ⎥ = 1−
α
( )
( )
α
α
(2.18)
⎣
,n−1
,n−1
2
2
⎦
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
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Hartley-Ross Ratio-Type Estimation
Y
Instead of using
n
Rn =
,we use
X n
n
1
r = ∑ri (sample mean of ratio)
(3.1)
n
n i=1
Where:
y
r = i ;( individual sample ratio)
(3.2)
i
xi
y
r = 2
1
x2
y
r = 2
2
x2
:
:
y
r = n
n
xn
Note that E( r n ) ≠ Rn ⇒ r n is inconsistent, therefore Y r = r n X N ⇒ Y r is biased
and consistent. Bias estimator of Y r is given as
N −1
Bias(Y r ) =
(s )
(3.3)
rx
N
Proof:
∑ y
Bias(Y
i
r ) = ( X N r N ) − Y N = ( X N r N ) −
N
1
=
[∑y − NX NrN
i
]
N
1
=
[∑r x − NX NrN
i i
]
N
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⎡
1
X
r ⎤
∑
=
⎢∑
i ∑
r x − N
i
i i
⎥
N ⎢⎣
N
N ⎥⎦
⎡
1
X
r ⎤
∑
=
⎢∑
∑
r x − N
i
i
i i
⎥
N ⎢⎣
N
⎥
⎦
Multiplying by N-1
N −1 ⎡
X
r ⎤
= -
⎢∑
∑ i∑
r x −
i
i i
⎥
N ⎢⎣
N
⎥
⎦
N −1
= -
(S
rx )
N
N −1
Bias(Y r ) = -
(s
rx )
N
Wherein the unbiased estimator of srx
1
S
[y −rn xn (3.4)
n
]
rx = n −1
Therefore,
N −1
Bias(Y r ) = -
(y −rn xn (3.5)
n
)
N
To get the unbiased estimator which make use of r n ,subtract the bias from
y = rx , therefore,
r
N
(N − )1n
y = r X
(y −rn xn
(3.6)
n
)
N +
r
N (n − )
1
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As cited by Hartley and Ross, the formula of the variances of the ratio
type estimation for large sample is given by
2
1 (N − )
1
1
N −1
V ( y ) =
S − r S +
S + S S
r
( 2
2
2 N
y
x )
(
) ( 2 2 2
rx
r
x )
2
n
N
n(n − )
1
N
The unbiased estimator is given by
2
1 (N − )
1
1
N −1
Vˆ ( y ) =
S − r S +
S + S S (3.7)
r
( 2
2
2 N
y
x )
(
) ( 2 2 2
rx
r
x )
2
n
N
n(n − )
1
N
Where
∑(x − xn
i
)2
2
S =
(3.8)
x
n −1
∑(y − y
i
n )2
2
S =
(3.9)
y
n −1
∑(r − rn
i
)2
2
S =
(3.10)
r
n −1
n
S =
y − r x (3.11)
rx
[
n
n
n
]
n −1
∑ x − xn y − y
2
( i
)( i n)
S =
(3.12)
xy
n −1
Estimated standard error is,
SE( y ) = V y (3.13)
r
( r)
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This is the square root of the variance. The estimated coefficient of
variation of the estimate is shown below,
MSE(yr )
CV ( y ) =
100 (3.14)
r
y r
Where,
MSE( y
= V y + Bias y
(3.15)
r
( r) ( ( r)2
)
All these result apply to the estimation of the population total Y N .The
unbiased estimator of Y N = NYn and the variance of standard error(SE) and bias
of NYn is defined by
V(NY r =N2V(Y r ) (3.16)
SE(NY r )= V (N y (3.17)
r )
N −1
Bias(NY r )=N(Bias(Y r ))=-N
(s ) (3.18)
rx
N
The confidence interval for the mean and total are shown below confidence
interval for the mean ( Y r )
Confidence Interval of the Mean
⎡
⎤
P⎢Y r − t
SE Y r ≤ Y r ≤ Y r + t
SE Y r ⎥ = 1−
α
( )
( )
α
α
(3.19)
⎣
,n−1
,n−1
2
2
⎦
Confidence Interval of the Total
⎡
⎤
P⎢NY r − t
SE NY r ≤ NY r ≤ NY r + t
SE NY r ⎥ = 1−
α
( )
( )
α
α
(3.20)
⎣
,n−1
,n−1
2
2
⎦
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
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Regression Estimation
The linear regression estimator is designed to increase precision by the use
of auxiliary variant xi that is correlated with yi. Suppose yi and xi are obtained for
every unit in the sample that the population mean X N of xi is known then the
regression estimator of Y N is
Y reg = y + β X − x (4.1)
n
( N n)
Where the subscript reg denote as regression and β is the estimate of the
change in y and x when x is increased by unity. The model is shown,
∑(x − xn y y
i
)( −
i
n )
β =
(4.2)
∑n(x − x 2
i
n )
i=1
Where,
n Y i
∑
Y
i=1
n =
(Sample mean for the production of strawberry) (4.3)
n
n
∑xi
X
i=
n =
1
(Sample mean of the number of strawberry plant ) (4.4)
n
Variance of regression estimation is estimated using the variance of residual as
⎛
n ⎞ se2
V Y
( reg ) = ⎜1−
⎟
(4.5)
⎝
N ⎠ n
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Where
e
ˆ
i = yi - Y n + β
(residual sample) (4.6)
1 (x − x n
i
)
where,
ˆ
n∑ x − y −
x y
i
i
∑( i )( i )
β =
(slope) (4.7)
1
n∑ −
x
i
(∑ i)2
2
1
en = ∑e (Mean of the residual) (4.8)
i
n
∑(e − en
i
)2
2
s =
(Variance of the residual) (4.9)
e
n −1
Standard error is defined by
SE(Y reg ) = V (Y reg ) (4.10)
And the bias of (Y reg ) is estimated as
1− n
2
N e x
x
N
i (
− n
i
)
bias (Y reg ) =
(4.11)
2
∑
n
1
N 1
sx
i=
−
The coefficient of variation of the estimator (Y reg ) which measures the variability
of the estimate is,
CV (
MSE Y
Y reg )
( reg )
=
100 (4.12)
Y reg
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where
MSE(Y reg ) = V (Y reg )+ (Bias(Y ) 2
reg
(4.13)
To estimate the population mean, Y N , we used the unbiased estimator
Y N =N (Y reg ) (4.14)
And estimate variance of the total N (Y reg ) is
V (NY reg ) = N2 V (Y reg ) (4.15)
SE(NY reg ) = V (NY reg ) (4.16)
And its bias is given by
1− n
2
N e x
x
N
i (
− n
i
)
bias(NY reg ) = N (bias(Y reg )) = -N
(4.17)
2
∑
n
1
N 1
sx
i=
−
At the regression estimation in the measure of accuracy of the estimate is
approximately 95% confidence interval(C.I) where half width of the C.I is the
margin error of an estimate that is,
Confidence Interval of the mean:
⎡
⎤
P⎢Y reg − t
SE Y reg ≤ Y reg ≤ Y reg + t
SE Y reg ⎥ = 1−
α
( )
( )
α
α
(4.18)
⎢
⎣
2,n−1
2,n−1
⎥
⎦
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Confidence Interval of the total
⎡
⎤
P ⎢N Y reg − t
SE N Y reg ≤ N Y reg ≤ N Y reg + t
SE N Y reg ⎥ = 1 −
α
(
)
(
)
α
α
⎢
⎣
2,n−1
2,n −1
⎥
⎦
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
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Magnitude of Efficiency
This is used to determine the efficiency of each estimator with respect to
the y-only estimator. The magnitude of efficiency can be computed using the
formula bellow:
V(Y n )
RE(Classical ratio over “Y-only”, in %) =
(5.1)
V (Y )
)
100
(
R
V(Y n )
RE (Hartley-Ross over “Y-only”, in %) =
(5.2)
V (Y )
)
100
(
r
V(Y n )
RE (Regression over “Y-only”, in %) =
(5.3)
V (Y )
)
100
(
reg
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
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25
Definition of Terms
Strawberry. Pulpy red fruit with a seeded-studded surface plant with
runners and white flower bearing this.
Production. Something produced.
Estimate. Approximate judgment, especially of cost, value, size,
ect.statement of approximate charge for work to be undertaken
Estimation. The process of providing a numerical value for population
parameter based on information collected from a sample.
Estimator. A statistic used to provide an estimate for a parameter. A
sample mean for example is of the population mean.
Ratio Estimation. The use of the ratio estimator when the relationship
between the response y and the subsidiary variable x which is proportional to y.
Regression Estimation: It is the relationship between the mean value of a
random variable and the corresponding values of one or more independent
variables.
Variable .Used to estimate the relationship between the y’s and the x’s not
through the origin but is a straight line.
Subsidiary variable. Used to estimate the mean of the response y.
Dependent variable. It is the variable to be determined or explained by one
or more explanatory variable.
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
26
METHODOLOGY
Sampling Design
Simple random sampling was employed in the selection of sample plots
using random numbers. The population consisted of 899 plots from 11,488 square
meters of the Sariling Sikap area and 22 farmers. The plots were planted with
strawberries.195 plots were drawn as samples. Each plot has different number of
strawberry plants and sizes of plots.
Data Gathering
The data was gathered through harvesting strawberries. The researchers
asked permission from the farmers to harvest the strawberry from the selected
plots and weighted it. The researchers gathered data until the sixth harvesting
season of the strawberry. Data gathering started on the 29th day of December,
2007 and ended on the 3rd day of February, 2008.Harvest is done twice a week
since farmers must spray and exposed the plants for two days before the
harvesting.
Data Analysis
The data gathered were summarized using the Microsoft excel program.
The mean, variance, mean square error, and bias of ratio, regression and
simple random sampling estimates were computed and compared from each other
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
27
through their relative efficiency with the strawberry production and number of
plants as auxiliary variable used.
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
28
RESULTS AND DISCUSSION
The Estimates of Strawberry Production/grams
in Strawberry Farm, La Trinidad, Benguet
The estimates of the mean and total production of strawberry using the
four estimators are presented in Table 1. It showed that simple random sampling
had the least estimated mean that is about 7,208.718 grams and a total of
6,480,637.482 grams in estimating strawberry production. Hartley-Ross
estimation had the largest mean and total estimation that is about 9,888.170 grams
and 8,889,464.170 grams. Classical estimation had 9,844.239 grams estimated
mean and 8,849,970.861 grams estimated total production while regression
estimation had the largest mean and total estimation that is about 9,888.170 grams
and 8,889,464.170 grams. Classical estimation had 9,844.239 grams estimated
mean and 8,849,970.861 grams estimated total production while regression
estimation has 8,613.167 grams and 7, 743.133 grams estimated mean and total
production of strawberry per plot in Strawberry Farm.
According to Cochran (1977) simple random sampling is a method of
selecting n units out of the N such that every one of the C distinct samples has
N
n
an equal chance of being drawn. He stated also that Ratio estimate is consistent. It
is biased except for some special types of population, although the bias is
negligible for large samples.
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
29
Table1. Estimates of mean and total of Strawberry Production using four methods
of estimation.
ESTIMATION ESTIMATOR ESTIMATE
± STANDARD BIAS
PROCEDURE
ERROR
for average
SRS
Y n
7,208.72
± 209.331 0
Classical Ratio Y
207.501 7.218
R
9,844.239
±
Hartley-Ross
Y
239.867 121.99
r
9,888.170
±
Regression
Y
8,613.167
± 201.426 0.0223
ir
for total
SRS NY
188,188.469 0
n
6,480,637.482 ±
Classical Ratio NY
186,548.308 6,488.982
R
8,849,970.861 ±
Hartley-Ross
NY
516,943.809 109,669.01
r
8,889,464.170 ±
Regression
NY
7,743,237.133 ± 181,082.317
20.048
ir
Consistency of the Estimates
Table 2 shows the Confidence Interval of the true mean and total of the
strawberry production using the different estimation procedures. Result shows
that Hartley-Ross ratio-type has the largest 95% Confidence Interval for the true
mean with a value of 9,598.167 grams to 10,357.957 grams and true total with a
value indicated to be 8,467,126.872 grams to 9,311,802.788. Simple random
sampling, a biased estimator has the least confidence interval for true mean with a
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
30
values of 6,798.429 to 7,619.007 grams and total production of 6,111,788.083
grams to 6,849,486.881 grams. Moreover, for the bias estimators, regression
Table 2. Confidence Interval of the true mean and total strawberry production
using three methods of estimation
ESTIMATION
LOWER LIMIT
UPPER LIMIT
PROCEDURE
for average
SRS 6,798.429
7,619.007
Classical Ratio
9,473.537
10,250.941
Hartley-Ross 9,598.167
10,421.065
Regression 8,218.372
9,007.962
for total
SRS
6,111,788.083
6,849,486.881
Classical Ratio
8,484,345.977
9,215,595.743
Hartley-Ross 8,628,750.649
10,011,902.3
Regression 7,388,315.792
8,098,158.474
estimator has the least confidence interval with 8,218.372 to 9,007.962 for the
true mean and 7,388,315.880 to 8,098,158.386 for the true total production.
Precision of the Estimates
The study of Damoslog and Tomin (2007) showed that classical ratio
estimator with the least variability is the more precise estimator. Table 3 showed
that classical ratio estimation had the least variability with 2.108 is the more
precise estimation but regression estimator also had a close value of coefficient of
variation which is 2.339. Specifically, simple random sampling has the highest
coefficient of variation with a value of 2.904 compared to the other estimators. It
was also found that the different estimators have small variability with Hartley
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
31
Ross ranked second from the highest variability obtained followed by regression
with 2.339 coefficient of variation.
Table3. Coefficient of Variability of the four methods of estimation.
ESTIMATION
COEFFICIENT OF
PROCEDURE VARIATION
SRS 2.904
Classical Ratio
2.108
Hartley-Ross 2.720
Regression 2.339
Efficiency of the Estimates
This is supported by Cochran (1977) as cited by Beligan (2004). If ˆo and
1
ˆo are unbiased estimators for the parametersθ , that is E( ˆo ) = E( ˆo ) = θ , then
2
1
2
ˆo is more efficient than ˆo if V( ˆo ) < V( ˆo ).
1
2
1
2
In the study of Damoslog and Tomin (2007), an estimator with higher
relative efficiency is considered an efficient Thus, in the said study classical ratio
was efficient compared to simple random sampling.
Based from the mentioned criterion for efficient estimators, regression is
the most efficient estimators in estimating strawberry production with a variance
of 40,572.592 and higher relative efficiency as shown in Table 4. Hartley-Ross
ratio-type having a value of 57,449.827 has the largest variance in estimating
strawberry production and classical ratio is next to the regression estimation with
a variance of 43,189.421.
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
32
And this result is the same with the study of Barrios (1995) where
regression estimator was considered efficient in estimating socio-economic
indicators.
Table 4. Relative Efficiency and Variance of the different estimates compared to
simple random sampling.
RELATIVE
VARIANCE
PROCEDURE EFFICIENCY
SRS
Classical Ratio
101.77
43,189.421
Hartley-Ross 76.274
57,449.827
Regression 108
40,572.592
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
33
SUMMARY, CONCLUSION AND RECOMMENDATION
Summary
The study was conducted to determine the efficiency and consistency of
ratio and regression estimation in estimating the strawberry production in
Strawberry farm at La Trinidad, Benguet over the simple random sampling
method.
The data was gathered from the 29th day of December 2007 until 3rd day of
February 2008 wherein the researchers ask permission from the owner of the
strawberry that was selected as one of the study site. The sample plots were
determined using simple random sampling and there were about 195 plots
considered. The data gathering ended until the sixth harvesting of strawberries.
The production was estimated using simple random sampling classical
ratio estimation, Hartley-Ross ratio-type and regression estimation. It was found
that each estimated values obtained due to different estimation schemes utilized.
The classical ratio, Hartley–Ross ratio-type and regression estimators have
greater estimated mean and total than the simple random sampling. The regression
estimator was said to have the least bias while the classical ratio was considered
to have the most bias. Though simple random sampling estimator is not bias, it is
considered as a consistent estimator since its 95 % confidence interval for the true
mean and true total of the strawberry production is narrow. Among the considered
bias estimator, regression was more consistent with its 95 % confidence interval
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
34
which is limited or having the least confidence interval. Classical ratio estimate is
also considered as the most precise among the estimates since it has the lowest
coefficient of variation. Result of the relative efficiency shows that regression
estimator is the more efficient compared to simple random sampling.
Conclusion
It is therefore concluded that classical ratio estimator is the most precise
among the four different estimators used in estimating strawberry production with
its least coefficient of variability. Moreover, simple random sampling was a
consistent and unbiased estimator.
However, regression estimator is the most efficient and most consistent
among the bias estimators as applied to strawberry production.
Recommendation
It is recommended that some study has to be conducted using the
independent variable that is the number of plants per plots to determine the
efficiency of estimator for more meaningful estimate .To maintain a high degree
of precision, the variability among sampling units within a plot should be kept
small. It is also recommended that in estimating strawberry production it is
efficient to use the regression estimator over simple random sampling.
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
35
LITERATURE CITED
BARRIOS, E. B. 1998. Small area Estimator of Employment Rate.7th National
Convention on Statistics. Manila, Philippines.
BELIGAN, S.Z. 2004. Survey and Sampling Techniques Syllabus
COCHRAN, G. W. 1997.Sampling Techniques.3rd Edition. John Wiley and Sons,
United States of America.
DAMOSLOG, F.T. AND TOMIN, J.T. 2007. Efficiency of Ratio and Regression
Estimator s in Estimating Farmers’ Income in La Trinidad Benguet.
DENG, LI-YUAN AND CHHIKARA.R.S (1990). On the Ratio and Regression
Estimation in Finite Population Sampling
DORFMAN, J.H. AND SANDERS, D.R. 2004. Generalized Hedge Ratio
Estimation with an unknown model
EVERITT, B.S. 1998. Dictionary of Statistics.
FAEDI W., MOURGUES F AND ROSATI. Strawberry Breeding and Varieties:
Situation and Perspective
HSU AND KUO. 2000. Household Solid Waste Recycling Included Production.
LOHR,S.L.1999. Sampling:Design and Analysis.Brooks/Cole
PublishingCompany
M.GOSSOP, J.STRANG, P.GRIFFITHS, B.POWIS AND C.TAYLOR. Drug
Transitions Study, National Addiction Centre, Maudsley Hospital, London
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
36
ONATE, B.T.1967. Ratio Estimators in Field Research. The Philippine
Statistician
Vol.VXI, No.3-4:62-74
(Oxford Quick Reference)Dictionary
PALANISWAMY, U.R AND PALANISWAMY, K.M.(2006). Handbook of
Statistics for teaching and research in plant and crop sciences.
RAND CORP SANTA MONICA CALIF. Some Finite Population Unbiased
Ratio and Regression Estimators.
SUKHATME, B.V. Hartley-Ross Unbiased Ratio-Type Estimator
TOTEMIC N. All season Strawberry Growing with day-neutral Cultivars.
WALPOLE, R.E.ET.AL.1998. Probability and Statistics for Engineers and
Scientists.Prentice-Hall, Inc.6th edition.
WEI CHEN, YOSHINAO OOEDA AND TOMONORI SUMI. 2005. A study on
a shopping Frequency Model with the Consideration of Individual
Difference. Kyushu University, Japan.
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
37
APPENDICES
Appendix A. Application for Oral Defense
Benguet State University
College of Arts and Sciences
DEPARTMENT OF MATHEMATICS-PHYSICS-STATISTICS
APPLICATION FOR ORAL DEFENSE
Date: March 17, 2008
Group Members:
Emily June C. Alos Major: Statistics
Eleonor B. Bat-oy Minor: Information Technology
Degree: BACHELOR OF SCIENCE AND APPLIED STATISTICS
Title of Thesis: “Efficiency of Ratio and Regression Estimation in Estimating the
Strawberry Production.”
ENDORSED BY : MARYCEL H. TOYHACAO
Adviser
Date and time of Defense: March 18, 2008 @ 5 PM
Place of Defense: CAS Annex 210
NOTED BY : MARIA AZUCENA B.LUBRICA
Department Chairman
REPORT OF RESULT ON ORAL DEFENSE
Name and signature *Remarks
Marycel H. Toyhacao
_______________________
Adviser
Salvacion Z. Beligan
_______________________
Member
Maria Azucena B.Lubrica
_______________________
Member
Cristina B.Ocden
_______________________
Member
*passed or failed
38
Appendix B. The raw data of strawberry production
Harvesting
plots
1st
2nd
3rd
4th
5th
6th
Production(g)
# of plants
1 1000 700 700 1000 1700 1100
6200
120
2 1000 600 800 500 1700 1100
5700
116
3 1000 900 500 600 1100 1200
5300
114
4 1000 500 1300 900 1900 1800
7400
122
5 700 700 800 500 900 1000
4600
110
6 800 700 900 600 1600
1250
5850
110
7 1000 500 1100 900 1700 1600
6800
122
8 1000 300 1000
1000 1300 1400
6000
118
9 1000 600 1100
1500 1200 1100
6500
116
10 700 500 600 1500 1200 1500
6000
116
11 700 400 700 1000 1100 1600
5500
118
12 700 600 800 1200 1300 1300
5900
120
13 600 400 1000
1000 1300 1250
5550
136
14 600 600 600 500 1400 1100
4800
110
15 1200 600 600 900 1200 1600
6100
112
16 600 600 700 500 1400 1750
5550
112
17 600 600 600 1100 1250 1900
6050
101
18 500 500 900 1200 1700 1100
5900
112
19 700 200 900 1500 2300 1150
6750
118
20 500 600 800 800 1700 1900
6300
120
21 600 400 800 700 1300 1800
5600
118
22 700 400 900 900 1400 1400
5700
112
23 600 400 900 1700 1450 1450
6500
110
24 700 400 900 1700 1950 1000
6650
108
25 600 300 800 1400 1850 1100
6050
110
26 600 500 900 1800 1500 1150
6450
112
27 500 300 900 1400 1900 1250
6250
106
28 600 500 600 1700 1500 1000
5900
130
29 700 500 900 1700 1850 900
6550
114
30 600 600 600 1600 1700 1800
6900
112
31 500 500 800 1600 1700 1800
6900
114
32 500 500 500 1600 1450 1900
6450
110
33 700 300 1200
1300 1700 1100
6300
114
34 500 300 700 1350 1450 1200
5500
106
35 600 300 800 1250 1500 1000
5450
110
36 400 300 700 1000 1700 1700
5800
130
37 400 500 800 1600 1200 1600
6100
108
38 900 500 1100 600 2250 1850
7200
110
39 600 400 1000
1400 1900 1550
6850
104
40 300 800 900 600 1300 1250
5150
106
41 500 200 700 600 900 1900
4800
124
42 300 100 400 800 1100 1800
4500
106
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
39
43 600 400 800 400 300 1000
3500
118
44 400 600 600 400 500 1150
3650
120
45 400 500 900 500 500 1100
3900
106
46 500 400 400 300 500 1000
3100
108
47 600 300 700 300 400 1500
3800
106
48 700 800 100 400 400 1000
3400
108
49 300 600 1000 400 500 800
3600
106
50 500 300 1200 500 500 1100
4100
94
51 400 400 700 400 300 1250
3450
108
52 300 500 900 300 300 1200
3500
104
53 400 700 900 500 500 800
3800
108
54 700 200 600 200 300 900
2900
110
55 500 600 500 200 300 1150
3250
106
56 400 400 700 300 400 800
3000
108
57 400 400 600 400 400 850
3050
118
58 400 600 700 200 400 1050
3350
116
59 600 700 600 300 300 850
3350
110
60 500 300 700 400 500 700
3100
108
61 800 900 800 500 500 1350
4850
108
62 400 400 600 400 400 800
3000
106
63 400 200 400 400 500 1250
3150
110
64 400 300 400 200 300 1000
2600
112
65 200 200 300 200 200 1000
2100
120
66 400 500 300 300 300 1100
2900
130
67 150 200 200 300 400 1150
2400
118
68 250 250 300 100 200 800
1900
114
69 200 300 400 200 200 1150
2450
116
70 400 300 200 100 300 1000
2300
122
71 200 300 400 300 400 1250
2850
120
72 500 300 300 300 500 800
2700
120
73 400 300 300 300 400 800
2500
124
74 300 400 500 200 100 900
2400
126
75 200 500 200 400 300 950
2550
130
76 300 500 200 100 100 850
2050
126
77 300 300 200 300 400 800
2300
128
78 200 400 500 400 300 1000
2800
129
79 300 300 200
200 400
1100
2500
132
80 600 500 700 700 500 1150
4150
130
81 500 900 900 900 900 800
4900
128
82 300 900 900 900 800 700
4500
122
83 1300 17000 900 1100 1200 2000
23500
115
84 1100 8000 900 1400 1400 2100
14900
96
85 1600 1000 500 1000 1600 2000
7700
116
86 1200 900 300 1000 1000 1500
5900
114
87 11100 1000 1200 1500 1900 2200
18900
123
88 1200 1000 600 1400 1600 2000
7800
120
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
40
89 1700 11000 1100 1500 1800 2300
19400
115
90 2200 1500 1700 2100 1900 2350
11750
117
91 1200 1000 1000 1600 2100 2500
9400
116
92 1700 1500 1200 1350 2100 2100
9950
107
93 1600 1500 1600 1700 1800 1500
9700
103
94 800 1300 1600
1900 1900 2200
9700
112
95 11000 1500 1500 2000 1300 2000
19300
105
96 1100 1500 800 1500 1600 1800
8300
106
97 1100 1100 500 1600 1800 1800
7900
112
98 1300 900 800 1400 1700 2000
8100
120
99 1000 900 1000 1500 1600 1800
7800
93
100 1400 900 900 1800 1700 2100
8800
112
101 12000 1200 1000 1900 2000 1200
19300
117
102 1100 1000 900 2200 2300 1250
8750
110
103 1300 1200 1000 1300 1500 2000
8300
119
104 1100 1300 1100 1600 1800 1400
8300
115
105 1400 1200 1000 1200 1600 1400
7800
114
106 1700 1800 1500 1400 1500 1450
9350
117
107 1200 1300 1500 1500 1300 1000
7800
115
108 1600 1200 1100 1600 1800 1100
8400
114
109 1300 1000 900 1800 1900 1000
7900
117
110 1500 1500 1300 1900 1600 1600
9400
115
111 1000 1500 1400 1800 1700 1900
9300
115
112 1500 1000 800 1300 1400 2000
8000
120
113 1300 1300 1100 1400 1600 2200
8900
120
114 1500 1600 1500 1700 1000 2250
9550
113
115 1600 1500 1200 1600 2000 2000
9900
118
116 1800 1500 1500 1800 2100 2200
10900
120
117 2200 2000 1800 2200 2200 2400
12800
134
118 1300 1500 1100 2100 2300 2000
10300
124
119 1200 1100 1000 1400 2100 1500
8300
123
120 1300 1400 1300 1500 1900 1600
9000
116
121 1000 1000 800 1600 1700 1800
7900
118
122 1000 1200 1500 1800 2000 1600
9100
101
123 1500 1400 1300 2100 2100 2000
10400
121
124 1500 1600 1500 2400 2100 1800
10900
112
125 1400 1200 1300 1400 1500 2200
9000
124
126 1500 1300 1000 1600 1900 2300
9600
105
127 1000 1000 800 1800 2000 2000
8600
100
128 1300 1300 1000 1000 1200 2250
8050
102
129 1400 2500 2400 2600 2300 2000
13200
150
130 1300 1100 1100 1300 1400 1250
7450
145
131 1600 1000 1600 1700 1400 1400
8700
142
132 1200 1100 1500 1800 1900 1800
9300
132
133 1300 1300 1600 1400 1500 1200
8300
131
134 1600 1200 1400 1200 1100 1000
7500
142
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
41
135 1700 700 1900 2000 1800 1500
9600
144
136 1400 700 1500 1300 1400 1900
8200
145
137 1200 750 1600 1350 1400 1400
7700
138
138 1300 1000 1400 1500 1300 1100
7600
138
139 1000 750 1000 750 800 1300
5600
138
140 700 800 600 600 800 1100
4600
120
141 1000 1100 900 1200 1300 1150
6650
140
142 1000 1100 1200 1100 1100 1300
6800
142
143 1200 2000 1800 1600 1900 1600
10100
152
144 800 500 700 900 1000 1800
5700
120
145 700 800 800 700 600 1900
5500
122
146 1000 700 1000 1800 1300 2000
7800
120
147 900 750 800 1300 1600 1400
6750
140
148 400 750 1400 1300 1500 1800
7150
150
149 300 750 400 900 1000 1200
4550
140
150 600 800 900 800 1000 1000
5100
152
151 400 1000 1000 500 600 900
4400
120
152 500 700 800 1500 1600 1100
6200
124
153 700 1000 1800 800 1900 1800
8000
130
154 1000 1100 1600 1700 1800 1200
8400
150
155 1300 1200 1400 1800 1800 1000
8500
131
156 1200 850 1200 1400 1400 1200
7250
142
157 700 850 900 1400 1500 1300
6650
145
158 500 800 1600 1000 1100 1900
6900
138
159 750 1600 1000 3500 2100 2250
11200
201
160 750 1550 1500 1400 2400 2500
10100
192
161 750 700 1600 1250 2200 2250
8750
189
162 500 800 700 2700 2000 2750
9450
189
163 1500 1000 800 1200 2200 2250
8950
201
164 1750 1000 800 1250 2250 2250
9300
219
165 1500 1100 1000 1250 2000 2200
9050
207
166 1500 700 1600 1400 2100 2250
9550
222
167 1300 800 800 1500 2600 2500
9500
210
168 1250 850 700 2500 2000 2300
9600
228
169 1000 850 750 2250 2100 2400
9350
207
170 650 900 800 2500 2000 2800
9650
180
171 650 650 850 1500 2600 2250
8500
240
172 600 600 800 2250 2100 2500
8850
189
173 500 550 800 3050 3000 2200
10100
168
174 1000 700 750 1500 2000 2250
8200
207
175 800 1000 1000 2000 2100 2600
9500
204
176 1000 800 1100 1200 2200 2250
8550
201
177 1100 1100 1150 1750 2150 2200
9450
177
178 1000 1150 1100 1750 2200 2400
9600
168
179 700 1600 1000 1500 1600 2250
8650
171
180 700 900 1150 1100 2200 2600
8650
168
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
42
181 750 850 1500 1500 2100 2500
9200
168
182 750 700 1600 2000 1900 3000
9950
174
183 800 750 1800 1900 1800 2500
9550
189
184 850 900 1600 2000 1950 2750
10050
192
185 600 1000 1650 2000 1100 2750
9100
189
186 600 950 1000 2200 2000 2000
8750
183
187 1000 750 1100 2000 1800 2000
8650
186
188 750 1100 1500 2000 1900 2100
9350
192
189 850 1000 1000 2500 1950 2000
9300
186
190 800 950 950 2250 2000 2800
9750
180
191 800 850 950 1200 2200 2800
8800
180
192 800 1000 950 1700 1500 2250
8200
177
193 750 1500 850 1500 2600 2500
9700
186
194 500 450 500 500 900 1300
4150
100
195 700 350 400 500 700 1250
3700
120
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
43
Appendix C. Sample Computation
Simple Random Sampling (SRS)(for average)
N = 899
n = 195
n
∑Yi 1405700
Y
i=1
n =
=
n
195
= 7208.718
∑(Y −Y n
i
)2
2
s =
=
596
,
911
,
10
n −1
2
V (Y n ) (N − n)⎛ s ⎞
=
⎜
⎜
⎟
⎟
N
⎝ n ⎠
899 −195 ⎛10911596 ⎞
=
⎜
⎟
899
⎝
195
⎠
= 43,819.4209
SE (Y n ) = V (Y n ) = 209 331
.
SE(Y n )
CV (Y n ) =
Y n
331
.
209
=
(100)
718
.
7208
= 2.904
Confidence Interval of the mean estimate:
⎡
⎤
P⎢Y n − t
SE Y n ≤ Y n ≤ Y n + t
SE Y n ⎥ = 1−
α
( )
( )
α
α
⎣
,n−1
,n−1
2
2
⎦
= 7,208.718 ± 1.96(209.331)
= 6,798.429 ≤ Y n ≤ 7,619.007
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
44
Simple Random Sampling(for total)
NY n = 899(7208.718)
= 6,480,637.482
N (N − n)
Vˆ (N Y n ) =
2
s
n
= 35,414,899,810
SE (NY n ) = V (Y n )
= 188,188.469
Confidence interval of the total estimate:
⎡
⎤
P⎢NY n − t
SE NY n ≤ NY n ≤ NY n + t
SE NY n ⎥ = 1−
α
( )
( )
α
α
⎣
,n−1
,n−1
2
2
⎦
= 6,480,637.482 ± 1.960 (188188.469)
= 6,111,788.082 ≤ NY n ≤ 6849486.881
Classical Ratio Estimation
Y
Y n
718
.
7208
R =
X N =
)
180
(
X n
81
.
131
= 9844.239
⎛
⎞
N − n
bias ( R
⎜
⎟
2
n ) = ⎜
{Rns − s
x
xy }
2 ⎟
⎝ Nnxn ⎠
⎛
899 −195
⎞
=
[54 69
. ( 186
,
3
596
.
)− 411
]
⎜
⎜
⎝ 899(195)(131 )
161
.
81
.
2 ⎟⎟⎠
= 0.0401
Y n
718
.
7208
Rn =
=
X n
81
.
131
= 54.69
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
45
∑(x − xn)2
s 2 =
i
= 3,186.596
x
n −1
∑(x − xn
i
)(y − y
i
n )
s 2
xy =
= 169,053.579
n −1
Bias (Y R ) = X N bias ( Rn )
= 180 (.0401)
= 7.218
N − n
2
V (Y R) =
∑[y − Rnx
i
i ]
Nn(n − )
1
899 −195
=
( 08
.
2
E09)
)
194
)(
195
(
899
= 43,056.623
SE(Y R ) = V (Y R ) = 207.501
MSE (Y R ) = V (Y R) + bias ( (Y ) ) 2
R
= 43,056.623 + 7.2182
= 43,108.723
MSE(Y R )
43061 605
.
CV(Y R ) =
100 =
100
(
)
Y
9844 239
.
R
= 2.109
Classical Ratio Estimation (Total)
NY R = 899 (9844.239)
= 8,484,346.662
V (NY R ) = N2 V (Y R) = (899)2 (43056.623)
= 34,798,405,770
SE (NY R ) = V (NY R ) = 186548.308
bias (NY R ) = N(bias (Y R))
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
46
= 899(7.218)
= 6,488.982
Confidence Interval of the mean estimate:
⎡
⎤
P⎢Y n − t
SE Y n ≤ Y n ≤ Y n + t
SE Y n ⎥ = 1−
α
( )
( )
α
α
⎣
,n−1
,n−1
2
2
⎦
= 9,844.239 ± 1.960(207.501)
= 9,437.537 ≤ Y n ≤ 10250.941
Confidence Interval of the total estimate:
⎡
⎤
P⎢NY n − t
SE NY n ≤ NY n ≤ NY n + t
SE NY n ⎥ = 1−
α
( )
( )
α
α
⎣
,n−1
,n−1
2
2
⎦
= 8,484,346.662 ± 1.960(186,543.308)
= 8,118,721.778 ≤ NY n ≤ 8,849,971.546
Hartley-Ross Ratio-Type Estimation(average)
n
1
1
r = ∑ri =
(
.
10844 39)
n
n i=1
195
= 55.612
N −1
899 −1
Bias(Y r ) =
(s ) =
(−
.
122
)
126
rx
N
899
= 121.99
1
S
[y −rnxn
n
]
rx = n −1
1
=
[7208 718
.
− 55 612
.
131
(
)
81
.
]
195 −1
=-122.126
(N − )1n
y = r X
(y −rn xn
n
)
N +
r
N (n − )
1
(899 − )1
= 55.612(180)+
(7208 718
.
− 55 62
.
131
(
)
81
.
)
195
(
899
− )
1
= 9,888.170
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
47
2
1 (N − )
1
1
N −1
Vˆ ( y ) =
S − r S +
S + S S
r
( 2
2
2 N
y
x )
(
) ( 2 2 2
rx
r
x )
2
n
N
n(n − )
1
N
1 899
(
− )
1
2
=
(
−
,
11
,
249
.
454 99 −
.
55
(
2
)(
612
.
411
)
161 )
1
(899 )1
+
195
899
195
(
195
− )
1
8992
(
102
.
1220
+
.
689
.
186
,
3
(
582
596)
=57,449.827
∑(x − xn
i
)2
2
S =
= 3,186.596
x
n −1
∑(y − y
i
n )2
2
S =
= 11,249,454.99
y
n −1
∑ r − r
2
( n
i
)2
S =
= 1220.102
r
n −1
∑ x − xn y − y
2
( i
)( i n)
S =
=169,053.580
xy
n −1
SE( y ) = V y = 209.950
r
( r)
MSE( y = V y + Bias y
= 44070.356 + (121.99)2
r
( r) ( ( r)2
)
= 72,331.387
MSE(y
.
72331 387
r )
CV ( y ) =
100 =
)
100
(
r
y
.
9888 170
r
= 2.720
Confidence Interval of the Mean
⎡
⎤
P⎢Y r − t
SE Y r ≤ Y r ≤ Y r + t
SE Y r ⎥ = 1−
α
( )
( )
α
α
⎣
,n−1
,n−1
2
2
⎦
= 9,888.170 ± 1.96( 239.687)
= 9,418.383 ≤ Y r ≤ 10,357.957
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
48
Hartley-Ross Ratio-Type Estimation(total)
NY r = 899(9,88.170)
V(NY r ) = N2 V(Y r ) = 8992 ( 9888.170)
= 4.643100763 E 10
SE(NY r ) = V (N y = 215,478.555
r )
Bias (NY r ) = N (Bias (Y r )) = 899(121.99)
=109669.01
Confidence Interval of the Total
⎡
⎤
P⎢NY r − t
SE NY r ≤ NY r ≤ NY r + t
SE NY r ⎥ = 1−
α
( )
( )
α
α
⎣
,n−1
,n−1
2
2
⎦
= 8,889,464.83 ± 1.96( 215,478.555)
= 8,467,126.872 ≤ NY r ≤ 9,311,802.788
Regression Estimation(average)
Y reg = y + β X − x = 7208.718 + 29.144 (180-131.81)
n
( N n)
= 8613.167
∑(x − xn y y
i
)( −
i
n )
β =
5570391
=
∑n(
1
.
191136
x − x 2
i
n )
i=1
= 29.144
⎛
n ⎞ se2
⎛
195 ⎞10103092.782
V Y
( reg ) = ⎜1−
⎟
= ⎜1−
⎟
⎝
N ⎠ n
⎝
899 ⎠
195
= 40572.587
ˆ
n∑ x − y −
x y
(
195 .
1 92E09) − (
(
25703 1405700)
i
i
∑( i )( i )
β =
=
1
n∑ −
x
(
195 3596429)− (
)2
25703
i
(∑ i)2
2
=32.201
e
ˆ
i = yi - Y n + β
= 2962.6841
1 (x − x n
i
)
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
49
1
1
en = ∑e =
(
.
2862 684)
i
n
195
= 15.193
∑(e − en
i
)2 96
.
1
E09
2
s =
=
e
n −1
194
= 10103092.78
SE(Y reg ) = V (Y reg ) = 40572.587
1− n
2
N e x
x
N
i (
− n
i
)
bias (Y reg ) =
2
∑
n
1
N 1
sx
i=
−
195
1899 ⎛12591 27
. (
)
6941
.
2962
−
86
.
131
⎞
=
⎜
⎟
)
238
.
985
(
195
⎝
898
⎠
= 0.0223
MSE(Y reg ) = V (Y reg )+ (Bias(Y ) 2
reg
= 40572.587 +( .0718)2
= 40572.568
CV (
MSE Y
Y reg )
( reg )
=
100
Y reg
= 40,572.568
Confidence Interval of the mean:
⎡
⎤
P⎢Y reg − t
SE Y reg ≤ Y reg ≤ Y reg + t
SE Y reg ⎥ = 1−
α
( )
( )
α
α
⎢
⎣
2,n−1
2,n−1
⎥
⎦
= 8,613.167 ± 1.960(201.426)
= 8,218.372 ≤ Y reg ≤ 9,007.962
Regression Estimation(total)
Y N =N (Y reg ) = 899(8613.167 )
= 7,743,237.133
V (NY reg ) = N2 V (Y reg ) =8992 ( 40572.587 )
= 3.27908922 E 10
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
50
SE(NY reg ) = V (NY reg ) = 181,082.272
bias(NY reg ) = N (bias(Y reg )) = 899(0.0223)
= 20.048
Confidence Interval of the total
⎡
⎤
P ⎢N Y reg − t
SE N Y reg ≤ N Y reg ≤ N Y reg + t
SE N Y reg ⎥ = 1 −
α
(
)
(
)
α
α
⎢
⎣
2,n−1
2,n −1
⎥
⎦
= 7,743,237.133 ± 1.960 (181,082.317)
= 7,388,315.88 ≤ NY reg ≤ 8,098,158.386
Magnitude of Efficiency
V(Y n )
RE(Classical ratio over “Y-only”, in %) =
V (Y )
)
100
(
R
421
.
43819
=
(100)
623
.
43056
= 101.77
V(Y n )
RE (Hartley-Ross over “Y-only”, in %) =
V (Y )
)
100
(
r
421
.
43819
=
)
100
(
827
.
57449
= 76.274
V(Y n )
RE (Regression over “Y-only”, in %) =
V (Y )
)
100
(
reg
= 108.00
Efficiency of Ratio and Regression Estimat ors in Estimating Strawberry Production
/ Emily June C. Alos & Eleonor B. Bat-Oy . 2008
Document Outline
- Efficiency of
Ratio and Regression Estimators in Estimating Strawberry Production.
- BIBLIOGRAPHY
- ABSTRACT
- TABLE OF CONTENTS
- INTRODUCTION
- REVIEW OF LITERATURE
- THEORETICAL FRAMEWORK
- METHODOLOGY
- RESULTS AND DISCUSSION
- SUMMARY, CONCLUSION AND RECOMMENDATION
- LITERATURE CITED
- APPENDICES