程序代写案例-STA304

欢迎使用51辅导,51作业君孵化低价透明的学长辅导平台,服务保持优质,平均费用压低50%以上! 51fudao.top
STA304
Chapter 6: Ratio Estimation
Dr. Luai Al Labadi
Fall 2021
Dr. Luai Al Labadi STA304 Fall 2021 1 / 51
Outline
1 Introduction r>2 Ratio Estimation
Estimation of R
Estimation of τy
Estimation of µy
Sample Size Determination
3 Regression (Linear) Estimation
4 Difference Estimation
5 Comparison between Estimators
Dr. Luai Al Labadi STA304 Fall 2021 2 / 51
Introduction
Ratio Estimation
• The estimators of µ, τ and p were based on a SRS of responses,
y1, y2, . . . , yn, selected from population (Chapter 4) or L SRSs se-
lected from L strata (Chapter 5).
• The emphasis in Chapter 4 and Chapter 5 was placed on sample
selection (design of the sample survey).
• In Chapter 6, a new estimation technique will be presented,
which, under certain conditions yield “better” estimates (i.e., es-
timates with smaller variances).
• The new estimation technique is called the Ratio Estimation.
Dr. Luai Al Labadi STA304 Fall 2021 3 / 51
Introduction
Ratio Estimation
• Ratio estimator makes use of auxiliary (subsidiary or ancillary) in-
formation to improve estimation of the population parameters.
• An ancillary variate xi, correlated with yi, is obtained for each ele-
ment in the sample.
• We must also be able to obtain the population total for x, τx.
Dr. Luai Al Labadi STA304 Fall 2021 4 / 51
Introduction
Motivating Example
• The wholesale price paid for oranges in large shipments is based on
the sugar content of the load.
• The exact sugar content cannot be determined prior to the purchase
and extraction of the juice from the entire load.
• It can be estimated (how)?
• Estimate the mean sugar content per orange, µy, and then to multiply
by the number of oranges N in the load.
• Thus, we could randomly sample n oranges from the load to deter-
mine the sugar content y for each.
• Take a sample: yl, y2, . . . , yn.
• An estimate of the total sugar content for the load is τˆy = Ny¯.
Dr. Luai Al Labadi STA304 Fall 2021 5 / 51
Introduction
Motivating Example
• But how to determine N?
• Count the total number of oranges in the load.
• This method is not feasible because it is too time-consuming and
costly.
• We can avoid the need to know N by noting the following two facts.
1 The sugar content of an individual orange, y, is closely related to its
weight x.
2 The ratio of the total sugar content τy to the total weight of the
truckload τx is equal to the ratio of the mean sugar content per
orange, µy, to the mean weight µx.
µy
µx
=
Nµy
Nµx
=
τy
τx
.
Dr. Luai Al Labadi STA304 Fall 2021 6 / 51
Introduction
Motivating Example
• Solve for τy, we have
τy =
µy
µx
(τx).
• We can replace µy and µx by y¯ (the average of the sugar contents
in the sample) and x¯ (the average of the weights in the sample).
• We get
τˆy =


(τx).
• Note that
τˆy =


(τx) =
ny¯
nx¯
(τx) =
∑n
i=1 yi∑n
i=1 xi
(τx).
Dr. Luai Al Labadi STA304 Fall 2021 7 / 51
Introduction
Another Example
• If y is the expenditure on textbooks by a college student then x
could be the number of courses the student is taking.
Dr. Luai Al Labadi STA304 Fall 2021 8 / 51
Introduction
Ratio Estimation
• For each member of the population, two variables are measured xi
and yi.
• Ratio estimation is used when the relationship between y and x
is linear and the line passes through the origin (y = 0, x = 0).
Dr. Luai Al Labadi STA304 Fall 2021 9 / 51
Ratio Estimation
Ratio Estimation
• There are two cases that may be of interest to the researcher to use
ratio estimator:
1 To estimate the ratio of two population characteristics. The most
common case is the population ratio R of means or totals:
R =
τy
τx
=
µy
µx
.
2 To use the relationship between x and y to improve estimation of the
µy or τy.
Dr. Luai Al Labadi STA304 Fall 2021 10 / 51
Ratio Estimation
Ratio Estimation
• Examples:
• If y is the total income earned by all adults in the household and x
is the total number of adults in the household, then R is the average
income per adult in a household.
• If y is weekly food expenditure and x is number of inhabitants, then
R is weekly food cost per inhabitant.
• If y is the number of motor vehicles and x is the number of inhabitants
of driving age, then R is the number of motor vehicles per inhabitant
of driving age.
Dr. Luai Al Labadi STA304 Fall 2021 11 / 51
Ratio Estimation Estimation of R
Ratio Estimation of the Population Ratio R
In this sampling plan we take a simple random sample of size n from a
population of size N and measure both yi and xi.
• Let
R =
τy
τx
=
µy
µx
be the population ratio.
• A sample-based estimator of R is given by
r =
∑n
i=1 yi∑n
i=1 xi
=


Dr. Luai Al Labadi STA304 Fall 2021 12 / 51
Ratio Estimation Estimation of R
Is r an unbiased estimator of R?
• r is not unbiased estimator of R but it is approximately unbiased
(for large sample size) since
E(Rˆ) = E(r) = E
(
y
x
)
≈ E
(
y
µx
)
=
1
µx
E(y) =
µy
µx
= R.
• Here the population is
(u1, v1), . . . , (uN , vN ).
µx =
1
N
N∑
i=1
ui and µy =
1
N
N∑
i=1
vi.
Dr. Luai Al Labadi STA304 Fall 2021 13 / 51
Ratio Estimation Estimation of R
Question
1 Show that
bias = E(Rˆ−R) = E(r −R) = −cov(r, x¯)
µx
.
2 Show that |E(r −R)|
σr
≤ σx¯|µx| .
Dr. Luai Al Labadi STA304 Fall 2021 14 / 51
Ratio Estimation Estimation of R
Question, continue
Dr. Luai Al Labadi STA304 Fall 2021 15 / 51
Ratio Estimation Estimation of R
Question, continue
Dr. Luai Al Labadi STA304 Fall 2021 16 / 51
Ratio Estimation Estimation of R
Question, continue
Dr. Luai Al Labadi STA304 Fall 2021 17 / 51
Ratio Estimation Estimation of R
Question, continue
Dr. Luai Al Labadi STA304 Fall 2021 18 / 51
Ratio Estimation Estimation of R
Ratio Estimation of the Population Ratio R
• Estimated variance of r:
Vˆ (r) =
(
1− nN
)
1
µ2x
s2r
n ,
where
s2r =
∑n
i=1(yi − rxi)2
n− 1 .
• If µx is unknown, we estimate it by x¯.
• Bound on the error of estimation:
B = 2

Vˆ (r).
Dr. Luai Al Labadi STA304 Fall 2021 19 / 51
Ratio Estimation Estimation of R
Discussion
Dr. Luai Al Labadi STA304 Fall 2021 20 / 51
Ratio Estimation Estimation of R
Discussion, continue
Dr. Luai Al Labadi STA304 Fall 2021 21 / 51
Ratio Estimation Estimation of R
Discussion, continue
Dr. Luai Al Labadi STA304 Fall 2021 22 / 51
Ratio Estimation Estimation of R
Discussion, continue
Dr. Luai Al Labadi STA304 Fall 2021 23 / 51
Ratio Estimation Estimation of R
Other Forms of Vˆ (r)
• The estimated variance of r can be written in many forms.
• One that is particularly useful is the one that involves the the cor-
relation coefficient ρ between x and y.
• This correlation ρ can be estimated by
ρˆ =
sxy
sxsy
,
where
sxy =
1
n− 1
n∑
i=1
(xi − x)(yi − y¯)
s2x =
1
n− 1
n∑
i=1
(xi − x)2
s2y =
1
n− 1
n∑
i=1
(yi − y¯)2.
Dr. Luai Al Labadi STA304 Fall 2021 24 / 51
Ratio Estimation Estimation of R
Other Forms of the Vˆ (r)
• Thus,
Vˆ (r) = 1−fn
1
µ2x
(
s2y + r
2s2x − 2rρˆsxsy
)
,
where f = n/N .
• If µx is replaced by x¯, then
Vˆ (r) = 1−fn r
2
(
s2y
y¯2
+ s
2
x
x¯2
− 2ρˆ sxsysxy
)
.
Dr. Luai Al Labadi STA304 Fall 2021 25 / 51
Ratio Estimation Estimation of R
Question
Show that
s2r =
∑n
i=1(yi − rxi)2
n− 1 = s
2
y + r
2s2x − 2rρˆsxsy.
Dr. Luai Al Labadi STA304 Fall 2021 26 / 51
Ratio Estimation Estimation of R
Question, continue
Dr. Luai Al Labadi STA304 Fall 2021 27 / 51
Ratio Estimation Estimation of R
Example
Suppose that 100 people who recently bought houses are surveyed, and
the monthly mortgage payment and gross income of each buyer are de-
termined. Let y denote the mortgage payment and x the gross income.
Suppose that
x¯ = $3100 y¯ = $868
sx = $1200 sy = $250
ρˆ = 0.85 n = 100
(a) Estimate the ratio of the mortgage payment to the gross income
and place a bound on the error of estimation.
(b) Find a 95% confidence interval for the ratio of the mortgage payment
to the gross income.
Dr. Luai Al Labadi STA304 Fall 2021 28 / 51
Ratio Estimation Estimation of R
Dr. Luai Al Labadi STA304 Fall 2021 29 / 51
Ratio Estimation Estimation of R
Dr. Luai Al Labadi STA304 Fall 2021 30 / 51
Ratio Estimation Estimation of R
Exercise- Try it!
The Toyota Company wants to estimate the ratio of the number of man-
hours lost due to sickness of its employees at one of its branches. It has
N = 7000 employees and takes a sample of n = 10 employees and obtains
the following data:
Employee 1 2 3 4 5 6 7 8 9 10
Previous year 15 18 30 25 10 20 16 12 13 2
Current year 14 20 34 18 15 25 20 15 10 5
(a) Plot the data and describe the main features of the plot.
(b) Obtain an estimate of the desired ratio and set up a 95% confidence
interval for it.
Dr. Luai Al Labadi STA304 Fall 2021 31 / 51
Ratio Estimation Estimation of τy
Ratio Estimation of the Population Total τy
• Recall: R = τyτx =
µy
µx
.
• Ratio estimator of the population total τˆY :
τˆy =
∑n
i=1 yi∑n
i=1 xi
(τx) = rτx.
• Note: We do not need to know N or µx but we must know τx.
• Estimated variance of τˆy:
Vˆ (τˆy) = (τx)
2Vˆ (r) = (Nµx)
2
(
1− n
N
) 1
µ2x
s2r
n
= N2
(
1− n
N
) s2r
n
.
Dr. Luai Al Labadi STA304 Fall 2021 32 / 51
Ratio Estimation Estimation of τy
Example 6.2 page 176
In a study to estimate the total sugar content of a truckload of oranges,
a random sample of n = 10 oranges was juiced and weighted. The total
weight of all the oranges, obtained by first weighing the truck loaded
and then unloaded, was found to be 1800 pounds.
Orange 1 2 3 4 5 6
Sugar content: y 0.021 0.025 0.030 0.022 0.033 0.027
Orange weight: x 0.40 0.48 0.43 0.42 0.50 0.46
Orange 7 8 9 10
Sugar content: y 0.019 0.021 0.023 0.025
Orange weight: x 0.39 0.41 0.42 0.44
Estimate τy, the total sugar content for the oranges, and place a bound
on the error of estimation. Take x¯ = 0.435 and sr = 0.0024.
Dr. Luai Al Labadi STA304 Fall 2021 33 / 51
Ratio Estimation Estimation of τy
Dr. Luai Al Labadi STA304 Fall 2021 34 / 51
Ratio Estimation Estimation of τy
Exercise- Try it!
The Toyota Company wants to estimate the ratio of the number of man-
hours lost due to sickness of its employees at one of its branches. It has
N = 7000 employees and takes a sample of n = 10 employees and obtains
the following data
Employee 1 2 3 4 5 6 7 8 9 10
Previous year 15 18 30 25 10 20 16 12 13 2
Current year 14 20 34 18 15 25 20 15 10 5
(a) Assuming that in the previous year the company lost 120,000 man-
hours, obtain a 95% confidence interval for the number of man-hours
which will be lost this year.
(b) Compute again a 95% confidence interval for the number of man-
hours which will be lost this year, but this time assuming that the
data from the previous year were unavailable. Compare the result
with that obtained in (a).
Dr. Luai Al Labadi STA304 Fall 2021 35 / 51
Ratio Estimation Estimation of µy
Ratio Estimation of the Population Mean µy
• Ratio estimator of the population mean µY :
µˆy =
∑n
i=1 yi∑n
i=1 xi
(µx) = rµx.
• Note: We do not need to know τx or N to estimate µy when using
the ratio procedure; however, we must know µx.
• Estimated variance of µˆy:
Vˆ (µˆy) =
(
1− n
N
) s2r
n
Dr. Luai Al Labadi STA304 Fall 2021 36 / 51
Ratio Estimation Estimation of µy
Example
A company wishes to estimate the average amount of money µy paid
to employees for medical expenses during the first three months of the
current calendar year. Average quarterly reports are available in the
fiscal reports of the previous year. A random sample of 100 employee
records is taken from the population of 1000 employees. The sample
results are summarized below. Use the data to estimate µy and place a
bound on the error of estimate.∑100
i=1 yi = 1750
∑100
i=1 xi = 1200
sx = $12.3 sy = $10.4
ρˆ = 0.947 τx = 12, 500
Dr. Luai Al Labadi STA304 Fall 2021 37 / 51
Ratio Estimation Estimation of µy
Dr. Luai Al Labadi STA304 Fall 2021 38 / 51
Ratio Estimation Sample Size Determination
Sample size determination
• The sample size required to estimate R, µy, and τy is given as
n =
Nσ2
ND + σ2
,
where
• D = B2µ2x4 for estimating R,
• D = B24 for estimating µy, and
• D = B24N2 for estimating τy.
• If no past information is available to calculate s2r as an estimate
of σ2, we take a preliminary sample (pilot study) of size n′ and
compute
σˆ2 =
∑n′
i=1(yi − rxi)2
n′ − 1 .
• If µx is unknown, we estimate it by x¯, calculated from the n′ pre-
liminary observations.
Dr. Luai Al Labadi STA304 Fall 2021 39 / 51
Ratio Estimation Sample Size Determination
Exercise 6.4 page 180
A manufacturing company wishes to estimate the ratio of change from
last year to this year in the number of worker-hours lost due to sickness.
A preliminary study of n′ = 10 employee records is made, and the results
are given in next table. The company records show that the total number
of worker-hours lost due to sickness for the previous year was τx = 16300.
Use the data to determine the sample size required to estimate R, the
rate of change for the company, with a bound on the error of estimation
of magnitude B = 0.01. Assume the company has N = 1000 employees.
Employee 1 2 3 4 5 6 7 8 9 10
Previous year,x 12 24 15 30 32 26 10 15 0 14
Current year,y 13 25 15 32 36 24 12 16 2 12∑10
i=1 yi = 187
∑10
i=1 xi = 178 σˆ
2 = 3.46
Dr. Luai Al Labadi STA304 Fall 2021 40 / 51
Ratio Estimation Sample Size Determination
Dr. Luai Al Labadi STA304 Fall 2021 41 / 51
Ratio Estimation Sample Size Determination
More Examples
• Try Example 6.4 page 180 and Example 6.5 page 182.
Dr. Luai Al Labadi STA304 Fall 2021 42 / 51
Regression (Linear) Estimation
Regression Estimation
Regression estimation is used when the relationship between y and x is
linear but the line does not pass through the origin.
• Regression estimator of the population mean µy:
µˆyL = y¯ + b(µx − x¯),
where
b =
∑n
i=1(yi−y¯)(xi−x¯)∑n
i=1(xi−x¯)2 =
∑n
i=1 xiyi−nx y∑n
i=1 x
2
i−nx2
.
• Estimated variance of µˆyL :
Vˆ (µˆyL) = (1− nN )( 1n)
∑n
i=1(yi−(a+bxi))2
n−2 = (1− nN )( 1n)MSE,
where MSE is the mean square error.
• Another form of the estimated variance of µˆyL is:
Vˆ (µˆyL) = (1− nN )( 1n)
∑n
i=1(yi−y¯)2−b2
∑n
i=1(xi−x¯)2
n−2 .
Dr. Luai Al Labadi STA304 Fall 2021 43 / 51
Regression (Linear) Estimation
Exercise 6.9 page 191
A mathematics achievement test was given to 486 students prior to their
entering a certain college. From these students a simple random sample
of n = 10 students was selected and their progress in calculus observed.
Final calculus grades were then reported in the following table. It is
known that µx = 52 for all 486 students taking the achievement test.
Estimate µy for this population and place a bound on the error of esti-
mation.
Student 1 2 3 4 5 6 7 8 9 10
Achievement test score, x 39 43 21 64 57 47 28 75 34 52
Final calculus grade, y 65 78 52 82 92 89 73 98 56 75∑10
i=1 xi = 460
∑10
i=1 yi = 760∑10
i=1 x
2
i = 23634
∑10
i=1 y
2
i = 59816∑10
i=1 xiyi = 36854
∑10
i=1(xi − x)2 = 2474∑10
i=1(yi − y)2 = 2056
Dr. Luai Al Labadi STA304 Fall 2021 44 / 51
Regression (Linear) Estimation
Dr. Luai Al Labadi STA304 Fall 2021 45 / 51
Difference Estimation
Difference Estimation
• Write µy = µx + (µy − µx).
• We can estimate µy − µx. This is called Difference Estimation.
• The difference method of estimating a population mean similar to
the regression method. However, in the difference method we set
b = 1.
• Thus, the difference method is easier to employ than the regression
method.
• It works well when the x values are highly correlated with the y
values and both are measured on the same scale.
Dr. Luai Al Labadi STA304 Fall 2021 46 / 51
Difference Estimation
Difference Estimation
• Difference estimator of the population mean µy:
µˆyD = y¯ + (µx − x¯) = µx + d¯, where, d¯ = y¯ − x¯.
• Estimated variance if µˆyD :
Vˆ (µˆyD) = (1− nN )
(
1
n
) ∑n
i=1(di−d¯)2
n−1 ,
where di = yi − xi.
Dr. Luai Al Labadi STA304 Fall 2021 47 / 51
Difference Estimation
Exercise 6.10 page 194
Auditors are often interested in comparing the audited value of items
with the book value. Suppose a population contains 180 inventory items
with a stated book value of $13,320. Let xi denote the book value and
yi the audit value of the ith item. A simple random sample of n = 10
items yields the following:
Sample 1 2 3 4 5 6 7 8 9 10
Audit value, yi 9 14 7 29 45 109 40 238 60 170
Book value, xi 10 12 8 26 47 112 36 240 59 167
di = yi − xi −1 +2 −1 +3 −2 −3 +4 −2 +1 +3
x = 71.7 y = 72.1 µx = 74.0
∑10
i=1(di − d)2 = 56.43
Estimate the mean audit value of µy by the difference method and esti-
mate the variance of of µˆyD .
Dr. Luai Al Labadi STA304 Fall 2021 48 / 51
Difference Estimation
Dr. Luai Al Labadi STA304 Fall 2021 49 / 51
Comparison between Estimators
Comparison between Estimators
• We have seen four estimators of the population mean µy:
1 SRS Estimator: µˆy = y¯ [Unbiased]
2 Ratio Estimator: µˆ
yR
= rµx [Biased]
3 Linear Estimator: µˆ
yL
= y¯ + b(µx − x¯) [Biased]
4 Difference Estimator: µˆ
yD
= y¯ + (µx − x¯) = µx + d¯ [Unbiased]
• Which one should we use?
The answer is ”It depends.” Different estimators perform differently
under different conditions.
Dr. Luai Al Labadi STA304 Fall 2021 50 / 51
Comparison between Estimators
Summary: See Section 6.8
• ρˆ: the sample correlation coefficient between x and y.
• More efficient means smaller variance.
1 Ratio estimators are more efficient than SRS estimators when |ρˆ| >
1
2 (when the line passes through the origin).
2 Regression estimators are always more efficient than SRS estima-
tors.
3 Difference estimators are more efficient than SRS estimators when
|ρˆ| > 12 × sysx
4 Regression estimators are always more efficient than ratio estimators
(except when line passes through the origin).
5 Regression estimators are better than difference estimators when
b 6= 1.
Dr. Luai Al Labadi STA304 Fall 2021 51 / 51

欢迎咨询51作业君
51作业君

Email:51zuoyejun

@gmail.com

添加客服微信: abby12468