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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 = y¯ x¯ (τx). • Note that τˆy = y¯ x¯ (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 = y¯ x¯ 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
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