Elementary Survey Sampling 7 E Scheaffer Mendenhall Ott

Elementary Survey Sampling, 7 E, Scheaffer, Mendenhall, Ott and Gerow STT 350: SURVEY SAMPLING DR. CUIXIAN CHEN Chapter 6: Ratio, Regression and Difference Estimation 1

6. 2 Ratio Estimators: examples 2 Wholesale price paid for oranges in large shipments is based on sugar content in load. Try to estimate exact sugar content. One method: first estimate mean sugar content per orange, my, and then to multiply by # of oranges N in load. So randomly sample n oranges from load to determine sugar content y for each. Average of these sample measurements, yl, y 2, . . . , yn, will estimate my; will estimate total sugar content for load, ty. Unfortunately, this method is not feasible because it is too time-consuming and costly to determine N (i. e. , to Elementary Survey Sampling, 7 E, Scheaffer, Mendenhall, Ott and Chapter 6 count total # of oranges in the load). Gerow

6. 2 Ratio Estimators: examples 3 We can avoid knowing N by noting two facts: First, sugar content of an individual orange, y, is closely related to its weight x; Second, ratio of total sugar content ty to the total weight of the truckload tx is equal to the ratio of the mean sugar content per orange, my, to the mean weight mx. my/mx= Nmy/Nmx= ty / tx Elementary Survey Sampling, 7 E, Scheaffer, Mendenhall, Ott and Gerow Chapter 6

6. 2 Ratio Estimators: examples 4 Ratio estimation is used in the analysis of data from many important and practical surveys used by government, business, and academic researchers. For instance, the CPI is actually a ratio of costs of purchasing a fixed set of items of constant quality and quantity for two points in time. Currently, the CPI compares today’s prices with those of the 1982– 1984 period. The CPI is based, in part, on data collected every month or every other month from approximately 24, 000 establishments (stores, hospitals, filling stations, and so on) selected from many areas around the country. The CPI is used mainly as a measure of inflation (see Chapter 1). Elementary Survey Sampling, 7 E, Scheaffer, Mendenhall, Ott and Gerow Chapter 6

6. 2 Ratio Estimators Looking at two measurements of a population, call these x and y (we will only focus on the SRS sampling strategy) Note that the ratio of population means is the same as the ratio of population totals!! my/mx= Nmy/Nmx= ty / tx Therefore, we will use R = ratio of the means = ratio of the totals

6. 3 Ratio estimator for SRS 6 Elementary Survey Sampling, 7 E, Scheaffer, Mendenhall, Ott and Gerow Chapter 6

Eg 6. 1, page 173 7 Elementary Survey Sampling, 7 E, Scheaffer, Mendenhall, Ott and Gerow Chapter 6

Eg 6. 1, page 173 An essential rule of data analysis is to plot the data first. A scatter plot of 2002 versus 1994 data shown in Fig 6. 1. if For ratio estimation to work well: Strong, positive linear trend here 8 is important. None of data pts deviate sharply from the linear pattern. Elementary Survey Sampling, 7 E, Scheaffer, Mendenhall, Ott and Gerow Chapter 6

Eg 6. 1, page 173 9 Elementary Survey Sampling, 7 E, Scheaffer, Mendenhall, Ott and Gerow Chapter 6

Rewrite estimated variance of r using coefficient of variation (CV) 10 Elementary Survey Sampling, 7 E, Scheaffer, Mendenhall, Ott and Gerow Chapter 6

Ratio estimator of total 11 Elementary Survey Sampling, 7 E, Scheaffer, Mendenhall, Ott and Gerow Chapter 6

Eg 6. 2, page 176 12 Elementary Survey Sampling, 7 E, Scheaffer, Mendenhall, Ott and Gerow Chapter 6

Ratio estimator of Population Means 13 Elementary Survey Sampling, 7 E, Scheaffer, Mendenhall, Ott and Gerow Chapter 6

Eg 6. 3, page 178 14 Elementary Survey Sampling, 7 E, Scheaffer, Mendenhall, Ott and Gerow Chapter 6

Eg 6. 3, page 178 15 Elementary Survey Sampling, 7 E, Scheaffer, Mendenhall, Ott and Gerow Chapter 6

Ratio estimator of total and mean For population total: thaty = rtx Estimated variance of thaty N 2(1 -n/N) For population mean: mhaty sr 2 /n (where sr 2 is defined previously) = rmx Estimated variance of mhaty (1 -n/N) sr 2 /n (where sr 2 is defined previously)

17 Elementary Survey Sampling, 7 E, Scheaffer, Mendenhall, Ott and Gerow Chapter 6

EX 6. 1, page 204 18 Elementary Survey Sampling, 7 E, Scheaffer, Mendenhall, Ott and Gerow Chapter 6

EX 6. 1, page 204 mean of x (square-foot) = 0. 558333 mean of y ( volume) = 11. 83333 thaty = (11. 8333/. 558333)*75=1589. 552204 sr 2 = 1. 750299 Estimated variance = 250^2*(1 -12/250)* 1. 750299/12 = 8678. 566 Bound=2*sqrt(8678. 566) = 186. 3176 Without fpc (don’t really need it since 1 -n/N = 0. 952), B = 190. 957

EX 6. 1: R codes R-code: ## After download the Excel file as EX 6. 1. csv to the desktop dat=read. csv(file. choose()) attach(dat) x=Basal. ar; y=Volume xbar=mean(x); ybar=mean(y) r=ybar/xbar Tau_hat=75*r Sr 2=var(y-r*x) N=250; n=12 B=2*sqrt(N^2*(1 -n/N)*Sr 2/n) __________________________ For EX 6. 2 rhat =250* 11. 83333 = 2958. 333 Variance for that=250^2*(1 -12/250)*26. 87879/12 = 133274 Bound = 730. 1342 Without fpc, bound = 748. 3146

EX 6. 2, page 204 21 Elementary Survey Sampling, 7 E, Scheaffer, Mendenhall, Ott and Gerow Chapter 6

EX 6. 6, page 205 22 Elementary Survey Sampling, 7 E, Scheaffer, Mendenhall, Ott and Gerow Chapter 6

More examples Do problem 6. 6 mean(x)=16. 47273, mean(y) = 16. 84545, so rhat = 16. 84545/16. 47273=1. 022627; sr 2=0. 2049424

6. 4 Sample size estimation n = Ns 2/(ND + s 2) where we can estimate s 2 as Si(yi-rxi)2/(n’-1) with n’ a small preliminary sample With D=B 2 mx 2/4 for estimating R D = B 2/4 for estimating my D = B 2/(4 N 2) for estimating ty

6. 4 Sample size estimation for SRS

Eg 6. 4, page 180 26 n’ Elementary Survey Sampling, 7 E, Scheaffer, Mendenhall, Ott and Gerow Chapter 6

Eg 6. 5, page 182 27 Elementary Survey Sampling, 7 E, Scheaffer, Mendenhall, Ott and Gerow Chapter 6

Eg 6. 6, page 184 28 Elementary Survey Sampling, 7 E, Scheaffer, Mendenhall, Ott and Gerow Chapter 6

EX 6. 13, page 207 29 Elementary Survey Sampling, 7 E, Scheaffer, Mendenhall, Ott and Gerow Chapter 6