Chapter 6

Ratio, Regression and Difference Estimation

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!!– x/y = Nx/Ny = x/y

• Therefore, we will use R = ratio of the means

6.3 Ratio estimator for SRS

• Calculating an estimate and bound on r:– r = ybar/xbar

• Estimated variance of r:– (1-n/N)*(1/x

2)*(sr2/n)

– where sr2 = i(yi-rxi)2/(n-1)

– Use xbar to estimate x when necessary

Ratio estimator of total and mean

• For population total:– haty = rx

• Estimated variance of haty

– N2(1-n/N) sr2 /n (where sr

2 is defined previously)

• For population mean:– haty = rx

• Estimated variance of haty

– (1-n/N) sr2 /n (where sr

2 is defined previously)

Examples

• 6.1: mean of x (square-foot) = 0.558333 mean of y ( volume) = 11.83333haty = (11.8333/.558333)*75=1589.552204sr

2 = 1.750299Estimated variance = 250^2*(1-12/250)*

1.750299/12 = 8678.566Bound=2*sqrt(8678.566) = 186.3176Without fpc (don’t really need it since 1-n/N =

0.952), B = 190.957

More examplesRcode:x=dat$Basal.ary=dat$Volumeybar=mean(y)xbar = mean(x)rhat =(ybar/xbar)t.hat.y = rhat*75n=length(dat$Volume)s.r.2 = sum((y-rhat*x)^2)/(n-1)var.that = (250^2)*(1-12/250)*s.r.2/12___________________________________________________• 6.2: that =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

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

• Coefficient of variation: CV(x) = sx/xbar, where sx is the standard deviation of x. It is desired that the CV is lower than 1. Find the CV for x and the CV for y in problem 6.6.

6.4 Sample size estimation

• n = N2/(ND + 2)where we can estimate 2 as

i(yi-rxi)2/(n’-1) with n’ a small preliminary sample

With D=B2x2/4 for estimating R

D = B2/4 for estimating y

D = B2/(4N2) for estimating y

Do problem 6.13