100 25 Again the standard error is less than that for Y 25 sy z SEY 1 0522 100

# 100 25 again the standard error is less than that for

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100 25 Again, the standard error is less than that for Y: 25 sy z SE[Y] = 1 - 0.522. 100 25 We expect regression estimation to increase the precision in this example because the
SCI 2-, ACC, :.d ,fl L]. ''' 3.3 Estimation in Domains 11 variables photo and field are positively correlated (r = 0.62). To estimate the total number of dead trees, use tvrcg = (100)(11.99) = 1199; SE[t3.reg] _ (100)(0.408) = 40.8. An approximate 95% confidence interval for the total number of dead trees is given by 1199 ± (2.07)(40.8) = [1114, 12831. Because of the relatively small sample size, we used the t-distribution percentile (with n - 2 = 23 degrees of freedom) of 2.07 rather than the normal distribution percentile of 1.96. 3.2.2 Difference Estimation Difference estimation is a special case of regression estimation, used when the inves- tigator "knows" that the slope B1 is 1. Difference estimation is often recommended in accounting when an SRS is taken. A list of accounts receivable consists of the book value for each account-the company's listing of how much is owed on each account. In the simplest sampling scheme, the auditor scrutinizes a random sample of the accounts to determine the audited value-the actual amount owed-in order to estimate the error in the total accounts receivable. The quantities considered are yj = audited value for company i x, = book value for company i. Then, y - .x is the mean difference for the audited accounts. The estimated total difference is t,. - tx = N(y -z); the estimated audited value for accounts receivable is tydiff = tx + (tp - tx) Again, define the residuals from this model: Here, e1 = yt - xi. The variance of tvdiff is V(tydiff) = V [tx + (t,. - tx)] = V (te), where t, = (N/n) Y'tcs e;. If the variability in the residuals ei is smaller than the variability among the yj's, then difference estimation will increase precision. Difference estimation works best if the population and sample have a large fraction of nonzero differences that are roughly equally divided between overstatements and understatements, and if the sample is large enough so that the sampling distribution of (y -x) is approximately normal. In auditing, it is possible that all audited values in the sample are the same as the corresponding book values. Then, y = x, and the standard error of t,, would be calculated as zero. In such a situation, where most of the differences are zero, more sophisticated modeling is needed. 3.3 Estimation in Domains Often we want separate estimates for subpopulations; the subpopulations are called domains or subdomains. We may want to take an SRS of visitors who fly to New York City on September 18 and to estimate the proportion of out-of-state visitors who
CD. R>> X12 .22 18 Chapter 3: Ratio and Regression Estimation intend to stay longer than I week. For that survey, there are two domains of study: visitors from in-state and visitors from out-of-state. We do not know which persons in the population belong to which domain until they are sampled, though. Thus, the number of persons in an SRS who fall into each domain is a random variable, with value unknown at the time the survey is designed.

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