Nn EY./X, = SLOPE vi / / The approximate confidence interval for R, (assuming R, 1s normally distributed) is Ry + 2NV'(R,) , where z_ (3) is the value of the standard normal deviate corresponding to the appropriate probability a. The assumption of normality is reason- ably good_if the coefficients of variation of both ¥ and x, sy/(/n Y) and 8y/(/n X), respectively, are less than 0.1 and the number of obser- vations is greater than 30 (Cochran, 1963, p. 164). If these conditions do not apply, V'(R,) (equation 2) will tend to be ton small. Cochran (1963, p. 164) gives an alternative method for computing confidence limits of Ry that should be more accurate than the above method (equation 3), but is also more difficult to compute. When the sample size is small, the t distribution rather than the standard normal is recommended for calculating the confidence interval. When the straight line is restricted to pass through the origin, as in our case to estimate an average ratio, and both X and Y¥ are normally distributed, then the maximum likelihood estimate of the slope (common X FIGURE 5. ratio} is also Ry = Y/X (Creasy, 1956). Ratio Estimation When Var(¥) is Proportional to X?, Recall that these estimators are based on the assumption that X is known with little error. In that case, each estimate (R,, R,, R) is statistically unbiased and the choice between them depends on whether the variance of Y is constant, proportional to X, or proportional to X* (Snedecor and Cochran, 1967 pp. 166-170). But what 1f X is not "constant", the usual case in transuranic field studies? For example, Y and X might be 229-2"0py and 24! 4m con- centrations, respectively, in the same aliquot of soil. Cochran (1963, p. 161) shows that the ratio estimate R, = Y/X, even though Y and KX are both random variables, is an unbiased estimate of the ratio of the true means if the regression of Y on X is a straight line through the origin.Hence, regardless of whether X is constant or variable, if the data forms a straight line through the origin, Y/X is an unbiased estimate of the ratio of true means. However, when X is a variable, the vartance of R, is only approximate and equation (1) may not be appropriate for estimating Var(R,). Cochran's (1963, p. 31) approximate variance for R, is VICR,) = DT CEH R)X,)?/n(n-1) x? i=l n — . The variance of the observa- tions from the regression line (line with slope Rj) was approximated by Creasy (1956) as n (¥,-Y)? - n n Y ¥°x x n-1 n . (X,-X) ¥ (4). The variance of Ro for this case has not been completely derived, but Ricker (1973) recommends n V"(Ry) = ex/ (X,-%)? i=l (5) (where 32. is computed using equation 4) as being as good an approxi~ mation a’ She computationally involved method proposed by Creasy. The confidence interval is then given by (2) R2 + t vv" (Ro) 608 609