Lindley et al.: Application of statistical power analysis to recovery of an endangered species 



761 



type-I error rate a. This approach differs from the conven- 

 tional approach to hypothesis testing in which a is fixed 

 at, say, 0.05, and the resulting power-level is either toler- 

 ated, or the sample size n is increased sufficiently to pro- 

 vide an acceptable level of power Increasing n is not an 

 option in this instance, because in any given year of appli- 

 cation the sample size of the (r^l data set will be fixed, and 

 a procedure is required that can be applied in each and 

 every year. In the context of monitoring winter chinook 

 salmon, a failure to reject //„ may be used to justify "busi- 

 ness as usual." By specifying the power n of the test in 

 advance, resource managers will know that if the mean 

 growth rate is falling seriously short of the goal, they will 

 be able to detect this with specified probability n. 



Calculation of a given n 



In this section, we formulate the relation between n and 

 a. This relation allows one to determine what values of a 

 should be used for the test in order to achieve a specified 

 level of power k. 



With the previously stated distributional assumptions, 

 t has a central ^-distribution if p = p,,,^/, with cumulative 

 distribution function (cdf) T and inverse cdf 7" '. If p ?i 

 Penal- ' h^^ ^ noncentral ^distribution with cdf T^, inverse 

 cdf T^ ', and noncentrality parameter 



 Pfi.,< 



;/V" 



(2) 



which is the difference between p and p , in standard 

 error units (Johnson et al., 1994). Given a particular criti- 

 cal value t^., the associated type-I error rate and power of 

 the ^test are 



a = Pr{reject//„|//„true} = T(t^ ), 

 n = Pr{reject//,-,|//„false} = T,{t, ), 



(3) 



(4) 



a by definition being the largest value of T^l/^ I under //„, 

 which occurs at p = p i where T]^,,!?,.) = T(t^). Solving 



