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Sample size can be determined by iteration using the following equation (Zar 1996, 107): n = (t2 x S2 )  d2 = (t x S  d )2 In this equation, n is the estimated sample size; t is Student’s t with n – 1 degrees of freedom for a particular alpha; S is an estimated standard deviation (may be the sample standard deviation of an initial sample); and d is the half width of the desired (1 – alpha) 100 percent confidence interval. If you are estimating biomass of vegetation and desire to estimate the true population mean with a 95 percent confidence interval no wider than 200 kg/ ha, then d in the equation would be 100 kg/ha, and the t-value for  = 0.05 would be used for t. If the objective is to obtain an adequate sample that will detect a 10 percent change in vegetation parameters, such as biomass or cover, from one sampling period to the next, (kx -)2 can be used for d, where k = 0.10 and x - is the mean of the presample values (Bonham 1989). Example: A researcher wants to determine carrying capacity of a ranch for white-tailed deer. The researcher obtains an initial sample consisting of twenty 1 m2 sampling frames in which the standing crop of forbs and browse are clipped, oven dried, and weighed. The mean weight is 1,000 kg/ha, and the sample standard deviation is 800 kg/ha. The researcher desires a 90 percent confidence interval with width of 200 kg/ha to estimate the population mean. The following is the initial equation to obtain an estimate of the adequate sample size: n = (1.7292 x 8002 )  1002 = (1.729 x 800  100)2 = 191 In this equation, t = 1.729 with 20 – 1 = 19 degrees of freedom, for an initial sample size estimate of 20, and alpha = 0.10 (the final estimated n obtained does not depend on what initial estimated n is used to determine the degrees Appendix 3 Determining Adequate Sample Sizes 252 verso runninghead ▼ of freedom for Student’s t). Repeating the equation for t =1.653 with 190 degrees of freedom, the estimated n is 175. If the equation is repeated for t = 1.654 with 174 degrees of freedom, the estimated n is again 175 (i.e., the iterative solution for n is 175). Therefore, to be at least 90 percent confident that the resulting sample mean differs from the population mean by less than 10 percent of the preliminary sample mean (d = 0.1 x 1,000 = 100), an estimated sample size of at least 175 is required. 252 appendix 3 ...

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