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Sample Size Chapter 13 Two Ways to Classify Techniques • Fixed or sequential sampling • Sample size logic is based on traditional (such as Neyman-Pearson) or Bayesian inferential methods Sample Size Affected by Practical Issues • Time pressure – Need for immediate results can reduce sample size • Cost constraint – Lower funds can lower sample size • Study objective – A decision not requiring great precision can make do with smaller sample size • Data analysis procedures – Single-variable analysis requires smaller sample size than two or more variable analyses Fixed vs. Sequential Sampling • Fixed-size sampling involves deciding the number of items in advance; size is chosen to achieve balance between sample reliability and sample cost • Sequential sampling does not pre-select the number of items; a decision rule is set up in advance that includes the choice of stopping the sampling process or collecting more information before making a final decision Sequential Sampling • We take a sample without knowing the sample size before we start. • In classical sampling we have two possibilities (accept or reject the null hypothesis). In sequential sampling we have three or more possibilities (accept, reject, continue or uncertainty about the null hypothesis). • When you are in the uncertainty condition you usually continue to sample. In Krebs (1989), he presents an example of rainbow trout put in the effluent coming from a coal processing plant. We are know from previous studies that the mean survival time should be less than 36 hours. We follow these steps: 1. Set up the alternatives H0: mean survival time <= 36 hours H1: mean survival time >= 40 hours 2. Determine the acceptable risk of type I and II error (alpha and beta error) 3. Estimate the threshold It is possible that by following this process correctly, the needed information about various ecological situations may be acquired with fewer samples than a classical fixed size procedure will require. Krebs, C. J. Environmental Methodology (2nd Ed., 1999, p. 304). Sampling Distribution • The probability distribution of a specified sample statistic (e.g. sample mean) for all possible random samples of a given size n drawn from the specified population Standard Errors • The standard error of the statistic is the standard deviation of the specified sampling distribution Sampling Distribution Properties • Arithmetic mean of the 1) sampling distribution of the mean ( ) or 2)proportion (p) for any given size sample equals parameter values μ (population mean) and ∏ (population proportion regarding some attribute) • The sampling distribution of the means of random sample will tend toward the normal distribution as sample size n increases • The finite multiplier is not required unless the sample contains an appreciable fraction (ex. >10%) of the population. By doing so, random sampling error can be reduced by 5% N–n N = total number of items in the universe N-1 n = number of items in sample Standard Error Formulas σ = σ σ = π (1-π) n n Mean Proportion Methods of Estimating Sample Size 4 Traditional Approaches: 1. Judgmentally based criterion 2. Minimum cell size needed 3. Budget 4. Specifying desired precision, then applying standard error formula Confidence Hypothesis Interval Testing Approach Approach Confidence Interval Approach 1. Specify the amount of error (E) that can be allowed. 2. Specify the desired level of confidence. 3. Determine the number of standard errors (Z) associated with the confidence level. 4. For sample mean, estimate the standard deviation of the population; for sample proportion, estimate the population proportion (π). 5. Calculate the sample size using the formula for the standard error of the mean. 6. Solve for n. 7. If resulting sample size represents a significant proportion of the population, the finite multiplier is required. Confidence Interval Approach For example, suppose that we wished to set up a 95 percent confidence interval around the sample mean of 2.6 pints. We would proceed by first computing the standard error of the mean: From Table A.1 in Appendix A we find that the central 95 percent of the normal distribution lies within 1.96 Z variates (2.5 percent of the total area is in each tail of the normal curve). With this information we can then set up the 95-percent confidence interval as and we note that the 95-percent confidence interval ranges from 2.54 to 2.66 pints. Thus, the preassigned chance of finding the true population mean to be within 2.54 and 2.66 pints is 95 percent. More Than One Interval Estimate From the Same Sample • To adhere to allowable error and confidence levels requires one to take the largest sample size calculated • When in doubt regarding error levels or standard deviations, using devices such as the nomograph provides sufficient accuracy Hypothesis-Testing Approach • Type I error (alpha risk) • Type II error (beta risk) • Based on two hypotheses: – H 0 : the null hypothesis – H 1 : the alternative hypothesis Action H0 is true H0 is false Accept H0 No error Type II error (β) Reject H0 Type I error ( ) No error Checklist Involving Means 1. Specify values for the null and alternate hypotheses in terms of population means. 2. Specify the allowable probabilities of Type I and Type II errors. 3. Determine number of standard errors associated with each of the error probabilities. 4. Estimate the population standard deviation. 5. Calculate the sample size that will meet the alpha and beta error requirements. 6. Solve for n. Checklist Involving Proportions 1. Specify values for the null and alternate hypotheses in terms of population proportions. 2. Specify the allowable probabilities of Type I and Type II errors. 3. Determine number of standard errors associated with each of the error probabilities. 4. Calculate the desired sample size n from the formula: 2 Zα π (1 – π ) + Z β π (1 – π ) 0 0 1 1 n = π 1 - π0 Determining Sample Size for Other Probability – Sample Designs • Sizes for other types of random-sample designs (systematic, stratified, cluster, area, multistage) uses the same general procedures, but the formulas for the standard errors differ Advantages of Stratified Sampling Over Simple Random Sampling • For the same level of precision, one would need a smaller sample size in total, and this leads to lower cost • For the same total sample size, one would gain a greater precision for any estimates made Evaluation of Traditional Approach Ideally, a method for determining sample size would be: 1. Logically complete 2. Adaptable to a wide range of sampling situations 3. Simple to use • Traditional approach is rated low for logical completeness and high for both adaptability and simplicity