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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
• 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

            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
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:
               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
Evaluation of Traditional Approach
Ideally, a method for determining sample
size would be:

   1. Logically complete
   2. Adaptable to a wide range of sampling
   3. Simple to use

• Traditional approach is rated low for logical
completeness and high for both adaptability and

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