sample size by jpl7986

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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
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

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
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

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