Sampling and Sample Sizes

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Sampling and Sample Sizes

• Dr. John T. Drea
• Professor of Marketing
• Western Illinois University

2
Some Basic Sampling Terminology

• Population any complete group that shares a
  common set of characteristics.
• Sample a subset of a larger population
• Population Element an individual member of a
  population.
• Incidence the of the population that qualifies
  for inclusion in the sample.
• Incidence has a direct bearing on cost
• Census an investigation of all elements of a
  population - a total enumeration rather than a
  sample.
• Additional source Zikmund (1999), Essentials of
  Marketing Research

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Procedure for Drawing a Sample
Define the population
Identify the sampling frame
Select a sampling procedure
Determine the sample size
Select the sample elements
Collect data from designated elements
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Define the population

• Who is the population for each project?
• Is it everyone in a radius of Milwaukee, everyone
  who attends Brewer games, or everyone who
  purchased the Brewers Bonus Package?
• Remember, the population is the group you want to
  infer to from the sample - define it carefully so
  it is clear who is in, and who is out.

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Identify the sampling frame

• Sampling frame the list of elements from which
  the sample may be drawn.
• It is sometimes referred to as the working
  population.
• e.g., to sample physicians, my sampling frame
  might be a mailing list from the American Medical
  Association.
• Sample frame error occurs when certain pop.
  elements are skipped or over-represented in the
  sample frame.
• Sample unit the element or group of elements
  selected for the sample (ex if we select every
  fifth train between Chicago and Milwaukee for a
  survey, the fifth train would be the sample unit)

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Select a sampling procedure

• Random sampling error
• the difference between the result of a given
  sample and the result of a census conducted using
  identical procedures.
• It is a statistical fluctuation due to chance
  variations in the elements selected for a sample.
• It is typically a function of sample size.

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Select a sampling procedure

• Systematic (nonsampling) error
• Error resulting from factors not due to chance
  fluctuations
• Includes the nature of a studys design,
  imperfections in execution.
• Ex highly educated people are more likely to
  fill out a mail survey than poorly educated ones.
• Ex doing a mall intercept at a grocery store on
  Saturday morning may underrepresent seniors.

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Select a sampling procedure Probability and
Nonprobability Samples

• Probability sample every member of the
  population has a known, non-zero chance of being
  included in the sample.
• Nonprobability sample the probability of any
  particular member of a population being chosen is
  unknown.

There are no appropriate statistical techniques
for measuring random sampling error from a
nonprobability sample. Projecting the data
beyond the sample is statistically inappropriate.
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Selecting a sampling procedure Nonprobability
samples

• Convenience sample obtaining those people/units
  that are most conveniently available (ex college
  students)
• Judgment sample selected by an experienced
  researcher based on judgment about appropriate
  characteristics of the sample members.
• Quota sample ensures that various subgroups of
  the population will be represented (ex setting a
  quota of 50 people from Milewaukee and 50 from
  outside Milwaukee)
• Quota samples have a tendency to produce people
  that can be easily found.

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Selecting a sampling procedure Probability
samples

• Simple random sample a procedure that assures
  that each element in the population of an equal
  chance of being included in the sample.
• Systematic sample a starting point is selected
  at random and then every nth number on the list
  is selected.
• Need to be careful of periodicity (ex collecting
  retail information every 7th day, or once per
  month)
• Multistage area sample involves a combination of
  two or more probability sampling techniques.
• Ex randomly choosing counties within a state,
  then randomly choosing census blocks within each
  county, then interviewing everyone in that block.

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Basic statistics underlying sample size
determination

• Central Limit Theorem As sample size, n,
  increases, the distribution of the mean, X, of a
  random sample taken from practically any
  population approaches a normal distribution.
• Sampling distribution If you took repeated
  samples from the same population, the sampling
  distribution is the distribution of these sample
  means.
• Standard error of the mean is a measure of the
  standard deviation of the sampling distribution -
  it is the standard deviation of the population
  divided by the square root of the sample size.

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Estimating sample size

• To estimate a sample size, a researcher must
• estimate the standard deviation of the population
  (a good rule of thumb is 1/6th of the range)
• make a judgment about allowable amounts of error
• determine a confidence interval
• Once these are known, the formula for calculating
  sample size is

where... Z standardized value that corresponds
to the confidence level S sample standard
deviation E acceptable magnitude of error
Z2S2 E2
n
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Estimating sample size

• Suppose a researcher studying annual expenditures
  on lipstick wishes to have a 95 confidence
  interval (Z1.96) and a range of error (E) of
  less than 2, and an estimate of the standard
  deviation is 29.

(1.962292)/22 808
If we change the range of acceptable error to 4,
sample size falls
n 202
Source Zikmund (1999), Essentials of Marketing
Research
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Estimating sample size

• Suppose you wanted to estimate the same size for
  a survey which contains the following question
• What is your overall attitude towards Miller
  Park?
• Very Good 7 6 5 4 3 2 1 Very Poor
• The range of acceptable error is 0.1 points, the
  confidence level is 95, and the estimated
  standard deviation is 1/6 of the range.

(1.96212)/0.12 a sample size of 384
If you increase the acceptable error to 0.2, the
sample size drops to n 96!
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Sample size determination when a proportion is
present
Sp estimate of the std. error of the
proportion p proportion of successes q (1 -
p), or the proportion of failures
pq n
Sp
Suppose that 20 of a sample of 1,200 recall
seeing an ad.
(0.2)(0.8) 1200
Thus, the population proportion who see the ad is
between 17.8 and 22.2, w/ 95 confidence.
0.0115
Sp
Confidence interval p ZclSp .2
(1.96)(0.0115) .2 .022
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Sample size determination when a proportion is
present (cont.)
To determine the sample size for a proportion, we
need to know or estimate the following Z2cl
square of the confidence level in standard error
units (i.e., typically 1.962, or
3.8416) p estimated proportions of successes q
(1 - p), or the proportion of failures E2
square of the maximum allowance for error
We insert this information into the following
formula
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Sample size determination when a proportion is
present (cont.)
Example We estimate that 60 of respondents will
describe the Brewer Bonus Package as positive,
with a confidence level of 95, and the
allowable error is 4.
(1.96)2(.6)(.4) 0.042
(3.8416)(0.24)/0.0016 n 576
If we assume a 70/30 split and if we increase the
maximum allowable error to 5, what would be n?
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Overall Estimating sample size when a
proportion is the characteristic of interest

• When the split is hypothesized to be 70/30 (95
  CI)
• 1 7,939
• 2 2,009
• 3 895
• 5 322

• When the split is hypothesized to be 85/15 (95
  CI)
• 1 4,850
• 2 1,222
• 3 544
• 5 306

Small population sizes typically require a
slightly smaller sample size If population
10,000, the 70/30 split sample sizes would be
14,465 21,678 3823 and 5313