Consider this
question:
 "Does it make any
practical difference how large a sample with repect to
the population from which it was drawn?"
Consider this
rejoinder:
 " Do you mean to sit
there, with that smug smile on your face, and tell me
that to make equally accurate estimates for the
population of Evanston, Illinois (population around
80,000), and for the United States (poplation around
280,000,000) you would require the same sample
sizes?"
Consider the quandry of
the apoplectic questioner:
 "How can this be
so?"
 The appropriate
reply:
 "Take a course in
statistics and find out."
 You are about to find
out.
Review of point estimates, confidence intervals, and
sample sizes in predicting to population
proportions
 A point estimate
is the prediction of the population proportion based on
the observed proportion in a sample
 If the sample shows
that 33% of the population favors abolishing the death
penalty
 The researcher
estimates that .33 of the population favors the death
penalty.
 A confidence
interval is a range of values around a point
estimate that expresses the probability that the interval
contains the population parameter between the upper and
lower limits of the interval.
 A confidence interval
is computed by adding and subtracting standard error
units around the mean.
 A confidence interval
is always associated with a level of confidence
that the estimate will be correct.
 Example:
 A point estimate
and confidence interval in a 11/15/85 article on
the Geneva summit in the New York Times,
based on a NYT/CBS telephone poll of 1,659 adults
in 48 continental states.
 Question:
"U.S. should try to reduce tensions with
Russians"
 Point
estimate: 49% agreed, 41% disagreed
 Statement on
confidence interval:
 "In theory, in
19 cases out of 20 the results based on such
samples will differ by no more than 3 percentage
points in either direction from what would have
been obtained by interviewing all adult
Americans. The error for smaller subgroups is
larger. For example, the margin of sampling
error for blacks is plus or minus 8 percentage
points, and for college graduates it is plus or
minus 5 percentage points."
 Computing a
confidence for a population parameter from
these data:
 Sampling error for
continuous data:
 Sampling error for
proportions:
 2 times .012
= .024 (or almost 3 percentage points)
 Confidence
interval:
 49% + or  3% =
46% to 52%
 Thus, we are
95% confident that the population parameter (49%
figure) lies between 46% and 52%.
What factors figure
into the confidence interval estimate of a population
mean?
 Variability in the
sampling distribution of the mean, which is based
on
 variation of values
in the population:
 size of the
sample:
 both factors are
combined in the standard error of the
mean:
 Degree of confidence in
making the estimate, which is decided on by the
researcher.
 Level of confidence:
e.g., .95, .99, .999
 Complement of the
alpha value: e.g., .05, .01, .001
 Note that the
proportion that the sample is of the population is
not a major factor in the estimate's accuracy.
 This is
counterintuitive (i.e., it goes against
reason).
 To explain this
requires discussion of sampling with and without
replacement.
There are two forms of
simple random sampling:
 Sampling with
replacement
 After each case is
selected, it is replaced in the sample.
 If replacement is not
done with small populations (e.g., a deck of 52
cards), probability calculations can be materially
affected.
 Sampling without
replacement.
 If the population is
very large (e.g, thousands of cases), probability
calculations will not be materially
affected.
 The population
decreases by one each time a case is
drawn.
 This is because
accuracy of inferences from samples to populations
 is due primarily
to the amount of information (i.e., the size
of the sample)
 and not the
proportion of information (i.e., the percent
the sample is of the population).

Correction for sampling without replacement
(see Leslie Kish, Survey Sampling (New York:
John Wiley, 1965), pp. 4345.
 Recall that the accuracy
of a sample depends on the size of its standard
error,
=
 If you can
reduce the standard error, you can
increase the accuracy of the
estimate
 You can't manipulate
the population variance, so a researcher cannot reduce
the standard error that route.
 But the research can
increase the sample size, which does reduce standard
error.
 What about the
proportion that the sample is of the
population? Need the researcher consider
that?
 In truth, the s.e. of
the mean can be lowered considering that respondendents
are drawn for samping without replacement
 The researcher is
entitled to multiply the standard error by a
fractional correction factor
 The correction
factor is always less than 1.
 Therefore, the
s.e. is reduced by being multiplied by a fator less
than 1.
 This correction
factor is based on p (p = the proportion the
sample is of the population)

 where is
the correction factor
 Assume the sample
were 20% of the population
 Then the
correction factor, ,
would equal the square root of 1.2 = sr rt of .8 =
.895
 So you would be
entitled to reduce the s.e. by multiplying it by
.895
 Unfortunately,
this is not much of a reduction.
 Assume the sample
were 10% of the population
 Then the
correction factor, ,
would equal the square root of 1.1 = sr rt of .9 =
.95
 So you would be
entitled to reduce the s.e. by multiplying it by
.895
 Unfortunately,
this is not much of a reduction.
 So this correction
factor has little effect unless a sample if 20% or more
of the population, which is rare.
 Evanston has a
population of 80,000.
 A 20% sample needs
16,000 respondents for the correction factor to "kick
in" at .895.
 This huge sample
only slightly reduces the s.e. and thus only
slightly increases accuracy in the
estimate.
 A 10% sample needs
8,000 respondents for the correction factor to "kick
in" at .95.
 This very large
sample hardly affects the s.e. at all.
 Essentially, neither
sample reduces the s.e. in any substantial
way
 Hence, the proportion
that the sample is of the population has little
practical effect on its accuracy .
 Most samples do not
approach proportions that would produce a meaningful
correction factor.
 Hence, it is usually
ignored in computing the s.e. of the
mean.
 Ergo: It is
unimportant what proportion the sample is of the
population .
 What is important
is the raw size of the sample.
 Period.
