IV.POPULATION DISTRIBUTION, SAMPLE DISTRIBUTION, AND SAMPLING
DISTRIBUTION
Three additional types of distribution must be defined:
1. population distribution
2. sample distribution
3. sampling distribution
A frequency distribution of the population elements is called a population distribution.
The population distribution has its mean and standard deviation represented by the Greek
letters
m
and
s
.
A frequency distribution of a sample is called the sample distribution.
The sample mean is designated with
X
, and the sample standard deviation is designated S.
However, we must now introduce another distribution: the sampling distribution of the
sample mean.
A sampling distribution is a theoretical probability that shows the functional relation
between the possible values of some summary characteristic of n cases drawn at random and
the probability (density) associated with each value over all possible samples of size n from a
particular population.
The sampling distribution’s mean is called the expected value of the statistic.
The expected value of the mean of the sampling distribution is equal to
m
.
The standard deviation of the sampling distribution is called the standard error of the mean
(
S
X
) and is approximately equal to
s n
.
Exhibit 17.13 shows the relationship among a population distribution, the sample distribution,
and three sampling distributions of varying sample size.
As sample size increases, the spread of the sample mean around decreases (i.e., larger
samples will have a skinnier sampling distribution).
V. CENTRAL-LIMIT THEOREM
The central-limit theorem states: As the sample size, n, increases, the distribution of the
mean,
X
, of a random sample taken from practically any population approaches a normal
distribution (with a mean
m
and a standard deviation,
s n
).
The central-limit theorem works regardless of the shape of the original population
distribution (see Exhibit 17.14).
This theoretical knowledge about distributions can be used to solve two very practical
business research problems:
1. estimating parameters
2. determining sample size
VI. ESTIMATION OF PARAMETERS
Point Estimates
Our goal in using statistics is to make an estimate about the population parameters.
The population mean,
m
, and standard deviation,
s
, are constants, but in most instances
of business research they are unknown.
To estimate the population values, we are required to sample.
Example: To estimate the average number of people participating in racquetball in one
week we may take a sample of 300 racquetball players throughout the area where our
researcher is thinking of building club facilities. If the sample mean,
X
, equals 2.6 days
per week, we may use this figure as a point estimate.
This single value, 2.6, is the best estimate of the population mean.
However, we would be extremely lucky if the sample estimate were exactly the
same as the population value.
A less risky alternative would be to calculate a confidence interval.
Confidence Intervals
A confidence interval estimate is based on the knowledge that
m = X ±
a small
sampling error.
After calculating an interval estimate, we can determine how probable it is that the
population mean will fall within this range of statistical values.
In the racquetball example the researcher, after setting up a confidence interval, would be
able to make a statement such as “with 95 percent confidence, I think that the average
number of days played per week is between 2.3 and 2.9.”
The researcher has a certain confidence that the interval contains the true value of the
population mean.
The crux of the problem for the researcher is to determine how much random sampling
error to tolerate.
The confidence level is a percentage or decimal that indicates the long-run probability
that the results will be correct.
Traditionally, researchers have utilized the 95 percent confidence level.
The confidence interval gives the estimated value of the population parameter, plus or
minus an estimate of the error:
m = X ±
a small sampling error (E)
where E =
Z
c . 1 .
S
X
Z
c . 1 .
= the value of Z, our standardized normal variable at a specified confidence level.
S
X
= the standard error of the mean.
The following is a step-by-step procedure for calculating confidence intervals:
1. Calculate
X
from the sample.
2. Assuming
s
is unknown, estimate the population standard deviation by finding
S, the sample deviation.
3. Estimate the standard error of the mean, using the following formula:
S
X = S n
.
4. Determine the Z-value associated with the confidence level desired. The
confidence level should be divided by 2 to determine what percentage of the area
under the curve must be included on each side of the mean.
5. Calculate the confidence interval.
Sample statistics, such as the sample means,
X
s, can provide good estimates of
population parameters such as
m
.
There will be a random sampling error, which is the difference between the survey results
and the results of surveying the entire population.
VII. SAMPLE SIZE
Random Error and Sample Size
Random sampling error varies with samples of different sizes.
Increasing the sample size decreases the width of the confidence interval at a given
confidence level.
When the standard deviation of the population is unknown, a confidence interval is
calculated by using the following formula:
X ± Z S
n
Observe that the equation for the plus or minus error factor in the confidence interval
includes n, the sample size.
If n increases, E is reduced (see Exhibit 17.18).
Increases in sample size reduce sampling error at a decreasing rate.
More technically, random sampling error is inversely proportional to the square root of n.
Thus, the main issue becomes ones of determining the optimal sample size.
Factors in Determining Sample Size for Questions Involving Means
Three factors are required to specify sample size:
1. the variance, or heterogeneity, of the population
2. the magnitude of acceptable error
3. the confidence level
The variance, or heterogeneity, of the population characteristic in statistical terms refers
to the standard deviation of the population parameter.
Only a small sample is required if the population is homogeneous.
As heterogeneity increases, so must sample size.
The magnitude of error, or the confidence interval, is defined in statistical terms as E, and
indicates how precise the estimate must be.
It indicates a certain precision level.
From a managerial perspective, the importance of the decision in terms of
profitability will influence the researcher’s specifications of the range of error.
The third factor of concern is the confidence level.
We will typically use the 95 percent confidence level.
This, however, is an arbitrary decision based on convention.
Estimating Sample Size for Questions Involving Means
The researcher must follow three steps:
1. Estimate the standard deviation of the population.
2. Make a judgment about the desired magnitude of error.
3. Determine a confidence level.
The only problem is estimating the standard deviation of the population.
Ideally, similar studies conducted in the past will be used as a basis for judging the
standard deviation.
In practice, researchers without prior information conduct a pilot study to estimate the
population parameters so that another, larger sample, of the appropriate sample size, may
be drawn.
This procedure is called sequential sampling, because researchers take an initial look at
the pilot study results before deciding on a larger sample to provide more precise
information.
A rule of thumb for estimating the value of the standard deviation is to expect it to be
one-sixth of the range.
In a general sense, doubling sample size will reduce error by only approximately
one-quarter.
The Influence of Population Size on Sample Size
In most cases the size of the population does not have a major effect on the sample size.
The variance of the population has the largest effect on sample size.
However, a finite correction factor may be needed to adjust the sample size if that size is
more than 5 percent of a finite population.
If the sample is large relative to the population, the above procedures may overestimate
sample size, and there may be a need to adjust sample size.
Factors in Determining Sample Size for Proportions
When the question involves the estimation of a proportion, the researcher requires some
knowledge of the logic for determining a confidence interval around a sample proportion
(p) of the population proportion ().
For a confidence interval to be constructed around the sample proportion (p), an estimate
of the standard error of the proportion (Sp) must be calculated and a confidence
coefficient specified.
The plus or minus estimate of the population proportion is:
Confidence interval = p
±
Zc.l.Sp
Where Sp =
pq n
p = proportion of successes
q = 1 – p, or proportion of failures
To determine sample size for a proportion, the researcher must make a judgment about
the confidence level and the maximum allowance for random sampling error.
Furthermore, the size of the proportion influences sampling error, so an estimate of the
expected proportion of successes must be made based on intuition or prior information.
The formula is:
n = Z
c . 1
2 pq
( )
E
2
Calculating Sample Size for Sample Proportions
In practice, a number of tables have been constructed for determining sample size.
Exhibit 17.20 illustrates a sample size table for problems that involve sample proportions
(p).
Determining Sample Size on the Basis of Judgment
Sample size may also be determined on the basis of managerial judgments.
Using a sample size similar to those used in previous studies provides the inexperienced
researcher with a comparison of other researchers’ judgments.
Another judgmental factor is the selection of the appropriate item, question, or
characteristics to be used for the sample size calculations.
Often the item that will produce the largest sample size will be used to determine the
ultimate sample size.
However, the cost of data collection becomes a major consideration, and judgment must
be exercised regarding the importance of such information.
Another consideration stems from most researchers’ need to analyze the various
subgroups within the sample.
There is a judgmental rule of thumb for selecting minimum subgroup sample size:
Each subgroup to be separately analyzed should have a minimum of 100 or more
units in each category of the major breakdowns.
With this procedure, the total sample size is computed by totaling the sample size
necessary for these subgroups.
Determining Sample Size for Stratified and Other Probability Samples
Stratified sampling involves drawing separate samples within the subgroups to make the
sample more efficient.
With a stratified sample, the sample variances are expected to differ by strata.
This makes the determination of sample size more complex.
Increased complexity may also characterize the determination of sample size for cluster
sampling and other probability sampling methods.
The formulas are beyond the scope of this book.
Determining Level of Precision after Data Collection
After we have collected the data, we also want to determine our level of precision, given
the size of the sample, the variance, and the confidence level.
Rather than solving for n in the sample size equation, we solve for E2, and then take the
square root of this to determine our level of precision.
VIII. A REMINDER ABOUT STATISTICS
Learning the terms and symbols defined in this chapter will provide you with the basics of the language of
statisticians and researchers.