3. INTRODUCTION
Sampling is a process of selecting representative units
from an entire population of a study.
Sample is not always possible to study an entire
population; therefore, the researcher draws a representative
part of a population through sampling process.
A simple random sample is a subset of a statistical
population in which each member of the subset has an
equal probability of being chosen. A simple random
sample is meant to be an unbiased representation of a
group.
4. A Proportion from Random Sample
A good analysis should provide two outputs :
A point estimate of the population mean or
proportion.
A quantitative measure of uncertainty associated
with the point estimate (e.g., a margin or
error and/or a confidence interval).
5. Analyze Survey Data
Any good analysis of survey sample data includes the
same seven steps:
Estimate a population parameter.
Estimate population variance.
Compute standard error.
Specify a confidence level.
Find the critical value (often a z-score or a t-
score).
Compute margin of error.
Define confidence interval.
6. Estimating a Population Mean or Proportion
The first step in the analysis is to develop a point
estimate for the population mean or proportion. Use
this formula to estimate the population means:
Sample mean = x = Σx / n
where Σx is the sum of all the sample observations,
and n is the number of sample observations.
7. Estimating Population Variance
The variance is a numerical value used to measure
the variability of observations in a group. If
individual observations vary greatly from the group
mean, the variance is big; and vice versa.
Given a simple random sample, the best estimate of
the population variance is:
s2 = Σ ( xi - x )2 / ( n - 1 )
where s2 is a sample estimate of population
variance, x is the sample mean, xi is the ith element
from the sample, and n is the number of elements in
the sample.
8. CONT…
where s2 is a sample estimate of population
variance, x is the sample mean, xi is the ith element from
the sample, and n is the number of elements in the
sample.
With a proportion, the population variance can be
estimated from a sample as:
s2 = [ n / (n - 1) ] * p * (1 - p)
where s2 is a sample estimate of population variance, p
is a sample estimate of the population proportion, and n
is the number of elements in the sample.
9. Computing Standard Error
The standard error is possibly the most important
output from our analysis. It allows us to compute
the margin of error and the confidence interval.
When we estimate a mean or a proportion from a
simple random sample, the standard error (SE) of the
estimate is:
SE = sqrt [ (1 - n/N) * s2 / n ]
where n is the sample size, N is the population size,
and s is a sample estimate of the population standard
deviation.
10. CONT…
Think of the standard error as the standard
deviation of a sample statistic. In survey sampling,
there are usually many different subsets of the
population that we might choose for analysis.
Each different sample might produce a different
estimate of the value of a population parameter.
The standard error provides a quantitative
measure of the variability of those estimates.
11. Specifying Confidence Level
In survey sampling, different samples can be randomly
selected from the same population; and each sample can
often produce a different confidence interval. Some
confidence intervals include the true population
parameter; others do not.
A confidence level refers to the percentage of all
possible samples that produce confidence intervals that
include the true population parameter. For example,
suppose all possible samples were selected from the
same population, and a confidence interval were
computed for each sample.
A 95% confidence level implies that 95% of the
confidence intervals would include the true population
parameter.
12. Often expressed as a t-score or a z-score, the critical value is a
factor used to compute the margin of error. To find the critical
value, follow these steps:
Compute alpha (α): α = 1 - (confidence level / 100)
Find the critical probability (p*): p* = 1 - α/2
To express the critical value as a z-score, find the z-score
having a cumulative probability equal to the critical
probability (p*).
To express the critical value as a t statistic, follow these
steps:
Find the degrees of freedom (df). When you estimate a
mean or proportion from a simple random sample,
degrees of freedom is equal to the sample size minus one.
The critical t statistic (t*) is the t statistic having degrees
of freedom equal to df and a cumulative probability equal
to the critical probability (p*).
Finding Critical Value
13. Computing Margin of Error
The margin of error expresses the maximum
expected difference between the true population
parameter and a sample estimate of that parameter.
Here is the formula for computing margin of error
(ME) :
ME = SE * CV
where SE is standard error, and CV is the critical
value.
14. Confidence Interval
Statisticians use a confidence interval to express the degree of
uncertainty associated with a sample statistic. A confidence
interval is an interval estimate combined with a probability
statement.
Here is how to compute the minimum and maximum values for a
confidence interval.
In the table above, x is the sample estimate of the population
mean, p is the sample estimate of the population proportion, SE is
the standard error, and CV is the critical value (either a z-score or
a t-score). And, the confidence interval is an interval estimate that
ranges between CImin and CImax.
Mean Proportion
CImin = x - SE * CV
CImax = x + SE * CV
CImin = p - SE * CV
CImax = p + SE * CV
15. CONCLUSION
Sampling is a general home that important for
a research project, many sampling start with
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