Sample Size: Simple Random Samples
This lesson describes how to find the smallest sample size that provides
the desired precision, when the sampling method is simple random sampling.
Factors That Influence Sample Size
The "right" sample size for a particular application depends on many factors,
including the following:
These factors interact in complex ways. Although a consideration of all the
variations is beyond the scope of this tutorial, the remainder of this lesson
covers a situation that commonly occurs with simple random samples: How to find
the smallest sample size that provides the required precision.
The next lesson covers two situations that commonly occur with stratified random
samples:
For these, and for other situations not covered in the tutorial, consider using
the Sample Size Calculator (described below).
Sample Size Calculator
Stat Trek's Sample Size Calculator can help you find the right sample size quickly,
easily, and accurately. You specify your main goal - maximize precision, minimize
cost, stay within budget, etc. Based on your goal, the calculator prompts you for the
necessary inputs and handles all computations automatically, allowing you to
compare alternative designs and sample sizes. The calculator creates a summary report
that lists key findings and describes analytical techniques. And the calculator
is free. You can find the Sample Size Calculator in Stat Trek's
main menu under the Stat Tools tab. Or you can tap the button below.
Sample Size Calculator
How to Choose Sample Size for a Simple Random Sample
Consider the following problem. You are conducting a survey to estimate a population mean or proportion. The
sampling method is
simple random sampling,
without replacement. You want your survey to provide a
specified level of precision.
To choose the right sample size for a simple random sample,
you need to define the following inputs.
- Specify the desired
margin of error ME. This is your measure of
precision.
- Specify
alpha.
- Find the critical
standard score z.
- Unless the population size is very large relative to sample size (e.g., 20 times larger), you need to specify
the size of the population (N).
You will also need to know the variance of the population, σ2.
Given these inputs, the following formulas find the smallest sample
size that provides the desired level of precision.
Sample statistic
|
Population size |
Sample size |
Mean
|
Known |
n = { z2 * σ2
* [ N / (N - 1) ] }
/ { ME2
+ [ z2 * σ2 / (N - 1) ] } |
Mean
|
Unknown |
n = ( z2 * σ2 ) / ME2 |
Proportion
|
Known |
n = [ ( z2 * p * q ) + ME2 ]
/ [ ME2 + z2 * p * q / N ] |
Proportion
|
Unknown |
n = [ ( z2 * p * q ) + ME2 ]
/ ( ME2 ) |
This approach works when the sample size is relatively large (greater than or
equal to 30). Use the first or third formulas when the population size is
known. When the population size is large but unknown, use the second
or fourth formulas.
For proportions, the sample size requirements vary, based on the value
of the proportion. If you are unsure of the right value to use, set
p equal to 0.5. This will produce a conservative sample
size estimate; that is, the sample size will produce at least
the precision called for and may produce better precision.
Sample Problem
At the end of every school year, the state administers a reading test to a
simple random sample drawn without replacement from a population of 100,000
third graders. Over the last five years, students
who took the test correctly answered 75% of the test questions.
What sample size should you use to achieve a margin of error equal to plus or
minus 4%, with a confidence level of 95%?
Solution: To solve this problem, we follow the steps outlined above.
- Specify the margin of error. This was given in the problem definition. The
margin of error is plus or minus 4% or 0.04.
- Specify the confidence level. This was also given. The confidence level is
95% or 0.95.
- Compute alpha. Alpha is equal to one minus the confidence level. Thus,
alpha = 1 - 0.95 = 0.05.
- Determine the critical standard score (z). Since this is an
estimation problem,
the critical standard score is the value for which the cumulative probability is
1 - alpha/2 = 1 - 0.05/2 = 0.975.
To find that value, we use the
Normal Calculator. Recall that the distribution
of standard scores has a mean of 0 and a standard deviation of 1. Therefore,
we plug the following entries into the normal calculator: Value = 0.975;
Mean = 0; and Standard deviation = 1. The calulator tells us that the value of
the standard score is 1.96.
- And finally, we assume that the population proportion p is
equal to its past value over the previous 5 years. That value
is 0.75. Given these inputs, we can find the smallest sample size
n that will provide the required margin of error.
n = [ (z2 * p * q ) + ME2 ]
/ [ ME2 + z2 * p * q / N ]
n = [ (1.96)2 * 0.75 * 0.25 + 0.0016] /
[ 0.0016 + (1.96)2 * 0.75 * 0.25 / 100,000 ]
n = (0.7203 + 0.0016) / ( 0.0016 + 0.0000072)
n = 449.2
Therefore, to achieve a margin of error of plus or minus 4 percent, we
will need to survey 450 students, using simple random sampling.