Sanderson M. Smith
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SAMPLE SIZE NEEDED FOR
SPECIFIED MARGIN OF ERROR
The general formula for a confidence interval is
estimate plus/minus (critical value )(standard deviation of the estimate )
A 95% confidence interval for proportions has the form
p(hat) plus/minus 1.96 ÷[((p(hat))(1-p(hat))/N]
where N is the sample size and p(hat) is the sample proportion.
Since 1.96 is approximately 2, we will use 2 in what follows to simply computations.
If the population proportion parameter is p, the margin of error, m, for a 95% confidence interval can be calculated using the formula
m = 2 ÷[p(1-p)/N]
When sampling, p is replaced by p(hat), the sample proportion, to compute m.
We now ask the question:
What sample size is needed if one wants a specific margin of error?
Solving the above equation for N yields m2/4 = p(1-p)/N ==> N = 4p(1-p)/m2.
YIKE! We face a "Catch 22" situation. We want N, and we know m, but we don't know a value for p, and we can't get such a value until we actually take a sample.
We get around this dilemma by finding the value of p that will maximize N. Since 4 and m2 are known constants, we need only maximize y = p(1-p) = p - p2. This is simply a parabola that opens downward. We need only find the vertex. We can take a derivative and note that dy/dp = 1 - 2p which has value of 0 when p = 1/2. In other words, looking at the equation
N = 4p(1-p)/m2
we will get the largest possible value of N when we substitute p = 1/2. Note that is the substitution is made, we get N = 4(1/2)(1/2)/m2 = 1/m2, a very simple formula. In other words, if we want a 95% confidence interval and know m, margin of error, we can determine the sample size needed for the specified m. For instance, if we want a margin of error = 2%, then the sample size required is 1/(.02)2 = 2,500.
What is shown in the box below is a published survey related to the Persian Gulf War some years ago.
Would you support or oppose U.S. forces resuming action to force Saddam from power?
For this Newsweek Poll, the Gallop Organization interviewed a national sample of 751 adults by telephone April 4-5. The margin of error is plus or minus 4 percentage points. Some "Don't Know" and other responses not shown.
Let's do some computations:
If we were to compute the margin of error using 54%, we would get 2 ÷[(.54)(.46)/751] = 0.0363736. Rounding "out" to the nearest integer percent, we would get the 4% stated in the survey results. If one calculates the margin of error using 37%, one obtains 2 ÷[(.37)(.63)/751] = 0.0352356. Again, if we round "out," we get 4%.
If we wanted a margin of error = 4%, the sample size needed would be 1/(.04)2 = 625. A margin of error of 3% would require a sample size = 1/(.03)2 = 1,111. What is reported in the survey "jives" with these calculations.
While published surveys such as the one above do not generally talk about a 95% confidence interval, the reported margin of error does relate to such an interval, as has been demonstrated. Using the information provided in the survey above, the 95% confidence interval for those support using action to remove Saddam from power is [50%, 58%]. The corresponding 95% confidence interval for those who oppose is [33%,41%].
"Numbers rule the universe." -PYTHAGORAS (around 550 B.C.)
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