Sample Size equation that is predicated on Central Limit Theorem
Correction Factor for Sample Size based on finite Population Size
For a 95% Level, Z = 1.96. Let us say that we want the result to be within 5% error--Confidence Interval and lets have P varying
from 0.1 to 0.9. Then inserting the above numbers into the equation, we get:
ConfidenceInterval = 0.05;Z = 1.96;
P = 0.1:0.1:0.9;
SampleSize = [Z^2 .* P .* (1-P)]/(ConfidenceInterval^2);
SampleSizeRequired = [P ;SampleSize].'
SampleSizeRequired =
0.1000 138.2976
0.2000 245.8624
0.3000 322.6944
0.4000 368.7936
0.5000 384.1600
0.6000 368.7936
0.7000 322.6944
0.8000 245.8624
0.9000 138.2976
Plot of SampleSize Required vs Probability:
figure()
plot(SampleSizeRequired(:,1),SampleSizeRequired(:,2))
xlabel('Probability P')
ylabel('Sample Size Required')
T-distribution table VS Sample Size and closeness to Standard Normal:
(Source: Jeffrey Russell NOTES)
T distribution with n-1 degrees of freedom at 0.05 level( 95%) It becomes normal when T is Approx 2 ( 1.96 to be exact)
[4.303 3
2.228 11
2.086 21
2.042 31
2.00 61];
Survey Example: Some tests for Sample Sizes:
NULL HYPOTHESIS:
IT IS TIE (50-50). I.e Even though 60% of the people say A, we want to make sure it is different from being a TIE (50-50)
when a bigger population is considered.
T-stat formula ( Any Basic Stats Textbook is the source here)
T_Stat = (P - p0)/sqrt(p0 * (1-p0)/n); Where p0 is NULL hypothesis. (0.50 in our case)
P = 0.60; p0 = 0.50;
n = 21;
t_stat = (P - p0)/sqrt(p0 * (1-p0)/n)
t_stat = 0.9165
n=30;
t_stat = (P - p0)/sqrt(p0 * (1-p0)/n)
t_stat = 1.0954
n = 30:10:100;
t_stat = (P - p0)./sqrt(p0 * (1-p0)./n);
figure()
plot(n,t_stat)
xlabel('Sample Size')
ylabel('T stat')
Reverse Experiment: calculating Percentages Needed with a given Sample Size to have statistical significance.
n = 21:30;
P = p0 + [2 * sqrt(p0 * (1-p0)./n)];
figure()
plot(n,P)
xlabel('Sample Size')
ylabel('Probability Needed to have statistical siginificance')
n = 30:1000;
P = p0 + [2 * sqrt(p0 * (1-p0)./n)];
figure()
plot(n,P)
xlabel('Sample Size')
ylabel('Probability Needed to have statistical siginificance')
S
ee How the PLOT flattens out after certain number of Samples( >450).
That is why most of the proffessional Pollsters pick a sample size of Approximately 500
History behind "Rules-of-thumb" (Why 22? 25? 30?)
(Source: iSixSigma)
n=22 was proposed by Fisher in Statistical Mehthod, p. 44, when he reviewed the impact of the the exeeding of the standard deviation once in evey three trials. Twice the standard deviation is exceeded in about 22 trials "For p-value = 0.05, or 1 in 20 and 1.96 or nearly 2; it is convenient to take the point as a limit in judging whether a deviation is to be condisered significant or not. Deviations exceeding twice the standard deviation are thus formally regarded as significant. Using this criterion we should be led to follow up a false indication only once in 22 trials even if the statsitics were the only guide. Small effects will still escape notice if the data are insufficiently numerous to bring them out, but lowering of the standard of signficicance meet this difficulty.
n = 25 has a truly statistical justification. At n = 25 the Law of Large numbers will start to show a pronounced symmetric/normal distribution of the sample means around the population mean. This normal distribution becomes more pronounced as n is increased.
n = 30 comes from a quote from Student (Gosset) in a 1908 paper "On the probable error of a Correlation" in Biometrika. In this paper he reviews the error associated with drawing of two independent samples from infinitely large population and their correlation (not the individual errors of each sample relative to the sample mean and the population mean!). The text reviews
different corrections to the correlation coefficient given various forms of the joint distribution. In a few sentences, Student says that at n = 30 (which is his own experience) the correction factors don't make a big difference.
Later, Fisher showed that the sample for a correlation needs to be determined based on a z-transformation of the correlation. So, Student's argument is only interesting historically. Also, Student wrote his introduction of the t-test in Biometrika during the same year (his
prior article). Historically, the n = 30 discussed in his correlation paper has been confused with the t-test paper, which only introduced the t-statistic up to sample size 10.
2. http://www.isixsigma.com/
3. http://www.itl.nist.gov/div898/handbook/ppc/section3/ppc333.htm
4. http://www.itl.nist.gov/div898/handbook/prc/section2/prc222.htm
5. http://www.itl.nist.gov/div898/handbook/prc/section2/prc243.htm
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