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I regularly encounter the following scenario: I am presented with a set, $\mathcal{S}$, of $i$ items.

I am assured that $d$ of the $i$ items possess a particular property, $d'$ - and that the rest, $(i-d)$, do not. So the items are in 2 piles: $d'$ and non-$d'$.

(For context, let's say that $i$ is known to be 1,000,000 and $d$ is said to be 30,000. )

I stipulate that the probability of any one member of $i$ also being a member of $d$ is independent of other members of $i$.

Consider the case where I wish to test this assurance by taking a sample. (For practical and financial reasons, I cannot test all of $\mathcal{S}$. )

Clearly I don't know how common the property of $d'$ actually is within $\mathcal{S}$.

  1. What size of sample from $\mathcal{S}$ do I need, for a given confidence and margin of error to test whether the provided value of $d$ is correct or not? E.g., if I was told $d$ was 30,000 and my sample indicated it was 30,100 I would be happy. If the indication was that the real value was 40,000 I would not be happy.

  2. Suppose I only have access to the items marked $d'$ and cannot sample the rest. I believe then I can only test for non-$d'$ items in the pile marked $d'$, what I call false positives. Can I then infer anything about false negatives, e.g. $d'$ items in the non-$d'$ pile which should not be there?

  3. Suppose further that in fact some proportion of the items in $\mathcal{S}$ do have a dependency, such that if a particular item $p$ has $d'$ then item $p+1$ has $d'$ also . Could I reasonably remove the dependency by considering i and i+1 as a single item?

If someone could point me at the relevant area of statistics, I would be very grateful!

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If the population size were infinite (in your example of i=1000,000 this is a reasonable practical assumption). Then if you randomly select N elements for the population the number k that have property d is binomially distributed. Its mean is the true proportion p and its variance is p(1-p)/N. So you can pick N large enough to make this variance as small as you please. Given the binomial distribution you can obtain confidence limits for an upper bound on the proportion or total number of cases having property d. If i is not large the actual variance can be obtained by applying the finite population correction. There is a large theory of survey sampling (also called sampling from finite populations) that covers all this. Classical texts on this subject are "Survey Sampling" by Leslie Kish and "Sampling Techniques" by William Cochran.

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  • $\begingroup$ Thank you for pointing me in the right direction! I had thought the binomial distribution would be the right one to pick but I was wondering whether I ought to be looking at a confidence interval for a population proportion based on a normal distribution. I shall go and refresh myself in the right chapters of my old school textbook! thanks again. $\endgroup$ – Nick Rich May 29 '12 at 12:15

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