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Let's say there are N chunks of metal, each are 4 inches thick. They are to be hammered and reduced to following best possible sizes: 1, 2, 3 or 4 inches. If we were to use sampling to get an estimate of possible reduction in the total number of inches in all the N chunks combined, how would we determine the sample size, and the formula for estimation, variance and accuracy?

I am thinking of this like a polling problem, where M/N people are asked to answer a multiple choice question. For eg. each chunk is asked, how many inches can be reduced from you? There are 4 answers. Most of the sampling theory stuff on the web talks about answering yes/no questions. Is there some theory which talks about multiple choice questions?

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Are you saying that every chunk of metal could be hammered into a 1, 2, 3, or 4 inch piece? Are you interested in the amount of metal that will be lost? If so, we need to know the proportion of each of the smaller pieces that you will make. In short, can you present the problem in more detail? – Joel W. Jul 11 '12 at 17:48
@JoelW. Yes, the goal is to find the total amount of metal lost in 'inches'. The problem is that after the hammering a chunk, there are 4 possible outcomes, i.e we have either not removed anything from the chunk, or removed 1 inch, 2 inches or 3 inches. If it was a binary outcome, I could have treated this problem like a polling problem, if each question had yes/no answer. And used the standard formulas for calculating sample size, std err, etc. like ones given here: link – Gang Liu Jul 11 '12 at 19:57
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An example of the kind of probability model needed and how to use it for computing standard errors of estimates appears in a closely related thread at stats.stackexchange.com/a/18609. One expresses the SE in terms of the sample size and solves for the sample size. Note that in order to do so, you must make assumptions about the population distribution. If you cannot do that, you can look at the worst case (which would be a population where half the chunks had values of $0$ and the other half had values of $3$). – whuber♦ Jul 11 '12 at 21:33
How do you decide whether you will make a chosen piece into a 1, 2, 3, or 4 inch object? Without knowing that, the question is not answerable. – Joel W. Jul 13 '12 at 14:51
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Are the 4 possible compressibility outcomes equally probable? – Joel W. Jul 13 '12 at 22:48
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closed as not a real question by whuber♦ May 9 at 19:11

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, see the FAQ.

1 Answer

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Sure. For yes/no you have only two outcomes. So you can compare one group to another on the basis of proportion answered yes (possibly assuming a binomial distributional form). If you have k possible choices with k>=3 for each group you can look at the distribution of responses and see if the distributions differ in any way. This can be done with contingency tables using Fisher's test or the approximate chi square test. In the one sample case case when k=2 you simply compare a binomial proportion to a hypothesized proportion (sometimes 0.5). For k>=3 you come the observed distribution for the sample to some specific hypothesized distribution (possible a uniform distribution). Sample size can be determined based on the power that you would like the test to have at some specific alternative distribution.

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I think I partly understand what you are suggesting here, but I don't quite get how the sample size is calculated. My current understanding is that, mathematically we can derive an optimal sample size for any given population(in this case, N) size, given the required margin of error/confidence interval. I am trying to formulate a formula to do that. Can you point me to some literature which discusses such a problem. Thanks – Gang Liu Jul 11 '12 at 20:05
Sorry but somehow the comment I meant to give to a question on the math stackexchange site wound up here by mistake. Ignore it. To answer the question in your comment. Very similar to picking a sample size to get a specified width for say a 95% confidence interval for a parameter, if you test a null hypothesis that your true proportion is 0.5 you specify power (the probability that you correctly reject the null hypothesis when the true proportion is say 0.7. – Michael Chernick Jul 11 '12 at 23:13
So for the power of the test to be say 95% when the true proportion is 0.7 there is a fixed minimum sample size necessary to acheive that when the significance level has been fixed at some alpha level (say 0.05). – Michael Chernick Jul 11 '12 at 23:13

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