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i just started working as a quality control engineer at a poultry company, one of my first duties is to design a mechanism to ensure the quality of the end product (the Chick) and processes leading up to it (incubation...etc...etc), so naturally i suggested that a sample of the eggs going from a process to the next is taken for quality checks, the problem is the sample size, there are around 200000 eggs going through the production process and only one veterinarian, a sample size of 100000 or 50000 or even 500 is way too much to be examined. what should be done in this situation?

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    $\begingroup$ It doesn't seem to me that the population size is relevant. What are you trying to find out about the process. Is it a matter of detecting when a change in performance occurs or are just trying to the total proportion of failures? If you sample randomly you can approximately estimate the proportion of failure using the binomial distribution since the population size is so large that it can be treated as infinite. $\endgroup$ – Michael R. Chernick Feb 5 '18 at 20:37
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    $\begingroup$ There are so many issues to address here that you will not be well served by any single answer, no matter how lucid or correct. You really need to study statistical quality control or retain the services of someone who knows it. $\endgroup$ – whuber Feb 5 '18 at 20:44
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One of the lovely things about sampling theory is that it demonstrates that you can make good inferences from sampling, even with a very large population and a correspondingly small proportion of sampled items. (Hell, you can even get good inferences with an infinitely large population!) It also demonstrates that it is incorrect to assume that you will require some minimal proportion of the population to obtain reasonable inferences. For example, if you have a population of $N$ units, and you take a sample of $n \leqslant N$ units, then a standard confidence interval (at confidence level $1-\alpha$) to infer the population mean would be:

$$\text{CI}(1-\alpha) = \left[ \bar{x}_n \pm \underset{\text{pop corr.}}{\sqrt{\frac{N-n}{N}}} \cdot \underset{\text{sampling}}{\frac{t_{n-1, \alpha/2}}{\sqrt{n}}} \cdot s_n \right] .$$

(In this equation $\bar{x}_n$ and $s_n$ are the sample mean and sample standard deviation, and $t_{n-1, \alpha/2}$ is the critical point of the Student's t-distribution with $n-1$ degrees-of-freedom and tail area $\alpha/2$.) As you can see, the width of this confidence interval (and thus, the accuracy of the inference) is determined by a product of terms; one of these is affected by the population size $N$ but the others are not. In the case where $N$ is large, the "population correction" term approaches one, and is therefore removed from the equation.

There are two aspects to the information taken from a sample of a finite population. One source of information is the direct information about the sampled units, which form part of the population. This source of information occurs in proportion to the amount of the population you have sampled. It is reflected in the above formula by population correction term, which goes to zero as $n \rightarrow N$. The other source of information is the indirect inferential information that the sampled units give about the non-sampled units. This source of information is determined by the raw size of the sample, and occurs independently of the size of the population. It is reflected in the "sampling" term, which decreases as $n$ gets larger.

Even if $N = \infty$ it is possible to reduce the width of this confidence interval (and thus increase the accuracy of the inference) as narrowly as you want by taking sufficiently large $n$. It is possible to obtain a very narrow confidence interval (i.e., a very accurate inference) with a sample that is only a tiny proportion of the full population (or even zero proportion of an infinite population). In sampling problems where you have a large (or even infinite) population, it is possible to get good inferences with samples that are large in raw size, but small as a proportion of the population (even infinitesimally small). In such cases there is little to no direct information from the sample, but a large amount of indirect inferential information.

This is really the core lesson of sampling theory, so if you are unfamiliar with it, you should try taking a basic course on sampling the mean of a population. The belief that a particular proportion of the population is required to obtain a good inference is a common sampling fallacy; it is based on ignoring the inferential information provided by the sample.

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