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As I understand it, Student's t will asymptotically approximate the Z statistic at sample sizes approaching infinity (see the infinite degrees of freedom row for degrees of freedom in most introductory statistics books). However, Z tests are applicable when populations are assessed in total.

Thus, when the population size is finite, known, and relatively small, it seems that a t-statistic may be overly conservative. Specifically, if the population is 50 and you sampled 49, it seems that the appropriate critical value would be closer to the associated critical value for Z (given the alpha criterion) than it would be to t (given the alpha and degrees of freedom).

Is it true that t is conservative in these small population cases? If so, is there a correction to the t critical values to account for cases where the population is small?

russellpierce
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2 Answers2

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When you have a finite population and the sample size is more than 5-10% of the population then you should use the Finite Population Correction, that is you multiply your standard error times $\sqrt{\frac{N-n}{N-1}}$, see this link.

You should still use the t-distribution, but the standard error will be smaller to account for the large portion of the population in the sample.

mpiktas
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Greg Snow
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The difference between Z-statistics and t-statistics is that Z statistic is asymptoticaly normal, when t-statistic has exact Student's distribution (with appropriate degrees of freedom) if we assume that the errors are normal. In small sample sizes the difference can be quite substantial (actually this is why Student "invented" t-statistic, he had very small sample sizes to work with). If the sample sizes are larger the critical values for Z statistic and t-statistic are closer, since Student's distribution tends to normal when degrees of freedom goes to infinity. But z-statistic is still only an approximation, so it is still beneficial to use the exact distribution. Of course if normality assumption does not hold, then Z statistic should be used, since then Student's distribution is exactly not the distribution of the statistic.

mpiktas
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  • -1, a bit off target: The Z-statistic is normal when the entire population has been sampled to find the population mean and standard deviation. Which I think I acknowledged in the question. The accepted answer (Greg Snow's) provides the sort of answer I was looking for. That is, an answer that talked about the insufficiency of Student's estimation when large proportions of the population are sampled. – russellpierce Aug 17 '11 at 17:34
  • @drknexus, hm, this is a question from the survey sampling, it was not clear for me that this is the case. On the other hand I stand behind my statement. Note that Greg Snow also suggests using t-distribution, but with standard error corrected. So the problem is not with distributions, but with incorrect standard errors. I am not that familiar with survey sampling, but I find your claim that Z-statistic is normal when entire population has been sampled a bit suspicious. When you sample entire population, Z-statistic is not random, so there is no point in discussing its distribution. – mpiktas Aug 18 '11 at 07:26
  • ... Furthermore since Z statistic is mean divided by standard error, so I fail to see why it should be exactly normal, except in very special circumstances, so I would be glad if you gave a reference to the result you cite. – mpiktas Aug 18 '11 at 07:30
  • I'm pretty sure we are not on the same page. I'm also pretty sure that the issue I raised would not be limited to survey sampling. I would imagine that the correction to standard errors is a fix to the t-distribution which is indeed incorrect given the situation described. However, the extent to which it is 'incorrect' is difficult if not impossible to talk about. – russellpierce Aug 19 '11 at 07:37
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    Specifically, distributions can presumably can have infinite samples drawn from them. However, when the population becomes finite the number of samples which can be drawn from it also become finite and fail to really be a distribution in the mathematical sense that seems to be implied by Z and t. So maybe the question itself (in that sense) is irreparably flawed. – russellpierce Aug 19 '11 at 07:41
  • Anyway, I acknowledge that I may be wrong in my critique of your answer simply because I did not understand it. If that is the case, I apologize. – russellpierce Aug 19 '11 at 07:44
  • Hm, we maybe have different definitions of Z-statistic in mind. The statistic I am talking about is generaly of the form $\frac{1}{n}\sum X_i/\sqrt{\frac{1}{n}\sum_{i=1}^n(X_i-\frac{1}{n}\sum_{j=1}^nX_j)^2}$. It is called Z-statistic if we are talking about its asymptotic distribution, and t-statistic if we are talking about exact distribution. – mpiktas Aug 19 '11 at 07:45
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    The [wikipedia page](http://en.wikipedia.org/wiki/Z_statistic) gives slightly different definition, but it does not change my point. No offense taken on the critique by the way. I realized that the bold face I used might seem offensive, but that was not my intention. – mpiktas Aug 19 '11 at 07:50