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I am just starting a self learning beginner course on statistics (majored in applied math but had no background on stats). Here is one naive question I encountered when trying to understand point estimation.

Confusion for point estimates of a certain model arises when one obtains different parameter estimates for different sample sizes. Suppose an example: if one is to find the incidence of lung cancer in mice and want do develop a model for it, and in one experiment, the researchers were to take 100 mice as the sample size, in which 10 of the mice has cancer, and assuming a Binomial Distribution, the estimate parameter can be $p = 10/100 = 0.1$, what if another sample was taken, say the sample size is now 80, in which 20 mice has cancer, and our new estimate is $p = 20/80 = 0.25$ - which is quite different from our previous estimates. In this case, we can get many different estimates for our model's true parameter, which one should we choose? Or am I asking a question that will be answered in my later chapters (of my self study materials)?

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Different point estimates are bound to happen given different data. The entire concept of the sampling distribution for the statistic is one way to resolve this.

Given your first example, the confidence interval for the mean would be between approximately 4 and 16. This means that true incidence of lung cancer between approximately 4 and 16 is consistent with the data we observed and can explain variation from sample to sample.

If we observed a much larger incidence than what we calculated above, then there are several explanations as to why. Our first sample calculated above could be uncharacteristically healthy, leading to a bias which is undetectable to us, for example. We could pool our estimates of incidence from both out samples to get a different estimate of incidence by leveraging all the data available to us.

So far as your question goes, the estimate you should choose should depend on the model you want to build. A simple binomial model is different that a hierarchical bayesian model and thus will need different kinds of data.

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