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The plot below shows the sample size needed to detect a proportion with a precision 0.01 for various true proportions:

Plot showing the sample size needed increases until the true proportion is 0.5 then it decreases again

This assumes an infinite population size, and the confidence intervals are fixed at 0.95.

If the true proportion is very small (or equivalently, very large) then we need a sample size of under 1000 to detect the proportion. However, if the true proportion is 0.5 then we need a sample size of almost 10,000.

I understand mathematically why this is from using the formula to calculate population size. But I would like an intuitive understanding of why this is the case.

It seems more intuitive to me that small proportions would need larger samples to detect them. Is this because I have kept the precision fixed (1% ± 1% is a much larger percentage error than 50% ± 1%)?

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    $\begingroup$ Somebody informs you of the truth, "red and green balls in this sack are very disproportional" and asks, "tell me which colour dominates". How many draws of a ball from the sack do yo think you will need before you can answer which colour prevails? Many draws or few draws? And what if they inform you of the truth "red and green balls are about balanced, yet one colour is a bit ahead"."Which is it?" Will you draw few or many times before you can answer it with confidence? $\endgroup$
    – ttnphns
    Aug 24, 2022 at 11:52
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    $\begingroup$ (in this example, the "information about the truth" corresponds to the assumption of the population proportion; and the question "who dominates" is like asking "is it far away from 50%", i.e. in a way akin to estimating a proportion) $\endgroup$
    – ttnphns
    Aug 24, 2022 at 12:06
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    $\begingroup$ Suppose the actual $p$ was very close to $0$: you could reach the conclusion it is likely to be less than $0.01$ with about $300$ all failures since $0.99^{300}\approx 0.049$. That is related to absolute precision. But finding the proportion with a $1\%$ relative precision might be much harder in that case and need a much larger sample size than if the proportion is close to $0.50$ $\endgroup$
    – Henry
    Aug 24, 2022 at 12:59
  • $\begingroup$ Thanks for the comments - Henry, yours in particular really cleared things up for me. Makes sense! $\endgroup$
    – Mhairi McNeill
    Aug 25, 2022 at 8:47

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The critical part is "assuming the true proportion is 0.5", had the hypothesis been that the true proportion is 0.1 then the image would look different and centered around 0.1. The logic is relatively simple.

If you assume that the true difference is 0.5 and you get a result from your sample far from this number then you can be relatively confident that the true proportion is in fact not equal to 0.5. However if the result from your sample is close 0.5 then you are less confident, because of the inherent variability associated with this estimate (standard error). Hence, in this case, you need a bigger sample to decrease the standard error and be equally confident that the difference is not equal to 0.5.

To give a concrete example, assume a coin that you flip 10 times. If the coin is fair then the likelihood of you getting 9 tails in 10 tosses is very low, so you would probably conclude that the coin is in fact not fair. However if you obtain 6 tails from 10 tosses you cannot be as sure in this statement. You would need to throw the coin many more times to see if it is slightly biased.

As you put it, because you keep the precision fixed, the sample size will vary depending on how "close" you get to the hypothesized difference.

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  • $\begingroup$ This was really helpful! In this case variability would be p(1 - p), right? So the closer we are to 0.5 the higher the variability? Makes sense, thanks. $\endgroup$
    – Mhairi McNeill
    Aug 25, 2022 at 8:49

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