Sample size needs depend upon population standard deviation and usually assume negligible bias, or at least an idea of its likely magnitude. This does not allow for any such 10% rule of thumb. Further, in general, the larger the population, the smaller the necessary "percent" will be. However, you also may need to consider the finite population correction (fpc) factor, sample design, and possibly auxiliary/regressor data, if available.
In short, no such 10% rule exists. It depends upon the application.
Also, the 'speed' with which the Central Limit Theorem becomes effective is dependent upon the population distributions.
Finally, I think that the word "independence" is not used meaningfully in the question, as stated.
I suppose what you, Joy, refer to is that sampling without replacement leads to dependent draws, but the CLT actually requires independent values.
The rule of thumb that you have heard of or read about, basically means that if the sample/population ratio is small enough (e.g. 10%), sampling without replacement may (approximately) be treated like sampling with replacement.
Sampling with replacement is convenient, because it leads to independent draws and thus the CLT can directly be applied.
Now why this rule of 10%? Think of a population of 5 individuals, and draw 1 unit - clearly, the distribution of the next draws depends on this first sampled value/unit.
In contrast, if you have a population of 5000 individuals, and you draw 1 unit, the distribution of the next draws is generally not really influenced by the first sampled value.
The influence of earlier draws becomes substantial only when a relevant percentage of the population is already drawn - at least this is what the rule of thumb states.
I also agree with James Knaub, however, that the actual behaviour of the sample and sample mean depends on much more than just the sample/population ratio. The rule of thumb you mentioned can only be what it is - a rule of thumb, that may fail in particular situations.
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