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I am in high school learning about sample proportions and they say that $n$ is the sample size. The example they gave is you spin a spinner board where the chance of landing on a 1 in 0.6, and 0 is 0.4.

and the formula for the variance is: $$\sqrt{\frac{p(1-p)}{n}}$$

And go on to say that $n$ is the number of spins. Then they go on to repeat this experiment a number of times and then make some claims. They say the above formula is because of the central limit theorem.

And because the variance is inversely proportional to $$\sqrt{n}$$ that the larger the number of spins in the experiment, the better the CLT applies. My question is:

Why isn't $n$ the number of experiments?

Wouldn't I be more sure that my estimate is closer to the true proportion if I repeat the experiment a large number of times? Because even if the number of spins in one experiment is large, it could be luck that I get a certain outcome. So how come they say that $n$ is the number of spins?

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    $\begingroup$ Welcome to crossvalidated, Venkat :-) n is indeed the number of experiments. A sample of size 1 would be just 1 experiment, so you can see n spins as n samples of size 1, or one sample of size n, or any combination of samples that add up in size to n. The formula for the variance comes from estimating the proportion from all n experiments taken together, and you can see that the variance gets smaller when n increases. This means your estimate gets more precise. $\endgroup$
    – Ute
    Aug 27, 2023 at 10:24

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In addition to the comment by user Ute:

They say the above formula is because of the central limit theorem.

That is FALSE. The formula have nothing to do with the central limit theorem (which can only give approximations). The formula is exact, and is due to the binomial distribution. To prove the formula, you only need independence plus simple properties of expectation and variance.

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