When do I use a z-score vs a t-score for confidence intervals?
I have a set of 1000 data points. I would like to estimate their mean using a confidence interval. I read somewhere that if the sample size, $n$, is bigger than 30 you should use a t-score, and else use a z-score.
Is that true?
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$\begingroup$Usually you use a t-test when you do not know the population standard deviation $\sigma$, and you use the standard error instead. You usually use the z-test when you do know the population standard deviation. Although it is true that the central limit theorem kicks in at around $n=30$. I think that formally, the convergence in distribution of a sequence of $t's$ to a normal is pretty good when $n>30$.
$\endgroup$ $\begingroup$If you don't know the variance of the population, then you should formally always use the $t$-distribution. If you do know the population variance, you can use the standard normal distribution.
However, as $n \to \infty$, the $t$-distribution becomes the same as the standard normal distribution. Even for relatively small samples, the distributions are virtually the same. Therefore, it is common to approximate the $t$-distribution using the normal distribution for sufficiently large samples (e.g. $n>30$ as you indicate).
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