What conditions must be fulfilled to apply the Central Limit Theorem? And in what cases would it be applied?
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The Central Limit Theorem states that, provided n (the number in the sample) is at least 30, the distribution of sample means is Normal, no matter what the distribution of the population from which the sample was taken. This theorem justifies the use of confidence intervals for sample means and proportions.
Not exactly n at least 30, that would still be the "t" distribution (already close to Normal for n = 30), but the limit as n--> infinity for sample means is Normal.
You're also only supposed to use Normal if you know the population variance (st. deviation).,
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If X bar is the mean of a sample size n from a population which is not Normally distributed, then the Central Limit Theorem states that X bar will be approximately Normally distributed if the sample size is sufficiently large.\[(a\ common\ value\ given\ is\ n \ge30)\]
Key word: "approximately". In general, use the t distribution (it's not much harder, same basic procedures), which approaches normal for larger and larger n.