Central Limit Theorem Slight Misunderstanding ?

[michael001]

I am reviewing some basic statistics.
From a youtube lecture: "What is the Central Limit Theorem in Statistics? - Part 1"
The instructor quoted the following theorem:
"If the population is normally distributed, then the sample means will have a normal distribution regardless of size"
1. Does that mean that for each individual sample size distributions will themselves be normally distributed ?
2. Or does it mean that the sampling distribution of the sample means will be normally distributed regardless of the sample size n taken.

Another theorem he quoted:

In a population distribution that is is not normally distributed, the sampling distribution of the means will be normally distributed
provided that the sample size n > 30

1. Does that mean a bi-modal or a uniform distribution or any kind of distribution will have sampling distributions that appear very normally distributed provided that the sample size is > 30 ?
2. Does the Central Limit Theorem still hold when sampling a large population with replacement ?

THANKS !

 

[Dr.Peterson]

2. Or does it mean that the sampling distribution of the sample means will be normally distributed regardless of the sample size n taken.

Yes.

1. Does that mean a bi-modal or a uniform distribution or any kind of distribution will have sampling distributions that appear very normally distributed provided that the sample size is > 30 ?

I'm not sure about "very"; this is just a rule of thumb saying when it's reasonable to assume normality for practical purposes, and I wish he didn't say "will be" as if it were definitely true. But, basically, yes.

2. Does the Central Limit Theorem still hold when sampling a large population with replacement ?

What does it say?