Central Limit Theorem and . Under what conditions do they apply?

Question

So, I'm a beginner statistics student who's confused a bit about conditions for Central Limit Theorem and confidence intervals.

CLT

I just read that the Central Limit Theorem (CLT) says that the distribution of sample statistics are nearly normal, centered at the population mean, and with a standard deviation equal to the population standard deviation divided by the square root of the sample size. The formula is something like this:

$$ \overline{x} \sim N(mean = \mu, SE = \frac{\sigma}{\sqrt{n}})$$

_{*If the population SD is unknown which is often the case, then use the standard deviation of the sample as your best estimate.}

Additionally, it looks like there are some conditions for ClT:

1. Independence.

   • The sampled obervsations must be independent
   • random sampling should be done.
   • if sampling without replacement, the sample should be less than 10% of the population.

2. Sample skew

   • The population distribution should be normal
   • But if the distribution is skewed, the sample must be large (greater than 30)

Why is this? If we take many samples of the population, why does CLT only hold true if the sampled observations are independent? What's the intuition behind this? Why does skewness of the population matter? Why is it that when the population is more skewed, we need larger samples? What's the intuition here? I'm not looking for too deep of a math proof at the moment.

My hunch is that with larger samples, we can start overcoming the skewness of the population distribution and that the sample distributions start looking normal, abiding by the CLT again. I guess we just need more observations when there's skewed data in order for our sample mean to be close to the population mean when there's a skewed population?

Confidence Intervals

So the video I'm watching says that the confidence interval is based on CLT so the conditions for confidence intervals are similar. How is the confidence interval based on CLT?

The confidence interval I see as defined as: $$ \overline{x} \pm z * \dfrac{s}{\sqrt{n}}$$

The conditions looks similar. In fact, they are identical.

1. Since the method is based on the CLT, the conditions are the same as CLT (again, how is this based on CLT?)