>    I am relatively new to statistics. I am trying to see if I get the meaning of both of the above; the CLT and the LLN, specifically with respect to using the sampling mean to estimate the actual population mean (assuming we know the true pop. standard deviation.)
>  I would appreciate it if you could critique this post.

>   Here is what I understand:

>  CLT:
> We draw random samples X_1,..X_n , each of large-enough size N (I think N>30 is usually the cutting point.). For each sample X_i, we find the mean m_i of the values obtained in the sample, i.e., we define:

>      m_i = (X_i1+ X_i2+...+X_iN)/N  

>   Then, as the size n of the sample grows, the distribution of the m_i is approximately normal, or approaches a normal distribution ( in different ways of convergence).

>   And, by the LLN, the mean of the distribution of the m_i approaches the true population mean mu, and the standard deviation of the m_i approaches sigma/(N^1/2).

> In terms of using the sampling mean for estimating
> the pop. mean: we cannot tell exactly where the actual pop. mean mu is, but we can use the above distribution properties of m_i to estimate or give an interval where mu may be, with certain degree of probability(confidence):

>   We take a random sample of size N, and find the mean x^ of the sample values.

>   We then   use a normal distribution
>  N(x^, sigma/N^1/2)), where sigma is the true pop. mean.

>   Then the probability that the true population mean is in an interval (x^-a,x^+a) is given by (the z-table value of ):

>          Z_a=(a-x^)/N^1/2  

>     Conversely, a confidence interval of confidence level C will give us a probability of C that the mean
> will be found in the interval. Also: if we were to take
> many samples , the true population mean would fall inside of the interval with probability C.

>  Thanks For your Comments.