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| Science Forum Index » Statistics - Math Forum » Interpretation of CLTheorem, LLN ;Law of Large... |
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| Bacle... |
 Posted: Wed Nov 04, 2009 6:47 am |
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Hi, everyone:
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. |
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| Paige Miller... |
| Posted: Wed Nov 04, 2009 7:48 am |
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On Nov 4, 11:47 am, Bacle <ba... at (no spam) yahoo.com> wrote:
[quote]Hi, everyone:
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.
[/quote]
While there are a few things in your description that I might word a
little differently, in general your description of CLT and LLN seems
good.
But ... you can use the sample mean to estimate the population mean
without ever invoking CLT or LLN, and whether we know the true
population standard deviation or not. Sample mean is an unbiased
estimate of population mean, and you don't need CLT or LLN or
population standard deviation to prove that.
--
Paige Miller
paige\dot\miller \at\ kodak\dot\com |
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