SOLUTION: I am trying to understand The Central Limit Theorem and do not quite understand the meaning where it states "the distribution of the mean will be approximately normal." Why is it a

Question 261280: I am trying to understand The Central Limit Theorem and do not quite understand the meaning where it states "the distribution of the mean will be approximately normal." Why is it advantageous to know that a distribution is approximately normal in The Central Limit Theorem? Thank you for your knowledge and help.
Found 2 solutions by drk, stanbon:
Answer by drk(1908) (Show Source): You can put this solution on YOUR website!
The basic idea is that the more people you include the more normal the curve will appear. This is the idea of the law of large numbers. If I remember correctly, we want at least 30 people as our n. There might be pictures that show if n = 5, n = 10, n = 15, n = 30, n = 50 in your book.If we know that a distribution is normal, then we can apply our z - and t -scores as well as use confidence intervals, and degrees of freedom. This leads to accept or fail to accept the null hypothesis in some tests.
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hope that helps.
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
I am trying to understand The Central Limit Theorem and do not quite understand the meaning where it states "the distribution of the means will be approximately normal."
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Example you have a set 999 people whose weights are normally distributed.
Say it has a mean of 100 lbs with a standard deviation of 8 lbs.
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You decide to randomly break the set into 333 groups of size 3,
Then you weigh each group and find the mean weight of each group.
You list all those sample means and you look at how they are
distributed.
The Central Limit Theormem says:
1. The mean of those sample means = the mean of the population (100)
2. The standard deviation of those sample means = (the std of the population)
divided by (the square root of the sample size) (8/sqrt(3)).
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The CLT has certain conditions on it but for problem purposes what
I listed are the relevant facts.
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Why is it advantageous to know that a distribution is approximately normal in The Central Limit Theorem? Thank you for your knowledge and help.
If the population is normal, sampling will tell you approximately where
the population mean is and what its standard deviation is.
That is what the CLT guarantees.
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Cheers,
Stan H.

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