Posted by ami on Sunday, October 14, 2012 at 12:14pm.
a) This one should state that the mean of the observed values will be approximately Normal, not that the distribution will be approximately Normal.
b) I don't know what the 68-95-99.7 rule actually is, but it'll probably say something about 68%, 95% and 99.7% of the distribution lying within one, two and three standard deviations of the mean respectively. It certainly won't say that the sample mean should lie within +/- 2 standard deviations of the mean of the actual mean about 95% of the time (even though it's actually true). It sounds to me as though the wrong rule is being quoted here.
c) The central limit theorem states that, given certain conditions, the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed. I'll guess that the problem with the statement given is that it doesn't say anything about how large is "large" needs to be, and it doesn't say anything about m being the mean of a number of independent random variables with finite mean and variance.
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