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Explain what is wrong in each of the following statements.
(a) For large sample size n, the distribution of observed values will be approximately Normal.
(b) The 68-95-99.7 rule says that $\bar x$ should be within µ ± 2σ about 95% of the time.
(c) The central limit theorem states that for large n, µ is approximately Normal.
(a) For a large sample size, the distribution of observed values will be approximately the actual underlying distribution of the random process. If they come from a normal distribution, it'll look normal. If they comes from a uniform distribution, it'll look uniform.
(b) According to the 68-95-99.7 rule, the sample average should be within $\mu\pm2\frac{\sigma}{\sqrt{n}}$ about 95% of the time. Note that as $n$, the number of samples, goes up, the sample average is contained in a closer and closer ball to the theoretical average.
(c) The sample average of $n$ values will be approximately normally distributed. The sample average, or observed average, is often called $\hat{\mu}$. However, if you knew the actual distribution, then $\mu$, the theoretical average or expectation of the distribution, is a fixed, non-random number that is a property of the distribution.
