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.

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#### Best Answer

(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 $mupm2frac{sigma}{sqrt{n}}$ 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.

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