Asked what the central limit theorem says, a student replies, "as you take larger and larger samples from a population, the histogram of the sample values looks more and more Normal". Is the student right? Explain your answer.
My answer the student is wrong because the histogram of the sample values will look like the population distribution, whatever that distribution might look like as the sample size increases. The CLT says the sample mean follows a normal distribution with mean $u$ and variance $σ^2/n$ as the sample size goes to infinity. But CLT fails to distribution that has fat tails such as Cauchy Distribtion.
Is this right? If so, do you think I could add a little more?