Chapter 6, Problem 5LC

Discuss each of the following topics in class or review the topics on your own. Then write a brief but complete essay in which you summarize the main points. Please include formulas and graphs as appropriate.

In a way, the central limit theorem can be thought of as a kind of “grand central station.” It is a connecting hub or center for a great deal of statistical work. We will use it extensively in Chapters 7, 8, and 9. Put in a very elementary way, the central limit theorem states that as the sample size n increases, the distribution of the sample mean $\bar{x}$ will always approach a normal distribution, no matter where the original $\bar{x}$ variable came from. For most people, it is the complete generality of the central limit theorem that is so awe inspiring: It applies to practically everything. List and discuss at least three variables from everyday life for which you expect the variable x itself not to follow a normal or bell-shaped distribution. Then discuss what would happen to the sampling distribution $\bar{x}$ if the sample size were increased. Sketch diagrams of the $\bar{x}$ distributions as the sample size n increases.