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.
1. Most people would agree that increased information should give better predictions. Discuss how sampling distributions actually enable better predictions by providing more information. Examine Theorem 7.1 again. Suppose that x is a random variable with a normal distribution. Then the sample mean based on random samples of size n, also will have a normal distribution for any value of n  1, 2, 3, . . .
What happens to the standard deviation of the X̅ distribution as n (the sample size) increases? Consider the following table for different values of n.

In this case, “increased information” means a larger sample size n. Give a brief explanation as to why a large standard deviation will usually result in poor statistical predictions, whereas a small standard deviation usually results in much better predictions. Since the standard deviation of the sampling distribution X̅ is σ/√n, we can decrease the standard deviation by increasing n. In fact, if we look at the preceding table, we see that if we use a sample size of only n = 4, we cut the standard deviation of by 50% of the standard deviation σ of x. If we were to use a sample of size n = 100, we would cut the standard deviation of to 10% of the standard deviation s of x. Give the preceding discussion some thought and explain why you should get much better predictions for μ by using X̅ from a sample of size n rather than by just using x. Write a brief essay in which you explain why sampling distributions are an important tool in statistics.

2. 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 8, 9, and 10. Put in a very elementary way, the central limit theorem states that as the sample size n increases, the distribution of the sample mean will always approach a normal distribution, no matter where the original 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 does not follow a normal or bell-shaped distribution. Then discuss what would happen to the sampling distribution X̅  if the sample size were increased. Sketch diagrams of the X̅ distributions as the sample size n increases.

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Π σινη, 1 Ισ 2 0.710 3 0.580 4 0.500 10 50 0.32σ 0.140 100 0.100