4. 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 6.1 again. Suppose that x is a random variable with a normal distribution. Then x, the sample mean based on random samples of size n, also will have a normal distribution for any value of n 5 1, 2, 3, . . . .

What happens to the standard deviation of the x distribution as n (the sample size) increases? Consider the following table for different values of n.

n 1 2 3 4 10 50 100 s/1n, 1s 0.71s 0.58s 0.50s 0.32s 0.14s 0.10s 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 x is s/1n, 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 5 4, we cut the standard deviation of x by 50% of the standard deviation s of x. If we were to use a sample size of n 5 100, we would cut the standard deviation of x 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 m by using x 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.