Chapter 7, Problem 3LCWP

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 $\bar{x}$, 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 $\bar{x}$ distribution as n (the sample size) increases? Consider the following table for different values of n.

n	1	2	3	4	10	50	100
$\sigma /\sqrt{n,}$	1σ	0.71σ	0.58*7	0.50σ	0.32σ	0.14σ	0.10σ

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 $\bar{x}$ is $\sigma /\sqrt{n}$. 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 $\bar{x}$ by 50% of the standard deviation σ of x. If we were to use a sample size of n = 100. we would cut the standard deviation of $\bar{x}$ to 10% of the standard deviation σ of x.

Give the preceding discussion some thought and explain why you should get much better predictions for µ by using $\bar{x}$ 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.