Sampling Distributions

394

image

Overview

  1. 7.1 Central Limit Theorem for Means
  2. 7.2 Central Limit Theorem for Proportions
  3. Chapter 7 Formulas and Vocabulary
  4. Chapter 7 Review Exercises
  5. Chapter 7 Quiz

395

image Trial of the Pyx: How Much Gold Is in Your Gold Coins?

The kings of bygone England had a problem: How much gold should they put into their gold coins? After all, the very commerce of the kingdom depended on the purity of the currency. How did the lords of the realm ensure that the coins floating around the kingdom contained reliable amounts of gold?

From the year 1282, the Trial of the Pyx has been held annually in London to ensure that newly minted coins adhere to the standards of the realm. It is the responsibility of the presiding judge to ensure that the trial proceeds lawfully and to inform Her Majesty's Treasury of the verdict. Six members of the Company of Goldsmiths compose the jury, who are given two months to test the coins. It works like this: A ceremonial boxwood chest, called the Pyx, is brought forth, and a sample of 100 of the coins cast that year at the mint is put into it. The Pyx is then weighed. In times past, each gold coin, called a guinea, had an expected weight of 128 grams, so the total weight of the guineas in the Pyx was expected to be 12,800 grams.

If the weight of the coins in the Pyx was much less than 12,800 grams, the jury concluded that the Master of the Mint was cheating the crown by pocketing the excess gold, and he was severely punished. On the other hand, if the coins in the Pyx weighed much more than 12,800 grams, that wasn't good either, because it cut down on the profits produced by the kings' coin-minting monopoly.

By how much could the Master of the Mint debase the coinage before getting caught? We shall see in this chapter's Case Study, Trial of the Pyx, which unfolds in Section 7.1.

THE BIG PICTURE

Where we are coming from and where we are headed …

  • In Chapters 1–4, we learned ways to describe data sets using numbers, tables, and graphs. Then, in Chapters 5 and 6, we learned the tools of probability and probability distributions that allow us to quantify uncertainty.
  • Here, in Chapter 7, “Sampling Distributions,” we will discover that seemingly random statistics, such as the sample mean , have predictable behaviors. The special type of distribution we use to describe these behaviors is called the sampling distribution. This leads us to perhaps the most important result in statistical inference: the Central Limit Theorem.
  • The sampling distributions we learn about in this chapter form the basis for most of the statistical inference we perform in the remainder of the book. For example, in Chapter 8, “Confidence Intervals,” we will learn how to estimate an unknown parameter with a certain level of confidence.