2. Excursions in Modern Mathematics, 7e: 13.1 - 2Copyright © 2010 Pearson Education, Inc. 13 Collecting Statistical Data 13.1The Population 13.2Sampling 13.3 Random Sampling 13.4Sampling: Terminology and Key Concepts 13.5The Capture-Recapture Method 13.6Clinical Studies

3. Excursions in Modern Mathematics, 7e: 13.1 - 3Copyright © 2010 Pearson Education, Inc. Every statistical statement refers, directly or indirectly, to some group of individuals or objects. In statistical terminology, this collection of individuals or objects is called the population. The first question we should ask ourselves when trying to make sense of a statistical statement is, “what is the population to which the statement applies?” The Population

4. Excursions in Modern Mathematics, 7e: 13.1 - 4Copyright © 2010 Pearson Education, Inc. In an ideal world, the specific population to which a statistical statement applies is clearly identified within the story itself. In the real world, this rarely happens because the details are skipped (mostly to keep the story moving along but sometimes with an intent to confuse or deceive) or, alternatively, because two (or more) related populations are involved in the story. The Population

5. Excursions in Modern Mathematics, 7e: 13.1 - 5Copyright © 2010 Pearson Education, Inc. Because of its iconic importance–the bald eagle is the national bird and the emblem of the United States–much has been said and written about bald eagle populations in North America, from their near extinction to their miraculous recovery (the details are explained briefly in the excerpt on the next slide, taken from a 2002 National Geographic article). Example 13.1The Return of the Bald Eagle

6. Excursions in Modern Mathematics, 7e: 13.1 - 6Copyright © 2010 Pearson Education, Inc. Example 13.1The Return of the Bald Eagle

7. Excursions in Modern Mathematics, 7e: 13.1 - 7Copyright © 2010 Pearson Education, Inc. In the context of populations, this is, in spirit, a story about the bald eagle population in the United States–more specifically the contiguous 48 states. Unlike many other animals and birds, bald eagles stay within the confines of a reasonably small geographical area, so it is possible to discuss bald eagle populations in a regional context such as the contiguous 48 states, or even populations within a specific state. Example 13.1The Return of the Bald Eagle

8. Excursions in Modern Mathematics, 7e: 13.1 - 8Copyright © 2010 Pearson Education, Inc. Interestingly enough, the case about the comeback of the bald eagle population is often made through the use of a proxy–the number of bald eagle breeding pairs (400 in the 1960s; over 6000 in 2000). Thus, upon closer scrutiny the story ties together two different but closely related populations: the overall bald eagle population within the contiguous 48 states (including chicks, adolescent birds, etc.) and the population of breeding pairs. Example 13.1The Return of the Bald Eagle

9. Excursions in Modern Mathematics, 7e: 13.1 - 9Copyright © 2010 Pearson Education, Inc. The former is the population of interest, but the latter is the population of convenience because breeding pairs are much easier to identify, track, and count. Example 13.1The Return of the Bald Eagle

10. Excursions in Modern Mathematics, 7e: 13.1 - 10Copyright © 2010 Pearson Education, Inc. Given a specific population, an obviously relevant question is, “How many individuals or objects are there in that population?” This number is called the N-value of the population. (It is common practice in statistics to use capital N to denote population sizes.) It is important to keep in mind the distinction between the N-value– a number specifying the size of the population–and the population itself. The N Value

11. Excursions in Modern Mathematics, 7e: 13.1 - 11Copyright © 2010 Pearson Education, Inc. Over a period of many years, the United States Fish and Wildlife Service was able to keep a remarkably accurate tally of the number of bald eagle breeding pairs in the contiguous 48 states. (As we discussed in Example 13.1, breeding pairs are used as a useful proxy for the health of the overall population.) A tremendous amount of effort has gone into collecting and verifying these N-values, which, for a wildlife population, are of remarkable accuracy. Example 13.2The Return of the Bald Eagle: Part 2

12. Excursions in Modern Mathematics, 7e: 13.1 - 12Copyright © 2010 Pearson Education, Inc. The figure on the next slide summarizes the population numbers over the period 1963– 2000. (No tallies were conducted in 1964–1973, 1975–1980, 1983, and 1985.) Since 2000 the bald eagle population has grown to the point that the U.S. Fish and Wildlife Service has discontinued the annual tallies. Example 13.2The Return of the Bald Eagle: Part 2

13. Excursions in Modern Mathematics, 7e: 13.1 - 13Copyright © 2010 Pearson Education, Inc. Example 13.2The Return of the Bald Eagle: Part 2

14. Excursions in Modern Mathematics, 7e: 13.1 - 14Copyright © 2010 Pearson Education, Inc. Andy has a coin jar full of quarters. He is hoping that there is enough money in the jar to pay for a new baseball glove. Dad says to go count them, and if there isn’t enough, he will lend Andy the difference. Andy dumps the quarters out of the jar, makes a careful tally, and comes up with a count of 116 quarters. Example 13.3N Is in the Eye of the Beholder

15. Excursions in Modern Mathematics, 7e: 13.1 - 15Copyright © 2010 Pearson Education, Inc. What is the N-value here? The answer depends on how we define the population. Are we counting coins or money? To Dad, who will end up stuck with all the quarters, the total number of coins might be the most relevant issue. Thus, to Dad, N = 116. Andy, on the other hand, is concerned with how much money is in the jar. If he were to articulate his point of view in statistical language, he would say that N = 29(dollars). Example 13.3N Is in the Eye of the Beholder

16. Excursions in Modern Mathematics, 7e: 13.1 - 16Copyright © 2010 Pearson Education, Inc. The word data is the plural of the Latin word datum, meaning “something given,” and in ordinary usage has a somewhat broader meaning than the one we will give it in this chapter. For our purposes we will use the word data as any type of information packaged in numerical form, and we will adhere to the standard convention that as a noun it can be used both in singular (“the data is…”), and plural (“the data are…”) forms. Data

17. Excursions in Modern Mathematics, 7e: 13.1 - 17Copyright © 2010 Pearson Education, Inc. The process of collecting data by going through every member of the population is called a census. The idea behind a census is simple enough, but in practice a census requires a great deal of “cooperation” from the population. For larger, more dynamic populations (wildlife, humans, etc.), accurate tallies are inherently difficult if not impossible, and in these cases the best we can hope for is a good estimate of the N- value. Census

18. Excursions in Modern Mathematics, 7e: 13.1 - 18Copyright © 2010 Pearson Education, Inc. The most notoriously difficult N-value question around is,”What is the N-value of the national population of the United States?” This is a question the United States Census tries to answer every 10 years–with very little success. The 2000 U.S.Census was the largest single peacetime undertaking of the federal government–it employed over 850,000 people and cost about $6.5 billion– and yet it missed counting between 3 and 4 million people (see story excerpt-next slide). Example 13.42000 Census Undercounts

19. Excursions in Modern Mathematics, 7e: 13.1 - 19Copyright © 2010 Pearson Education, Inc. Example 13.42000 Census Undercounts

20. Excursions in Modern Mathematics, 7e: 13.1 - 20Copyright © 2010 Pearson Education, Inc. Given the critical importance of the U.S. Census and given the tremendous resources put behind the effort by the federal government, why is the head count so far off? How can the best intentions and tremendous resources of our government fail so miserably in an activity that on a smaller scale can be carried out by a child trying to buy a base- ball glove? Example 13.42000 Census Undercounts

21. Excursions in Modern Mathematics, 7e: 13.1 - 21Copyright © 2010 Pearson Education, Inc. Article 1, Section 2, of the Constitution of the United States mandates that a national census be conducted every 10 years. The original intent of the census was to “Count heads” for a twofold purpose: taxes and political representation. Like everything else in the Constitution, Article 1, Section 2, was a compromise of many competing interests: The count was to exclude “Indians not taxed” and to count slaves as “three-fifths of a free Person.” CASE STUDY 1 THE U.S. CENSUS

22. Excursions in Modern Mathematics, 7e: 13.1 - 22Copyright © 2010 Pearson Education, Inc. Since then, the scope and purpose of the U.S. Census have been modified and expanded by the 14th Amendment and the courts in many ways: CASE STUDY 1 THE U.S. CENSUS ■ Besides counting heads, the U.S.Census Bureau now collects additional information about the population: sex, age, race, ethnicity, marital status, housing, income, and employment data. Some of this information is updated on a regular basis, not just every 10 years.

23. Excursions in Modern Mathematics, 7e: 13.1 - 23Copyright © 2010 Pearson Education, Inc. ■ Census data are now used for many important purposes beyond the original ones of taxation and representation: the allocation of billions of federal dollars to states, counties, cities, and municipalities; the collection of other important government statistics such as the Consumer Price Index and the Current Population Survey; the redrawing of legislative districts within each state; and the strategic planning of production and services by business and industry. CASE STUDY 1 THE U.S. CENSUS

24. Excursions in Modern Mathematics, 7e: 13.1 - 24Copyright © 2010 Pearson Education, Inc. ■ For the purposes of the Census, the United States population is defined as consisting of “all persons physically present and permanently residing in the United States.” Citizens, legal resident aliens, and even illegal aliens are meant to be included. CASE STUDY 1 THE U.S. CENSUS

25. Excursions in Modern Mathematics, 7e: 13.1 - 25Copyright © 2010 Pearson Education, Inc. Nowadays, the notion that if we put enough money and effort into it, all individuals living in the United States can be counted like coins in a jar is unrealistic. In 1790, when the first U.S. Census was carried out, the population was smaller and relatively homogeneous, as people tended to stay in one place, and, by and large, they felt comfortable in their dealings with the government. Under these conditions it might have been possible for census takers to count heads accurately. Taking a Census

26. Excursions in Modern Mathematics, 7e: 13.1 - 26Copyright © 2010 Pearson Education, Inc. Today’s conditions are completely different. People are constantly on the move. Many distrust the government. In large urban areas many people are homeless or don’t want to be counted. And then there is the apathy of many people who think of a census form as another piece of junk mail. Taking a Census

27. Excursions in Modern Mathematics, 7e: 13.1 - 27Copyright © 2010 Pearson Education, Inc. If the Census undercount were consistent among all segments of the population, the undercount problem could be solved easily. Unfortunately, the modern U.S. Census is plagued by what is known as a differential undercount. Ethnic minorities, migrant workers, and the urban poor populations have significantly larger undercount rates than the population at large, and the undercount rates vary significantly within these groups. Taking a Census

28. Excursions in Modern Mathematics, 7e: 13.1 - 28Copyright © 2010 Pearson Education, Inc. Using modern statistical techniques, it is possible to make adjustments to the raw Census figures that correct some of the inaccuracy caused by the differential undercount, but in 1999 the Supreme Court ruled in Department of Commerce et al. v. United States House of Representatives et al. that only the raw numbers, and not statistically adjusted numbers, can be used for the purposes of apportionment of Congressional seats among the states. Taking a Census