SAMPLES AND STATS(Managing Data) Grades 9-10 Skills and Objectives: * Students will identify different sampling methods. * Students will design and conduct surveys using sampling methods. Suggested Groupings-Partners, small groups Getting Started: * Have students share any prior knowledge of sampling they may have. Discuss the idea that sampling makes it possible to gather information about a population when surveying every member is impossible or impractical. Educators, advertisers, and policymakers all use information gathered through sampling. The U.S. Census Bureau also develops and uses sampling techniques to gather information about the U.S. population. For example, while the Census Bureau distributes both a short and long census questionnaire, only one in six households will get a long form for Census 2000. This sample is large enough to provide the data to accurately describe the U.S. population. * Explain to students that this activity will introduce them to a variety of sampling methods and the multiple steps involved in the process of conducting a survey to gather statistical information. Make sure they understand that the sampling process they will use to obtain data is much simpler than the methods used by the U.S. Census Bureau and others. Using the Activity Workshe0s: * Distribute copies of pages 16 and 17 to students. Have them read and complete the activity on page 16. Alternatively, you may wish to do this activity as a class. * Then have students read and discuss the section on bias at the top of page 17. Ask: Can you come up with your own example of a biased sample? (For example, if you conduct a survey of your classmates by e-mail, you will automatically exclude all class members who do not have access to e-mail.) What steps can researchers take to ensure that the studies they design are not biased? * Before students begin, discuss the difference between the type of survey the Census Bureau conducts and a poll. The Census Bureau uses surveys to collect and analyze social, economic, and geographic data. A poll is a survey that is used to measure attitudes and opinions. Go over these steps with them. 1. Choose a survey question. Make sure students choose a question that asks for factual information, like age or education level, rather than an attitude or opinion. 2. Identify the target population and sample size. 3. Decide on the sample method. 4. Conduct the survey and interpret, tabulate, and graph or map results. * To demonstrate, choose your own question and do a quick survey with your students. Wrapping Up: * Have each group present their surveys and results. Ask a spokesperson for each group to discuss the target population, sample size, and sample method used in the survey. Have students share their conclusions. * Have students conduct further research about the sampling methods presented here. Have the class agree on one survey question. Divide the class into three groups, and have each group use a different sampling method. Be sure each group uses the same size sample. Then invite the groups to compare results. Alternatively, student groups could use the same sampling method on different sample sizes. * Students can visit the U.S. Census Bureau Web site (www.census.gov) to get information from surveys conducted on such subjects as computer use, crime, education, etc. Click on "Subjects A-Z" and choose "S" then "Surveys. Answers: Page 16: 1. Cluster sampling. 2. Random sampling. 3. Systematic sampling. Chalkboard Definitions sampling: using a finite part of a statistical population for study, in order to gain information about the whole survey: a set of questions asked of a specific population to collect data for analysis. poll: a survey that measures attitudes and opinions. Lesson 5 Activity Worksheet SAMPLES AND STATS * Sampling is a scientific technique used to obtain as accurate a figure or measurement as possible, when an exact count cannot be taken. By measuring a scientifically selected portion of a population, it is possible to describe the characteristics of the entire population. Below is a chart describing three different scientific sampling methods. The U.S. Census Bureau's long form is an example of systematic sampling. For Census 2000, a systematic sampling of approximately 1 in every 6 households will receive the long form, and an average of 5 out of every 6 households will receive the short form. Although the long form doesn't go to every household, information from these forms can be used to accurately describe the entire U.S. population. Here are three different sampling methods: Random Sampling-Each individual in the population has an equal chance of being selected. Example: To take a random sample of students in your school, you could write the name of each student on a slip of paper, then choose slips at random. Cluster Sampling-Groups, rather than individuals, are randomly selected. Example: You might randomly select certain classes, then interview every student in only those classes. Systematic Sampling-A rule, or pattern, that applies to a population is used to make selections. Example: Using an alphabetical list of students, count off by 6, and select every 6th student on the list. * Test your understanding of different sampling techniques. Draw lines to match the sampling methods with their types. 1. Choose any three pages from the telephone book at random, and call everyone on those pages. 2. Choose 100 telephone numbers at random from the entire book. 3. Choose every 100th listing in the telephone book. a. Random Sampling b. Systematic Sampling c. Cluster Sampling * When choosing a sampling method, you need to beware of hidden biases. For example, imagine that you want to know if teenagers today are taller than teenagers in the past. You've found information about the average height of students in your school in 1940 and 1970. Now you need to find out the average height of students in your school today. You probably don't want to get the height data from a sample consisting of members of the school basketball team! Why not? * Design your own sample survey. 1. Acting as your school's census bureau, identify a characteristic of interest or importance to your school and choose a survey question. (Topic examples- transportation to and from school, team sports or other extracurricular activities, foreign languages studied, etc.) For some of these topics, you may be able to check the accuracy of your survey results against actual tallies your school keeps. Be sure not to ask questions about attitudes or opinions. Write your topic and survey question here: 2. Choose your target population. The target population is the group of people to whom you want the sample survey to apply. For instance, a survey about a school-related question could apply to the students in your grade or to the whole student body. Make sure you survey a good sample of your target population. (For example, if your survey applies to a student body of 400, you might want to talk to at least 10%, or 40 people.) Write your target population and sample size here: 3. Based upon the steps above, which sampling method would you choose for your survey? Why? 4. Now conduct your sample survey and tabulate the results. Then organize your results into a graph or table and add a narrative summary. Share your graph, or table and summary, with the class. FORECASTING THE FUTURE(Managing Data) Grades 11-12 Skills and Objectives: * Students will learn about population estimates and population projections. * Students will compare population projections based on numerical (arithmetic) growth and on percent (geometric) growth. Getting Started: * Introduce the lesson by discussing the following terms that are defined in the lesson as they relate to population: enumerations, estimates, projections, components of population change, births, deaths, and net migration. Help the students understand that information about the U.S. population is important for a variety of purposes, including planning in both the public sector (e.g., where to build schools and hospitals) and the private sector (e.g., store location and marketing), and that population figures are used in determining federal and state fund allocations. Using the Activity Worksheets: Distribute copies of pages 19 and 20 to students and discuss the problems with them. Have students individually, or in pairs, calculate the answers to questions 1 through 11. Then with the entire class, discuss answers to questions 12 through 16. Population estimates and projections: Discuss with students how U.S. Census Bureau population estimates and projections are actually done, and explain that the methodology used by Census Bureau demographers is more complicated than the hypothetical examples given here. There can be many assumptions and variables involving the set of components (fertility, mortality, and net migration) that contribute to the population growth estimates and projections the U.S. Census Bureau publishes. For further information on population estimates: www.census.gov/population/www/estimates/concepts.html For further information on population projections: www.census.gov/population/www/projections/aboutproj.html Answers: 1. 32,621,613. 2. 254,899 and 8.4 percent. 3. 568,996 and 14.9 percent. 4. 895,990 and 34.6 percent. 5. 1,889,829 and 106.4 percent. 6. Answers will vary. 7. 3,542,015 and 3,563,234. 8. 4,944,095 and 5,026,989. 9. 4,382,693 and 4,693,102. 10. 5,555,057 and 7,565,031. 11. Answers will vary. 12. Because the percent increase is applied to a larger population in 1990 than in 1970. 13. Arizona. Because Arizona had the highest percent increase in population during the 1970-1990 period, it has the largest proportionate difference between a population projection for the year 2010 based on numerical growth versus percent growth. 14. The population projection based on percent change would be larger because the percent decline would be applied to the smaller 1990 population. 15. Calculate one-half the numerical growth of the 1970-1990 period and then add it to the 1990 population. 16. Calculate the ratio of the 1990 to the 1970 population (to six decimal places to minimize rounding error), then take the square root of the ratio and convert it to a percent increase. Multiply the percent increase by the 1990 population, then add the product to the 1990 population. You can't assume one-half of the percent growth for the 1970-1990 period because of the compounding effect of a geometric rate of increase - an analogy would be compound interest rates. Taking South Carolina as an example, the ratio of its 1990 to its 1970 population is 1.345847. The square root of 1.345847 is 1.160, yielding a 16 percent increase in population in the 1990-2000 decade. The increase of 557,872 added to the 1990 population of 3,486,703 yields a population projection for the year 2000 of 4,044,575. Chalkboard Definitions rate: a standard amount used to calculate a total, as in a percentage change in population over the course of a year. population estimate: a conclusion about the past or present population based on existing data. population projection: computation of future changes in population size based on assumptions about births, deaths, and migration. Lesson 6 Activity Worksheet FORECASTING THE FUTURE * Enumerations, estimates, and projections of population The U.S. Census Bureau produces three basic types of information about the U.S. population: enumerations, estimates, and projections. Enumerations are counts of the population such as in the 1990 census of population. Estimates are calculations of the population for a recent date and are usually based on the last census as well as on information about population change since the last census. Projections are calculations of the population for a future date and are usually based on the last census or estimate, and on assumptions about future population growth or decline. * Population Estimates The three basic components of population change between two dates are births, deaths, and net migration. For population estimates for states, net migration may be divided into net international migration (immigration to the United States minus emigration from the United States) and net domestic migration (in-migration from other states minus out-migration to other states). For California, the population in 1990 was 29,785,857. For the 1990-1998 period, data on the components of population change show the following: births (B) = 4,708,894, deaths (D) = 1,810,698, net international migration (NIM) +2,019,488, and net domestic migration (NDM) = -2,081,928. Calculate the 1998 population estimate for California using the following formula: 1. P(1998)=P(1990) + B - D + NIM + NDM * Population Projections To make population projections for the United States or for individual states, demographers make assumptions about future trends in the components of population change. These assumptions, which reflect professional judgment and take into account past trends, are made in terms of rates for births and deaths, and in terms of rates or numbers for migration. For simplicity, the population projections discussed below are based on assumptions about past trends in total population, not on assumptions about each component of population change. Table 1 shows the 1970 and 1990 census populations for four states, all with populations that increased between 1970 and 1990. Calculate numerical growth (1990 population minus 1970 population) and percent growth (population growth as a percent of 1970 population, with percent change rounded to one decimal place). Table 1. Population of Selected States: 1970 and 1990 2. Connecticut in 1970 had 3,032,217, and had 3,287,116 people in 1990. 3. Minnesota had 3,806,103 in 1970, and had 4,375,099 people in 1990. 4. South Carolina had 2,590,713 in 1970, and had 3,486,703 in 1990. 5. Arizona had 1,775,399 in 1970, and in 1990 had 3,665,228. 6. Your State (Also pictured: two empty columns; "Population growth, 1970-1990", "Numerical", "Percent") Calculate population projections for each state for the year 2010 assuming a continuation of trends for the 1970-1990 period: first based on numerical change (an arithmetic rate of change), then based on percent change (a geometric rate of change) with the results rounded to the nearest integer. Table 2. Population Projections for Selected States: 2010 7. Conneticut(based on numerical, based on percent change) 8. Minnesota(based on numerical, based on percent change) 9. South Carolina(based on numerical, based on percent change) 10. Arizona(based on numerical, based on percent change) 11. Your State(based on numerical, based on percent change) Questions about population projections 12. Why are the population projections for the year 2010 larger when based on percent change than when based on numerical change for the 1970-1990 period? 13. For which of the first four states is the proportionate difference between the two projections the largest and why? 14. If the population of a state had declined between 1970 and 1990, which population projection-numerical change or percent change - would be larger for the year 2010 and why? 15. How would you use the data in Table 1 to project population for states for the year 2000 assuming past trends in numerical population change? 16. How would you use the data in Table 1 to project population for states for the year 2000 assuming past trends in percent population change? Selected Census 2000 Short Form Questions 1. What is this person's sex? Male/Female 2. What is this person's age and date of birth? (Print numbers in boxes) Age on April 1, 2000 __ Month of Birth __ Day of Birth __ Year of Birth ____ Note: Please answer BOTH questions 3 and 4. 3. Is this person Spanish/Hispanic/Latino? * No, not Spanish/Hispanic/Latino * Yes, Mexican, Mexican American, Chicano * Yes, Puerto Rican * Yes, Cuban * Yes, other Spanish/Hispanic/Latino-Print group below. 4. What is this person's race? Mark one or more races to indicate what this person considers himself/herself to be. * White * Black, African American, or Negro * American Indian or Alaska Native-Print name of enrolled or principal tribe below. * Asian Indian * Japanese * Chinese * Korean * Filipino * Vietnamese * Other Asian-Print race below. * Native Hawaiian * Guamanian or Chamorro * Samoan * Other Pacific Islander-Print race below. * Some other race-Print race below. View the Census 2000 questionnaire on the U.S. Census Bureau Web site (www.census.gov).