Excursions in Modern Mathematics Sixth Edition

1. Excursions in Modern MathematicsSixth Edition Peter Tannenbaum

2. Chapter 13Collecting Statistical Data Censuses, Surveys, and Clinical Studies

3. Collecting Statistical DataOutline/learning Objectives • To identify whether a given survey or poll is biased. • To list and discuss the quality of several sampling methods. • To identify components of a well-constructed clinical study.

4. Collecting Statistical DataOutline/learning Objectives • To define key terminology in the data collection process. • To estimate the size of a population using the capture-recapture method.

5. Collecting Statistical Data 13.1 The Population

6. Collecting Statistical Data • Population Every statistical statement refers, directly or indirectly, to some group of individuals or objects. • N-value Given a specific population, an obviously relevant question is, “How many individuals or objects are there in that population?”

7. Collecting Statistical Data • Census The process of collecting data by going through every member of the population.

8. Collecting Statistical Data Over the last 45 years, the United States Fish and Wildlife Service has been able to keep a remarkably accurate tally of the number of bald eagle breeding pairs in the lower 48 states.

9. Collecting Statistical Data A tremendous amount of effort has gone into collecting and verifying these N-values, which, for a wildlife population, are of remarkable accuracy. The above figure summarizes the population numbers over the period 1963-2000.

10. Collecting Statistical Data 13.2 Sampling

11. Collecting Statistical Data • Survey The practical alternative to a census is to collect data only from some members of the population and use that data to draw conclusions and make inferences about the entire population. • Poll When the data collection is done by asking questions.

12. Collecting Statistical Data • Sample The subgroup chosen to provide the data. • Sampling The act of selecting a sample.

13. Collecting Statistical Data • Target population The most important step in a survey is to distinguish the population for which the survey applies. • Sampling frame The actual subset of the population from which the sample will be drawn.

14. Collecting Statistical Data Public Opinion Polls Selection bias When the choice of the sample has a built-in tendency to exclude a particular group or characteristics within the population. • Response rate The percentage of respondents out of the total sample. • Nonresponse bias When the response rate to a survey is low.

15. Collecting Statistical Data Convenience Sampling In convenience sampling the selection of which individuals are in the sample is dictated by what is easiest for the data collector. A classic example is when interviewers set up at a fixed location such as a mall or outside a supermarket and ask passersby to be a part of a public opinion poll.

16. Collecting Statistical Data Quota Sampling Quota sampling is a systematic effort to force the sample to be representative of a given population through the use of quotas– the sample should have so many women, so many men, so many blacks, so many whites, so many people living in urban areas, so many people living in rural areas, and so on.

17. Collecting Statistical Data 13.3 Random Sampling

18. Collecting Statistical Data • Random sampling Sampling methods that use randomness as part of their design. • Random sample Any sample obtained through random sampling.

19. Collecting Statistical Data Simple Random Sampling It is based on the same principle a lottery is. Any set of numbers of a given size has an equal chance of being chosen as any other set of numbers of that size.

20. Collecting Statistical Data Stratified Sampling The alternative to simple random sampling used nowadays for national surveys and public opinion polls. The basic idea of stratified sampling is to break the sampling frame into categories, called strata, and then randomly choose a sample from these strata.

21. Collecting Statistical Data 13.4 Sampling: Terminology and Key Concepts

22. Collecting Statistical Data • Statistic To describe any kind of numerical information drawn from a sample. • Parameter An estimate for some unknown measure of the population. • Sampling error To describe the difference between a parameter and a statistic used to estimate that parameter.

23. Collecting Statistical Data • Chance error The result of the basic fact that a sample, being just a sample, can only give us approximate information about the population. • Sampling variability Different samples are likely to produce different statistics for the same population, even when the samples are chosen in exactly the same way.

24. Collecting Statistical Data • Sample bias The result of choosing a bad sample and is a much more serious problem than chance error. • Sample proportion The size of the sample, denoted by n (to contrast with N, the size of the population). The ratio n/N is the sample proportion.

25. Collecting Statistical Data 13.5 The Capture-Recapture Method

26. Collecting Statistical Data The Capture-Recapture Method • Step 1. Capture (sample): Capture (choose) a sample of size n1, tag (mark, identify) the animals (objects, people), and release them back into the general population. • Step 2. Recapture (resample): After a certain period of time, capture a new sample of size n2, and take an exact head count of the tagged individuals. Let’s call this number k.

27. Collecting Statistical Data Small Fish in a Big Pond A large pond is stocked with catfish. As part of a research project we need to estimate the number of catfish in the pond. • Step 1. For our first sample we capture a predetermined number n1 of catfish, say n1 = 200. The fish are tagged and released unharmed back in the pond.

28. Collecting Statistical Data Small Fish in a Big Pond • Step 2. After giving enough time for the released fish to mingle and disperse throughout the pond, we capture a second sample of n2catfish. While n2 does not have to equal n1, it is a good idea for the two samples to be of approximately the same order of magnitude. Let’s say that n2 = 250. Of the 250 catfish in the second sample, 35 have tags (were part of the original sample).

29. Collecting Statistical Data Small Fish in a Big Pond • The ratio of tagged fish in the second sample is the same as the ratio of tagged fish in the pond. 35/250  200/N which in turn gives N 200 X 250/35  1428.57 A sensible conclusion is that there are approximately N = 1400 catfish in the pond.

30. Collecting Statistical Data Clinical Studies Terminology • Clinical study (trial). Studies concerned with determining whether a single variable or treatment can cause a certain effect. • Confounding variables. All other possible contributing causes that could produce the same effect in a clinical study.

31. Collecting Statistical Data Clinical Studies Terminology • Controlled study. The subjects are divided into two different groups. • Treatment group. Subjects receiving the actual treatment. • Control group. Subjects that are not receiving any treatment.

32. Collecting Statistical Data Clinical Studies Terminology • Randomized controlled study. The subjects are assigned to the treatment group or the control group randomly. • Placebo effect. A critical confounding variable from the generally accepted principle that just the idea that one is getting a treatment, can produce positive results.

33. Collecting Statistical Data Clinical Studies Terminology • Placebo. A make-believe form of treatment– a harmless pill, an injection of saline solution, or any other fake type of treatment intended to look like the real treatment. • Controlled placebo study. A controlled study in which the subjects in the control group are given a placebo.

34. Collecting Statistical Data Clinical Studies Terminology • Blind. A study in which neither the members of the treatment group nor the members of the control group know to which of the two groups they belong. • Double-blind study. A controlled placebo study in which neither the subjects nor the scientist conducting the experiment know which subjects are in the treatment group and which are in the control group.

35. Collecting Statistical Data Conclusion • Census • Sample/ Survey/ Sample Bias • Simple Random/Stratified Sampling • Confounding Variables • Controlled Study