1. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 11- 1

2. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Unit III AP Statistics Test Review

3. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 11- 3 A Simulation A simulation consists of a collection of things that happen at random. The most basic event is called a component of the simulation. Each component has a set of possible outcomes, one of which will occur at random.

4. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 11- 4 A Simulation (cont.) The sequence of events we want to investigate is called a trial. Trials usually involve several components. After the trial, we record what happened—our response variable. There are seven steps to a simulation…

5. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 11- 5 Simulation Steps 1.Identify the component to be repeated. 2.Explain how you will model the outcome. 3.Explain how you will simulate the trial. 4.State clearly what the response variable is. 5.Run several trials. 6.Analyze the response variable. 7.State your conclusion (in the context of the problem, as always).

6. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 12- 6 Idea 1: Examine a Part of the Whole The first idea is to draw a sample. We’d like to know about an entire population of individuals, but examining all of them is usually impractical, if not impossible. We settle for examining a smaller group of individuals—a sample—selected from the population.

7. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 12- 7 Idea 1: Examine Part of the Whole (cont.) Opinion polls are examples of sample surveys, designed to ask questions of a small group of people in the hope of learning something about the entire population. Professional pollsters work quite hard to ensure that the sample they take is representative of the population. If not, the sample can give misleading information about the population.

8. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 12- 8 Bias Samples that don’t represent every individual in the population fairly are said to be biased. Bias is the bane of sampling—the one thing above all to avoid. There is usually no way to fix a biased sample and no way to salvage useful information from it. The best way to avoid bias is to select individuals for the sample at random. The value of deliberately introducing randomness is one of the great insights of Statistics.

9. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 12- 9 Does a Census Make Sense? Why bother determining the right sample size? Wouldn’t it be better to just include everyone and “sample” the entire population? Such a special sample is called a census.

10. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 12- 10 Populations and Parameters Models use mathematics to represent reality. Parameters are the key numbers in those models. A parameter that is part of a model for a population is called a population parameter. We use data to estimate population parameters. Any summary found from the data is a statistic. The statistics that estimate population parameters are called sample statistics.

11. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 12- 11 Notation We typically use Greek letters to denote parameters and Latin letters to denote statistics.

12. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 12- 12 Simple Random Samples (cont.) We will insist that every possible sample of the size we plan to draw has an equal chance to be selected. Such samples also guarantee that each individual has an equal chance of being selected. With this method each combination of people has an equal chance of being selected as well. A sample drawn in this way is called a Simple Random Sample (SRS). An SRS is the standard against which we measure other sampling methods, and the sampling method on which the theory of working with sampled data is based.

13. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 12- 13 Stratified Sampling (cont.) Designs used to sample from large populations are often more complicated than simple random samples. Sometimes the population is first sliced into homogeneous groups, called strata, before the sample is selected. Then simple random sampling is used within each stratum before the results are combined. This common sampling design is called stratified random sampling.

14. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 12- 14 Stratified Sampling (cont.) Stratified random sampling can reduce bias. Stratifying can also reduce the variability of our results. When we restrict by strata, additional samples are more like one another, so statistics calculated for the sampled values will vary less from one sample to another.

15. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 12- 15 Cluster and Multistage Sampling Sometimes stratifying isn’t practical and simple random sampling is difficult. Splitting the population into similar parts or clusters can make sampling more practical. Then we could select one or a few clusters at random and perform a census within each of them. This sampling design is called cluster sampling. If each cluster fairly represents the full population, cluster sampling will give us an unbiased sample.

16. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 12- 16 Cluster and Multistage Sampling (cont.) Cluster sampling is not the same as stratified sampling. We stratify to ensure that our sample represents different groups in the population, and sample randomly within each stratum. Strata are homogeneous, but differ from one another. Clusters are more or less alike, each heterogeneous and resembling the overall population. We select clusters to make sampling more practical or affordable.

17. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 12- 17 Cluster and Multistage Sampling (cont.) Sometimes we use a variety of sampling methods together. Sampling schemes that combine several methods are called multistage samples. Most surveys conducted by professional polling organizations use some combination of stratified and cluster sampling as well as simple random sampling.

18. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 12- 18 Systematic Samples Sometimes we draw a sample by selecting individuals systematically. For example, you might survey every 10th person on an alphabetical list of students. To make it random, you must still start the systematic selection from a randomly selected individual. When there is no reason to believe that the order of the list could be associated in any way with the responses sought, systematic sampling can give a representative sample.

19. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 12- 19 Systematic Samples (cont.) Systematic sampling can be much less expensive than true random sampling. When you use a systematic sample, you need to justify the assumption that the systematic method is not associated with any of the measured variables.

20. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 12- 20 Who’s Who? The Who of a survey can refer to different groups, and the resulting ambiguity can tell you a lot about the success of a study. To start, think about the population of interest. Often, you’ll find that this is not really a well- defined group. Even if the population is clear, it may not be a practical group to study.

21. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 12- 21 Who’s Who? (cont.) Second, you must specify the sampling frame. Usually, the sampling frame is not the group you really want to know about. The sampling frame limits what your survey can find out.

22. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 12- 22 Who’s Who? (cont.) Then there’s your target sample. These are the individuals for whom you intend to measure responses. You’re not likely to get responses from all of them—nonresponse is a problem in many surveys.

23. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 12- 23 Who’s Who? (cont.) Finally, there is your sample—the actual respondents. These are the individuals about whom you do get data and can draw conclusions. Unfortunately, they might not be representative of the sample, the sampling frame, or the population.

24. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 12- 24 Who’s Who? (cont.) At each step, the group we can study may be constrained further. The Who keeps changing, and each constraint can introduce biases. A careful study should address the question of how well each group matches the population of interest.

25. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 12- 25 Who’s Who? (cont.) One of the main benefits of simple random sampling is that it never loses its sense of who’s Who. The Who in an SRS is the population of interest from which we’ve drawn a representative sample. (That’s not always true for other kinds of samples.)

26. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 12- 26 Who’s Who? (cont.)

27. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 12- 27 What Can Go Wrong?—or, How to Sample Badly Sample Badly with Volunteers: In a voluntary response sample, a large group of individuals is invited to respond, and all who do respond are counted. Voluntary response samples are almost always biased, and so conclusions drawn from them are almost always wrong. Voluntary response samples are often biased toward those with strong opinions or those who are strongly motivated. Since the sample is not representative, the resulting voluntary response bias invalidates the survey.

28. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 12- 28 What Can Go Wrong?—or, How to Sample Badly (cont.) Sample Badly, but Conveniently: In convenience sampling, we simply include the individuals who are convenient. Unfortunately, this group may not be representative of the population. Convenience sampling is not only a problem for students or other beginning samplers. In fact, it is a widespread problem in the business world—the easiest people for a company to sample are its own customers.

29. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 12- 29 What Can Go Wrong?—or, How to Sample Badly (cont.) Sample from a Bad Sampling Frame: An SRS from an incomplete sampling frame introduces bias because the individuals included may differ from the ones not in the frame. Undercoverage: Many of these bad survey designs suffer from undercoverage, in which some portion of the population is not sampled at all or has a smaller representation in the sample than it has in the population. Undercoverage can arise for a number of reasons, but it’s always a potential source of bias.

30. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 12- 30 What Else Can Go Wrong? Watch out for nonrespondents. A common and serious potential source of bias for most surveys is nonresponse bias. No survey succeeds in getting responses from everyone. The problem is that those who don’t respond may differ from those who do. And they may differ on just the variables we care about.

31. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 12- 31 What Else Can Go Wrong? (cont.) Don’t bore respondents with surveys that go on and on and on and on… Surveys that are too long are more likely to be refused, reducing the response rate and biasing all the results.

32. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 12- 32 What Else Can Go Wrong? (cont.) Work hard to avoid influencing responses. Response bias refers to anything in the survey design that influences the responses. For example, the wording of a question can influence the responses:

33. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 12- 33 What have we learned? A representative sample can offer us important insights about populations. It’s the size of the same, not its fraction of the larger population, that determines the precision of the statistics it yields. There are several ways to draw samples, all based on the power of randomness to make them representative of the population of interest: Simple Random Sample, Stratified Sample, Cluster Sample, Systematic Sample, Multistage Sample

34. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 13- 34 Observational Studies In an observational study, researchers don’t assign choices; they simply observe them. The text’s example looked at a student of the relationship between music education and grades. Since the researchers did not assign students to get music education and simply observed students “in the wild,” it was an observational study.

35. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 13- 35 Observational Studies (cont.) Because researchers in the text example first identified subjects who studied music and then collected data on their past grades, this was a retrospective study. Had the researchers identified subjects in advance and collected data as events unfolded, the study would have been a prospective study.

36. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 13- 36 Observational Studies (cont.) Observational studies are valuable for discovering trends and possible relationships. However, it is not possible for observational studies to demonstrate a causal relationship.

37. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 13- 37 Randomized, Comparative Experiments An experiment is a study design that allows us to prove a cause-and-effect relationship. An experiment: Manipulates factor levels to create treatments. Randomly assigns subjects to these treatment levels. Compares the responses of the subject groups across treatment levels. In an experiment, the experimenter must identify at least one explanatory variable, called a factor, to manipulate and at least one response variable to measure.

38. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 13- 38 Randomized, Comparative Experiments (cont.) In an experiment, the experimenter actively and deliberately manipulates the factors to control the details of the possible treatments, and assigns the subjects to those treatments at random. The experimenter then observes the response variable and compares responses for different groups of subjects who have been treated differently.

39. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 13- 39 Randomized, Comparative Experiments (cont.) In general, the individuals on whom or which we experiment are called experimental units. When humans are involved, they are commonly called subjects or participants. The specific values that the experimenter chooses for a factor are called the levels of the factor. A treatment is a combination of specific levels from all the factors that an experimental unit receives.

40. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 13- 40 The Four Principles of Experimental Design 1.Control: We control sources of variation other than the factors we are testing by making conditions as similar as possible for all treatment groups. 2.Randomize: Randomization allows us to equalize the effects of unknown or uncontrollable sources of variation. It does not eliminate the effects of these sources, but it spreads them out across the treatment levels so that we can see past them. Without randomization, you do not have a valid experiment and will not be able to use the powerful methods of Statistics to draw conclusions from your study.

41. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 13- 41 The Four Principles of Experimental Design (cont.) 3.Replicate: Repeat the experiment, applying the treatments to a number of subjects. The outcome of an experiment on a single subject is an anecdote, not data. When the experimental group is not a representative sample of the population of interest, we might want to replicate an entire experiment for different groups, in different situations, etc.

42. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 13- 42 The Four Principles of Experimental Design (cont.) 4.Block: Sometimes, attributes of the experimental units that we are not studying and that we can’t control may nevertheless affect the outcomes of an experiment. If we group similar individuals together and then randomize within each of these blocks, we can remove much of the variability due to the difference among the blocks. Note: Blocking is an important compromise between randomization and control, but, unlike the first three principles, is not required in an experimental design.

43. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 13- 43 Diagrams of Experiments It’s often helpful to diagram the procedure of an experiment. The following diagram emphasizes the random allocation of subjects to treatment groups, the separate treatments applied to these groups, and the ultimate comparison of results:

44. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 13- 44 Experiments and Samples Both experiments and sample surveys use randomization to get unbiased data. But they do so in different ways and for different purposes: Sample surveys try to estimate population parameters, so the sample needs to be as representative of the population as possible. Experiments try to assess the effects of treatments, and experimental units are not always drawn randomly from a population.

45. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 13- 45 Control Treatments Often, we want to compare a situation involving a specific treatment to the status quo situation. A baseline (“business as usual”) measurement is called a control treatment, and the experimental units to whom it is applied is called the control group.

46. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 13- 46 Blinding When we know what treatment was assigned, it’s difficult not to let that knowledge influence our assessment of the response, even when we try to be careful. In order to avoid the bias that might result from knowing what treatment was assigned, we use blinding.

47. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 13- 47 Blinding (cont.) There are two main classes of individuals who can affect the outcome of the experiment: those who could influence the results (subjects, treatment administrators, technicians) those who evaluate the results (judges, treating physicians, etc.) When every individual in either one of these classes is blinded, an experiment is said to be single-blind. When everyone in both classes is blinded, the experiment is called double-blind.

48. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 13- 48 Placebos Often simply applying any treatment can induce an improvement. To separate out the effects of the treatment of interest, we can use a control treatment that mimics the treatment itself. A “fake” treatment that looks just like the treatment being tested is called a placebo. Placebos are the best way to blind subjects from knowing whether they are receiving the treatment or not.

49. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 13- 49 Placebos (cont.) The placebo effect occurs when taking the sham treatment results in a change in the response variable. This highlights both the importance of effective blinding and the importance of comparing treatments with a control. Placebo controls are so effective that you should use them as an essential tool for blinding whenever possible.

50. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 13- 50 The Best Experiments… are usually: randomized. comparative. double-blind. placebo-controlled.

51. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 13- 51 Blocking When groups of experimental units are similar, it’s often a good idea to gather them together into blocks. Blocking isolates the variability due to the differences between the blocks so that we can see the differences due to the treatments more clearly. When randomization occurs only within the blocks, we call the design a randomized block design.

52. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 13- 52 Blocking (cont.) Here is a diagram of a blocked experiment:

53. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 13- 53 Blocking (cont.) In a retrospective or prospective study, subjects are sometimes paired because they are similar in ways not under study. Matching subjects in this way can reduce variability in much the same way as blocking.

54. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 13- 54 Blocking (cont.) Blocking is the same idea for experiments as stratifying is for sampling. Both methods group together subjects that are similar and randomize within those groups as a way to remove unwanted variation. We use blocks to reduce variability so we can see the effects of the factors; we’re not usually interested in studying the effects of the blocks themselves.

55. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 13- 55 Confounding When the levels of one factor are associated with the levels of another factor, we say that these two factors are confounded. When we have confounded factors, we cannot separate out the effects of one factor from the effects of the other factor.

56. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 13- 56 Lurking or Confounding A lurking variable creates an association between two other variables that tempts us to think that one may cause the other. This can happen in a regression analysis or an observational study. A lurking variable is usually thought of as a prior cause of both y and x that makes it appear that x may be causing y.

57. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 13- 57 Lurking or Confounding (cont.) Confounding can arise in experiments when some other variables associated with a factor has an effect on the response variable. Since the experimenter assigns treatments (at random) to subjects rather than just observing them, a confounding variable can’t be thought of as causing that assignment. A confounding variable, then, is associated in a noncausal way with a factor and affects the response. Because of the confounding, we find that we can’t tell whether any effect we see was caused by our factor or by the confounding factor (or by both working together).

58. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 13- 58 What have we learned? We can recognize sample surveys, observational studies, and randomized comparative experiments. These methods collect data in different ways and lead us to different conclusions. We can identify retrospective and prospective observational studies and understand the advantages and disadvantages of each. Only well-designed experiments can allow us to reach cause-and-effect conclusions. We manipulate levels of treatments to see if the factor we have identified produces changes in our response variable.

59. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 13- 59 What have we learned? (cont.) We know the principles of experimental design: Identify as many other sources of variability as possible so we can be sure that the variation in the response variable can be attributed to our factor. Control the sources of variability we can, and consider blocking to reduce variability from sources we recognize but cannot control. Try to equalize the many possible sources of variability that cannot be identified by randomly assigning experimental units to treatments. Replicate the experiment on as many subjects as possible.

60. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 13- 60 What have we learned? (cont.) We’ve learned the value of having a control group and of using blinding and placebo controls. We can recognize problems posed by confounding variables in experiments and lurking variables in observational studies.