REVIEW FOR THE PREVIOUS LECTURE

Transcription

1 Slide 2-1

2 Calculator: The same calculator policies as for the ACT hold for STT 315: It is highly recommended that you have a TI-84, as this is the calculator with statistical functions and would save you a lot of work in the assignments and exams. Some explanations how to use these functions on the TI-84 will be provided in recitation. FYI, the TI-83 does not have an inverse function for the t-distribution to calculate percentiles.

3 REVIEW FOR THE PREVIOUS LECTURE

4 What are statistics? A. Large databases kept by the government B. Tricks using data to make claims C. Particular calculations made from data D. Data used to prove anything you want

5 What are statistics? A. Large databases kept by the government B. Tricks using data to make claims C. Particular calculations made from data D. Data used to prove anything you want

6 Statistics A. Assesses risk B. Predicts results C. Helps us understand our world D. All of the above

7 Statistics A. Assesses risk B. Predicts results C. Helps us understand our world D. All of the above

8 Variables that are numbers are always quantitative. A. True B. False

9 Variables that are numbers are always quantitative. A. True B. False

10 A(n) holds information about the same characteristic for many cases. A. Element B. Population C. Variable D. Sample

11 A(n) holds information about the same characteristic for many cases. A. Element B. Population C. Variable D. Sample

12 What are data? A. A list of numbers B. Numbers or labels together with context A. Context only B. A list of labels

13 What are data? A. A list of numbers B. Numbers or labels together with context A. Context only B. A list of labels

14 A(n) is an individual about whom or which we have data. A. Sample B. Population C. Variable D. Case

15 A(n) is an individual about whom or which we have data. A. Sample B. Population C. Variable D. Case

16 The cases we actually examine to understand a larger group is the: A. Element B. Population C. Sample D. Variable

17 The cases we actually examine to understand a larger group is the: A. Element B. Population C. Sample D. Variable

18 All the cases we wish we knew about is the: A. Element B. Population C. Sample D. Variable

19 All the cases we wish we knew about is the: A. Element B. Population C. Sample D. Variable

20 The SPCA collects data about the dogs they house. Which is categorical? A. Breed B. Age C. Weight D. Veterinary costs

21 The SPCA collects data about the dogs they house. Which is categorical? A. Breed B. Age C. Weight D. Veterinary costs

22 School administrators collect data on the students attending the school. Which of the following is quantitative? A. Class ( freshman, sophomore, etc.) B. Grade point average C. Whether the student is in AP class D. Whether the student has taken the SAT

23 School administrators collect data on the students attending the school. Which of the following is quantitative? A. Class ( freshman, sophomore, etc.) B. Grade point average C. Whether the student is in AP class D. Whether the student has taken the SAT

24 Which of the following is not one of the W s? A. Who B. Whether C. How D. Why

25 Which of the following is not one of the W s? A. Who B. Whether C. How D. Why

26 Without which of the following two W s should we consider that we do not have data or useful information? A. Who and When B. Who and Why C. Why and How D. Who and What

27 Without which of the following two W s should we consider that we do not have data or useful information? A. Who and When B. Who and Why C. Why and How D. Who and What

28 What is it about chance outcomes being random that makes random selection seem fair? I. Nobody can guess the outcome before it happens. II. When we want things to be fair, usually some underlying set of outcomes will be equally likely. III. Random outcomes display personal stakes in a particular outcome. A. I and II B. II and III C. I and III D. All are not true.

29 What is it about chance outcomes being random that makes random selection seem fair? I. Nobody can guess the outcome before it happens. II. When we want things to be fair, usually some underlying set of outcomes will be equally likely. III. Random outcomes display personal stakes in a particular outcome. A. I and II B. II and III C. I and III D. All are not true.

30 Examples of random selection are: A. Flipping a fair coin B. Rolling a fair die C. Using a spinner D. All of the above

31 Examples of random selection are: A. Flipping a fair coin B. Rolling a fair die C. Using a spinner D. All of the above

32 Numbers generated from a computer are generally: A. Only integers B. Random C. Pseudorandom D. Biased

33 Numbers generated from a computer are generally: A. Only integers B. Random C. Pseudorandom D. Biased

34 A simulation always models imaginary situations. A. True B. False

35 A simulation always models imaginary situations. A. True B. False

36 Which of the following is not a step in building a simulation? A. Explain how you will model the component s outcome. B. State clearly what the response variable is. C. Run only one trial. D. State the conclusion.

37 Which of the following is not a step in building a simulation? A. Explain how you will model the component s outcome. B. State clearly what the response variable is. C. Run only one trial. D. State the conclusion.

38 Which of the following is the first step in building a simulation? A. Explain how you will model the component s outcome. B. State clearly what the response variable is. C. State the conclusion. D. Identify the component to be repeated.

39 Which of the following is the first step in building a simulation? A. Explain how you will model the component s outcome. B. State clearly what the response variable is. C. State the conclusion. D. Identify the component to be repeated.

40 Among a dozen eggs, three are rotten. A cookie recipe calls for two eggs; they will be selected randomly from that dozen. Which plan could be used to simulate the number of rotten eggs that might be chosen? I. Let 0, 1, and 2 represent the rotten eggs, and 3 11 the good eggs. Generate two random numbers 0 11, ignoring repeats. II. Randomly generate a 0, 1, or 2 to represent the number of rotten eggs you get. III. Since 25% of the eggs are rotten, let 0 = rotten and 1, 2, 3 = good. Generate two random numbers 0 3 and see how many 0 s you get. A. I only B. II only C. III only D. I or III E. I, II, and III

41 Among a dozen eggs, three are rotten. A cookie recipe calls for two eggs; they will be selected randomly from that dozen. Which plan could be used to simulate the number of rotten eggs that might be chosen? I. Let 0, 1, and 2 represent the rotten eggs, and 3 11 the good eggs. Generate two random numbers 0 11, ignoring repeats. II. Randomly generate a 0, 1, or 2 to represent the number of rotten eggs you get. III. Since 25% of the eggs are rotten, let 0 = rotten and 1, 2, 3 = good. Generate two random numbers 0 3 and see how many 0 s you get. A. I only B. II only C. III only D. I or III E. I, II, and III

42 We call each time we obtain a simulated answer to our question a. A. random variable B. event C. trial D. response

43 We call each time we obtain a simulated answer to our question a. A. random variable B. event C. trial D. response

44 Chapter 12 Sample Surveys

45 Background To make decisions, we need to go beyond the data at hand and to the world at large. Let s investigate three major ideas that will allow us to make this stretch

46 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.

47 Idea 1: Examine a Part of the Whole (cont.) Sampling is a natural thing to do. Think about sampling something you are cooking you taste (examine) a small part of what you re cooking to get an idea about the dish as a whole.

48 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.

49 Bias Selecting a sample to represent the population is more difficult than it sounds. Sampling methods that, by their nature, tend to over- or under- emphasize some characteristics of the population 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.

50 Idea 2: Randomize Randomization can protect you against factors that you know are in the data. It can also help protect against factors you are not even aware of. Randomizing protects us from the influences of all the features of our population, even ones that we may not have thought about. Randomizing makes sure that on the average the sample looks like the rest of the population.

51 Randomizing (cont.) Not only does randomizing protect us from bias, it actually makes it possible for us to draw inferences about the population when we see only a sample. Such inferences are among the most powerful things we can do with Statistics. But remember, it s all made possible because we deliberately choose things randomly.

52 Idea 3: It s the Sample Size How large a random sample do we need for the sample to be reasonably representative of the population? The fraction of the population that you ve sampled doesn t matter. It s the sample size itself that s important.

53 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.

54 Does a Census Make Sense? (cont.) There are problems with taking a census: It can be difficult to complete a census there always seem to be some individuals who are hard to locate or hard to measure. Populations rarely stand still. Even if you could take a census, the population changes while you work, so it s never possible to get a perfect measure. Taking a census may be more complex than sampling.

55 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.

56 Notation We typically use Greek letters to denote parameters and Latin letters to denote statistics.

57 Simple Random Samples We draw samples because we can t work with the entire population. We need to be sure that the statistics we compute from the sample reflect the corresponding parameters accurately. A sample that does this is said to be representative.

58 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.

59 Simple Random Samples (cont.) To select a sample at random, we first need to define where the sample will come from. The sampling frame is a list of individuals from which the sample is drawn. Once we have our sampling frame, the easiest way to choose an SRS is with random numbers. Individuals who may be in the population of interest, but who are not in the sampling frame, can not be included in any sample.

60 Simple Random Samples (cont.) Samples drawn at random generally differ from one another. Each draw of random numbers selects different people for our sample. These differences lead to different values for the variables we measure. We call these sample-to-sample differences sampling variability.

61 Stratified Sampling Simple random sampling is not the only fair way to sample. More complicated designs may save time or money or help avoid sampling problems. All statistical sampling designs have in common the idea that chance, rather than human choice, is used to select the sample.

62 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.

63 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.

64 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.

65 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.

66 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.

67 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.

68 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.

69 Example We need to survey a random sample of the 300 passengers on aflight from San Francisco to Tokyo. Name each sampling method described below: a) Pick every 10th passenger as people board the plane. Systematic Sampling b) From the boarding list, randomly choose 5 people flying first class and 25 of the other passengers Stratified Sampling c) Randomly generate 30 seat numbers and survey the passengers who sit there. SRS d) Randomly select a seat position (right window, right aisle, etc..) and survey all the passengers sitting in those seats. Cluster Sampling

70 Ws Why determines the population What identifies the parameter of the interest Who is the sample we actually draw The How, When and Where are given by the sampling plan.

71 The Valid Survey It isn t sufficient to just draw a sample and start asking questions. Before you set out to survey, ask yourself: What do I want to know? Am I asking the right respondents? Am I asking the right questions? What would I do with the answers if I had them; would they address the things I want to know?

72 The Valid Survey (cont.) These questions may sound obvious, but they are a number of pitfalls to avoid. Know what you want to know. Understand what you hope to learn and from whom you hope to learn it. Use the right frame. Be sure you have a suitable sampling frame. Time your instrument. The survey instrument itself can be the source of errors.

73 The Valid Survey (cont.) Ask specific rather than general questions. Ask for quantitative results when possible. Be careful in phrasing questions. A respondent may not understand the question or may understand the question differently than the way the researcher intended it. Even subtle differences in phrasing can make a difference.

74 The Valid Survey (cont.) Be careful in phrasing answers. It s often a better idea to offer choices rather than inviting a free response.

75 The Valid Survey (cont.) The best way to protect a survey from unanticipated measurement errors is to perform a pilot survey. A pilot is a trial run of a survey you eventually plan to give to a larger group.

76 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.

77 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.

78 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.

79 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.

80 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. Slide 1-80

81 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. Make sure the question wording is neutral. Many surveys, especially those conducted by special-interest groups, present one side of an issue before the question itself. Slide 1-81

82 Chapter 13 Experiments and Observational Studies

83 Observational Studies In an observational study, researchers don t assign choices; they simply observe them. The text s example looked at 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.

84 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.

85 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.

86 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.

87 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.

88 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.

89 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.

90 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. Replication of an entire experiment with the controlled sources of variation at different levels is an essential step in science.

91 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.

92 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:

93 Does the Difference Make a Difference? How large do the differences need to be to say that there is a difference in the treatments? Differences that are larger than we d get just from the randomization alone are called statistically significant. We ll talk more about statistical significance later on. For now, the important point is that a difference is statistically significant if we don t believe that it s likely to have occurred only by chance.

94 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.

95 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.

96 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.

97 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 all individuals 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.

98 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.

99 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.

100 Placebos (cont.t) The best experiments are usually: randomized. comparative. double-blind. placebo-controlled.

101 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.

102 Blocking (cont.) Here is a diagram of a blocked experiment:

103 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.

104 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.

105 Adding More Factors It is often important to include multiple factors in the same experiment in order to examine what happens when the factor levels are applied in different combinations.

106 Adding More Factors (cont.) For example, the following diagram shows a study of the effects of different fertilizer/water combinations on the juiciness and tastiness of tomatoes:

107 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.

108 What Can Go Wrong? Don t give up just because you can t run an experiment. If we can t run an experiment, often an observational study is a good choice. Beware of confounding. Use randomization whenever possible to ensure that the factors not in your experiment are not confounded with your treatment levels. Be alert to confounding that cannot be avoided, and report it along with your results.

109 What Can Go Wrong? (cont.) Bad things can happen even to good experiments. Protect yourself by recording additional information. Don t spend your entire budget on the first run. Try a small pilot experiment before running the full-scale experiment. You may learn some things that will help you make the full-scale experiment better.