AP Stats- Unit 3 How to use this Study Guide: 1.) PRINT IT OUT 2a.) If you don’t understand this material, read through the notes 2b.) If you have a foundational understanding of the material, start on Part 2 Questions and refer to the notes if you don’t understand 3.) Write “Why” each of the 10 answers is what it is 4.) Go over Part 3 (the exams) until you’re comfortable with the answers Part I.) Unit 3 Notes: This unit is all about the APPROPRIATE way to collect data. If you don’t collect data in an appropriate way it is not at all useful. This unit is full of vocabulary o “the vocabulary of statistics” Know the difference between an Experiment and an Observational Study… o the vocabulary word they are looking for is RANDOM placement into TREATMENT groups. When you talk about BIAS… o talk about the groups in the population that will be over-represented and underrepresented in your sample as a result of the bias. o There is a difference between bias that comes from the sampling TECHNIQUE and bias that results from the QUESTION. Questioning bias is known as RESPONSE bias. Make sure you can run a simulation and design an experiment. Make sure you know how to create a plan for random selection A well-designed experiment with random placement into treatment groups can show CAUSE and EFFECT. Randomness and Simulation (Ch. 11): Random: the heart of inferential statistics Everything has to be random to properly evaluate Table of Random Digits They are constructed so that every digit has an equal chance of appearing as the next digit. That doesn’t mean that you will find an equal number of each digit in an actual table—it means that the process that was used should produce equal proportions in the long run. They are constructed so that each digit is independent of the others. o Knowing the current digit doesn’t give you any information (change the probability) about the digit in the next position. Usually, the digits are written in groups of five with a space between them. o There is no meaning in this—it’s just there to keep your eyes from losing their place. o The lines of a table are often numbered. These lines have no meaning o It doesn’t matter how you read it, technically, but read it left to right for AP Simulation: models random events by using random numbers to specify outcomes with relative frequencies that correspond to the true real-world relative frequencies we are trying to model Sometimes collecting actual, real data is either difficult or impossible. Thus, we sometimes have to simulate events. Describing One Trial of a Simulation 1.) Describe how you will assign groups of digits to each component (“event”). Every event needs a number with the same number of digits as every other event o this is often handled by “padding” a number with leading zeroes. o In some cases, this may be very simple and straightforward…in others, you may need to think carefully. o Be clear and thorough when describing how you are assigning digits. 2.) Describe how you will read digits from the table. o This is usually just a statement like “I’ll read two digits from the table.” 3.) Describe what you will do with the digits you read. o How will you decide whether to accept or discard these digits? 4.) Describe how you’ll know when to stop o we call this a Stopping Rule 5.) Describe what will be measured from your trial o Ex: Number of missed shots? Average waiting time? 6.) Be sure to state how many trials you will complete. o The question should tell you how many to do… o Usually, it will be three. Implementing the Simulation o Basically, just do what you described o Remember to show your work—usually, this will include marking on some digits that are provided in the question. Examples [1.] Let’s simulate dealing cards from a standard deck until two red cards are drawn. In particular, how many cards does it take for this to happen? Use three trials of your simulation. 1. The event is drawing one card. 2. I’ll model drawing a card by reading two digits. o The numbers 01 through 52 will represent cards. o Even numbers will represent red cards and odd numbers will represent black cards. o Any number that is not between 01 and 52 will be discarded. o Repeated numbers will also be discarded (you can’t draw the same card twice!). 3. I’ll read from a table of random digits until I’ve read numbers that represent two red cards. 4. What I’ll measure is the number of cards that were drawn. 5. Now for the table…here’s a sample of one, followed by how I read it: o 81294 47895 20282 58439 55920 o 48258 72810 38198 12210 30524 Trial 1: o 81: discard (not a card) o 29: a black card o 44: a red card o 78: discard o 95: discard o 20: a red card. o This ends trial 1. I measured three cards needed to get one red. Trial 2: 28 (red), 25 (black), 84 (discard), 39 (black), 55 (discard), 92 (discard),04 (red). o That was 4 cards. Trial 3: 82(discard), 58(discard), 72(discard), 81(discard), 03 (black), 81(discard), 98 (discard), 12 (red), 21 (black), 03 (black), 05 (black), 24 (red). o That was 6 cards. That was an average of 4.33 cards drawn in order to find two red cards. [2.] Let’s simulate rolling a die three times and finding the sum of the digits. Again, we’ll conduct three trials. 1. The event is rolling a die. 2. I’ll model rolling a die by reading a single digit. o One through six indicate an actual roll; o all other digits will be discarded. o Repeated digits are allowed and accepted. 3. I’ll read digits until I’ve accepted three rolls. 4. What I’ll measure is the sum of the rolls. 5. Here are some more random digits. o 18820 57491 90844 98386 61939 o 55427 44188 57080 69890 80117 Trial 1: o 1: I rolled a 1. o 8: discard o 8: discard o 2: I rolled a 2. o 0: discard o 5: I rolled a 5. o That’s the third roll, so the simulation is over. The sum is 8 Trial 2: 7(x), 4, 9(x), 1, 9(x), 0(x), 8(x), 4. o Those dice added up to 9. Trial 3: 4, 9(x), 8(x), 3, 8(x), 6. o That’s a sum of 13. 9 + 8 + 13 = 30 o Average = 10 Surveys and Bias (Ch. 12): Sample survey- a study that asks questions of a sample drawn from some population in the hope of learning something about the entire population. What does it mean for a sample to be representative of a population? This means that the small sample basically covers what the entire population thinks or does. The best way to get a sample to represent the population is by picking the sample at random. Bias: a sample that does not represent the population in some way, it overlooks an important group It results from a poor sampling method Randomization helps reduce bias Increasing the sample size does NOT reduce bias A sample that is bias fails to represent its population accurately Common errors include (that result in bias): o relying on voluntary response o undercoverage of the population o nonresponse bias o response bias Randomization protects us from influences of all the features of a population, even effects we weren't aware of. o It makes sure that on the average the sample looks like the rest of the population. Randomization: process by which each individual is given a fair chance of selection into a sample o Necessary for samples to be unbiased sampling variability: the natural tendency of randomly drawn samples to differ, one from another o If I break up a class into 5 random groups and measure their average weight, it will be different for each team o Amount that it’s different = sampling variability Types of Biases voluntary response bias: type of bias that is problematic because those who volunteer tend to have strong negative opinions o Sampling design where individuals can choose on their own whether to participate in the sample o Bias introduced to a sample when individuals can choose on their own whether to participate in the sample o Voluntary Response Samples: they are usually biased towards those with strong opinions or who are strongly motivated. People with very negative opinions tend to respond more often than those with equally strong positive opinions. Samples based on voluntary response are always invalid and cannot be recovered, no matter how large the sample size. nonresponse bias: Bias introduced to a sample when a large fraction of those sampled fail to respond. o those who do respond are likely to not represent the entire group o Voluntary response is a form of nonresponse bias...but it may occur for many other reasons. o Nonresponse is a problem in many surveys Calling during dinnertime many ppl say “fuck that” Also, people who are at work during the day won't respond to a telephone survey only conducted during the workday. response bias: bias that is caused by a survey design that influences responses o Ex: the words that are used to persuade people to answer questions more favorable to the cause, or if there is something that could be gained if you answer the survey undercoverage: type of bias that is problematic because some groups are not represented in the sample o a sampling scheme that does not include some part of the population and/or makes them have less representation than it does have in the population causes bias o An example is in a telephone survey, when a telephone survey is conducted and you often eat out, you are less likely to be surveyed (which is a source of undercoverage.) Measurement bias: introduced when the measurement process tends to give results that differ (systematically) from the population. o For example: if a light meter is not properly calibrated, then the measurements it gives will not be correct! o A common source of measurement bias is wording bias—the way in which a question is worded can often have an effect on the responses. o Some people use the terms measurement bias and response bias synonymously, but they don’t quite mean the same thing. Measurement bias: refers exclusively to problems with the measurement device Response bias: refers to anything that might affect the measured results For example, you are likely to get very different answers to a survey if the person conducting the survey is wearing a Darth Vader. Or is naked. Statistic vs Parameter Population: the entire group of individuals or instances about whom we hope to learn Sample: a (representative) subset of a population, examined in hope of learning about a population Statistics are from a set of sampled data, parameters are a numerically valued attribute of a model for a population o Population parameter represents whole population o sample statistic represents sample A parameter is something we hope to estimate from the data. o For example, from the mean salaries of the subjects in a sample (a statistic created from data) we hope to estimate the mean salaries of all workers in the U.S. We rarely will know the true parameter Difference btw population, sampling Frame, and sample o a population is an entire group of individuals in question. o a sampling frame is the list of individuals from whom the sample is drawn. o a sample represents a population, and is examined in hope of learning about that population. The subjects that actually gave us data. census: a sample that consists of the entire population o A census is a special sample where everyone is included and responses to surveys are received from the whole population. o A census is often impractical because: it can be difficult to complete there will always be some individuals too hard to locate and the cost of locating them may exceed your budget. populations rarely stand still babies are born and people die or move, a population changes while you work so it is never possible to get a perfect measure. taking a census can be more complex that sampling they require the cooperation of the population members, many people are accidentally counted twice, and many people are overlooked Sample Design: simple random sample: sampling design in which each set of n elements in the population has an equal chance of selection o Two Conditions: Every subject/unit/etc. must have an equal chance for being selected Each combination of subjects/units/etc. must have an equal chance of being selected stratified random sample: sampling design in which the population is divided into several strata, and random samples are then drawn from each stratum o strata-homogenous groups (the same) that are used to split up the population before a sample is chosen. o Stratified Random Sampling is useful when having two or more different groups may bias your results. It may be better to just split them and analyze them separately o To take a Stratified Sample, first divide the population into homogenous groups (strata). Then take an SRS from each group Combine these individual SRS’s into a single sample from the population. o For example, I could divide the student body into Freshmen, Sophomores, Juniors and Seniors; then take an SRS from each group; and finally combine these to make a single sample. cluster sample: sampling design in which entire groups(clusters) are chosen at random o Cluster Samples are similar to stratified samples, but the difference is that a cluster sample uses naturally occurring groups that are heterogeneous (“all mixed up”). o The idea is that each group (each cluster) is individually representative of the population—so we take a sample of those clusters (to help deal with random variation in the various clusters). o o o o Used for convenience, practicality, or cost Useful when the sample size is very large and it would be too tidious to pick out random samples. You would take one little part, or cluster, and look at those data very thoroughly. Example- If you wanted to check the reading level off of a textbook, you wouldn't want to go through the whole book and pick out a random word on every page or every other page. Instead, you would pick one page and use every word on that page as our sample. multistage sample: sampling schemes that combine several sampling methods o Makes more randomization o Ex: the combination of the cluster sampling and simple random samples, which is used by polling organizations systematic sample: a sample drawn by selecting individuals systematically from a sampling frame. o There is some “stream” of individuals from which to select. o The method involves selecting every kth individual (every 10th, or every 50th, for example). o When there is no relationship between the order of the sampling frame and the variables of interest, a systematic sample can be representative o In other words, it shouldn’t matter what order you choose. So you choose random rate to choose people convenience sample: sampling design where individuals are chosen based on who is easily available o Simply include the individuals who are at hand. An unreliable method of sampling. Observation Studies and Experiments (Ch. 13): Observational study: study based on data in which no treatments have been assigned to subjects Observational study – researchers don’t assign choices, they simply observe them; “in the wild” Valuable for discovering trends and possible relationships Two Types: retrospective study: observational study in which subjects are selected and then their previous conditions or behaviors are determined o Retrospective study – researches first identify subjects of a specific characteristic, then collect data on past information o Have a restricted view of the world because they are usually restricted to a small part of the entire population. o Because retrospective records are based on historical data, they can have errors. prospective study: observational study in which subjects are followed to observe future outcomes o Prospective study – identifying subjects in advance and collecting data as events unfold Experiment: an approach where you try to prove cause and effect. experimental units: individuals on which an experiment is done An experimenter must identify at least one explanatory variable, called a factor, to manipulate and at least one response variable to measure. Experimenters actively and deliberately manipulate the factors to control the details of the possible treatments o assigns the subjects to those treatments at random. o You then observe the response variables and compare responses for different groups of subjects who have been treated differently. Sample surveys try to estimate population parameters so the sample needs to represent the population Experiments try to assess the effects of treatments. o treatment: the process or intervention applied to randomly assigned experimental units Combination of specific levels from all the factors that an experimental unit receives Experiments are rarely performed on random samples from population. Don’t describe the subjects in an experiment as a random sample unless they REALLY ARE. More likely, the randomization was in assigning subjects to treatments. Assign participants at random Wouldn’t test prenatal treatments on randomly chosen people (not everyone is pregnant). Experiments draw stronger conclusions than survey because they cancel out many sources of bias. When you assign subjects randomly to treatment groups, all the groups are still biased, but in the same way, so their biases cancel out allowing us to see the differences due to the treatment more clearly. Designing a good experiment: o Control aspects of the experiment that we know may have an effect on the response Control sources of variation other than the factors we are testing by making conditions as similar as possible for all treatment groups. o Randomize subjects to treatments to even out effects that we cannot control – the subjects don’t need to be randomly chosen, just assigning them to the treatments Randomizing allows us to equalize the effects of unknown or uncontrollable sources of variation. Does not eliminate the effects of these sources, but rather spreads them out across the treatment levels to that we can see past them. If experimental units are not assigned to treatments at random, you will not be able to use the powerful methods of Statistics to draw conclusions from your study. o Reduces bias due to uncontrolled sources of variation. o Protects us even from effects we didn’t know about. o “Control what you can, and randomize the rest” o Replicate over as many subjects as possible Repeat the experiment, applying the treatments to a number of subjects. Estimate the variability of responses. The outcome of an experiment on a single subject is an anecdote, not data Experimental units are not a representative sample from the population of interest. We may believe that what we have is true for the entire population, but to be more confident in the results, replicate in another location with people of different countries, ages, times of year, etc. Replication of an entire experiment with the controlled sources of variation at different levels is an essential step in science. o Block to reduce the effects of identifiable attributes of the subjects that cannot be controlled Block – the ability of randomizing to equalize variation across treatment groups works best in the long run. Blocking variable is the characteristic being blocked on. 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. Factor vs Level factor: variable whose levels are controlled by the experimenter level: specific values that the experimenter chooses a factor Statistically Significant: when an observed difference is too large to believe that it is likely to have occurred naturally How do you know if it is statistically significant? Are the differences you get about as big as you would get just from the randomization along? o If the changes are bigger you might have something. • A difference is statistically significant if we don’t believe that it’s likely to have occurred only by chance. Control treatment – baseline measurement; control group – subjects that get the control treatment • Use a control group as another level of the factor to compare the treatment results to a situation where “nothing happens”. • Baseline treatment level for experimental units Control: aspects of the experiment that we know may have an effect on the response, but that are not the factors being studied Blinding – disguise or conceal the treatment 2 Types of individuals who can affect outcomes: o those who influence the results (the subjects) o those who evaluate the results (the administrators) Single blind – when all individuals in either one of these classes is blinded Double blind – when everyone in BOTH classes are blinded; not only the test subjects, but the experiment administrators too. matched pairs type of study in which subjects who are similar in ways not under study may be grouped together and then compared with each other on the variables of interest Placebo – a control treatment that mimics the treatment of the actual experiment. A “fake” treatment that looks just like the treatments being tested. Not unusual for 20% or more of subjects given a placebo to report improvement. The placebo effect highlights both the importance of effective blinding and the importance of comparing treatments with a control. placebo effect: the tendency of many human subjects to show a response even when administered a fake treatment Best experiments are usually: • Randomized • Comparative • Double-blind • Placebo-controlled Blocking/Matching When groups of experimental units are similar, it’s often a good idea to gather them together into blocks. Block: a subdivision of the population By blocking, we isolate the variability attributable to the differences between the blocks, so that we can see the differences caused by the treatments more clearly. Randomized block design – when the randomization occurs within the blocks. • Same as stratified random for sampling; • Randomized block is for experiments. Matching – when subjects are paired because they are similar in ways NOT under study. • Matching subjects in this way can reduce variation in much the same way as blocking. • Both methods group together subjects that are similar and randomize within those groups as a way to remove unwanted variation. • Do not say we stratify an experiment or block a sample. 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 block themselves.. confounding: when the levels of one factor are associated with the levels of another factor so their effects cannot be separated A confounding variable 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 variable –or even by both working together. 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 The Main Types of Experiments 1.) Completely Randomized Design: This is a design where all treatments are assigned at random. Completely Randomized two-factor experiment – any subject could end up assigned at random to any of the treatments and there are two types of factors. • With 2 factors, you can account for more of the variation. That lets us see the underlying patterns more clearly. • Experiments with more than one factor are both more efficient and provide more information than one-at-a-time experiments. There are many ways to design efficient multifactor experiments 2.) Randomized Block Design: subjects are first blocked by some variable, then assigned treatments (randomly) within each block. o In essence, the experiment is split up into lots of little experiments (one per block). 3.) Matched Pairs Design: o This could be thought of as a very special kind of block design, where each block consists of exactly two individuals that have been paired by one or more variable o in other words, they are similar or identical in some manner. Within each pair, one individual is (randomly) assigned to the treatment, and the other to the control. Part II.) Practice Questions: 1. A company sponsoring a new internet search engine wants to collect data on the ease of using it. Which is the best way to collect the data? a. Census b. Sample survey c. Observational study d. Experiment e. Simulation 2. The January 2005 Gallup Youth Survey telephone a random sample of 1,028 US teens and asked these teens to name their favorite movie from 2004. “Napoleon Dynamite” had the highest percentage with 8% of teens ranking it as their favorite movie. Which is true? I. The population of interest is all US teens II. 8% is a statistic and not the actual percentage of all US teens who would rank this movie as their favorite III. This sampling design should provide a reasonably accurate estimate of the actual percentage of all US teens who would rank this movie as their favorite a. I only b. II only c. III only d. I and II e. I, II, and III 3. Suppose your local school district decides to randomly test high school students for ADD. There are three high schools in the district, each with grades 9-12. The school board pools all of the students together and randomly samples 250 students. Is this a simple random sample? a. Yes, because the students were chosen at random. b. Yes, because each student is equally likely to be chosen. c. Yes, because they could have chosen any 250 students from throughout the district. d. No, because we can’t guarantee that there are students from each school in the sample. e. No, because we can’t guarantee that there are students from each grade in the sample. 4. A basketball players has a 70% free throw percentage. Which plan could be used to simulate the number of free throws she will make in her next five free throw attempts? I. Let 0, 1represent making the first shot, 2, 3 represent making the second shot, …, 8, 9 represent making the fifth shot. Generate five random numbers 0-9, ignoring repeats. II. Let 0, 1, 2 represent missing a shot and 3, 4, …, 9 represent making a shot. Generate five random numbers 0-9 and count how many numbers are in 3-9. III. Let 0, 1, 2 represent missing a shot and 3, 4, …, 9 represent making a shot. Generate five random numbers 0-9 and count how many numbers are in 3-9, ignoring repeats. a. I only b. II only c. III only d. II and III e. I, II, and III 5. More dogs are being diagnosed with thyroid problems than have been diagnosed in the past. A researcher identified 50 puppies without thyroid problems and kept records of their diets for several years to see if any developed thyroid problems. This is a(n): a. Randomized experiment b. Survey c. Prospective study d. Retrospective study e. Blocked experiment 6. A chemistry professor who teaches a large lecture class survey his students who attend his class about how he can make the class more interesting, hoping he can get more students to attend. This survey method suffers from: a. Voluntary response bias b. Nonresponse bias c. Response bias d. Undercoverage e. None of the above 7. Placebos are a tool for: a. Sampling b. Blocking c. Blinding d. Control e. Randomization 8. Double-blinding in experiments is important so that: I. The evaluators do not know which treatment group the participants are in. II. The participants do not know which treatment group they are in III. No one knows which treatment any of the participants are getting a. I only b. II only c. III only d. I and II e. I, II, and III 9. Which of the following is NOT required in an experimental design: a. Blocking b. Control c. Randomization d. Replication e. All are required in an experimental design 10. Which statement is true about bias? Bias results from a random variation and will always be present II. Bias results from a sampling method likely to produce samples that do not represent the population III. Bias is usually reduced when sample size is larger a. I only b. II only c. III only d. I and III e. II and III I. Part III.) Exams AP Statistics Test A – Gathering Data – Part III ___ 1. A company sponsoring a new Internet search engine wants to collect data on the ease of using it. Which is the best way to collect the data? A) census B) sample survey C) observational study D) experiment E) simulation ___ 2. The January 2005 Gallup Youth Survey telephoned a random sample of 1,028 U.S. teens and asked these teens to name their favorite movie from 2004. Napoleon Dynamite had the highest percentage with 8% of teens ranking it as their favorite movie. Which is true? I. The population of interest is all U.S. teens. II. 8% is a statistic and not the actual percentage of all U.S. teens who would rank this movie as their favorite. III. This sampling design should provide a reasonably accurate estimate of the actual percentage of all U.S. teens who would rank this movie as their favorite. A) I only B) II only C) III only D) I and II E) I, II, and III ___ 3. Suppose your local school district decides to randomly test high school students for attention deficit disorder (ADD). There are three high schools in the district, each with grades 9-12. The school board pools all of the students together and randomly samples 250 students. Is this a simple random sample? A) Yes, because the students were chosen at random. B) Yes, because each student is equally likely to be chosen. C) Yes, because they could have chosen any 250 students from throughout the district. D) No, because we can’t guarantee that there are students from each school in the sample. E) No, because we can’t guarantee that there are students from each grade in the sample. ___ 4. A basketball player has a 70% free throw percentage. Which plan could be used to simulate the number of free throws she will make in her next five free throw attempts? I. Let 0,1 represent making the first shot, 2, 3 represent making the second shot,…, 8, 9 represent making the fifth shot. Generate five random numbers 0-9, ignoring repeats. II. Let 0, 1, 2 represent missing a shot and 3, 4,…, 9 represent making a shot. Generate five random numbers 0-9 and count how many numbers are in 3-9. III. Let 0, 1, 2 represent missing a shot and 3, 4,…, 9 represent making a shot. Generate five random numbers 0-9 and count how many numbers are in 3-9, ignoring repeats. A) I only B) II only C) III only D) II and III E) I, II, and III ___ 5. More dogs are being diagnosed with thyroid problems than have been diagnosed in the past. A researcher identified 50 puppies without thyroid problems and kept records of their diets for several years to see if any developed thyroid problems. This is a(n) A) randomized experiment B) survey C) prospective study D) retrospective study E) blocked experiment ___ 6. A chemistry professor who teaches a large lecture class surveys his students who attend his class about how he can make the class more interesting, hoping he can get more students to attend. This survey method suffers from A) voluntary response bias B) nonresponse bias C) response bias D) undercoverage E) None of the above ___ 7. Placebos are a tool for A) sampling B) blocking C) blinding D) control E) randomization ___ 8. Double-blinding in experiments is important so that I. The evaluators do not know which treatment group the participants are in. II. The participants do not know which treatment group they are in. III. No one knows which treatment any of the participants is getting. A) I only B) II only C) III only D) I and II E) I, II, and III ___ 9. Which of the following is not required in an experimental design? A) blocking B) control C) randomization D) replication E) All are required in an experimental design. ___ 10.A researcher wants to compare the effect of a new type of shampoo on hair condition. The researcher believes that men and women may react to the shampoo differently. Additionally, the researcher believes that the shampoo will react differently on hair that is dyed. The subjects are split into four groups: men who dye their hair; men who do not dye their hair; women who dye their hair; women who do not dye their hair. Subjects in each group are randomly assigned to the new shampoo and the old shampoo. This experiment A) is completely randomized. B) has three factors (shampoo type, gender, whether hair is dyed). C) has two factors (gender and whether hair is dyed) blocked by shampoo type. D) has two factors (shampoo type and whether hair is dyed) blocked by gender E) has one factor (shampoo type), blocked by gender and whether hair is dyed. 11. Video games A headline in a local newspaper announced “Video game playing can lead to better spatial reasoning abilities.” The article reported that a study found “statistically significant differences” between teens who play video games and teens who do not, with teens who play video games testing better in spatial reasoning. Do you think the headline was appropriate? Explain. No, this was not a controlled experiment, so no determination of cause and effect can be made. Perhaps there is something about teens who play video games that make them good at video games and at spatial reasoning, or maybe teens with good spatial reasoning enjoy games more. 12. College students’ spending A consumer group wants to see if a new education program will improve the spending habits of college students. Students in an economics class are randomly assigned to three different courses on spending habits. a. What are the experimental units? The subjects are the students in the economics class. b. How many factors are there? There is one factor (course on spending habits) with three levels. c. How many treatments are there? Since there is only one factor at three levels, there are three treatments. d. What is the response variable? The response variable is the spending habits of the students 13. Good CDs Brian is a systems manager for a large company. In his work, he has found that about 5% of all CDs he orders are bad. He needs to give one of the executives at his company five good CDs. Conduct a simulation to estimate how many CDs Brian will have to check to get five good CDs for the executive. a. Describe how you will use a random number table to conduct this simulation. Let the digits 00-04 represent bad CDs and the digits 05-99 represent good CDs. Look at pairs of digits in the random number table to determine if the CD is good or bad. Continue this until you get five good CDs. . b. Show three trials by clearly labeling the random number table given below. Specify the outcome for each trial. Simulation Trial #1: 03242 50692 18977 28370 Trial #2: 78695 21402 85525 81183 Trial #3: 60809 06765 39996 81915 Let B = bad and G = good. c. State your conclusion. According to my simulation, it will take an average of 5.7 CDs to get five good CDs. 14. Bone Builder Researchers believe that a new drug called Bone Builder will help bones heal after children have broken or fractured a bone. The researchers believe that Bone Builder will work differently on bone breaks than on bone fractures, because of differences in initial bone condition. Bone Builder will be used in conjunction with traditional casts. To test the impact of Bone Builder on bone healing, the researchers recruit 18 children with bone breaks and 30 children with bone fractures. Design an appropriate experiment to determine if Bone Builder will help bones heal. Blocking is employed since breaks and fractures have different initial bone condition. This experiment can be double blind, if patients and bone evaluators don’t know whether or not the patient was given Bone Builder. 15. Military funding A college group is investigating student opinions about funding of the military. They phone a random sample of students at the college, asking each person one of these questions (randomly chosen): A: “Do you think that funding of the military should be increased so that the United States can better protect its citizens?” B: “Do you think that funding of the military should be increased?” Which question do you expect will elicit greater support for increased military funding? Explain. What kind of bias is this? The first question will elicit greater support for increased military funding. The wording of the question appeals to the feelings of safety of the respondent. The second question does not do this – it is more neutral and will elicit less response. o This is a form of response bias. AP Statistics Test B – Gathering Data – Part III ___ 1. Among a dozen eggs, three are rotten. A cookie recipe calls for two eggs; they’ll 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, 4, …, 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 may 0’s you get. A) I only B) II only C) III only D) I or III only E) I, II, or III ___ 2. We wish to compare the average ages of the math and science teachers at your high school. Which is the best way to collect the data? A) census B) sample survey C) observational study D) experiment E) simulation ___ 3. Hoping to get information that would allow them to negotiate new rates with their advertisers, Natural Health magazine phoned a random sample of 600 subscribers. 64% of those polled said they use nutritional supplements. Which is true? I. The population of interest is the people who read this magazine. II. “64%” is not a statistic; it’s the parameter of interest. III. This sampling design should provide the company with a reasonably accurate estimate of the percentage of all subscribers who use supplements. A) I only B) I and II only C) I and III only D) II and III only E) I, II, and III ___ 4. Suppose the state decides to randomly test high school wrestlers for steroid use. There are 16 teams in the league, and each team has 20 wrestlers. State investigators plan to test 32 of these athletes by randomly choosing two wrestlers from each team. Is this a simple random sample? A) Yes, because the wrestlers were chosen at random. B) Yes, because each wrestler is equally likely to be chosen. C) Yes, because stratified samples are a type of simple random sample. D) No, because not all possible groups of 32 wrestlers could have been the sample. E) No, because a random sample of teams was not first chosen. ___ 5. Which statement about bias is true? I. Bias results from random variation and will always be present. II. Bias results from a sampling method likely to produce samples that do not represent the population. III. Bias is usually reduced when sample size is larger. A) I only B) II only C) III only D) I and III only E) II and III only ___ 6. A researcher identified 100 men over forty who were not exercising and another 100 men over forty with similar medical histories who were exercising regularly. She followed all the men for several years to see if there was any difference between the two groups in the rate of heart attacks. This is a(n) … A) survey B) prospective study C) retrospective study D) randomized experiment E) matched pairs experiment. ___ 7. Of A-D, which is not a critical part of designing a good experiment? A) Control of known sources of variability. B) Random selection of subjects. C) Random assignment of subjects to treatments. D) Replication of the on a sufficient number of subjects. E) All of these are important. ___ 8. In an experiment the primary purpose of blinding is to reduce … A) bias. B) confounding. C) randomness. D) undercoverage. E) variation. ___ 9. Does donating blood lower cholesterol levels? 50 volunteers have a cholesterol test, then donate blood, and then have another cholesterol test. Which aspect of experimental design is present? A) randomization B) a control group C) a placebo D) blinding E) none of these ___ 10. A researcher wants to compare the performance of three types of pain relievers in volunteers suffering from arthritis. Because people of different ages may suffer arthritis of varying degrees of severity, the subjects are split into two groups: under 60 and over 60. Subjects in each group are randomly assigned to take one of the medications. Twenty minutes later they rate their levels of pain. This experiment … A) is completely randomized. B) uses matched pairs. C) has two factors, medication and age. D) has one factor (medication) blocked by age. E) has one factor (age) blocked by medication type. 11. Insulators Ceramics engineers are testing a new formulation for the material used to make insulators for power lines. They will try baking the insulators at four different temperatures, followed by either slow or rapid cooling. They want to try every combination of the baking and cooling options to see which produces insulators least likely to break during adverse weather conditions. a) What are the experimental units? Material for insulators b) How many factors are there? 2 – baking temp and cooling method c) How many treatments are there? 8 d) What is the response variable? Likeliness to break during adverse weather 12. Cloning A polling organization is investigating public opinion about cloning. They phone a random sample of 1200 adults, asking each person one of these questions (randomly chosen): A: “Do you favor allowing doctors to use cloned cells in attempts to find cures for such terrible diseases as Alzheimer’s, diabetes, and Parkinson’s?” B: “Should research scientists be allowed to use cloned human embryos in their experiments?” Which question do you expect will elicit greater support for cloning? Explain. What kind of bias is this? The first question will elicit greater support for cloning. o The wording of the question appeals to the emotions of the respondent. The second question conjures up images of scientists experimenting on humans and cloned embryos, and will elicit less support. o This is a form of response bias. 13. Sweet potato pie Sam is preparing sweet potato pies as his dessert for Thanksgiving. The store he shops at sells six sweet potatoes in a bag. He has found that each bag will contain 0, 1, or 2 bad sweet potatoes. Based on experience he estimates that there will be no bad sweet potatoes in 40% of the bags, one bad sweet potato in 30% of the bags, and two bad sweet potatoes in the rest. Conduct a simulation to estimate how many bags Sam will have to purchase to have three dozen sweet good potatoes. a) Describe how you will use a random number table to conduct this simulation. Use the digits 0–3 to represent bags with 6 good sweet potatoes (none bad), digits 4–6 to represent bags with five good sweet potato(one bad), and digits 7–9 to represent bags with four good sweet potatoes(two bad). Look at each single digit of the random number table to determine whether you have 4, 5, or 6 good sweet potatoes in the bag. Continue this until the cumulative count is at least 36 good sweet potatoes. b) Show two trials by clearly labeling the random number table given below. Specify the outcome for each trial. Answer: c) State your conclusion. According to my simulation, it will take (on average) 7.5 bags of sweet potatoes to get at least three dozen good sweet potatoes for the pies. 14. Moods A headline in the New York Times announced “Research shows running can alter one’s moods”. The article reported that researchers gave a Personality Assessment Test to 231 males who run at least 20 miles a week, and found “statistically significant personality differences between the runners and the male population as a whole.” Do you think the headline was appropriate? Explain. No, this was not a controlled experiment, so no determination of cause and effect can be made. Perhaps people in better moods are more likely to be runners, or perhaps something in their body chemistry makes some people enjoy running and also impacts their personality. 15. Property taxes Administrators of the fire department are concerned about the possibility of implementing a new property tax to raise moneys needed to replace old equipment. They decide to check on public opinion by having a random sample of the city’s population. a. Several plans for choosing the sample are proposed. Write the letter corresponding to the sampling strategy in the blank next to each plan. A. convenience B. stratified C. simple D. cluster E. systematic _____i. The city has five property classifications: single family homes, apartments, condominiums, temporary housing (hotel and campgrounds), and retail property. Randomly select ten residents from each category. B. stratified _____ii. Each property owner has a 5-digit ID number. Use a random number table to choose forty numbers. C. simple _____iii. At the start of each week, survey every tenth person who arrives at the city park. E. systematic _____iv. Randomly select a housing classification (say, apartments) and survey all the people who live in that property classification. D. cluster _____v. Have each firefighter survey 10 of his/her neighbors. A. convenience b. Briefly explain why plan iv suggested above, sampling an entire housing classification, might be biased. Be sure to name the kind(s) of bias you describe. The housing classification selected might not be a representative group of the population (undercoverage). In fact the housing classification might be made up of residents with similar viewpoints on property taxes, different from the views of the nonresidents. c. Name and briefly describe the kind of bias that might be present if the administration decides that instead of subjecting people to a random sample they’ll just… i. interview people about the new property tax at a fire station open house. ii. ask people who are willing to be taxed to sign a petition. i. Convenience sample: People attending a fire station open house are likely to be very supportive of the fire station and not be representative of the entire city (undercoverage). ii. Voluntary Response Bias: individuals with strong opinions or who have reason to need fire equipment would sign. (Or possibly no one volunteers to pay an extra tax.) 16. Grape juice and blood pressure Researchers who wanted to see if drinking grape juice could help people lower their blood pressure got 120 non-smokers to volunteer for a study. They measured each person’s blood pressure and then randomly divided the subjects into two groups. One group drank a glass of grape juice every day while the other did not. After sixty days the researchers measured everyone’s blood pressure again. They reported that differences in changes in blood pressure between the groups were not statistically significant. a. Was this an experiment or an observational study? Explain briefly. Experiment – there was an application of a treatment, grape juice, to groups containing randomly assigned subjects with a comparison of the blood pressure across the treatment groups. b. Briefly explain what “not statistically significant” means in this context. There was not a large reduction of blood pressure between the group who drank grape juice and the group who did not drink grape juice; the reduction could be reasonably attributed to random variation (or sampling error). c. Briefly explain why the researchers randomly assigned the subjects to the groups. Randomly assigning subjects to groups allows us to equalize the effects of unknown or uncontrollable sources of variation. d. Since everyone’s blood pressure was measured at the beginning and at the end of the study, the researchers could have simply looked at the grape juice drinkers to see if their blood pressure changed. Briefly explain why the researchers bothered to include the control group. The control group provided a basis for comparison after the sixty days to determine if the group drinking grape juice had lower blood pressure because of the juice. Maybe everyone’s blood pressure naturally drops at a certain time of the year. e. Briefly explain why the researchers studied only non-smokers. By studying only non-smokers the researchers were trying to reduce any impact the smoking might have had on blood pressure. The researchers were trying to remove any confounding the smoking might have made on blood pressure. The new study must be blocked by whether a person smokes or not. Half of each group would then be randomly assigned to the grape juice or no grape juice groups. f. Other researchers now plan to replicate this study using both smokers and non-smokers. Briefly describe the design strategy you think they should use. Block by smoking status, randomly assigning people in each block to the treatments. Compare blood pressure after 60 days within each block. AP Statistics Test C – Gathering Data – Part III ___ 1. If we wish to compare the average PSAT scores of boys and girls taking AP Statistics at this high school, which would be the best way to gather these data? A) census B) SRS C) stratified sample D) observational study E) experiment ___ 2. A factory has 20 assembly lines producing a popular toy. To inspect a representative sample of 100 toys, quality control staff randomly selected 5 toys from each line’s output. Was this a simple random sample? A) Yes, because the toys were selected at random. B) Yes, because each toy produced had an equal chance to be selected. C) Yes, because a stratified sample is a type of simple random sample. D) No, because not all combinations of 100 toys could have been chosen. E) No, because toys do not come off the assembly line at random. ___ 3. Which is true about sampling? I. An attempt to take a census will always result in less bias than sampling. II. Sampling error is usually reduced when the sample size is larger. III. Sampling error is the result of random variations and is always present. A) I only B) II only C) III only D) II and III E) all three ___ 4. The owner of a car dealership planned to develop strategies to increase sales. He hoped to learn the reasons why many people who visit his car lot do not eventually buy a car from him. For one month he asked his sales staff to keep a list of the names and addresses of everyone who came in to test drive a car. At the end of the month he sent surveys to the people who did not buy the car, asking them why. About one third of them returned the survey, with 44% of those indicating that they found a lower price elsewhere. Which is true? I. The population of interest is all potential car buyers. II. This survey design suffered from non-response bias. III. Because it comes from a sample 44% is a parameter, not a statistic. A) I only B) II only C) I and II only D) II and III only E) I, II, and III ___ 5. Does regular exercise decrease the risk of cancer? A researcher finds 200 women over 50 who exercise regularly, pairs each with a woman who has a similar medical history but does not exercise, then follows the subjects for 10 years to see which group develops more cancer. This is a: A) survey B) retrospective study C) prospective study D) randomized experiment E) matched experiment ___ 6. Which is important in designing a good experiment? I. Randomization in assigning subjects to treatments. II. Control of potentially confounding variables. III. Replication of the experiment on a sufficient number of subjects. A) I only B) I and II C) I and III D) II and III E) all three ___ 7. Can watching a movie temporarily raise your pulse rate? Researchers have 50 volunteers check their pulse rates. Then they watch an action film, after which they take check their pulse rates once more. Which aspect of experimentation is present in this research? A) a placebo B) blinding C) randomization D) a control group E) none of these ___ 8. In an experiment the primary purpose of blocking is to reduce A) bias. B) confounding. C) randomness. D) undercoverage. E) variation. ___ 9. To check the effect of cold temperatures on the battery’s ability to start a car researchers purchased a battery from Sears and one from NAPA. They disabled a car so it would not start, put the car in a warm garage, and installed the Sears battery. They tried to start the car repeatedly, keeping track of the total time that elapsed before the battery could no longer turn the engine over. Then they moved the car outdoors where the temperature was below zero. After the car had chilled there for several hours the researchers installed the NAPA battery and repeated the test. Is this a good experimental design? A) Yes B) No, because the car and the batteries were not chosen at random. C) No, because they should have tested other brands of batteries, too. D) No, because they should have tested more temperatures. E) No, because temperature is confounded by brand. ___ 10. Twenty dogs and 20 cats were subjects in an experiment to test the effectiveness of a new flea control chemical. Ten of the dogs were randomly assigned to an experimental group that wore a collar containing the chemical, while the others wore a similar collar without the chemical. The same was done with the cats. After 30 days veterinarians were asked to inspect the animals for fleas and evidence of flea bites. This experiment is… A) completely randomized with one factor: the type of collar B) completely randomized with one factor: the species of animal C) randomized block, blocked by species D) randomized block, blocked by type of collar E) completely randomized with two factors 11. Public opinion A member of the City Council has proposed a resolution opposing construction of a new state prison there. The council members decide they want to assess public opinion before they vote on this resolution. Below are some of the methods that are proposed to sample local residents to determine the level of public support for the resolution. Match each with one of the listed sampling techniques. 1 cluster 2 convenience 3 judgment 4 multistage 5 simple (SRS) 6 stratified 7 systematic 8 voluntary response ___ a) Place an announcement in the newspaper asking people to call their council representatives to register their opinions. Council members will tally the calls they receive. a) voluntary response ___ b) Have each council member survey 50 friends, neighbors, or co-workers. b) convenience ___ c) Have the Board of Elections assign each voter a number, then select 400 of them using a random number table. c) simple (SRS) ___ d) Go to a downtown street corner, a grocery store, and a shopping mall; interview 100 typical shoppers at each location. d) judgment ___ e) Randomly pick 50 voters from each election district. e) stratified ___ f) Call every 500th person in the phone book. f) systematic ___ g) Randomly pick several city blocks, then randomly pick 10 residents from each block. g) multistage ___ h) Randomly select several city blocks; interview all the adults living on each block. h) cluster 12. Telephone poll The City Council decides to conduct a telephone poll. Pollsters ask a carefully chosen random sample of adults this question: “Do you favor the construction of a new prison to deal with the high level of violent crime in our State?” In what way might the proportion of “Yes” answers fail to accurately reflect true public opinion? Explain briefly. What kind of bias is this? Response bias: the wording will induce people to answer “yes,” so the poll will overestimate the public’s support. 13. M&Ms The Mars candy company starts a marketing campaign that puts a plastic game piece in each bag of M&Ms. 25% of the pieces show the letter “M”, 10% show the symbol “&”, and the rest just say “Try again”. When you collect a set of three symbols “M”, “&”, and “M” you can turn them in for a free bag of candy. About how many bags will a consumer have to buy to get a free one? Use a simulation to find out. a. Explain how you will use the random numbers listed below to conduct your simulation. Check random numbers 2 digits at a time. Let 00-24 = “M”, 25-34 = “&”, 35-99 = “Try Again”. Go across the row until you have two M’s and an &. Count the number of bags. b. Carefully label your simulation for 2 trials. #1: 69 = TA, 07 = M, 49 = TA, 19 = M, 76 = M, 33 = &. 6 bags #2: 48 = TA, 32 = &, 47/79 = TA, 28/31 = &, 24 = M, 96/47 = TA, 10 = M. 10 bags c. State your conclusion. This simulation estimates you would have to buy an average of 8 bags to get a free one. 14. Preservative Leather furniture used in public places can fade, crack, and deteriorate rapidly. An airport manager wants to see if a leather preservative spray can make the furniture look good longer. He buys eight new leather chairs and places them in the waiting area, four near the south-facing windows and the other four set back from the windows as shown. He assigned the chairs randomly to these spots. a. Use the random numbers given to decide which chairs to spray. Explain your method clearly. 32219 00597 86374 Use one digit at a time, ignoring 0, 9 and any repeated numbers. Choose two chairs from each row. (Example: 3, 2, 5, and 7) b. Briefly explain why your assignment strategy is important in helping the manager assess the effectiveness of the leather preservative. Windows may play a role in the leather’s life (sun, heat, etc.). Blocking can reduce variability from that source. 15. Candy packaging Marketing researchers wonder if the color and type of a candy’s packaging may influence sales of the candy. They manufacture test packages for chocolate mints in three colors (white, green, and silver) and three types (box, bag, and roll). Suspecting that sales may depend on a combination of package color and type, the researchers prepare nine different packages, then market them for several weeks in convenience stores in various locations. In this experiment. a. what are the experimental units? Packages b. how many factors are there? 2 c. how many treatments are there? 9 d. what is the response variable? sales 16. Aggressiveness A recent study evaluated elementary age children for aggressiveness. This study found that the children who played video games were more likely to engage in aggressive or violent play at school. The researchers said the difference was statistically significant. a. Briefly explain what “statistically significant” means in this context. The difference in aggressiveness was greater than random variation might reasonably be expected to produce. b. The news media reported that this study proved that playing computer games causes children to be aggressive or violent. Briefly explain why this conclusion is not justified. This was not a controlled experiment, so no determination of cause and effect can be made. Perhaps only violent kids like to play video games c. But perhaps it is true. We wonder if playing computer games can lead to aggressive or violent behavior in elementary school children. We find 50 young children whose families volunteer to participate in our research. Design an appropriate experiment. (You need not explain how to randomize.) Randomly divide the kids into two groups of 25. Have one group play video games and prevent the other group from playing them for a period of time. Have observers evaluate the kids’ aggressiveness on the playground. Don’t let the kids know the purpose of the experiment or that they are being observed. And don’t let the evaluators know which kids are in each group. Compare the aggressiveness ratings for the two groups.