1 Preliminary Chapter P.1 Getting data from Jamie and her friends is convenient, but it does not provide a good snapshot of the opinions held by all young people. In short, Jamie and her friends are not a representative sample from the population of interest (young people). P.2 (a) This is an observational study. Patients were matched by age, sex, and race, but the investigators did not impose any treatment. They simply asked about cell phone use and recorded the responses. (b) No, the results of this study are encouraging, but cause-and-effect conclusions, like this one, must be based on experiments. P.3 (a) This is an experiment. The company uses animation for one group and a text for the other group in order to compare the performance of the two different groups. (b) The company could conclude that they have solid evidence that computer animation was more successful than a textbook for these students. If the company wants to generalize this conclusion to a broader population, e.g. all juniors at a school or all juniors in a state or all juniors in the country, then they must assume that these groups are representative samples of the population of interest. The information provided does not indicate whether or not these students were randomly selected from some larger population. If they were, then inferences to that population are reasonable. Similarly, the information provided does not indicate whether or not randomization was used to partition the students into two groups. If randomization was used, then cause and effect conclusions can be drawn. Finally, we must assume that the tests adequately assess biology concepts. P.4 (a) This type of question is best addressed with a survey by polling agencies like the Gallup Organization. They use a variety of methods to get representative samples from the entire country and have experts who can clarify the wording of particular questions to avoid bias. Wording which leads the respondent to think about economic issues would likely get a much different response than wording which leads the respondent to think about social, political, or global issues. (b) Since we are interested in comparing two different teaching methods and making an inference to all college students, an experiment is best. College students who are interested in taking accounting must be randomly selected and then those students must be randomly placed into two groups, one which will be taught in a classroom and another which will learn the same material online. Describing ideal experiments is easy, but think about the practical problems (and costs) associated with the experiment described above. (c) Since we are specifically interested in how long your teachers wait before asking a question, it would be easiest to use an observational study that is not a survey. Different lectures, labs, or discussions should be randomly selected for each teacher. During the class, simply record the amount of time the teacher waits to move on after asking each question. Since teachers have different styles, you will have to think about whether you want the same number of questions for each teacher, which would require you to observe some teachers longer than others, or whether you want to observe each teacher for the same amount of time. P.5 (a) In an observational study, people who drink alcohol would be randomly selected and then variables which measure health characteristics would be collected and compared. In an experiment, the researchers would randomly assign the treatment (type of alcohol) to the
2 Preliminary Chapter participants and they would be required to drink that type of alcohol. The variables which measure health characteristics would be collected and compared after a reasonable amount of time. (b) Wine drinkers tend to be wealthier, exercise more frequently, and have better eating habits than beer drinkers. P.6 Go to http://nces.ed.gov, type employment of college students in the search box, and click on the first link for Indicator 3: Employment of College Students. A graphical summary of the percentages from 197 to 23 is provided at http://nces.ed.gov/programs/youthindicators/indicators.asp?pubpagenumber=3 For a numerical summary of the percentages from 197 to 23 see (or click View Table): http://nces.ed.gov/programs/youthindicators/indicators.asp?pubpagenumber=3&showtablepage=tableshtml/3.asp As you would expect, the percentages vary considerably depending on whether the undergraduates are full-time or part-time students. According to the U.S. Department of Commerce, Census Bureau, Current Population Surveys, in 23 (the most recent data available in July, 26), 47.7% of full-time college students were employed. Of these students, 29.5% worked 2 or more hours per week and 8.8% worked 35 or more hours per week. 79% of parttime college students were employed, with 7.1% of them working 2 or more hours per week and 42.8% of them working 35 or more hours per week. To answer the question directly, we will consider full-time to be 35 or more hours per week. Thus, 8.8% of full-time students and 42.8% of part-time students work full-time while they are taking classes. The percentages of students working part-time are: 38.9% (47.7 8.8) for full-time students and 36.2% (79 42.8) for part-time students. P.7 (a) A bar graph is shown below. 2 Cool Car Colors Percent of vehicles 15 1 5 Silver White Black Medium/dark gray Color Light brown Medium/dark blue Medium red (b) The bar graph shows that consumers prefer lighter colors (silver and white) to darker colors. Silver is the most popular color (tallest bar) and medium red is the least popular color (shortest bar). The percent of vehicles with other colors is 1 2.1 18.4 11.6 11.5 8.8 8.5 6.9 = 14.2. P.8 (a) Two side-by-side bar graphs are shown below. Each graph presents a slightly different view of the same percentages.
What Is Statistics? 3 Comparing Car Colors Comparing Car Colors 3 3 Percent of vehicles 25 2 15 1 5 Vehicle type Color Black Light brown Medium/dark b lue Medium/dark gray Medium/dark green Mediu m red White Silv er Percent of vehicles 25 2 15 1 Color Vehicle type 5 Black Light brown Medium/dark blue Medium/dark gray Medium/dark green Medium red White Silver Black Light brown Medium/dark blue Medium/dark gray Medium/dark green Medium red White Silver (b) The graph above on the left provides an easy comparison of luxury cars with /truck/van percentages for each color. The heights of the bars are different for every color, with the biggest difference in Medium/dark gray. The graph above on the right shows the colors separately for each type of vehicle. Once again the differences in preferences are clear from the different shapes of the bars for the two types of cars. White was the most popular color for these two types of cars. The medium/dark colors were much less popular, with one exception. Medium/dark gray was a popular color (2 nd most popular) for luxury cars. P.9 (a) A dotplot for goal differential is shown below. Goal Differential (U.S. score - opponent's score) -2 2 4 Differential 6 8 (b) The dotplot is centered around 2 with a long right tail. Only two of the 34 differentials are negative, which indicates that the U.S. women s soccer team had a very good season. The team scored at least as many goals as their opponents in 32 of 34 games. In one game they beat the other team by 8 goals, a very unusual event in soccer. P.1 (a) A dotplot for the total number of gold medals for a sample of countries is shown below.
4 Preliminary Chapter Number of Gold Medals Won in 24 Summer Olympics 5 1 15 2 Gold medals 25 3 35 The overall distribution is skewed to the right with a mode of, which indicates that many countries did not win any gold medals. The United States, with 35 gold medals, is clearly unusual. The dot for the United States is way out in the right tail, and the number of U.S. gold medals is almost 4 times larger than the total for the country with the second highest number of gold medals. (b) Yes, it makes sense that the distribution is skewed to the right. Most countries win no medals or very few medals and a few countries will a lot of medals. For example, China was very close to the total U.S. count with 32 gold medals. See http://www.mapsofworld.com/olympic-trivia/olympic-games-results/athens24.html for complete medal counts in the 24 Olympics. P.11 Who? The individuals are the AP Statistics students who completed a questionnaire on the first day of class. What? The categorical variables are gender (female or male), handedness (right or left), and favorite type of music (classical, gospel, rock, rap, country, R&B, top 4, oldies, etc.). The quantitative variables are height (in inches), amount of time the student is expecting to spend on homework (in minutes per week), and the total value of coins in a student s pocket (in cents). Why? The data were collected for the teacher to learn more about her/his students and to provide an interesting data set for the students to analyze. Students will be able to make interesting comparisons for men versus women, lefties versus righties, rappers versus rockers, etc. When, where, how, and by whom? A teacher collected these data on the first day of class at a high school using a questionnaire. P.12 Categorical: Gender (a), Race (c), Smoker (d) Quantitative: Age (b), Systolic blood pressure (e), Level of calcium in the blood (f) P.13 Reality shows (yes or no indicating whether the student watches reality shows), Music (yes or no indicating whether the student watches music videos, concerts, or documentaries about musicians and singers), Time (the average amount of time, in minutes per day, spent watching television), Network (the average number of network programs shows, movies, sporting events, etc. watched per week on ABC, CBS, NBC, and FOX). The categorical variables are Reality shows and Music, and the quantitative variables are Time and Network. P.14 16 The word fair implies that all six sides are equally likely. The chance of getting a 6 is the same as getting any one of the other five sides (1, 2, 3, 4, or 5). The probability of 1/6 means that if you roll a fair die a large number of times, you will get a 6 about 1/6 th of the time.
What Is Statistics? 5 P.15 The result with 14 out of 21 correct identifications is further away from what we expect (7) if the students are guessing, so it would provide more convincing evidence. Alternatively, the chance of getting 14 or more correct would be smaller than.68, the chance of getting 13 or more correct, but the chance of getting 12 or more correct is larger than.68. P.16 The chance of getting 11 or more correct is.557. (Let X = the number of correct answers, then P X 11 = 1 P( X < 11) = 1 P( X 1) 1.9443 =.557.) Since the ( ) students have about a 6% chance of getting 11 or more correct by simply guessing, many people would say that this event could happen by chance. The evidence is not as clear that the students can distinguish between tap and bottled water. P.17 Results will vary from student to student. One possibility is provided here. (a) A graph of the proportion of heads against the toss number is shown below..5.4 Proportion of heads.3.2.1. 5 1 Number of tosses 15 2 (b) Eight of the 2 spins were heads, so the estimated proportion of heads is 8/2 =.4 or 4%. (c) The chance of getting 8 or fewer heads if the coin is balanced so that it lands heads half of the time is.2517. Even though this proportion is below 5%, it does appear as if this result could happen by chance. (d) For a class of 3 students, each spinning a coin 2 times, a graphical display of the results is shown below..5.4 Proportion of heads.3.2.1. 1 2 3 4 Number of tosses 5 6 The number of heads out of 6 spins was 248, so the estimated proportion of heads is 248/6 =.413 or about 41%. With this many trials it is very unlikely (.125) that we would get 248 or fewer heads if the coin is balanced so that it lands heads half of the time. Thus, we would
6 Preliminary Chapter change our conclusion from part (c) and conclude that the chance of getting a head when spinning a coin is less than.5. P.18 (a) No, there will be variation from sample to sample so we would not expect the sample percentage to be exactly equal to the population percentage. (b) Yes, it does appear that a higher percentage of girls thought young people should abstain from sex until marriage. A difference of 64% 48% = 16% or something larger would be extremely unlikely if boys and girls thought the same way about abstinence. CASE CLOSED 1. (a) Who? The individuals are the fifty polio patients who reported steady pain. What? The variables are treatment group (active or inactive) and pain ratings (a number between and 1). Why? The data were collected to see if medical magnets reduce the pain experienced by polio sufferers. The doctors would like to make inferences to all polio patients who suffer from chronic pain. If the magnets are successful, the doctors may suggest future research on patients who suffer from other types of pain. When, where, how, and by whom? The data from this experiment were collected on recruited patients by doctors and scientists. Even though the patients were recruited, and not randomly selected, it may be reasonable to view them as a representative sample of polio patients with chronic pain. This is often the way medical studies are conducted. (b) The dotplots are shown below. Dotplot of Active, Inactive Active Inactive 2 4 Data 6 8 1 (c) The center of the distribution is lower for patients wearing the active magnets. Overall, the ratings for patients with the active magnets tend to be lower than those for patients with the inactive magnets. The ratings have less variability for the patients who were wearing the inactive magnets. Notice that the ratings for the patients wearing the active magnets ranged from to 1, while the ratings for patients wearing the inactive magnets ranged from 4 to 1. The distribution of ratings for the inactive magnets is skewed to the left while the distribution of the ratings for the active magnets is roughly symmetric. (d) The mean pain rating for the active group is 4.379 and the mean pain rating for the inactive group is 8.429. The difference (inactive active) is 4.5. 2. (a) These ratings are new data to find out if the medical magnets reduce pain for these patients. (b) The doctor selected, hopefully at random, a sealed envelope from a box and the patients were required to wear the magnet selected by the doctor. Since a treatment was actively imposed, this is an experiment. (c) Chance is used to assign the patients to the treatment groups to avoid bias. Researchers do not do a good job forming identical groups when they use their
What Is Statistics? 7 subjective opinions to place individuals into treatment groups. (d) Yes. Double blind studies where the patients and the researchers do not know which treatment is being given are preferred. If a patient knew she had an inactive magnet, she would not expect her pain to be reduced. Similarly, if the doctor knew that a patient had an active magnet, she may suggest that the pain seems to be decreasing. (This sounds too obvious, but it happens in practice.) 3. (a) The graph in Figure P.7 is roughly symmetric about zero. Thus, half of the differences are above zero and half are below zero. About 5% of the time the difference will be positive. (b) It is very unlikely that the difference in mean pain ratings is greater than 4.5. In fact, none of the 1, simulated differences is greater than 4.5. 4. (a) Use the difference in the averages, 4.5. (b) Reject this claim. The medical magnets appear to substantially reduce pain for these patients. P.19 (a) Available data from family interviews and police records were used. No treatments were imposed in order to observe various responses. (b) Parental involvement, profession of parents, educational priorities, amount of reading, type of child care, participation in sports, and participation in other activities with peers are just a few other variables that may be related to the amount of TV watched. The effects of these other variables are mixed up with and cannot be separated from the effect of watching TV. This is known as confounding. P.2 From The National Highway Traffic Safety Administration web page (www.nhtsa.gov), click on the tab labeled Traffic Safety, then click New Drivers, and finally click Graduated Driver Licensing System. These selections will lead you to a pdf version of Traffic Safety Facts. (Note: These links were obtained in January, 27 and will almost certainly change. However, students will be able to find the information with a quick search.) The background information in this report states that A significant percentage of young drivers are involved in traffic crashes and are twice as likely as adult drivers to be in a fatal crash. Sixteen-year-old drivers have crash rates that are three times greater than 17-year-old drivers, five times greater than 18-year-old drivers, and twice the rate of 85-year-old drivers. The factors contributing to these higher crash rates include lack of driving experience and inadequate driving skills; excessive driving during night-time, higher-risk hours; risk-taking behavior; poor driving judgment and decision making; drinking and driving; and distractions from teenage passengers. Other discouraging facts are included in the report and a new licensing system is being implemented to prolong the learning process for new drivers. More statistics related to teen drivers can be found by searching www.google.com or www.ask.com. Try searches for teen driving statistics, teen fatalities, are teens safe drivers, etc. P.21 Who? The individuals are motor vehicles produced in 24. What? The categorical variables are: make and model; vehicle type; transmission type. The quantitative variables are: number of cylinders (integer count); city MPG (miles per gallon); highway MPG (miles per gallon). Why? The data were compiled to compare fuel economy. When, where, how, and by whom? A statement on www.fueleconomy.gov reveals that
8 Preliminary Chapter The data included in the Department of Energy's Fuel Economy Guide are the result of vehicle testing done at the Environmental Protection Agency's National Vehicle and Fuel Emissions Laboratory in Ann Arbor, Michigan and by vehicle manufacturers themselves with oversight by EPA. P.22 (a) A bar graph showing the percent of students in each grade who said they rarely or never wore bicycle helmets is provided below. The percent for 9 th graders is just a little lower than the percents for the older students, but all four percents are very high. Percent of students who said they rarely or never wore bicycle helmets 9 8 7 6 Total % 5 4 3 2 1 9 1 Grade 11 12 (b) A side-by-side bar graph showing the percent of males and females at each grade level who said they rarely or never wore bicycle helmets is provided below. Notice that the percent for males is higher than the percent for females in every grade level. However, all of the percents are above 8. Percent of students who said they rarely or never wore bicycle helmets 9 8 7 6 Percents 5 4 3 2 1 Female Male Female Male Female Male Female Male Grade 9 1 11 12 P.23 There are 13 different types of cards (A, 2, 3, 4, 5, 6, 7, 8, 9, 1, J, Q, K) in a standard deck of cards. The chance of getting exactly 3 of the same type, 1 of a second type, and 1 of a third type in a hand of 5 cards is 1/5. Thus, if we kept dealing 5 card hands over and over again, the proportion of hands that would contain three of a kind approaches.2. Alternatively, we could say the percentage of hands that would contain three of a kind approaches 2%. P.24 (a) Because they were interested in the opinions of all U.S. adults and not just MLB fans who support the athletes. (b) Although there will be variation from sample to sample, we can assume that the sample obtained by the Gallup organization was representative of the entire U.S. adult population. Thus, the population percentages would be about the same as the sample percentages obtained by Gallup, 42% for probably not and 33% for definitely not. (c) No, we can not conclude that Barry Bonds is lying. These percentages reflect the opinions of U.S.
What Is Statistics? 9 adults, but public opinion can be much different than fact. Some day we may find out if Barry Bonds lied, but right now only Barry and a few other people know the truth! P.25 From Figure P.7, roughly 5 of the simulated differences were 2.5 or larger. Thus, the chance of getting a difference of 2.5 or something larger if there is no difference in the magnets is approximately.5. Our conclusion would remain the same; we have evidence that the active magnets relieve pain in polio patients. P.26 (a) Dotplots for both groups are shown below. The differences for the patients in the active group follow a distribution with a gap between 1 and 4. We might even stretch a bit and say that the distribution is roughly symmetric with a mean of about 5. The differences ranged from to 1. (b) The distribution of the differences for the patients in the inactive group is skewed to the right with a center slightly above 1. Many patients reported no change in their pain ratings (a difference of ) and the largest difference was 5. Active Inactive 2 4 Data 6 8 1 (c) The average difference for the active group is 5.241 and the average difference for the inactive group is 1.95. The difference in these two means (inactive active) is 4.146. (d) Figure P.8 shows that none of the 1, simulated differences was smaller than 4.146. Thus, a difference of 4.146 would be extremely unlikely if both types of magnets provided the same level of relief. We would reject the hypothesis of no treatment effect and conclude that the active magnets do provide relief for polio patients. P.27 A dotplot for the highway gas mileage for 32 model year 24 midsize cars is provided below. The distribution of MPG is roughly symmetric with a center at about 27 miles per gallon. The range is 14, 34 2, with the Chevrolet Malibu and the Honda Accord both getting the best mileage of 34 MPG. The Mercedes-Benz E5 had the worst gas mileage, 2 MPG. 24 Midsize Cars 2 22 24 26 28 mpg24 3 32 34
1 Preliminary Chapter P.28 (a) These data were probably obtained from an observational study that wasn t a survey. The organizers of the program could easily collect information from volunteers who participated in the Mozart for Minors program and were willing to share their test scores. These scores could then be compared with average scores for all students that are typically reported to school districts and parents. (b) No, we can not conclude that the Mozart for Minors program caused an increase in the students test scores. Educational background of the parents, family income, neighborhood, and many other factors may influence test scores and participation in the music program. (c) Conduct a simple comparative experiment. A large group of students completes the exams and then is randomly split into two groups. One group participates in the Mozart for Minors program and the other group does not. Ideally, the two groups will have exactly the same experiences, except for participation in the music program. This is easier said than done! After a reasonable period of time, the students take exams to measure verbal and math skills again. The averages of the changes in the scores (second exam first exam) can be compared for the two groups.