Statistics I 5,6,2,2,2,7 Consider this list of numbers: ‘The mean of the list is the average: BHO+24I+2E7 _ 6 The median is the number in the middle when the list is in order. For example, the median for 1,2,3,4,5 is 3. For our particular list, which looks like 2,2,2,5,6,7 when ordered, there is no single middle number we can consider the median. When that happens, the median is the average the two middle numbers: 245 a = “BS Now what if the list were 100 numbers long? How would you determine the median? Take half to get 50. The 50th and Sist numbers would be the ones in the middle you would average. For an ordered list of 101 numbers, take half to get 50.5. Round up. The 5ist number is the median. Seems a little counterintuitive, right? If you find this hard to memorize, just keep the smallest case in your back pocket. For a list of 3 numbers, the second one is obviously the median. How would we get this ‘mathematically? Take half of 3 fo get 1.5. Round up to 2, which designates the second number. For a list of 4 numbers, the median is the average of the second and third numbers. Take half of 4 to get 2. This designates the second and third numbers In both cases, we “rounded up.” When there was an odd number of numbers, we rounded 1.5 up to 2. When there was an even number of numbers, we rounded 2 up to 3, which indicated that two numbers would. contribute to the median. This technique may seem a bit odd, but many students have found it helpful in quickly finding the median of a large batch of numbers. ‘The mode is the number that shows up the most often. In our particular list, it’s [2], The range is the difference between the biggest number in the list and the smallest number: 7-2=[5] 223 SAT & IG BOOK STORE 01227746409 223CHAPTER 25 STATISTICS I The standard deviation is a measure of how spread out a list of numbers is. In other words, how much they “deviate” from the mean. The standard deviation is lower when more numbers are closer to the mean. The standard deviation is higher when more numbers are spread out away from the mean. For example, our list 2,2,2,5,6,7 would have a higher standard deviation than the following list 5,5,5,5,6,7 because the second list is more tightly clustered around the mean. It tums out that the standard deviation of our list is 2.28 and the standard deviation of the second list is 0.83. Don’t worry about how we got these values—you'll never be asked to calculate the standard deviation on the SAT, Just know how to compare one list’s standard deviation with another's as we just did. EXAMPLE 1: Daily Hours Spent Playing Sports Number of students Hours The histogram above summarizes the daily number of hours spent playing sports for 80 students at a school PART 1: What is the mean daily number of hours spent playing sports for the 80 students? PART 2: What is the median daily number of hours spent playing sports for the 80 students? Part 1 Solution: Sum up the total number of hours for every student. Then divide that by the number of students. Total hours (0x5) + (1x35) + (2 * 15) + ( Teal hours 5) 10x25) _ MO _ p75) Number of students 80, 80 1125! Part 2 Solution: In a group of 80 students, the 40th and 41st students are the two in the middle (the histogram already orders the students by their hours so we don’t have to). The first 5 students spend 0 hours playing, sports each day. The next 35 students spend 1 hour. This group includes the 40th student, so the 40th stuclent spends 1 hour. The next 15 students spend 2 hours. Now this group includes the 41st student, so the 41st student spends 2 hours. Taking the average, Daily hours spent by 40th student + Daily hours spent by 4ist student _ 14-2 : 42 _[75) 224 SAT & IG BOOK STORE 01227746409 224THE COLLE PANDA EXAMPLE 2: . 0 The dot plot above summarizes the number of flights taken in a year by 19 college students. If the student who took 6 flights in a year is removed from the data, which of the following correctly describes the changes to the statistical measures of the data? ‘The mean decreases. I, The median decreases. IL. The rangp decreases. A) Ilonly B) Land Il only ©) land II only D) L.Iand The student who took 6 fights in a year is called an outlier, an extreme data point that is far outside where most of the daia lies. Because this outlier is greater than the rest of the data, it brings the average (mean) up. It also inereases the range since there isa larger gap between the minimum (0) and the maximum (6) When thisoutlicris removed, the mean decreases and the range decreases. The median, however, is unaffected. To confirm this, le’s calculate it. Before the outlier is removed, there are 19 students, and the median is represented by the 10th stuclent, who took one flight. After the outlier is removed, there are 18 students, and the median is represented by the Ith end 10th students, both of whom took one flight. So the median of 1 does not change. And in fact, outliers typically affect the mean but not the median. Answer| (C) EXAMPLE 3: ‘The average weight of a group of pandas is 200 pounds. Another panda, weighing 230 pounds, joins the group, raising the average weight of the entire group to 205 pounds. How manly pandas ‘were in the original group? (Once in a while, you will get a word problem that involves averages. ‘These questions have less to do with statistics and more to do with algebra, but because we cover averages in this chapter, we decided to cover these types of word problems here as well. ‘When dealing with average questions on the SAT, think in terms of sums or totals. You can always find the sum by multiplying the average with the number of subjects. Let the number of pandas in the original group be x. ‘The total weight of the original group is then 20x, When another panda joins the group, the number of pandas is x + 1 and the total weight is 205(x + 1) 25 SAT & IG BOOK STORE 01227746409 225(CHAPTER 25 Since that panda weighs 230 pounds, 20x +230 = 205(x +1) 20x + 230 = 208% + 205 bx = 5 ‘There were[5 | pandas in the original group. EXAMPLE 4: Neighborhood A ‘Number of residents os as Number of cars owned ‘Neighborhood B Number of residents 012445 6 Number of cars owned ‘The bar charts above summarize the number of cars that residents from two neighborhoods, A and B, ‘own, Which of the following correctly compares the standard deviation of the number of cars owned by residents in each of the neighborhoods? ‘A) The standard deviation of the number of cars owned by residents in Neighborhood A is larger. B) The standard deviation of the number of cars owned by residents in Neighborhood B is larger. ©) The standard deviation of the number of cars owned by residents in Neighborhood A and Neighborhood B is the same, 1D) The relationship cannot be determined from the information given, 26 SAT & IG BOOK STORE 01227746409 226Ee THE COLLEGE PANDA Most of the data for Neighborhood B are at the ends and are much more spread out from the mean, which, because the bar graph is symmetrical, we can estimate to be 3 cars, The data for Neighborhood A, on the other hand, are more clustered towards the low end, where the mean is. Therefore, the standard deviation for Neighborhood B is larger. Answer |(B) | CHAPTER EXERCISE: Answers for this chapter start on page 310. ‘A calculator is allowed on the following questions. TEE oer 6s Te Be: | ‘The average height of 14 students in one classis | g 63 inches. The average height of 21 students in gt another class is 68. If the two classes are 3 | combined, what is the average height, in inches, g | of the students in the combined class? g 2} A) 645 Zz B) 6 0 OQ 6 PS 0 Pm PP D) 665 SLPS Sf Books read EL | the histogram above shows the number of books read last year by 20 editors at a publishing Kristie has taken five tests in science class. The company. Which of the following could be the average of all five of Kristie's test scores is 94. ‘median number of books read by the 20 editors? The average of her last three test scores is 92. What is the average of her first two test scores? a a BR y ov B) 9% D) 2 ov D) 98 27 SAT & IG BOOK STORE 01227746409 227CHAPTER 25. STATISTICS es Miss World Titleholders eoeceeene 8 19 20 21 “2 Bh Age (years) The dotplot above shows the distribution of ages for 24 winners of the Miss World beauty pageant at the time they were crowned. Based on the data, which of the following is closest to the average (arithmetic mean) age of the winning Miss World pageant contestant? A) 19 B) 20 o 2 b) 2 BB 2 See Locks are sections of canals in which the water level can be mechanically changed to raise and lower boats. The table below shows the number of locks for 10 canals in France: Name | # Locks Aisne | 27 Alsace | 25 |" Rhone 5 Centre | 30 [Garonne | 23 | Lalinde | 27 | (3 32. Ea 93 29 Removing which ofthe following two canals from the data would result in the greatest decrease in the standard deviation of the number of locks in each canal? A) Aisne and Lalinde: B) Alsace and Garonne C) Centre and Midi D)_ Rhone and Vosges SAT & IG BOOK STORE 01227746409 228THE COLLEGE PANDA The tables below give the distribution of travel times between two towns for Bus A and Bus B ‘over the same 40 days. Bus A Travel time (minutes) | Frequency a 5 5 10 a [5 | so 10 BusB ‘Travel time (minutes) | Frequency | Bo | 5 | 30 10 35 15 40 10 Which of the following statements is true about the data shown for these 40 days? A) The standard deviation of travel times for Bus A is smaller. B) The standard deviation of travel times for Bus Bis smaller. C) The standard deviation of travel times is, the same for Bus A and Bus B. D) The standard deviation of travel times for Bus A and Bus B cannot be compared with the data provided. SAT & IG BOOK STORE 01227746409 229 10 Number of kayaks oH etuaroe 45 46 474849 Weight (in pounds) ‘The bar chart above shows the distribution of ‘weights (to the nearest pound) for 19 kayaks made by Company A and 19 kayaks made by ‘Company B. Which of the following correctly ‘compares the median weight of the kayaks made by each company? A) The median weight of the kayaks made by ‘Company A is smaller. B) The median weight of the kayaks made by ‘Company B is smaller. ©) The median weight of the kayaks is the same for both companies. D) The relationship cannot be determined from the information given. ‘Temperature (°F) | Frequency 60. 3 a 4 6 4 a 10 7 7 The table above gives the distribution of low temperatures for a city over 28 days. What is the median low temperature, in degrees Fahrenheit (°F), of the city for these 28 days? 229CHAPTER 25 STATISTICS 1 BS Sse Se A shoe store surveyed a random sample of 50 customers to better estimate which shoe sizes should kept in stock. The store found that the median shoe size of the customers in the sample is 10 inches. Which of the following statements must be true? A) The sum of all the shoe sizes in the sample is 500 inches. B) The average of the smallest shoe size and the largest shoe size in the sample is 10 inches C) The difference between the smallest shoe size and the largest shoe size in the sample is 10 inches. D) Atleast half of the customers in the sample have shoe sizes greater than or equal to 10 inches. A food company hires an independent research agency to determine its product's shelf life, the length of time it may be stored before it expires. | Using a random sample of 40 units of the product, the research agency finds that the product's shelf life has a range of 3 days. Which of the following must be true about the units in| the sample? A) Allthe units expired within 3 days. B) ‘The unit with the longest shelf life took 3 days longer to expire than the unit with the shortest shelf life. ©) The mean shelf life of the units is 3 more than the median D) The median shelf life of the units is3 more than the mean. 230 SAT & IG BOOK STORE 01227746409 1 , ee See 5 6 7 8 9 0 Integers ‘The graph above shows the frequency distribution of alist of randomly generated integers between 5 and 10. Which of the following correctly gives the mean and the range of the list of integers? A) Mean = 7.6, Range = 4 B) Mean = 7.6, Range =5 ©) Mean = 82, Range D) Mean = 82, Range CE a as Quiz [1 [2[3f4a[s{ol7] Score [87 | 75] 90 [85 | 98 [87 [91 ] ‘The table above shows the scores for Jay’s first seven math quizzes. Which of the following are true about his scores? 1. The mode is greater than the median, Il, The median is greater than the mean. I, The range is greater than 20. A) Tonly B) Mlonly ©) Mand tt D) 1,1 andl 230Eee = {mam School A Number of classes 192 3 4 «5 Number of films shown The bar chart above shows the number of films shown in class over the past year for 19 classes in School A and 15 classes in School B. Which of the following correctly compares the mean and median number of films shown in each class for the two schools? A) The mean and median number of films shown in each class are both greater in School A. B) ‘The mean and median number of films shown in each class are both greater in School B. ©) The mean number of films shown in each, lass is greater in School A, but the median is the same in both schools. D) The mean and median number of films shown in each class are the same in both, schools. Calories in Meals 500 | 500 | 520 | 550 | 55% '550 | 550 | 600 | 600 | 900 The table above lists the number of calories in each of Mary's last 10 meals. If 900-calorie ‘meal that she had today is added to the values listed, which of the following statistical measures of the data will not change? 1. Median Il, Mode IIL, Range A) Land ILonly B) [and Il only ©) Mand ttl only D) 1,1, and Mt 21 22 23 24 25 26 27 28 29 30 Gas mileage (miles per gallon) The dotplot above gives the gas mileage (in miles per gallon) of 15 different cars. If the dot representing the car with the greatest gas mileage is removed from the dotplot, what will happen to the mean, median, and standard deviation of the new data set? A) Only the mean will decrease. B) Only the mean and standard deviation will decrease C) Only the mean and median will decrease. D) The mean, median, and standard deviation will decrease. SAT & IG BOOK STORE 01227746409 231CHAPTER 25 STATISTICSI_ Snowfall (in inches) 45 | 48 | 49 | 50 | 52 | 54 55 | 57 | 57 | 57 | 58 | 59 60 | 60 [61 | 61 | 65 | 90 ‘The table above lists the amounts of snowfall, to the nearest inch, experienced by 18 different cities in the past year. The outlier measurement of 100 inches is an error. Of the mean, median, and range of the values listed, which will change the most if the 90-inch measurement is replaced by the correct measurement of 20 inches? A) Mean B) Median ©) Range D) None of them will change. 232 SAT & IG BOOK STORE 01227746409 23226 Statistics II ‘The goal of statistics is to be able to make predictions and estimations based on limited time and information. For example, a statistician might want to estimate the mean weight of all female raccoons in the United States. The problem is that it’s impossible to survey the entire female raccoon population. In fact, by the time that could be accomplished, not only would the data be out of date but there would be new females in the population. Instead, a statistician takes a random sample of female raccoons to make an estimation of what the actual mean might be. In other words, the sample mean is used to estimate the population mean. Using a sample to predict something about the entire population is a common theme in statistics and in SAT questions, EXAMPLE 1: A pet food store chose 1,000 customers at random and asked each customer how many pets 4ormore ‘There are a total of 18,000 customers in the store's database. Based on the survey data, whatis the expected total number of customers who own 2 pets? Using the sample data, we can estimate the total number who own 2 pets to be m 5g 18.000 2%, =f 233, SAT & IG BOOK STORE 01227746409 233CHAPTER 26 STATISTICS It EXAMPLE 2: i H os tH % 1000 Heart rate (beats per minute) vn Bio a Oxygen Uptake (liters per minute) ‘The scatterplot above shows the relationship between heart rate and oxygen uptake at 16 different points during Kyle's exercise routine. The line of best fit is also shown. PART 1: Based on the line of best fit, what is Kyle's predicted oxygen uptake at a heart rate of 110 beats per minute? PART 2: What is the oxygen uptake, in liters per minute, of the measurement represented by the data point that is farthest from the line of best fit? Part 1 Solution: Using the line of best fit, we can see that at a heart rate of 110 beats per minute (along the sani), the oxygen uptake is [15 liters per minute. Using the line of best fit to make a prediction can be dangerous, especially when + we are makinga prediction outside the scope of our data set (predicting the oxygen uptake ata heart rate ‘of 250 beats per minute, for example—you'd probably be dead). ‘* there are outliers that may heavily influence the line of best fit (see Part 2). + the data is better modeled by a quadratic or exponential curve rather than a linear one. In this case, a linear model looks tobe the right one, but something like compound interest may look linear at first even though it’s exponential growth, Part 2 Solution: From the scatterplot, we can see that the data point farthest away fom the line of best fits at 1iBalong the »-axis. The point represents an oxygen uptake of [25 liters per minute [Note that this data point is likely an outlier, which can heavily infiuence the line of best fit and throw off our predictions. Oulliers should be removed from the data if they represent special cases or exceptions. Not only will you be asked to make predictions using the line of best fit, but you'll also be asked to interpret its slope and y-intercept. We'll use the data from this example in the next one to show you how these concepts are tested. 234 SAT & IG BOOK STORE 01227746409 234THE COLLEGE PANDA EXAMPLE 3: ‘Oxygen Uptake versus Heart Rate ae LT ttt TT Sees ‘Oxygen Uptake (liters per minute) So 1000 Heart tate (beats per minute) ‘The scatterplot above shows the relationship between heart rate and oxygen uptake at 16 different points during Kyle's exercise routine. The line of best fit is also shown, PART 1: Which of the following is the best interpretation of the slope of the line of best fit in the context ofthis problem? ‘A) The predicted increase in Kyle’s oxygen uptake, in liters per minute, for every one beat per minute increase in his heart rate B) The predicted increase in Kyle's heart rate, in beats per minute, for every one liter per minute increase in his oxygen uptake ©) Kyle's predicted oxygen uptake in liters per minute at a heart rate of 0 beats per minute D) Kyle's predicted heart rate in beats per minute at an oxygen uptake of O liters per minute PART 2: Which of the following. is the best interpretation of the y-intercept of the line of best fit in the context of this problem? A) The predicted increase in Kyle's oxygen uptake; i ‘increase in his heart rate Titers pet minute, for every one beat per ininute B) The predicted increase in) Kyle's heart rate, in beats per minute, for every one liter per minute increase in his oxygen uptake ©) Kyle's predicted oxygen uptake in liters per minute at a heart rate of d beats per minute D) Kyle's predicted heart rate in beats per minute at an oxygen uptake of O liters per minute Part 1 Solution: As we learned in the linear model questions in the interpretation chapter, the slope is the increase in y (oxygen uptake) for each increase in x (heart rate). The only difference now is that it’s a predicted increase. The answer is[(A) | Part 2 Solution: The intercept isthe value of y oxygen uptake) when x (the her eat) 0, The answer [(©) Nove tat this value would have no significance in el ifesince you woul be deat a ear rate of This again illustrates the danger of predicting values outside the scope of the sample data. 235 SAT & IG BOOK STORE 01227746409 235CHAPTER 26 STATISTICS __ EXAMPLE 4: Malden is a town in the state of Massachusetts. A real estate agent randomly surveyed 50 apartments for sale in Malden and found that the average price of each apartment was $150,000. Another real estate agent intends to replicate the survey and will attempt to get a smaller margin of error. Which Of the following samples will most likely result in a smaller margin of error for the mean price of an apartment in Malden, Massachusetts? ‘A) 30 randomly selected apartments in Malden 'B) 20 randomily selectedt apartments in all of Massachusetts ©) 80 randomly selected apartments in Malden 1D) 80 randomly selected apartments in all of Massachusetts The answer is| (C)| The margin of error refers to the room for error we give to an estimate. For example, we could say the mean price of an apartment in Malden is $150,000 with a margin of error of $10,000. This implies that the true mean price ofall apartments in Malden is likely between $140,000 and $160,000. This interval is called a confidence interval (see Example 6). To get a smaller margin of error in Example 4, we should first only select from apartments in Malden. Selecting, apartments from all of Massachusetts not only introduces more variability to the data but also strays from the original intent of the survey, which is to find the average price of Malden apartments. Secondly, we should use larger sample size. This is common sense. The more apartments we survey, the more accurate our data and ‘our estimations are and the lower our margin of error is. In fact, the margin of error for any estimate from an experiment depends on two factors: + Sample size + Variability in the data (often measured by standard deviation) The larger the sample size and the less variable the data is, the lower the margin of error. We typically can’t control the standard deviation of the data (how spread out itis), but we can control the sample size. So why don’t researchers always use huge sample sizes? Because it’s too costly and time-consuming to gather data from everyone and everywhere. EXAMPLE 5: Researchers conducted an experiment to determine whether exercise improves stident exam scores. They randomly selected 200 students who exercise at least once a week and 200 students who do not exercise at least once a week. After tracking the students’ academic performances for a year, the researchers found that the students who exercise at least once a week performed significantly better ‘on the same exams than the students who do not, Based on the design and results of the study, which of the following is an appropriate conclusion? ‘A) Exercising at least once a week is likely to:improve exam scores. 1B) Exérvising three times a week improves exam scotes more than exercising just once a week. ©) Any student who starts exercising at least once a week will improve his or her exam scores. D) There is a positive association between exercise and student exam scores. This question deals with a classic case of association (also called correlation) vs. causation. Just because students who exercise got better exam scores doesn’t mean that exercise causes an improvement in exam scores, It’s ust associated with an improvement in exam scores. Perhaps students who exercise just have more discipline or they have more demanding parents who make them study harder. Due to the way the experiment was designed, we can’t tell what the underlying factor is. 236 SAT & IG BOOK STORE 01227746409 236THE COLLEGE PANDA ‘Therefore, answer (A) is wrong because it implies causation. Answer (B) is wrong because it not only implies ‘causation but also implies that the frequency of exercise matters, something that wasn’t tracked in the experiment Answer (C) is wrong because it suggests a completely certain outcome. Even if exercise DID improve exam scores, not every single student who starts exercising will improve their scores. There might be students for whom exercising makes their scores worse. Any conclusion drawn from sample data is a generalization and should not be regarded as a truth for every individual. ‘The answer is|(D) | There is a positive association between exercise and student exam scores. One of the things the researchers did correctly was to take random samples from each group. The key word is, random. If the samples weren’t random, we wouldn’t even have been able to conclude that there is a positive association between exercise and exam scores. Why? Let's say the researchers picked 30 students from the tennis team for the exercise group and 30 students who just play video games all day for the non-exercise ‘group. Definitely not random. Now, did the exercise group do better on their exams because they exercise or because they play tennis? Or was it the video games that made the non-exercise group perform worse? Because the selection isn’t random, we can't tell how each factor influences the result. When the selection is, random, all the factors except the one we're testing are “averaged out.” Now what f the researchers wanted to see whether exercise does indeed cause an improvement in exam scores. What should they have done differently? The answer is random assignment. Instead of randomly selecting 200 students from one group that already exercises regularly and 200 students from another group that does not, they should have just randomly selected 400 students. The next step would be to randomly assign each student to exercise or not. Everyone in the exercise group is forced to exercise at least once a week and everyone in the rnon-exercise group is not allowed to exercise. If the exercise group performs better on the exams, then we can conclude that exercise causes an improvement in exam scores, Of course, conducting this type of experiment can be extremely difficult, which is why proving, causation can be such a monumental task ‘The following list summarizes the conclusions you can draw from different experimental designs involving two variables (e.g. exercise and exam scores) 1. Subjects not selected at random & Subjects not randomly assigned ‘¢ Results cannot be generalized to the population. Cause and effect cannot be proven. © Example: Researchers want to see whether medication X is effective in treating the flu. People with the flu from Town A receive medication X. People with the flu from Town B receive a placebo (sugar pill). More people in the medication X group experience a reduction in flu symptoms. The generalization that medication X is associated with a reduction in flu symptoms cannot be made since it was only tested in Town A and Town B (sample was not randomly selected from the general, population). There may be something special about Town A and Town B. No cause and effect relationship can be established because the medication was not randomly assigned. Perhaps Town A experienced a less severe flu epidemic. 2. Subjects not selected at random & Subjects randomly assigned ‘+ Results cannot be generalized to the population. ‘* Cause and effect can be proven. ample: Researchers want to see whether medication X is effective in treating the flu. People with the flu from Town A and Town B are randomly assigned to either medication X or a placebo (sugar pill). More people in the medication X group experience a reduction in flu symptoms. The generalization that medication X is effective for everyone cannot be made since it was only tested in Town A and Town B (sample was not randomly selected from the general population). Perhaps only one particular strain of the flu exists in Town A and Town B. A cause and effect relationship can be established because the medication was randomly assigned. For the people in Town A and 237 SAT & IG BOOK STORE 01227746409 237CHAPTER 26 STATISTICS IL ‘Town B, we can conclude that medication X causes a reduction in flu symptoms. Note that this is still just a generalization—as with any other medicatio definitely get better, even if you live in Town A or Town B. nedication X does not guarantee you will 3. Subjects selected at random & Subjects not randomly assigned. + Results can be generalized to the population, '* Cause and effect ¢ rnot be proven. * Example: Researchers want to see whether medication X is effective in treating the flu. People with the flu from the general population are randomly selected. They are given the choice of a new medication (medication X) oF a traditional medication (really a sugar pill). More people in the ‘medication X group experience a reduction in flu symptoms. We can generalize that people who choose to receive medication X fare better than those who don’t. However, no cause and effect relationship can be established because the medication was not randomly assigned. We don't know whether the reduction in symptoms is due to the medication or a difference between those who volunteered and those who didn’t 4. Subjects selected at random & Subjects randomly assigned Results can be generalized to the population. ‘¢ Cause and effect can be proven. Fxample: Researchers want to see whether medication X is effective in treating the flu. People ith the flu from the general population are randomly selected. Using, a coin toss (heads or tails), researchers randomly assign each person to either medication X or a placebo (sugar pill). More people in the medication X group experience a reduction in flu symptoms. We can conclude that medication X causes a reduction in flu symptoms. This conclusion can be generalized to the entire Population of people with the flu. EXAMPLE 6; Environmentalists are testing pH levels ina forest that is being harmed by acid rain. They analyzed water samples from 40 rainfalls in the past year and found that the mean pH of the watersamples has a 95% confidence interval of 3.2 to 3.8, Which of the following conclusions is the most appropriate based on the confidence interval? ‘A) 95% of all the forest rainfalls in the past year havé a pH between 3.2.and 3.8. 'B) 95% ofall the forest rainfalls im the past decade have a pH between 3.2 and 3.8. ©) Itis plausible that the true mean pH of all the forest rainfalls in the past yearis between 3.2 and 3.8 1D) Tks plausible that the true mean pH of all the forest rainfalls in the past decade is between 3.2 and 38. If you don’t know what a confidence interval is, don’t worry. You'll never need to calculate one and the SAT makes these questions very easy. Alla confidence interval does is tell you where the true mean (or some other statistical measure) for the population is likely to be (e.g. between 3.2 and 3.8). Even though the SAT only brings up 95% confidence intervals, there are 97% and 99% (any percentage) confidence intervals. The higher the confidence, the more likely the true mean falls within the interval. So in the example above, we can be quite confident that the true mean pH of al the forest rainfalls in the past year is between 3.2 and 38. Answer (©] The answer is not (D) because we cannot draw conclusions about the past decade when all the samples were gathered from the past year. A confidence interval does NOT say anything about the rainfalls themselves. You cannot say that any one rainfall has a 95% chance of having a pH between 3.2 and 3.8, and you cannot say that 95% of all the forest rainfalls in the past year had a pH between 3.2 and 3.8. Always remember that a confidence interval applies 238 SAT & IG BOOK STORE 01227746409 238THE COLLEGE PANDA only fo the mean, which is a statistical measurement, NOT an individual data point or a group of data points. Secondly, a 95% confidence interval does not imply that there is a 95% chance it contains the true mean. Even. though confidence intervals are computed for the mean, you cannot say that the interval of 3.2 to3.8 has a 95% chance of containing the true mean pH. ‘So what does it mean in statistics to be 95% confident in something? If the experiment were repeated again and. again, each with 40 water samples, 95% of those experiments would give us a cor the true mean. In other words, the confidence idence interval that contains terval given in the example is the result of just one experiment Another run of the same experiment (another 40 samples) would produce a different confidence interval. Keep on getting these confidence intervals and 95% of them will contain the true mean. So the 95% pertains to al the confidence intervals generated by repeated experiments, NOT the chance that any one confidence interval contains the true mean. Again, don’t worry about how confidence intervals are calculated, but be aware that this is how “confidence” is defined in statistics. 239 SAT & IG BOOK STORE 01227746409 239CHAPTER 26 STATISTICS IL CHAPTER EXERCISE: Answers for this chapter start on page 312. A calculator is allowed on the following | BSS ee aquestons Bek eae Traffic Light Violations in Various Towns 100 Bes essa 8 t 30 40 50 60 70 80 9% 100 Number of traffic lights 10 11 12 13 14 15 16 17 18 19 20 Average weekly number of traffic light violations ‘Age (years) ‘The scatterplot above shows the relationship between age, in years, and shoe size for 24 males between 10 and 20 years old. The line of best fit is also shown. Based on the data, how many 19 year old males had a shoe size greater than the ‘one predicted by the line of best fit? The scatterplot above shows the number of traific lights in 15 towns and the average weekly ‘number of traffic light violations that occur in each town. The line of best fit is also shown. Based on the line of best fit, which of the following is the predicted average weekly number of traffic light violations in a town with Al | Brraffic lights? B2 A) 40 o3 i D4 | 240 SAT & IG BOOK STORE 01227746409A university wants to determine the dietary preferences of the students in its freshman class. Which of the following survey methods is most likely to provide the most valid results? A) Selecting a random sample of 600 students from the university B) Selecting a random sample of 300 students from the university's freshman class C) Selecting a random sample of 600 students from the university's freshman class D) Selecting a random sample of 600 students from one of the university's freshman dining halls ‘Two candidates are running for governor of a state. A recent poll reports that out of a random sample of 250 voters, 110 support Candidate A and 140 support Candidate B. An estimated {500,000 state residents are expected to vote on election day. According to the poll, Candidate B is expected to receive how many more votes than Candidate A? A) 60,000 B) 130,000 ©) 220,000 D) 280,000 241 SAT & IG BOOK STORE 01227746409 THE COLLEGE PANDA Consumer Behavior during Store Sales Sa8 5 0 ‘Average shopping time (minutes) 0 5 10 15 20 25 30 35 40 45 50 Store Discount (%) Shopping time refers to the time a customer spends in one store. The scatterplot above shows the average shopping time, in minutes, of customers at 26 different stores offering various discounts. The line of best fit is also shown. Which of the following is the best interpretation of the meaning of the y-intercept of the line of best fit? A) The predicted average shopping time, in minutes, of customers ata store offering no discount ‘The predicted average shopping time, in minutes, of customers at a store offering a 50% discount ‘The predicted increase in the average shopping time, in minutes, for each one percent increase in the store discount ‘The predicted average number of customers at a store offering no discount B) °) D) 241CHAPTER 26 STATISTICS I ES ee SS Advertising for 16 Companies ES ae Movie Length versus Box Office Sales zB z Revenue (in thousands of dollars) 0 10 20 30 40 50 60 70 80 90 100 Advertising Expenses (in thousands of dollars) ‘The scatterplot above shows the relationship between revenue and advertising expenses for 16 companies. The line of best fit is also shown. Which of the following is the best interpretation ‘of the meaning of the slope of the line of best fit? ‘A) The expected increase in revenue for every one dollar increase in advertising expenses B) The expected increase in revenue for every one thousand dollar increase in adve expenses ‘The expected increase in advertising ‘expenses for every one thousand dollar increase in revenue D) The expected revenue of a company that hhas no advertising expenses ° s Box Office Sales (in millions of dollars) o88aees 160 70 80 90 100 110 120 130 140 150 Movie Length (minutes) ‘The scatterplot above plots the lengths of 15, movies against their box office sales. The line of best fit is also shown. Which of the following is the best interpretation of the meaning of the slope of the line of best fit? A) The expected decrease in box office sales per minute increase in movie length B) The expected increase in box office sales per mute increase in movie length C) The expected decrease in box office sales per 10-minute increase in movie length D) The expected increase in box office sales per 10-minute increase in movie length 242 SAT & IG BOOK STORE 01227746409 242‘Mistakes Made in Incentive-based Task THE COLLEGE PANDA Fat and Calories of lee Cream. Boss 88 Number of mistakes made -. Total calories 0 0 100 200 300 400 500 600 Prize (in dollars) Ina psychological study, researchers asked participants to each complete a difficult task for a cash prize, the amount of which varied from. participant to participant. The results of the study, as well as the line of best fit, are shown in the scatterplot above. Which of the following is the best interpretation of the meaning of the grintercept of the line of best fit? A) The expected decrease in the number of iiistakes made per dollar increase in the cash prize B) The expected increase in the number of ‘iistakes made per dollar increase in the ‘cash prize ‘The expected dollar amount of the cash prize required for a person to complete the task with 0 mistakes ‘The expected number of mistakes a person makes in completing the task when no cash prize is offered ° Db) SAT & IG BOOK STORE 01227746409 283, ‘Total fat (grams) ‘The scatterplot above shows the fat content and calorie counts of 8 different cups of ice cream. Based on the line of best fit to the data shown, ‘what is the expected inerease in the number of calories for each additional gram of fat in a cup of ice cream? AS B) 8 ©) 20 D) 40 243bk eae ae z Food Courts in Various Malls > 3 100 T T g i ; 4 | R g ’ | 3 a 1 i i So i i 3 5 i é * 9 goo +. 0 100 200 300-400 500 2B 4 5 6 7 Amount of nitrogen applied (pounds per acre) Number of restaurants “The scatterplot above shows the amount of |The scatterplot above shows the distribution of nitrogen fertilizer applied to 8 oat fields and ie eo ne Hee irene nel ooe their yields. The line of best fit is also shown, fares ie MeO Dee A lee con Which of the following is closest to the amount | __ According to the data, what is the total number of nitrogen applied, in pounds per acre, the ‘of seats at the food court represented by the data cat field whose yield is best predicted by the line, _ Point that is farthest from the line of best fit? of best fit? |g) 200 A) 200 B) 240 B) 350 ©) 320 ©) 400 D) 560 D) 450 Researchers must conduct an experiment to see whether a new vaccine is effective in relieving cortain allergies. They have selected a random. sample of 100 allergy patients. Some of the patients are assigned to the new vaccine while the rest are assigned to the traditional treatment. Which of the following methods of assigning cach patient's treatment is most likely to lead to a reliable conclusion about the effectiveness of the new vaccine? A) Females are assigned to the new vaccine. B) Those who have more than one allergy are assigned to the new vaccine, ©) The patients divide themselves evenly into two groups. A coin is tossed to decide which group receives the new vaccine. D) Each patient is assigned a random number. ‘Those with an even number are assigned to the new vaccine, 244 SAT & IG BOOK STORE 01227746409 244A basketball manufacturer selects a random sample of its basketballs each week to ensure a consistent air pressure within them is maintained. In Week 1, the sample had a mean air pressure of 8.2 psi (pounds per square inch) and a margin of error of 0.1 psi. In Week 2, the sample had a mean air pressure of 7.7 psi and a margin of error of 0.3 psi. Based on these results, which of the following is a reasonable conclusion? A) Most of the basketballs produced in Week 1 hhad an air pressure under 8.2 psi, whereas most ofthe basketballs produced in Week 2 had an air pressure under 7.7 psi B) The mean air pressure of all the basketballs produced in Week 1 was 0.5 psi more than the mean air pressure of al the basketballs produced in Week 2. © Thenumber of basketballs in the Week 1 sample was more than the number of basketballs in the Week 2 sample D) tis very likely that the mean air pressure of all the basketballs produced in Week 1 was less than the mean air pressure ofall the basketballs produced in Week 2. A student is assigned to conduct a survey to determine the mean number of servings of vegetables eaten by a certain group of people each day. The student has not yet decided which group of people will be the focus of this survey. Selecting a random sample from which of the following groups would most likely give the smallest margin of error? A) Residents of the same city B) Customers of a certain restaurant ©) Viewers of the same television show D) Students who are following the same daily diet plan SAT & IG BOOK STORE 01227746409 [i | Ee ae 245 THE COLLEGE PANDA The length of a blue-spotted salamander’s tail can be used to estimate its age. A biologist selects 80 blue-spotted salamanders at random and finds that the average length of their tails has a 95% confidence interval of 5 to 6 inches. Which of the following conclusions is the most appropriate based on the confidence interval? A) 95% of all blue-spotted salamanders have a tail that is between 5 and 6 inches in length. B) 95% ofall salamanders have a tal that is, between 5 and 6 inches in length. C) The true average length of the tails ofall blue-spotted salamanders is likely between Sand 6 inches, D) The true average length of the tails of all salamanders is likely between 5 and 6 inches. a An economist conducted research to determine a relationship between the price of food and population density. He collected data from a random sample of 100 US. cities and found significant evidence that the price of food ower in places with a high population density. Which of the following conclusions is best supported by these results? A) InUS. cities, there isa positive association ‘between the price of food and population density, B) InUS. cities, there is a negative association between the price of food and population density. CC) InUS cities, a decrease in the price of food is caused by an increase in the population density. D) InUS. cities, an increase in the population density is caused by a decrease in the price of food. 245