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Exam Review Chapter 1 1. In 1997, there were 12,298,000 undergraduate students in U.S. colleges. According to the U.S. Department of Education, there were 127,000 American Indian or Alaskan Native, 737,000 Asian or Pacific Islander, 1,380,000 non-Hispanic black, 1,108,000 Hispanic, and 8,682,000 non-Hispanic white students. In addition, 265,000 foreign undergraduates were enrolled in U.S. colleges. (a) Each number, including the total, is rounded to the nearest thousand. Separate rounding may cause roundoff errors, so that the sum of the counts does not equal the total given. Are roundoff errors present in these data? Explain. (b) Present the data in a graph. 2. How much oil wells in a given field will ultimately produce is key information in deciding whether to drill more wells. Here are the estimated total amounts of oil recovered from 64 wells in the Devonian Richmond Dolomite area of the Michigan basin, in thousands of barrels. 21.7 53.2 46.4 42.7 50.4 97.7 103.1 51.9 43.4 69.5 156.5 34.6 37.9 12.9 2.5 31.4 79.5 26.9 18.5 14.7 32.9 196 24.9 118.2 82.2 35.1 47.6 54.2 63.1 69.8 57.4 65.6 56.4 49.4 44.9 34.6 92.2 37.0 58.8 21.3 36.6 64.9 14.8 17.6 29.1 61.4 38.6 32.5 12.0 28.3 204.9 44.5 10.3 37.7 33.7 81.1 12.1 20.1 30.5 7.1 10.1 18.0 3.0 2.0 Construct an appropriate graph of the distribution and describe its shape, center, and spread. 3. Create a set of fivepositive numbers (repeats allowed) that have median 10 and mean 7. What thought process did you use to create your numbers? 4. Of the 50 species of oaks in the United States, 28 grow on the Atlantic Coast and 11 grow in California. We are interested in the distribution of acorn sizes among oak species. Here are data on the volumes of acorns (in cubic centimeters) for these 39 oak species: Atlantic California 1.4 3.4 9.1 1.6 10.5 2.5 0.9 4.1 5.9 17.1 6.8 1.8 0.3 0.9 0.8 2.0 1.1 1.6 2.6 0.4 0.6 1.8 4.8 1.1 3.0 1.1 1.1 2.0 6.0 7.1 3.6 8.1 3.6 1.8 0.4 1.1 1.2 5.5 1.0 (a) Make a stemplot of the 39 acorn sizes. Describe the shape, center, and spread of the distribution. Choose appropriate numerical summaries. (b) Compare the Atlantic and California regional distributions with a graph and numerical summaries. What do you find? Chapter 2 I’m thinking about a density curve that consists of a straight line segment from the point (0, 2/3) to the point (1, 4/3) in the x–y plane. 1. Sketch this density curve. 2. What percent of the observations lie below 1/2? 3. What percent of the observations lie below 1? 4. What percent of the observations lie between 1/2 and 1? Chapter 1 Quiz 1.1A The scores of a reference population on the Wechsler Intelligence Scale for Children (WISC) are normally distributed with µ = 100 and = 15. 5. What score would represent the 50th percentile? Explain. 6. Approximately what percent of the scores fall in the range from 70 to 130? 7. A score in what range would represent the top 16% of the scores? In a study of elite distance runners, the mean weight was reported to be 63.1 kilograms (kg), with a standard deviation of 4.8 kg. 8. Assuming that the distribution of weights is normal, sketch the density curve of the weight distribution, with the horizontal axis marked in kilograms. 9. Jill scores 680 on the mathematics part of the SAT. The distribution of SAT scores in a reference population is normally distributed with mean 500 and standard deviation 100. Jack takes the ACT mathematics test and scores 27. ACT scores are normally distributed with mean 18 and standard deviation 6. Find the standardized scores for both runners. 10. Assuming that both tests measure the same kind of ability, who has the higher score, and why? Using Table A (table of standard normal probabilities) or your calculator, find the proportion of observations from a standard normal distribution that satisfies each of the following statements. In each case, sketch the normal curve and shade the area under the curve that is the answer to the question. 11. Z < –1.5 12. 1.5 < Z < 0.8 Chapter 3 The table below shows the Men’s 800 Meter Run World Records: Year Record 1905 113.4 1915 111.9 1925 111.9 1935 109.7 1945 106.6 1955 105.7 1965 104.3 1975 104.1 1985 101.73* 1995 101.73 * Sebastian Coe, GB, 1981 . Plot a scatter Plot. Chapter 1 Quiz 1.1A 1. Is there an explanatory–response relationship? If so, then which is which? Explanatory: Response: 2. Plot a scatterplot of the data. Don’t forget to label your axes and mark scales. 3. Is there a positive or negative association between the two variables? If so, which? If there is an association between YEAR and RECORD, describe it in a sentence. 4. Identify any regression outliers. 5. If there is an association, is it strong? Would you be willing to predict the world record for the men’s 800 meter run in the year 2005? 6. Enter the data into your calculator and then perform least squares regression. Write the LSRL equation and the correlation here: 7. Plot the regression line on your scatterplot above. Identify the points you use to plot the line. 8. Construct a residual plot. 9. Is this least squares line an acceptable model for this data? Explain. 10. Predict the world record in the year 2005. Prediction: Briefly describe any reservations you might have about this prediction. When water flows across farmland, some of the soil is washed away, resulting in erosion. Researchers released water across a test bed at different flow rates and measured the amount of soil washed away. The following table gives the flow (in liters per second) and the weight (in kilograms) of eroded soil: Flow Rate 0.31 0.85 1.26 2.47 3.75 Eroded Soil 0.82 1.95 2.18 3.01 6.07 Here is a plot of the data. 11. Is the association positive or negative? 12. Calculate the correlation r. 13. Explain why your r is reasonable based on the scatterplot. 6.0 o ErodSoil 3.0 o o o o 0.80 1.60 2.40 3.20 FlowRate You may NOT use your calculator for the following exercises. Chapter 1 Quiz 1.1A 14 15 16 17 Match the above scatterplots with the appropriate correlation from the list. Note that not all of the correlations are used. The viewing window is the same in all four plots. Correlations: r = –.48 r = .98 r = .82 r = –.17 r=1 r = .17 r = –1 Chapter 4 The number of motor vehicles registered in the U.S. has grown as follows: Year Vehicles Year Vehicles 1940 32.4 1965 90.4 1945 31.0 1970 108.4 1950 49.2 1975 132.9 1955 62.7 1980 155.8 1960 73.9 1985 171.7 1. Plot the number of vehicles against time. What kind of growth does this exhibit? 2. Use logs to transform the data into a linear association, and plot the transformed data. Clearly, the point (1945,1.49) is an outlier. Can you suggest an explanation? 3. Delete the outlier (x=1945) and use the remaining points to find the LRSL equation of Chapter 1 Quiz 1.1A Log Y on X. Write the equation below, and draw this line on your second (linear) graph. 4. Determine the exponential equation (model) for the original dataset. (Form: y = c•10^ kx ) A study of cognitive development in young children recorded the age (in months) at which 21 children spoke their first word, and their Gesell Adaptive Score, the result of an aptitude test taken much later. Here are the data, and a scatterplot of the data: Case Age Score Case Age Score 1 15 95 11 7 113 2 26 71 12 9 96 3 10 83 13 10 83 4 9 91 14 11 84 5 15 102 15 11 102 6 20 87 16 10 100 7 18 93 17 12 105 8 11 100 18 42 57 9 8 104 19 17 121 10 20 94 20 11 86 21 10 100 5. A scatterplot with the least squares line is shown to the right. There is an influential observation. Circle it, and then explain why this point is influential. 6. A residual plot for the Gesell data is also shown here, and one of the points is marked with an . What descriptive term have we studied that applies to this point, and why is it so called? 7. What case number is the right-most point in this residual plot? 8. In regressing Gesell score on age at first spoken word, the TI-83 reports the following information. Interpret the r-value to a layperson. 9. Without using any technical terms, what does the reported r2 tell us? In a 1980 study, researchers looked at the relationship between the type of college (public or private) attended by 3265 members of the class of 1960 who went into industry and the level of job each member had in 1980. The results were: Management Level Public Private High 75 107 Middle 962 794 Chapter 1 Quiz 1.1A Low 732 595 10. Compute the marginal counts. 11. Compute the conditional distributions of management level given college type (in percents). [Write the numbers next to the counts in the above table.] 12. A segmented bar graph allows a graphical comparison of distributions. Each bar describes one group, and the bar is divided into segments to show the distribution for that group. Each bar has height 100%. Show the conditional distributions from (2) as a segmented bar graph. Be sure to label both axes and provide a key to identify the segments. One group has been done for you. 13. Comment on the observed relationship. Chapter 5 1. A university’s financial aid office wants to know how much it can expect students to earn from summer employment. This information will be used to set the level of financial aid. The population contains 3,478 students who have completed at least one year of study but have not yet graduated. A questionnaire will be sent to an SRS of 100 of these students, drawn from an alphabetized list. (a) Describe how you will label the students in order to select the sample. (b) Use Table B, beginning at line 105, to select the first five students in the sample. 2. You’re in college now, and you want to investigate the attitudes of students at your school toward the faculty’s commitment to teaching. The Student Government will pay the costs of contacting about 500 students. (a) Specify the exact population for your study; for example, will you include part-time students? (b) Describe your sample design. Will you use a stratified sample? (c) Briefly discuss the practical difficulties that you anticipate; for example, how will you contact the students in your sample? A medical study of heart surgery investigates the effect of a drug called a beta-blocker on the pulse rate of the patient during surgery. The pulse rate will be measured at a specific point during the operation. The investigators will use 20 patients facing heart surgery as subjects. You have a list of these patients, numbered 1 to 20, in alphabetical order. 3. Outline as an algorithm (paragraph form) or in diagram form a randomized experimental design for this study. 4. Use the random digit table starting at line 125 to carry out the randomization required by your design and report the result. A game of chance is based on spinning a 1–10 spinner like the one shown two times in succession. The player wins if the larger of the two numbers is greater than 5. 5. What constitutes a single play of this game? What are the possible outcomes resulting in win or lose? 6. Describe a correspondence between the random digits from a random digit table and outcomes in the game. Chapter 1 Quiz 1.1A 7. Describe a technique using the randInt command on the TI-83 to simulate the result of a single play of the game. 8. Use the random digit table or your TI-83 to simulate 20 repetitions. If you use the table, begin at line 140; if you use the calculator, first enter 123rand to provide a seed. Report the proportion of times you win the game. Chapter 6 Suppose you toss a coin and roll a die. 1. Use a principle you’ve learned to determine how many outcomes there are. 2. List the outcomes in the sample space. 3. Find the probability of getting a head and an even number. 4. Find the probability of getting 1 head. 5. Find the probability of getting a 1, 2, or 3 on the die. 6. Suppose a person was having two surgeries performed at the same time. If the chances of success for surgery A are 85%, and the chances of success for surgery B are 90%, what are the chances that both will fail? 7. Suppose that you have torn a tendon and are facing surgery to repair it. The orthopedic surgeon explains the risks to you. Infection occurs in 3% of such operations, the repair fails in 14%, and both infection and failure occur together in 1%. What percent of these operations succeed and are free from infection? 8. Parking for students at Central High School is very limited, and those who arrive late have to park illegally and take their chances at getting a ticket. Joey has determined that the probability that he has to park illegally and that he gets a parking ticket is .07. He has kept data from last year and found that because of his perpetual tardiness, the probability that he will have to park illegally is .25. Suppose that he arrived late once again this morning and had to park in a no-parking zone. Find the probability that Joey will get a parking ticket. 9. Two cards are dealt. one after the other, from a shuffled 52-card deck. Why is it wrong to say that the probability of getting two red cards is (1/2)(1/2) = 1/4? What is the correct probability of this event? Chapter 7 The probabilities that a customer selects 1, 2, 3, 4, or 5 items at a convenience store are 0.32, 0.12, 0.23, 0.18, and 0.15, respectively. 1. Construct a probability distribution (table) for the data, and draw a probability distribution histogram. 2. Find P(X > 3.5). 3. Find P(1.0 < X < 3.0). 4. Find P(X < 5). A certain probability density function is made up of two straight line segments. The first segment begins at the origin and goes to the point (1, 1). The second segment goes from (1, 1) to the point (X, 1). 5. Sketch the distribution function, and determine what X has to be in order to be a legitimate density curve. 6. Find P(0 < X ≤ 0.5). 7. Find P(X = 1). 8. Find P(0 < X < 1.25). 9. Circle the correct option: X is an example of a ( discrete ) ( continuous ) random variable. The probabilities that a customer selects 1, 2, 3, 4, or 5 items at a convenience store are 0.32, 0.12, 0.23, 0.18, and 0.15, respectively. Chapter 1 Quiz 1.1A 9. Construct a probability distribution (table) for the data, and verify that this is a legitimate probability distribution. 10. Find the mean of the random variable, X. 11. Find the standard deviation of X. 12. Here’s a game: If a player rolls two dice and gets a sum of 2 or 12, he wins $20. If the person gets a 7, he wins $5. The cost to play the game is $3. Find the expected payout for the game. 13. The random variable X takes the two values µ – and µ + , each with probability 0.5. Use the definition of mean and variance for discrete random variables to show that X has mean µ and standard deviation . Chapter 8 “What do you think is the ideal number of children for a family to have?” A Gallup poll asked this question of 1006 randomly chosen adults. Almost half (49%) thought two children was ideal. Suppose that p = 0.49 is exactly true for the population of all adults. Let X = number of adults who thought that two children was ideal. 1. Is X binomial? Check the four conditions for a binomial setting. 2. Find the mean and standard deviation of X. 3. Find the probability that X = 493. 4. Find the probability that X is within two standard deviations of the mean. 5. Describe the distribution of the binomial random variable X. 6. Would it make sense to compare your results in Question 4 with the empirical rule? Explain. 7. Use the Normal approximation to find the area between X = 461 and X = 525. 8. The results in Questions 4 and 7 should be extremely close. Why would this be so? Government data reveal that approximately 65% of all divorce cases cite incompatibility as the underlying cause. Let X be the number of divorce cases that cite incompatibility. 1. You randomly select divorce cases to study. How many cases would you expect to see until you see one that cites incompatibility as the cause 2. What is the probability that it takes more than 4 cases to find one that cites incompatibility? 3. Construct a probability distribution table (out to n = 5) for the number of cases examined until you find one that cites incompatibility 4. Sketch a probability histogram (out to n = 5) for your table in Question 3. Chapter 1 Quiz 1.1A Chapter 1 Quiz 1.1A Chapter 1 Quiz 1.1A