Review Problems for Exam I
1. Sales tax in Los Angeles County is 8.25%, and in Orange County is 7.75%. You are contemplating buying a $875 video disk player.
a) If you buy the video disk player in Los Angeles County, how much would you pay in sales tax?
b) If you buy the video disk player in Orange County, how much would you pay in sales tax?
c) Suppose you are in Los Angeles County, and the salesperson offers you a 15% discount on the price of the vide disk player. With the discount, and sales, tax, how much would you have to pay for the video disk player?
2. Four months ago, a news report stated that 6% of the employees of the Mudville Register were laid off. Today, the Mudville Register announces that it will increase its work force by 6%. Has the size of the Mudville Register’s work force returned to its original size? Explain why or why not. What percent age of the original Mudville Register work force will there be?
3. Find the absolute change, and percent change in each statement:
a) In 1996, the average housing price in the Bixby neighborhood was $260,000 and in 1998, the average housing price was $251,000.
b) Your 1997 salary was $32,000, and your 1998 salary was $33,500.
4. According to the National Association of Colleges and Employers Annual Salary Survey, the national average starting salary for a 2004 college graduate with a major in Political Science/Government was $32,296, up 3.6% from the previous year. What was the average starting salary for a college graduate with a major in Political Science/Government the previous year?
5. State the number of significant digits and the implied precision in each given number.
a) 2.232 x 105 b) 42,000 c) 305,000 d) .0031
6. The following statement appeared in the Wall Street Journal. “Since 1949, when the government began keeping national energy-use statistics, Americans have increased their annual use of household electricity 17-fold, from 67 billion kilowatt-hours to 1.1 trillion kilowatt-hours.” What is the implied precision in 67 billion kilowatt-hours? In 1.1 trillion kilowatt-hours?
7. For each measurement described below, describe any likely sources of random errors and any likely sources of systematic errors.
a) Heights of 6th graders, measured as they arrive at school.
b) The GPA of LMU students, found by asking them the GPA during interviews at the lair.
c) A count of the number of people entering the subway system at a particular entrance located on a busy street between 7:15 and 7:45 am on a weekday.
8. A pharmaceutical company runs two trials of two treatments for an illness. In the first trial, Treatment A cures 35% of the cases (70 out of 200) and Treatment B cures 28% of the cases (56 out of 200). In the second trial, Treatment A cures 85% of the cases (85 out of 100) and Treatment B cures 82% of the cases (328 out of 400). Which treatment has the better cure rate in the two trials individually? Which treatment has the better overall cure rate? Explain why this is an example of Simpson’s paradox.
9. The five-digit zip code has become an integral part of the U.S. postal system.
a) How many different 5-digit zip codes are available for use by the postal service?
b) The first three digits of a Chicago zip code are 606. If no other city in the United States has these first three digits as part of its zip code, how many different zip codes may possibly exist in Chicago?
10. A county welfare agency employs thirty welfare workers that interview prospective food stamp recipients. Periodically, the supervisor selects, at random, the forms completed by two workers to audit for illegal deductions. Unknown to the supervisor, six of the workers have regularly been giving illegal deductions to applicants.
a) What is the probability that the first worker chosen has been giving illegal deductions?
b) What is the probability that both workers chosen have been giving illegal deductions?
c) What is the probability that neither of the two workers chosen has been giving illegal deductions?
11. Empirical data suggests that the probability that there is an earthquake this year in Sinclair County is 1/50 and that the probability that there is a drought this year in Sinclair County is 1/28. Assuming that these two events are independent, what is the probability that both an earthquake and a drought occur this year?
12. The names of 20 left-handed men, 150 right-handed men, 35 left-handed women and 140 right-handed women are placed into a jar. The names are mixed and one name is randomly drawn.
a) What is the probability that the name drawn will be that of a left or right-handed man?
b) What is the probability that the name drawn will be that of a left-handed person or of a woman?
13. What is the probability on a single card draw of drawing
a) either a queen or a king from a deck of 52 cards?
b) either a 10 or a diamond from a deck of 52 cards?
14. A store carries 8 styles of pants. For each style, there are 4 color choices and 20 different possible sizes. How many different types of pants does the store have?
15. Determine whether the study described below is observational or a controlled experiment. If it is observational, is it case-controlled? If it is experimental, was a control group used?
According to a study published in 1993, left-handed people do not die at an earlier age then right-handed people, contrary to the conclusion of a highly publicized report done two years prior. The investigation involved a 6-year study of 3800 people in East Boston older than age 65. Researchers at Harvard and the National Institute of Aging found that "lefties" and "righties" died at exactly the same rate.
16. In 1991, the Census Bureau released the results of a statistical study on U.S. citizens and health insurance. They contacted 20,000 persons and asked whether they were covered by health insurance. Of the 20,000 persons, 86.5% were covered by insurance and so they concluded 86.5% of U.S. citizens are covered by health insurance.
a) What is the population of the study?
b) What is the sample of the study?
c) What is the raw data for the study?
d) What is the sample statistic for the study?
e) What is their estimate of the population parameter?
17. An insurance company would like to determine the proportion of all medical doctors who have been involved in 1 or more malpractice suits. The company randomly selects 500 doctors to whom they mail a voluntary survey. Of those who responded, 42% had been involved in 1 or more malpractice suits. Evaluate the believability of this study.
18. Tastee-Food, a baby-food manufacturer, wanted to determine if infants find their peach flavored baby food tastes better than the competitor’s peach flavored baby food. They put babies in high chairs and gave each a bowl of Tastee-Food peach flavored baby food and a bowl of the competitor’s peach flavored baby food. Sixty-one percent of the babies ate from the bowl of Tastee-Food baby food. The Tastee-Food Company concluded 61% of babies think their baby food tastes better than the competitor’s. Evaluate the believability of this study.
19. According to American Consumer Credit Counseling, the total U.S. credit card debt in the first quarter of 2002 was approximately $750 billion. The U.S. adult population is 217 million people. Approximately how many dollars per adult was the U.S. credit card debt in the first quarter of 2002? Be sure to use the correct number of significant digits in your final answer.
20. The Motley Fool's Credit Center claims that the total U.S. consumer credit is $1.7 trillion. The U.S. adult population is 217 million people. Approximately how many dollars in credit per adult is this? Be sure to use the correct number of significant digits in your final answer.
21. Suppose a company wants to do random drug testing of its employees and that 2% of the 5,000 employees use illegal drugs. The test for detecting drug use is 96% accurate - it returns a positive result 96% of the time when given to a person who has used illegal drugs and returns a negative result 96% of the time when given to a person who has not used illegal drugs. What percent of those who test positive have used illegal drugs?
22. Suppose that a prostate cancer test is 98% accurate (it will correctly identify 98% of the patients who have prostate cancer and it will correctly identify 98% of the patients who do not have prostate cancer). Within a sample of 10,000 patients who took the test, the incidence rate of the disease is 2%. What percent of those who test positive actually have prostate cancer?
23. Researchers wished to determine the attitudes of persons living in Lincoln, Nebraska regarding a proposal to build a new town hall. Sampling was carried out by researchers standing outside of a Lincoln supermarket on Monday between 10 am and noon and collecting data from persons as they left the supermarket. Evaluate the believability of this study.
24. The data set below shows the yards gained by a high school running back over a 13-game football season.
48 31 85 14 –5 42 61 39 25 5 50 20 20
a) Find the mean number of yards gained by the running back.
b) Find the median number of yards gained by the running back.
c) Find the mode for the data set.
d) Find the 5-number summary of the data set.
e) Create a boxplot for the data
f) Find the standard deviation of the data set.
25. The mean cost of automobile insurance in the 5 most expensive states is shown below.
New Jersey $1,013
Hawaii $963
District of Columbia $958
New York $905
Masssachusetts $898
Find the mean and standard deviation of this data. Interpret your answers.
26. Two fifth grade classes have nearly identical mean scores on an aptitude test, but one class has a standard deviation twice as large as that of the other. Which class is easier to teach, and why?
27. Decide whether the graph below creates a visual distortion that exaggerates the true change in the unemployment rates between 1992 and 2000. Explain.
28. An advertisement for a speed-reading course claimed that the “average” reading speed for people completing the course was 530 words per minute. Shown below are the actual data for the reading speeds per minute for 10 people who completed the course.
400 350 300 400 550 950 350 1,250 350 400
Which measure of central tendency was given in the advertisement? Which measure of central tendency is the best indicator of the “average”?
29. Decide whether the graphic below creates a visual distortion that exaggerates the true change in the price per barrel of light crude leaving Saudi Arabia rates between 1973 and 1979. Explain.
30. If a set of test scores has a large range but a small standard deviation, describe what this means about students’ performance on the test.
Answers
1. a) $72.19 b) $67.81 c) $805.11
2. No. To see this, suppose that four months ago, there were 1,000 employees. Then 6%, or 60 employees were laid off. There are now 940 employees. If the workforce is increased by 6% today, .06(940) ≈ 56 employees will be added. This does not bring the total workforce back to 1,000.
There will be (.94)(1.06) = .996 or 96.4% of the original workforce.
3. a) absolute change: – $9,000; percent change: –3.46% b) absolute change: $1,500;
percent change: 4.69%
4. $31,174
5. a) number of significant digits: 4 implied precision: to the nearest 100
b) number of significant digits: 2 implied precision: to the nearest 1,000
c) number of significant digits: 3 implied precision: to the nearest 1,000
d) number of significant digits: 2 implied precision: to the nearest one ten thousandth.
6. To the nearest 1 billion, or 1,000,000,000, kilowatt hours. To the nearest .1 trillion, or 100,000,000,000, kilowatt hours.
7. a) Random error due to error in reading height, systematic error due to shoes and “calibration”.
b) Random error due to misrecording GPA, systematic error due to overreporting GPA during interviews.
c) Random error due to miscounting.
8. Treatment A does better in each trial individually. Treatment A has an overall cure rate of 48% and Treatment B has an overall cure rate of 64%. So Treatment B does better overall. This is an example of Simpson’s paradox because Treatment A appears to be better in each individual trial, but it not better overall.
9. a) 100,000; b) 100 10. a) 1/5 b) 1/29 c) 92/145
11. 1/1400 12. a) 34/69 b) 13/23
13. a) 2/13 b) 4/13 14. 640 15. observational; case - controlled
16. a) All U.S. citizens b) The 20,000 persons contacted
c) The 20,000 yes or no answers obtained d) The percentage of those sampled that are covered by health insurance e) 86.5%
17. There is a problem with participation bias, since the survey was voluntary. Only those doctors who have strong feelings are likely to respond.
18. a) There is potential for bias since the Tastee-Food company carried out the study themselves and has reason to want the study to show their baby food tastier.
b) There are difficulties in measuring which food a baby really thinks tastes better since the baby cannot talk. (e.g. How can we be sure that they didn’t eat from that bowl because it was a brighter color?)
19. $3,500. (round to 2 significant digits since $750 billion has just 2 significant digits).
20. $7,800. 21. 33% 22. 50%
23. The method of sampling is not likely to produce a representative sample because of the time and location of at which the sampling is done. Who is most likely to be at the supermarket on a Monday between 10 and noon?
24. a) 33.5 b) 31 c) 20 d) –5, 17, 31, 49, 85