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467
Key Concepts
blind study, 463
capture-recapture m e t h o d , 460
census. 450
chance error, 459
clinical study (clinical trial). 462
confounding variable, 462
control group, 462
controlled study, 462
controlled placebo study, 463
convenience sampling, 455
data, 447
double-blind study. 463
nonresponse bias. 454
parameter, 459
placebo, 463
placebo effect. 463
population. 448
quota sampling. 455
randomized controlled study. 463
r a n d o m sampling, 457
response rate, 454
sample, 452
sample bias. 459
sampling error, 459
sampling frame. 452
sampling proportion. 460
sampling variability. 459
selection bias, 454
simple random sampling, 457
statistic, 459
strata. 458
stratified sampling, 457
survey. 452
t r e a t m e n t group, 462
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, Surveys and Polls
Exercises 1 through 4 refer to the following story: As part of a
sixth-grade statistics project, the teacher brings to class a
candy jar full of gumballs of two different colors: red and
green. The assignment is to estimate the proportion of red
gumballs in the jar. To do this, the jar is shaken well, and one
of the students draws 25 gumballs from the jar. Of these, 8 are
red and 17 are green.
1. (a) Describe the population for this survey.
(b) Describe the sample for this survey.
(c) Give the sample statistic for the proportion of red
gumballs in the jar.
(d) Name the sampling method used for this survey.
2. Given that the total number of gumballs in the jar is 200,
(a) give the sampling proportion for this survey.
(b) give the sample statistic for the number of red gumballs in the jar.
3. Given that the total number of gumballs in the jar is 200
and the number of red gumballs is 50,
(a) give the parameter for the proportion of red gumballs in the jar.
(b) give the sampling error, expressed as a percent.
(c) Is the sampling error found in (b) a result of sampling variability or sampling bias? Explain.
4. (a) Compare and contrast the target population and
sampling frame for this survey.
(b) What data collection method could be used to find
the exact value of the parameter?
Exercises 5 through 8 refer to the following story: The city of
Cleansburg has 8325 registered voters. There is an election for
mayor of Cleansburg, and there are three candidates for the
position: Smith, Jones, and Brown. The day before the election, a telephone poll of 680 randomly chosen registered voters produced the following results: 306 people surveyed
indicated that they would vote for Smith, 272 indicated that
they would vote for Jones, and 102 indicated that they would
vote for Brown.
5. (a) Give the sampling proportion for this survey.
(b) Give the sample statistic estimating the percentage
of the vote going to Smith.
6. (a) Describe the population for this survey.
(b) Describe the sample for this survey.
(c) Name the sampling method used for this survey.
7. Given that in the actual election Smith received 42% of
the vote, Jones 43% of the vote, and Brown 15% of the
vote, find the sampling errors in the survey expressed as
percentages.
8. Do you think that the sampling error in this example is
due primarily to chance error or to sample bias? Explain your answer.
Exercises 9 through 12 refer to the following story: The 1250
students at Eureka High School are having an election for
Homecoming King. The candidates are Tomlinson (captain of
the football team), Garcia (class president), and Marsalis
(member of the marching band). At the football game a week
before the election, a pre-election poll is taken of students as
they enter the gates. Of the students that attended the game,
203 planned to vote for Tomlinson, 42 planned to vote for
Garcia, and 105 planned to vote for Marsalis.
9. (a) Describe the sample for this survey.
(b) Give the sampling proportion for this survey.
468
13 Collecting Statistical Data
10. Name the sampling method used for this survey.
11. (a) Compare and contrast the population and the sampling frame for this survey.
(b) Is the sampling error a result of sampling variability or sampling bias? Explain.
12. (a) Give the sample statistics estimating the percentage
of the vote going to each candidate.
(b) A week after this survey, Garcia was elected
Homecoming King with 5 1 % of the vote; Marsalis
got 30% of the vote, and Tomlinson came in last
with 19% of the vote. Find the sampling errors in
the survey expressed as percentages.
Exercises 13 through 16 refer to the following true story: In
1988, "Dear Abby" asked her readers to let her know whether
they had cheated on their spouses or not. The readers' responses are summarized in the following table.
Status
Women
16. If money were no object, could you devise a survey that
might give more reliable results than the "Dear Abby"
survey? Describe briefly what you would do.
Exercises 17 through 20 refer to the following story: The
Cleansburg Planning Department is trying to determine what
percent of the people in the city want to spend public funds to
revitalize the downtown mall. To do so, it decides to conduct a
survey. Five professional interviewers are hired. Each interviewer is asked to pick a street corner of his or her choice
within the city limits and every day between 4:00 and 6:00 P.M.
the interviewers are supposed to ask each passerby if he or she
wishes to respond to a survey sponsored by Cleansburg City
Hall. If the response is yes, the follow-up question is, "Are you
in favor of spending public funds to revitalize the downtown
mall?" The interviewers are asked to'return to the same street
corner as many days as are necessary until each one has conducted a total of 100 interviews. The results of the survey are
shown in the following table.
Men
Yesa
No6
A
35
65
321
B
21
79
208
C
58
42
103
D
78
22
87
Ed
12
63
594
Interviewer
Faithful
Unfaithful
Total
127,318
44,807
22,468
15,743
149,786
60,550
Based on the results of this survey, "Dear Abby" concluded
that the amount of cheating among married couples if much
less than people believe. (In her words, "The results were astonishing. There are far more faithfully wed couples than I
had surmised. ")
13. (a) Describe as specifically as you can the sampling
frame for this survey.
(b) Compare and contrast the target population and
the sampling frame for this survey.
(c) How was the sample chosen?
(d) Eighty-five percent of the women who responded
to this survey claimed to be faithful. Is 85% a
parameter? A statistic? Neither? Explain your
answer.
" In favor of spending public funds' to revitalize the downtown mall.
'Opposed to spending public funds to revitalize the downtown mall.
' Declined to be interviewed or had no opinion.
''Got frustrated and quit.
17. (a) Describe as specifically as you can the target population for this survey.
(b)
18. (a)
19. (a)
(b)
(c) How accurate do you think these estimates are?
Explain.
What is the size of the sample?
survey subject to nonresponse bias?
(b) Explain why this survey was subject to nonresponse bias.
(b) Based on the "Dear Abby" data, estimate the percentage of married people who are faithful to
their spouses.
Compare and contrast the target population and
the sampling frame for this survey.
lb) Calculate the response rate in this survey. Was this
14. (a) Explain why this survey was subject to selection bias.
15. (a) Based on the "Dear Abby" data, estimate the
percentage of married men who arc faithful to
their spouses.
'
Nonrespondents
Can you explain the big difference in the data from
interviewer to interviewer?
One of the interviewers conducted the interviews at a
street corner downtown. Which interviewer? Explain.
(c) Do you think the survey was subject to selection
bias? Explain.
(d) Was the sampling method used in this survey the
same as quota sampling? Explain.
20. Do you think this was a good survey? If you were a consultant to the Cleansburg Planning Department, could
you suggest some improvements? Be specific.
Exercises
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Systematic Sampling. Exercises 21 through 24 refer to the notion of systematic sampling, as illustrated by the following story:
The dean of students at Tasmania State University (TSU) wants
to determine the percent of undergraduates who tried but could
not enroll in Math 101 this semester because of insufficient
space. There are 15,000 undergraduates at TSU, so it is decided
that the cost of checking with each and every one would be prohibitive. The following method (called systematic sampling) is
proposed to choose a representative sample of undergraduates
to interview. Start with the registrar's alphabetical listing containing the names of all undergraduates. Randomly pick a number
between 1 and 100, and count that far down the list. Take that
name and every 100th name after it. (For example, if the random
number chosen is 73, then pick the 73rd, 173rd, 273rd, etc., names
on the list.) Assume that the survey has a response rate of 0.90.
21. (a) Compare and contrast the sampling frame and the
target population for this survey.
Give the exact N-value of the population.
22. (a) Find the size n of the sample.
(b) Find the sampling proportion.
23. (a) Explain why the method used for choosing the
sample is not simple random sampling.
(b)
(b)
24.
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If all those responding claimed they could not enroll in Math 101, is it more likely the result of sampling variability or sampling bias? Explain.
If eight of the students who responded said that they
were unable to enroll in Math 101, give a reasonable
estimate for the total number of students at the university that were unable to enroll in Math 101.
(b) Do you think the results of this survey will be reliable? Explain.
Exercises 25 and 26 refer to the following story: An orange
grower wishes to compute the average yield from his orchard.
The orchard contains three varieties of trees—50% of his trees
are of variety A, 25% of variety B, and 25% of variety C
25. (a) Suppose the grower samples randomly from 300
trees of variety A, 150 trees of variety B, and 150 trees
of variety G What type of sampling is being used?
(b) Suppose the grower selects for his sample a 10-by30 rectangular block of 300 trees of variety A, a 10by-15 rectangular block of 150 trees of variety B,
and a 10-by-15 rectangular block of 150 trees of variety C. What type of sampling is being used?
26. (a) Suppose that in his survey, the grower found that
each tree of variety A averages 100 oranges, each
tree of variety B averages 50 oranges, and each tree
of variety C averages 70 oranges. Estimate the average yield per tree of his orchard,
(b) Is the yield you found in (a) a parameter or a statistic? Explain.
27. Name the sampling method that best describes each situation. Choose your answer from the following list: simple random sampling, convenience sampling, quota
sampling, stratified sampling, census.
469
(a) George wants to know how the rest of the class did
on the last quiz. He peeks at the scores of a few students sitting right next to him. Based on what he
sees, he concludes that nobody did very well.
(b) On a day when every student is present, the students are given a Teacher Evaluation Questionnaire. Every student fills out the Questionnaire.
(c) Eureka High School has 400 freshmen, 300 sophomores, 300 juniors, and 200 seniors. The student
newspaper conducts a poll of student opinion regarding the rumor that the football coach is about
to be fired. In the poll 20 freshmen, 15 sophomores,
15 juniors, and 10 seniors are randomly selected to
be interviewed.
(d) For the last football game of the season the coach
chooses the three captains by putting the names of
all the players in a hat and drawing three names.
(Maybe that's why they are trying to fire him!)
(e) For the last football game of the season the coach
chooses the three captains by randomly selecting a
senior offensive player, a senior defensive player,
and a senior special teams player.
28. Name the sampling method that best describes each situation. Choose your answer from the following list: simple random sampling, convenience sampling, quota
sampling, stratified sampling, census.
(a) The shipper inspects every orange that comes from
the suppliers before packaging and shipping.
(b) A few randomly chosen crates of oranges are opened
and the oranges on the top layer are inspected.
(c) Of the 250 crates of oranges received by the shipper, 75 crates came from supplier A, 75 from supplier B, and 100 from supplier C. Six crates are
randomly chosen for inspection from A's shipment, six crates are randomly chosen from B's
shipment, and eight crates are randomly chosen
from C's shipment.
B. The Capture-Recapture Method
29. You want to estimate how many fish there are in a small
pond. Let's suppose that you capture n^ — 500 fish, tag
them, and throw them back in the pond. After a couple
of days, you go back to the pond and capture n¿ = 120
fish, of which k = 30 are tagged. Give an estimate of the
/V-value of the fish population in the pond.
30. To estimate the population in a rookery, 4965 fur seal
pups were captured and tagged in early August. In late
August, 900 fur seal pups were captured. Of these, 218
had been tagged. Based on these figures, estimate the
population of fur seal pups in the rookery to the nearest hundred.
[Source: Chapman and Johnson, "Estimation of Fur
Seal Pup Populations by Randomized
Sampling,"
Transactions of the American Fisheries Society, 97 (July
1968), 264-2701
470
13 Collecting Statistical Data
Exercises 31 through 34 refer to the following story: You have,
a very large coin jar full of nickels, dimes, and quarters. You
want to have an approximate idea of how much money you
have, but you don't want to go through the trouble of counting
all the coins, so you decide to use the capture-recapture
method. For the. first sample, you shake the jar well and randomly draw 50 coins. You get 12 quarters, 15 nickels, and 23
dimes. Using a black marker, you mark the 50 coins with a
black dot and put them back in the jar. For the second sample,
you shake the jar well and randomly draw another set of 100
coins. You get 28 quarters, 4 of which have black dots; 29 nickels, 5 of which have black dots; and 43 dimes, 8 of which have
black dots.
31. Estimate the total number of quarters in the jar.
[Source: Mastro et al, "Estimating the Number of HIVinfected Injection Drug Users in Bangkok: A CaptureRecapture Method," American Journal of Public Health,
(July 1994), 84-87]
37. (a) Estimate the number of drug users in Bangkok
during the time of the study.
(b) In 1991 it was estimated that the 89% of drug users
in Bangkok injected drugs, and that one-third of
these were infected with HIV. Using this information estimate the number of HIV-infected drug
users in Bangkok in 1991.
38. Discuss the potential pitfalls of this study. Is the general
approach used a reasonable approach for estimating
populations of drug users?
32. Eslimate the total number of nickels in the jar.
33. Estimate the total number of dimes in the jar.
34. Do you think the capture-recapture method is a reliable
way to estimate the number of coins in the jar? Explain
your answer. Discuss some of the potential pitfalls and
issues one should be concerned about.
35. Starting in 2004, a collaborative study between the
Canadian Ministry of Natural Resources, Minnesota's
Department of Natural Resources and the Rainy River
First Nations investigated the populations of lake sturgeon on the Rainy River and the Lake of the Woods on
the U.S.-Canadian border. During the capture phase of
the project 1700 lake sturgeon were caught, tagged, and
released. During the recapture phase of the project, 660
lake sturgeon were caught and released. Seven of the
660 were part of the original capture group that had
been tagged. Based on these figures, estimate the population of lake sturgeon in the Lake of the Woods to the
nearest thousand.
[Source: Gauthier, Dan, "Lake of the Woods Sturgeon
Population Recovering, " Daily Miner and News (Kenora, Ontario), June 11, 2005, p. 31]
36. Efforts at Utah Lake in 2004 show the carp population
dominates fish life in the lake. In 15 days, workers captured, tagged, and released 24,000 carp. Of the 10,300
carp that were later recaptured, 208 had tags. Give an
estimate for the N-value of the carp population in Utah
Lake in 2004.
[Source: Pretty man, Brett, "With Carp Cooking Utah
Lake, It's Time to Eat," Salt Lake Tribune (Salt Lake
City, Utah), July 15,2004, p. D3]
Exercises 37 and 38 refer to the following story: In 1991, a
study using the capture-recapture method was used to estimate the number of HIV-infected drug users in Bangkok,
Thailand. The "capture" consisted of a list of 4064 names of
individuals treated at 18 methadone treatment facilities in
Bangkok between April 17 and May 17. The "recapture" consisted of a list of 1540 persons held at 72 Bangkok police stations between June 3 and September 30 that tested positive for
methadone. There were 171 persons included on both lists.
v.
C. Clinical Studies
Exercises 39 through 42 refer to the following story: The manufacturer of a new vitamin (vitamin X) decides to sponsor a
study to determine its effectiveness in curing the common
cold. Five hundred college students in the San Diego area who
are suffering from colds are paid to participate as subjects in
this study. They are all given two tablets of vitamin X a day.
Based on information provided by the subjects themselves,
457 out of the 500 subjects are cured of their colds within three
days. The average number of days a cold lasts is 4.87 days. As
a result of this study, the manufacturer launches an advertising campaign, claiming that "vitamin X is more than 90% effective in curing the common cold."
46.
39. (a) Describe as specifically as you can the target population for this study.
(b) Compare and contrast the target population and
the sampling frame for this study.
(c) Is selection bias present in the sample?
40. (a) Was this study controlled?
(b) List three possible causes other than the effectiveness of vitamin X itself that could have confounded
the results of this study.
41. List four different problems with this study that indicate
poor design.
42. Make some suggestions for improving the study.
Exercises 43 through 46 refer to the following story: A team of
researchers and surgeons at the Houston VA Medical Center
randomly divided 180 potential knee surgery patients into
three groups. (There were 324 participants who met inclusion
criteria for the study, but 144 declined to participate.) The first
group received arthroscopic debridement. A second group received arthroscopic lavage. Patients in the third group received skin incisions and underwent a simulated procedure
("sham" surgery) without actual insertion of the arthroscope.
The patients in the study did not know which group they were
being divided into and therefore did not know if they were receiving the real or simulated surgery. All the patients who participated in the study were evaluated for two years after the
49.
471
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472
13 Collecting Statistical Data
56. Super Size Me II. Thinking that Spurlock's documentary
Super Size Me (see Exercise 55) unfairly targeted McDonald's, Merab Morgan set out to do a fast-food study
of her own. Morgan ate only at McDonald's for 90 days,
sticking to meal plans of no more than 1400 calories a
day and choosing to eat mostly burgers and salads (no
french fries!). By the end of her study Morgan had
dropped 37 pounds and felt well.
(a) Describe the treatment group in Morgan's study.
(b) List some of the possible confounding variables in
Morgan's study.
(c) Carefully state what a legitimate conclusion from
this study might be.
57. A study by the Center for Academic Transformation
showed that students using electronic learning tools
have seen marked progress in their test scores. At the
University of Alabama, the passing rate for an intermediate algebra class doubled from 40% to 80% when
the class was redesigned to rely heavily on supplemental materials, online practice exercises, and interactive
tutorials.
(a) Was this study a survey or a clinical trial? Explain.
(b) List some of the possible confounding variables in
this study.
58. There are 50 students in the Math 101 class at Tasmania
State University—30 females and 20 males. The professor chooses a sample of ten students from the class as
follows: Six students are randomly chosen from among
the females and four students are randomly chosen
from among the males.
(a) Does every student in the class have an equal
likelihood of being selected for the sample?
Explain.
(b) Does every set of 10 students in the class have an
equal likelihood of being selected as the sample?
Explain.
59. Determine in each case if the data is a population parameter or a sample statistic.
(a) In 2001,25% of students taking the SAT math test
scored above 590.
(b) In crash testing of a new automobile model, 20%
of the crashes would have caused severe neck
injury.
(c) Mr. Johnson's blood tested positive for Type II
diabetes.
(d) A recent Gallup poll shows that 6 in 10 Americans
have attempted to lose weight.
60. Choose the most appropriate concept for each statement from the following list: (i) association is not causation; (ii) sampling variability; (iii) selection bias; (iv)
placebo effect.
(a) "In the early part of the twentieth century it was
discovered that when viewed over time, the number of crimes increased with membership in the
Church of England."
(b) "Half of the subjects in the study were given sugar
pills. These subjects responded with fewer days ill
than those receiving the treatment."
(c) "Jack chose a simple random sample of 100 widgets, and 11 of the widgets in his sample were defective. Jill chose a simple random sample of 100
widgets, and 19 of the widgets in her sample were
defective."
(d) "Jay didn't understand how Jones could have been
elected mayor when almost every person he knows
voted for Brown."
JOGGING
61. Informal surveys. In everyday life, we are constantly involved in activities that can be described as informal
surveys, often without even realizing it. Here are
some examples.
(i)
Al gets up in the morning and wants to know what
kind of day it is going to be, so he peeks out the
window. He doesn't see any dark clouds, so he figures it's not going to rain.
(ii) Betty takes a sip from a cup of coffee and burns her
lips. She concludes the coffee is too hot and decides
to add a tad of cold water to it.
(iii) Carla got her first Math 101 exam back with a C
grade on it. The students sitting on each side of her
also received C grades. She concludes that the entire Math 101 class received a C on the first exam.
For each of the preceding examples,
(a) describe the population.
(b) discuss whether the sample is random or not.
(c) discuss the validity of the 'conclusions drawn.
(There is no right or wrong answer to this question.
but you should be able to make a reasonable case
for your position.)
62. Read the examples of informal surveys given in Exercise 61. Give three new examples of your own. Make
them as different as possible from the ones given m
Exercise 61 [changing coffee to soup in (ii) is not a
new example].
63. Leading-question bias. The way the questions in many
surveys are phrased can itself be a source of bias.
When a question is worded in such a way as to predispose the respondent to provide a particular response,
the results of the survey are tainted by a special type
of bias called leading-question bias. The following |S
an extreme hypothetical situation intended to drive
the point home.
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In an effort to find out how the American taxpayer feels about a tax increase, the institute
conducts a "scientific" one-question poll.
Are you in favor of paying higher taxes to bail
the federal government out of its disastrous economic policies and its mismanagement of the
federal budget?
Yes
. No
.
Ninety-five percent
answered no.
of
the
473
and (ii) using a computer to randomly generate sevendigit numbers to try that are compatible with the local
phone numbers.
(a) Briefly discuss the advantages and disadvantages
of each technique. In your opinion, which of the
two will produce the more reliable data? Explain.
(b) Suppose that you are trying to market burglar
alarms in New York City, Which of the two techniques for selecting the sample would you use? Explain your reasons.
respondents
(a) Explain why the results of this survey might be
invalid.
(b) Rephrase the question in a neutral way. Pay particular attention to highly charged words.
(c) Make up your own (more subtle) example of
leading-question bias. Analyze the critical words
that are the cause of bias.
64. Consider the following hypothetical survey designed to
find out what percentage of people cheat on their income taxes.
Fifteen hundred taxpayers are randomly selected from the Internal Revenue Service (1RS)
rolls. These individuals are then interviewed in
person by representatives of the 1RS and read
the following statement.
66. The following two surveys were conducted in January
1991 in order to assess how the American public
viewed media coverage of the Persian Gulf war. Survey 1 was an Area Code 900 telephone poll survey
conducted by ABC News. Viewers were asked to call a
certain 900 number if they felt that the media was
doing a good job of covering the war and a different
900 number if they felt that the media was not doing a
good job in covering the war. Each call cost 50 cents.
Of the 60,000 respondents, 83% felt that the media
was not doing a good job. Survey 2 was a telephone
poll of 1500 randomly selected households across the
United States conducted by the Times-Mirror survey
organization. In this poll, 80% of the respondents indicated that they approved of the press coverage of
the war.
(a) Briefly discuss survey 1, indicating any possible
types of bias.
(b) Briefly discuss survey 2, indicating any possible
types of bias.
This survey is for information purposes only.
Your answer will be held in strict confidence.
Have you ever cheated on your income taxes?
Yes
- No
.
(c) Can you explain the discrepancy between the results of the two surveys?
Twelve percent of the respondents answered
yes.
67. An article in the Providence Journal about automobile
accident fatalities includes the following observation:
"Forty-two percent of all fatalities occurred on Friday.
Saturday, and Sunday, apparently because of increased
drinking on the weekends."
(a) Explain why the 12% statistic might be unreliable.
(b) Can you think of ways in which a survey of this
type might be designed so that more reliable information could be obtained? In particular, discuss
who should be sponsoring the survey and how the
interviews should be carried out.
65. Listing bias. Today, most consumer marketing surveys
are conducted by telephone. In selecting a sample of
households that are representative of all the households
in a given geographical area, the two basic techniques
used are (i) randomly selecting telephone numbers to
call from the local telephone directory or directories,
(d) In your opinion, which of the two surveys gives the
more reliable data?
(a) Give a possible argument as to why the conclusion
drawn may not be justified by the data.
(b) Give a different possible argument as to why the conclusion drawn may be justified by the data after all.
68. (a) For the capture-recapture method to give a reasonable estimate of N, what assumptions about the
two samples must be true?
(b) Give reasons why in many situations, the assumptions in (a) may not hold true.