Section 14 Making Sense of Statistical Information Main Ideas

Part II Probability and Statistical Reasoning
Section 14 Making Sense of Statistical Information
Section 14
Making Sense of Statistical Information
Main Ideas
•Statistical Terminology
•Statistical Studies
•Errors in Statistical Studies
Data Versus Datum
The word data is plural. The singular
form, datum, is seldom used. A datum
is a single fact or number in context.
In statistics, we generally work with
sets of related information (data)
rather than a single fact (datum).
Remember to use plural verbs with
the plural noun “data.” For example,
you should write and say, “the data
are …” or “the data tell us …”
Ordinal Data
This book focuses on ­quantitative
and categorical variables. A third type
of variable is an ordinal ­variable.
Data on ordinal variables can be
ranked, but the ranks are not evenly
spaced. For example, in the Boston
Marathon, there may be 6 s between
the first- and second-place finishers, and 13 min between second and
third place. In this case, the time to
complete the race is quantitative, but
finishing rank is ordinal.
A former state senator … once claimed, “I don’t know whether we
need a bill on teen pregnancy because statistics show teen pregnancy
drops off significantly after age 25.”
The Colorado Independent, January 7, 2009
Statistics is the science of collecting, organizing, and interpreting data (numbers
in context) by describing a situation or drawing conclusions regarding claims. The
main goal of Sections 14–20 is for you to become an intelligent consumer of statistics. We hope that you will learn to think critically about arguments that use data
and statistics as evidence.
Statistical Terminology
Knowing some basic terms can help us evaluate statistical evidence. A data set
contains information on a collection of individuals. Individuals may be people, animals, or things. For each individual, the data give values for one or more variables. A
variable is an attribute or property of the individual (or case) that can be assigned
a value. Variables can be expressed as numbers or by groupings:
1.A quantitative variable assumes numerical values on which arithmetic operations can be performed sensibly.
•A continuous quantitative variable can take on any value in a given interval of real numbers (e.g., temperature or weight).
•A discrete quantitative variable can take on only particular values and no
other values in between (e.g., test scores, number of students in a class).
2.A categorical variable assumes named or coded values that represent groups
associated with the variable. Eye color, political party, and country of birth are
examples of a categorical variable. A categorical variable also can be called a
nominal variable.
The distribution of a variable involves the structure and arrangement of the variable’s data set viewed as a whole rather than as individual values. We will examine
distributions of variables in more detail in later sections.
Recognizing whether a given data value represents a categorical or quantitative
variable and identifying types of variables help in choosing appropriate methods
of describing, displaying, and analyzing data.
Quick Question 14.1 Individuals and Variables
Use the data in Table 14.1 to answer the following questions:
(a) What are the individuals (or cases) in this data set?
(b) For each individual, what variables are given? Which of these variables
are categorical, and which are quantitative?
Copyright © 2015 by Gregory D. Foley, Thomas R. Butts, Stephen W. Phelps, and Daniel A. Showalter
Advanced Quantitative Reasoning 117
Part II Probability and Statistical Reasoning
Section 14 Making Sense of Statistical Information
Table 14.1 Selected 2012 Automobile Ratings
Make and Model
Chevrolet Impala
Chevrolet Malibu 4-cylinder
Chevrolet Malibu V6
Ford Fiesta hatchback SES
Ford Fiesta sedan SE
Ford Focus AT
Transmission
auto 4
auto 6
auto 6
man 5
seq 6
auto 4
HP
211
169
252
120
120
132
Engine
3.5-L V6
2.4-L 4-cyl.
3.6-L V6
1.6-L 4-cyl.
1.6-L 4-cyl.
2.0-L 4-cyl.
City
MPG
13
16
13
23
22
18
Acc
(s)
9.5
9.4
6.5
10.7
10.9
10.1
Source: Consumer Reports
Note: HP = horsepower; MPG = miles per gallon; Acc (s) = time (in seconds) to accelerate
from 0 mph to 60 mph.
Quick Question 14.2 Variables: Categorical Versus Quantitative
(a) Give an example of a number that is a categorical variable.
(b) You are preparing to study the television-viewing habits of the students in
your class. Describe two categorical variables and two quantitative variables
that you might measure for each student. Give the units of measurement for
the quantitative variables. [Hint: Consider the use of DVRs, iPads, etc.]
Statistical evidence is presented in many ways, including graphs, charts, and
tables.
Figure 14.1 Students at Truman High School
who reported that they ride a bicycle.
Figure 14.2 Favorite sports of eighth grade
students at Madison Middle School.
Quick Question 14.3 Interpreting Graphs I
All students at Truman High School completed a questionnaire about bicycle
riding. Some of the results are shown in Figure 14.1. Does each statement
accurately describe the data in the circle graph (pie chart) in Figure 14.1?
­Explain your reasoning.
(a) Forty percent of the reported bicycle riders at Truman High were freshmen.
(b) Twenty-five percent of the sophomores at Truman High reported riding
bicycles.
(c) The percent of students at Truman High who reported riding a bicycle
and who were juniors is 20%.
(d) Freshmen at Truman High were twice as likely as juniors to report riding
bicycles.
Quick Question 14.4 Interpreting Graphs II
All of the eighth graders at Madison Middle School were asked whether they
most enjoyed basketball, badminton, or volleyball. Their responses are shown
in Figure 14.2.
(a) Roughly how many eighth grade girls chose basketball as their favorite
sport?
(b) Approximately how many eighth grade boys chose badminton as their
favorite sport?
(c) About what percent of eighth grade girls chose volleyball as their favorite
sport?
(d) Approximately what percent of eighth grade boys chose basketball as
their favorite sport?
118 Advanced Quantitative Reasoning
Copyright © 2015 by Gregory D. Foley, Thomas R. Butts, Stephen W. Phelps, and Daniel A. Showalter
Part II Probability and Statistical Reasoning
Section 14 Making Sense of Statistical Information
Quick Question 14.5 Interpreting Tables I
Use Table 14.2 to answer the following questions:
(a) What fraction of the students are Independents?
(b) What fraction of the females are Democrats?
(c) What percent of the Democrats are female?
(d) Write and solve a problem involving the entry “8” in Table 14.2.
(e) What percent of students are female or Democrat?
Table 14.2 Political Party Preference
Gender Democrat Republican
Female
7
8
Male
11
7
Total
18
15
Independent
1
2
3
Total
16
20
36
Analyzing statistical evidence frequently requires interpreting calculations involving percents or logical reasoning.
Quick Question 14.6 Interpreting Tables II
Use Table 14.3 to answer the following questions:
(a) What fraction of those who earned degrees were women?
(b) What fraction of the women who earned degrees received a professional
degree?
(c) What percent of men who earned degrees received a master’s degree?
(d) What percent of those receiving a bachelor’s degree were men?
(e) What percent of the degree recipients were female or received a doctorate?
Table 14.3
Numbers of Degrees Awarded During 2008 in the U.S. (in Thousands)
Gender
Bachelor’s
Master’s
Professional
Doctorate
Total
Female
918
381
46
33
1,378
Male
681
250
47
32
1,010
Total
1,599
631
93
65
2,388
Statistical Studies
Random Sample
Recall from Section 12 that, when a
sample is drawn from a population,
it is a random sample if each combination of members of the population is
equally likely to be chosen.
A statistical study requires a clearly stated research question. The researcher must
formulate a proper question before designing the study or collecting data. The
­research question describes what we want to know in simple terms and anticipates
using data that vary across some population—a group of people, animals, or
things.
It is usually not practical to conduct a census, that is, to collect data from every
individual in the population. So we typically try to get data from a representative
sample, or subset of individuals chosen from the population. Next, we organize,
summarize, and analyze the collected data to learn about their general features. Finally, we use the data to make inferences or draw conclusions. You will explore this
process in more detail in Section 15.
Copyright © 2015 by Gregory D. Foley, Thomas R. Butts, Stephen W. Phelps, and Daniel A. Showalter
Advanced Quantitative Reasoning 119
Part II Probability and Statistical Reasoning
Section 14 Making Sense of Statistical Information
more online
For the full article, go to http://nutri.jtf.
org.tw/en/doc/990803Nutritional%20
imbalance%20endorsed%20by%20
televised%20food%20advertisements..
pdf
Investigation 14.7 Analyzing a Statistical Study
Do you enjoy watching TV? Do you like food? A research team led by Dr.
­Michael Mink of Armstrong Atlantic State University watched nearly 100 hr
of television to observe the eating habits promoted by the media (84 hr of
prime-time shows and 12 hr of Saturday morning children shows). The team
noted every food ad and compared the nutritional values of the advertised
foods to the proportions suggested in the U.S. Food and Drug Administration’s food pyramid.
They found that a diet based on TV ads would exceed the daily recommendations of sugar by over 25 times, fat by over 20 times, and meat by almost 30%.
Moreover, the TV diet would only have 32% of the recommended amount of
dairy products, 40% of the recommended vegetable servings, and 27% of the
recommended fruit servings. Mink’s study did not examine how much people’s
eating habits are affected by the ads they watch. It did mention that the average
person watches over 2 million ads during their life, many of which involve food.
The researchers recommend that food producers work together with health
professionals to showcase healthier foods and nutritional information. They
also suggest that governmental policy follow the path of other countries such
as China, Thailand, and Romania, which have placed restrictions of food ads.
(a) What was the main question addressed by the statistical study?
(b) How were data collected to answer this question?
(c) What variables were measured? Label each variable as quantitative or categorical. Explain your choice.
(d) What were the conclusions drawn from this statistical study? Are these
conclusions valid, based on the information provided? Explain.
Source: Mink, M., Evans, A., Moore, C. G., Calderon, K. S., & Deger, S. (2010). Nutritional imbalance endorsed by televised food advertisements. Journal of the American
Dietetic Association, 110, 904–910.
Errors in Statistical Studies
Quantifying Uncertainty
The margin of error helps us estimate
the location of a population parameter, but there is no certainty that the
parameter is within this range. Unless otherwise stated, most margins
of error are stated at the 95% confidence level. For example, if a political
poll states that 27% of the population
favor a candidate, with a 3-point
margin of error, you can be 95% confident that the actual favoring rate is
between 24% and 30%—provided the
sample was random and unbiased.
more online
Go to www.aqrpress.com/sa1401 for
an interactive file that you can use with
­Exploration 14.8 to study the differences from one ­sample to the next.
Many types of errors can occur when conducting a statistical study. In this section,
we will briefly discuss two types of errors as a prelude to a more complete discussion of errors and misuses in Section 19.
Statisticians typically estimate the properties of a population by examining a random sample. The margin of error is a statistic that estimates the amount of sampling
error in the results of a statistical study. A margin of error does not mean that someone messed up the research. No sampling method can guarantee that the sample exactly represents the population. However, sampling techniques, when used correctly,
can be trusted to give results that are accurate within a certain range.
Exploration 14.8 Margin of Error
Based on a survey of 1,500 U.S. voters, 53% of voters prefer Smith to Jones
with a margin of error of ±3%.
(a) How would you describe the number of voters who preferred Smith according to this survey?
(b) If the respondents were a representative sample of U.S. voters, how might
the respondents have been chosen?
(c) If you did another survey of 1,500 registered voters, would you expect the
results to be the same? Why or why not?
(d) How could you conduct this survey to reduce the margin of error?
120 Advanced Quantitative Reasoning
Copyright © 2015 by Gregory D. Foley, Thomas R. Butts, Stephen W. Phelps, and Daniel A. Showalter
Part II Probability and Statistical Reasoning
Section 14 Making Sense of Statistical Information
Definition Convenience Sampling
Many of these biases are a result of
convenience sampling in which individuals are sampled only because
they are easily accessible.
more online
Have you ever heard of President
Landon? Go to http://www.math.
upenn.edu/~deturck/m170/wk4/
lecture/fdr-landon.html to learn about
the role bias played in the Literary ­Digest
Presidential poll of 1936.
Another source of error is having a bias (often unintentional) in choosing the
sample or framing the survey questions. Following are some common types of bias:
Undercoverage bias (or exclusion bias) occurs when part of the population is
excluded from the sampling process. For example, you want to estimate the size of
the moose population in a national park. Certain areas of the park are too remote to
access and so are not in the sample by choice. From the collected data, you provide
an estimate for the park (including the inaccessible parts).
Nonresponse bias occurs when individuals chosen for the sample are unwilling
or unable to participate in the survey. For example, a questionnaire respondent
refuses to provide certain information (e.g., regarding age, income, or gender) or
refuses to participate at all.
Self-selection bias (or voluntary response bias) occurs when individuals select
themselves (or volunteer) for the sample. For example, call-in radio shows or online polls that solicit viewpoints on controversial topics can lead to a sample that
overrepresents individuals with strong opinions.
Response bias occurs when respondents are influenced by poorly worded survey questions or by the behavior of the interviewer. Response bias can occur because respondents may lie when presented with questions of a personal nature or
questions about illegal or taboo activity.
more online
Quick Question 14.9 Wording
Go to http://stattrek.com/surveyresearch/survey-bias.aspx for more on
bias in surveys.
(a) Suppose two surveys asked Catholics whether contraceptives should be
made available to unmarried women. The first survey involved in-person
interviews, and 44% of the respondents answered yes. The second survey
was conducted by mail and telephone, and 75% of the respondents answered yes. Which of the two surveys do you think was more likely to be
accurate? Why?
(b) Suppose you write two versions of a question for a poll: (i) You can ask
people whether the government is spending too much on “assistance to
the poor.” (ii) You can ask people whether the government is spending
too much on “welfare.” Which of the two wordings do you think will produce a higher percent of those saying “Yes”? Why?
more online
Exploration 14.10 Bias
In each case, indicate whether you think the survey result was biased, misleading, or neither. Give reasons for your answer.
(a) Survey result: Drinking and driving is not considered a serious problem in
Mathville.
Method of survey: Researchers waited outside various randomly selected
bars. They asked every third patron who exited if he or she felt drinking
and driving was a serious problem.
(b) Survey result: Franklin High School students earn a B in mathematics.
Method of survey: Investigators found the mean grade (average) in mathematics of all students at Franklin High.
(c) Survey result: 90% of U.S. adults rarely use a cell phone.
Method of survey: Researchers called randomly selected homes and inquired as to how often they used a cell phone.
(d) Survey result: Two out of three dentists recommend Toothlover toothpaste.
Method of survey: Researchers polled 15 local dentists and asked them
which toothpaste they would recommend to their patients.
Go to http://www.math.upenn.
edu/~deturck/m170/wk4/lecture/
case1.html for more details on the
Literary Digest Presidential poll of 1936.
Copyright © 2015 by Gregory D. Foley, Thomas R. Butts, Stephen W. Phelps, and Daniel A. Showalter
Advanced Quantitative Reasoning 121
Part II Probability and Statistical Reasoning
Section 14 Making Sense of Statistical Information
Quick Review for Section 14
Do the calculation in your mind. Write down only the answer,
and be prepared to explain your thought process.
1. What is 25% of 1,200?
5. 300 is what percent of 1,000?
6. 1,400 is what percent of 7,000?
7. 6,000 is what percent of 3,000?
2. What is 20% of 5,000?
8. 4,000 is what percent of 800?
3. What is 12.5% of 1,600?
9. What is 3/5 of 6,000?
4. What is one-third of 7,500?
10. What is 2/3 of 6,600?
Exercises for Section 14
Using Percents and Ratios
1. John pays $50 for a calculator, and Tim pays $70 for a calculator. Which of the following statements correctly compares
Tim’s cost to John’s cost?
(A)Tim’s cost is 20% more than John’s cost.
7. Table 14.4 gives the average price for a gallon of milk for the
month of January in the years listed. If a price-of-milk index
is established with a base of 100 in 1996, what was the value
of the price-of-milk index in January 2012?
(C) Tim’s cost is 1.2 times John’s cost.
Table 14.4
Average Milk Prices in Dollars per
Gallon From 1996 to 2012
(E) Tim’s cost is 40% more than John’s cost.
1996
2.546
100
2000
2.785
109
(B) John’s cost is 40% less than Tim’s cost.
Year Price ($/gal) Price-of-Milk Index
(D)John’s cost is 20% less than Tim’s cost.
2. The participation rate of students in intramural athletics at
State College was 44% in 2000 and 55% in 2010. Based on this
information, which of the following statements is correct?
(A)About 11% more students participated in intramural
­athletics in 2010 than in 2000.
(B) The participation rate was 11% greater in 2010 than in 2000.
(C) There was a 20% increase in the participation rate from
2000 to 2010.
(D)The 2010 participation rate was 1.11 times the 2000 participation rate.
(E) The number of students attending State College did not
change from 2000 to 2010.
3. During a 5-year period, the level of methyl chloroform, a
man-made industrial solvent that is harmful to the ozone
layer, decreased from 150 parts per trillion (ppt) to 120 ppt.
By what percent did the level of methyl chloroform decrease
during this 5-year period?
4. On January 1, 2008, you placed $1,000 in an investment that
earned 10% annual interest, compounded annually. What
was the value of your investment at the end of 2011?
5. The fall 2011 enrollment at Midsize State University was
18,200, which was an increase of 4% over the fall 2010 enrollment. What was the fall 2010 enrollment?
6. A credit card has a $25 annual fee and charges 18% interest
per year on the unpaid balance. If you average $500 unpaid
balance each month for a year, what was your total payment
for use of the card for the year?
122 Advanced Quantitative Reasoning
2004
2.879
113
2008
3.871
152
2012
3.583
Source: U.S. Bureau of Labor Statistics
8. In 1990, according to Census data, one in four U.S. residents
over 18 years of age had never married, as compared to one
in six in 1970. What is the ratio of the fraction of residents
never married in 1990 to the fraction of never married residents in 1970? Simplify your answer.
9. Percent profit. Casa Tamale, a fast food restaurant with a big
carryout business, sells two-item breakfast tacos for $1.50
each. The total cost to produce each taco on average is 75¢. In
a typical morning, Casa Tamale sells 1,200 tacos.
(a) What is the daily profit on breakfast tacos as a percent of
the receipts?
(b) What is the daily profit on breakfast tacos as a percent of
the investment?
10.The national debt is the amount of money owed by the U.S.
government. As of December 2011, the U.S. national debt was
approximately $15 trillion.
(a) Suppose the debt were reduced by $500 billion. What
would the new debt be? By what percent would it have
been reduced? [Note: One trillion equals 1,000 billion.]
(b) As of December 2011, the approximate size of the U.S.
population was about 315 million. Suppose every person
in the U.S. contributed $10,000 toward paying off the
national debt. What would the remaining balance be? By
what percent would the debt have been reduced? [Note:
One billion equals 1,000 million.]
Copyright © 2015 by Gregory D. Foley, Thomas R. Butts, Stephen W. Phelps, and Daniel A. Showalter
Part II Probability and Statistical Reasoning
Section 14 Making Sense of Statistical Information
11. Data from medical studies often contain many related variables for each person in the study. Which of the following
variables are categorical, and which are quantitative?
(a) Gender (male, female)
(b) Age (years)
(c) Race
14. Figure 14.4 shows the results of a survey on a recent
­committee’s deficit reduction proposal.
(d) Systolic blood pressure
(e) Social Security number
12. Table 14.5 gives an excerpt from the grade book of mathematics
teacher Mr. Prez Ident.
Table 14.5
Final Grades for Selected Students in Mr. Prez Ident’s Class
Student
Points
Final
Name
Number
Class
Earned
Grade
Adams, J.
321465
Sophomore
543
A
Arthur, C.
173953
Junior
389
C
Buchanan, J.
456912
Sophomore
367
C
Bush, G.
786342
Junior
495
B
Carter, J.
357864
Senior
520
B
Identify the individuals and the variables in these data.
Which variables are categorical, and which are quantitative?
13. See Figure 14.3.
Figure 14.4 Opinions regarding the deficit reduction proposal.
(a) What percent of those surveyed either thought it was a
good idea or were mixed or unsure on the proposal?
(b) Which two groups made up 70% of those surveyed?
(c) What is the ratio of those who were mixed or unsure
about the proposal to those having no opinion?
(d) What is the ratio of those thinking it was a bad idea to
those thinking it was a good idea?
15. The bar chart in Figure 14.5 shows the profits of Java House
Coffee during the last quarter of 2009 and all four quarters of
2010 in millions of dollars. The profits rose 35.1% during this
time.
Figure 14.3 Favorite movie types of Monroe High School students.
(a) What percent of the Monroe High School students chose
either romance or science fiction as their favorite type of
movie?
(b) Which two movie types made up 45% of the favorite
movie types of the students?
(c) What is the ratio of those preferring horror to those
­preferring comedy?
(d) What is the ratio of those preferring action to those
­preferring foreign films?
Figure 14.5 Java House Coffee’s quarterly profits in millions of
dollars.
(a) Show how the profit increase of 35.1% was calculated.
(b) Find the approximate percent profit decrease from the
first quarter to the third quarter in 2010.
(c) Find the approximate percent profit increase from the
fourth quarter in 2009 to the first quarter in 2010.
(d) Find the approximate percent profit increase from the
second quarter to the third quarter in 2010.
Copyright © 2015 by Gregory D. Foley, Thomas R. Butts, Stephen W. Phelps, and Daniel A. Showalter
Advanced Quantitative Reasoning 123
Part II Probability and Statistical Reasoning
Section 14 Making Sense of Statistical Information
16. The graph in Figure 14.6 shows the average cost of a new
­automobile (in dollars) for the years 1950, 1960, …, 2000.
18. The graph in Figure 14.8 represents the U.S. sales of Patti
O’Furniture Industries by territory; last year’s sales are
stacked above year-to-date sales.
Figure 14.6 Average cost in dollars of a new automobile.
(a) The average cost in 1960 was approximately what percent
of the average cost in 1990?
(b) The average cost in 2000 was approximately what percent
of the average cost in 1980?
(c) The average cost of a car increased by approximately
what percent from 1960 to 1970?
(d) The average cost of a car increased by approximately
what percent from 1980 to 2000?
17. The stacked bar graph in Figure 14.7 represents the favorite
games of Briton Middle School students.
Figure 14.8 U.S. sales of Patti O’Furniture Industries by territory
(in millions of dollars). On the vertical scale: 2 = $2,000,000, 4 =
$4,000,000, etc.
(a) Which territories have had greater sales year-to-date than
last year?
(b) What is the approximate value of the greatest year-todate:last-year ratio? Explain.
(c) Which territories had greater sales last year than
year-to-date?
(d) In the Southeast, what percent of last year’s sales have
occurred so far this year?
19. Table 14.6 gives the number of students of each gender
­majoring in three areas at Iota College.
Table 14.6
Numbers of Students With Select Majors, By
Gender, Iota College
Major
Male Female All Students
Business
60
20
80
Economics
10
50
60
Management
30
30
60
All majors
100
100
200
(a) What percent of the students are female?
Figure 14.7 Favorite games of Briton Middle School students.
(a) Roughly how many of the girls chose chess as their
­favorite game?
(b) Approximately how many of the boys chose chess as
their favorite game?
(b) What percent of the females are management majors?
(c) What percent of the business majors are male?
(d) What percent of the economics majors are female?
(e) What percent of the students were either female or a business major?
(c) Roughly what percent of the girls chose cricket as their
favorite game?
(d) Approximately what percent of the boys chose hockey as
their favorite game?
124 Advanced Quantitative Reasoning
Copyright © 2015 by Gregory D. Foley, Thomas R. Butts, Stephen W. Phelps, and Daniel A. Showalter
Part II Probability and Statistical Reasoning
Section 14 Making Sense of Statistical Information
20. Table 14.7 gives the number of students of each ­gender
­majoring in three areas at Tinier College.
Table 14.7
Numbers of Students With Select Majors, By
Gender, Tinier College
Major
Male Female All Students
Business
30
20
50
Economics
10
25
35
Management
10
5
15
All majors
50
50
100
(d) What fraction of those who prefer raspberry sherbet are
Republicans?
Table 14.10 contains data on the age and marital status of adult
American women (in thousands) in 2010. Use these data to complete Exercises 23 and 24.
(b) What percent of the females are management majors?
(c) What percent of the business majors are male?
(d) What percent of the management majors are female?
(e) What percent of the students were either male or a
­management major?
21. Suppose that students at Earmuff Junction High School are
asked to identify their preferences in political affiliation
(Democrat, Independent, or Republican) and in frozen yogurt
flavors (chocolate, swirl, or vanilla). Their responses are presented in Table 14.8. (Some are left for you to calculate.)
Table 14.8
Frozen Yogurt Preferences and Political Affiliations of
Earmuff Junction Students
Democrat
Chocolate
Swirl
26
Independent
12
Republican
12
Total
43
Vanilla
Total
16
60
8
25
64
150
13
(a) What fraction of the respondents are Independents?
(b) What fraction of the respondents prefers chocolate
yogurt?
(c) What fraction of Independents prefers chocolate yogurt?
(d) What fraction of those who prefer chocolate yogurt are
Independents?
22. Suppose that students at Fairmont West High School are asked
to identify their preferences in political affiliation (Democrat, Independent, or Republican) and in sherbet flavors
(lime, orange, or raspberry). Their responses are presented in
Table 14.9. (Several are left for you to calculate).
Table 14.9
Sherbet Preferences and Political Affiliations of
Fairmont West Students
Affiliation
Lime
Orange
Democrat
360
280
Independent
100
120
Republican
220
Total
Raspberry
Total
900
750
530
790
(b) What fraction of the respondents prefers raspberry sherbet?
(c) What fraction of Republicans prefers raspberry sherbet?
(a) What percent of the students are female?
Affiliation
(a) What fraction of the respondents are Republicans?
Table 14.10
Age and Marital Status of U.S. Women (in
Thousands) in 2010
Age (years)
Marital Status
18–29
30–64
≥ 65
Total
Married
7,178 48,233
9,688
65,099
Never married 17,138
9,670
984
27,792
Widowed
62
2,601
8,699
11,362
Divorced
612 10,725
2,412
13,749
Total
24,988 71,228 21,784 118,000
Source: The 2012 Statistical Abstract, Table 57.
23. If one woman is chosen at random,
(a) What is the probability that the woman is at least 65 years
old?
(b) What is the (conditional) probability that the selected
woman was never married, given that she is at least 65?
(c) How many women are both never married and at least
65 years old? What is the probability that the selected
woman is both?
(d) Check that the probabilities you found in parts (a), (b),
and (c) satisfy the multiplication rule.
24. If one woman is chosen at random,
(a) What is the conditional probability that she is 18–29 years
old, given that she is divorced?
(b) What is the probability that she is divorced, given that
she is 18–29?
25. Which study gives better evidence that taking zinc will decrease the risk of a heart attack? Why?
(A)Some subjects chose to take supplements containing zinc,
and others did not. There were 30% fewer heart attacks
among those who took zinc than among those who did
not.
(B) Half the subjects were randomly assigned to take zinc;
the others got a placebo. Those who received zinc had
10% fewer heart attacks than those who received the
placebo.
26. Which study gives better evidence that taking zinc will decrease the risk of a heart attack? Explain your reasoning.
(A)Some subjects chose to take supplements containing zinc,
and others did not. Those who took the zinc had 20%
fewer heart attacks than those who did not.
(B) Half the subjects were randomly assigned to take zinc;
the others got a placebo. Those who received zinc had
20% fewer heart attacks than those who received the
placebo.
Copyright © 2015 by Gregory D. Foley, Thomas R. Butts, Stephen W. Phelps, and Daniel A. Showalter
Advanced Quantitative Reasoning 125
Part II Probability and Statistical Reasoning
Section 14 Making Sense of Statistical Information
27. A 2007 CNN poll reported that 51% of Nevada voters favored
Hillary Clinton. The poll indicated a margin of error of ±5%.
Which of the following statements best represents what this
sampling error means? Explain your reasoning.
(A)Between 48.5% and 53.5% of Nevada’s Democratic voters would vote for Clinton over any of the other listed
Democrats.
(B) Between 46% and 56% of Nevada’s Democratic voters would vote for Clinton over any of the other listed
Democrats.
(C) Five percent of the people polled may change their mind
before the election.
(D)Five percent of the population couldn’t be reached for a
response.
(E) About 5% of the poll results might have been recorded
incorrectly.
28. Indicate whether you agree or disagree with each statement.
Give a reason for your choice.
(a) In a scientific poll with a margin of error of ±3%, Candidate R has 45% support and Candidate D has 43% support, so R is supported by more voters than D.
(b) In a scientific poll with a margin of error of ±3%, Candidate D has 38% support and Candidate R has 33% support, so D has more support than R.
(c) In a scientific poll conducted a month ago, and with a
margin of error of ±3%, Proposition P received support
from 56% of respondents. In a similar poll with the same
margin of error conducted today, P received only 54%
support. So, popular support for P is falling.
(d) In past scientific polls with a margin of error of ±3%,
the lowest percentage of respondents who approved of
Governor G’s performance was 33%. In a current poll, the
Governor’s approval rating was 28%. Therefore, public
approval of Governor G is at a record low.
29. A newspaper headline describing a poll of registered voters
taken two weeks before a recent election read “Ringel leads
with 52%.” The accompanying article describing the poll
stated that the margin of error was 2% with 95% confidence.
The poll shows Ringel leading. But the newspaper article said
that the election was too close to call. Explain why.
30. Researchers polled 1,025 women and 472 men randomly selected from the U.S., excluding Alaska and Hawaii. The poll
found that 47% of the women said they do not get enough
time for themselves.
(a) The poll announced a margin of error of ±3 percentage
points. Explain to someone who knows no statistics why
we can’t just say that 47% of all adult women do not get
enough time for themselves.
Bias
31. The National Highway Traffic Safety Administration periodically reports data related to school bus accidents. According
to these reports, an average of 11–13 children die each year in
school bus accidents, and an average of 550–650 school-age
children die each year in auto accidents during school hours.
These numbers suggest that riding the bus is safer than driving to or from school with a parent. These numbers are not
fully convincing, however. What rates would you like to know
to compare the safety of riding a bus and using an auto?
32. Popular magazines often rank cities based on how desirable
it is to work and live there. What are five variables that you
would measure for each city if you were designing such a
study? Give reasons for each of your choices.
33. Sharon is a young systems analyst. She and all her friends
regularly use cell phones. Last year, two of Sharon’s friends
developed brain cancer. She wonders if the cancer is related
to use of cell phones. Explain briefly why the experience of
Sharon’s friends does not provide good evidence that cell
phones cause brain tumors.
34. Because of its high caffeine content, Jamie and all of his
friends prefer Rushing River to either Kooky cola or Pixy
cola. Explain why Jamie’s experience is not sufficient evidence that most young people prefer Rushing River to Kooky
or Pixy.
35. When discussing the pros and cons of wearing seat belts,
Herman always brings up the case of a friend who survived
an accident because he was not wearing a seat belt. The
friend was thrown out of the car and landed on a grassy
bank, suffering only minor injuries, and the car burst into
flames and was destroyed. Explain briefly why this anecdote
does not provide adequate evidence that it is safer not to
wear seat belts.
36. A researcher calls 1,800 randomly chosen residential telephone numbers, and then asks to speak with an adult member of the household. The researcher asks, “How many 3D
movies have you watched in a movie theater in the past 12
months?”
(a) What was the intended population?
(b) In all, 1,350 people responded. What was the rate (percent)
of nonresponse?
(c) What source of response error is likely for the question
asked?
37.On American Idol, viewers are asked to choose their favorite singer among those competing on a show by dialing a
number such as 866-IDOLS01 to vote for one contestant and
866‑IDOLS02 to vote for another. Explain why this poll is almost certainly biased.
(b) The margin of error for results concerning men was ±4
percentage points. Why is this error larger than the margin of error for women?
126 Advanced Quantitative Reasoning
Copyright © 2015 by Gregory D. Foley, Thomas R. Butts, Stephen W. Phelps, and Daniel A. Showalter
Part II Probability and Statistical Reasoning
Section 14 Making Sense of Statistical Information
38. A survey asked 26,000 doctors, “How do current changes in
the medical system affect your desire to practice medicine?”
Over 16,000 responded and the results were “I’m energized”:
4.6%; “Makes me think about quitting”: 82.6%; Unsure/No
Opinion: 12.8%. Many newspapers reported the results by
saying “Eighty-three percent of American physicians have
considered leaving their practices over President Barack
Obama’s health care reform law.”
(a) Why is this survey probably biased?
(b) Do you think the percentage of doctors who would consider quitting is actually more or less than 83%? Why?
39. Here are two wordings for the same question used in 1993 by
candidate Ross Perot in a political poll.
(a) Should laws be passed to eliminate all possibilities of special
interests giving huge sums of money to candidates?
(b) Should laws be passed to prohibit interest groups from contributing to campaigns, or do groups have a right to contribute to
the candidates they support?
One of these questions drew 40% favoring banning contributions; the other drew 80% with this opinion. Which question
produced the 40% and which got 80%? Why were the results
so different? (W. Mitofsky, “Mr. Perot, you’re no pollster,”
New York Times, March 27, 1993.)
40. Comment on each of the following as a potential survey
question. Is the question clear? Is it slanted toward a desired
response?
(a) “Some cell phone users have developed brain cancer.
Should all cell phones come with a warning label explaining the danger of using cell phones?”
(b) “Do you agree that a national system of health insurance should be favored because it would provide health
insurance for everyone and would reduce administrative
costs?”
(c) “Do you believe that Congressional Republicans will seek
to obstruct further progress by gumming up the gears of
government as they did after the 1994 elections?” (From a
survey by the Democratic party in February, 2011.)
41. From time to time the government budget shows a surplus
and people are sometimes polled on how to use this “extra”
money. Here were two suggested questions:
42. A teacher asks her class, “How many children are there in
your family, including yourself?” The mean response is about
3 children. According to the 2010 census, the average household has approximately 1.9 children. Why is this teacher’s
sample biased toward higher outcomes?
Writing Poor Survey Questions
43. Write a biased question designed to get one answer rather
than another.
44. Write a question that is confusing, so that it is hard to answer.
Investigations
45. Use the following link to read an article on the health effects
of omega-3 oil: http://www.nature.com/news/fish-oilsupplement-research-remains-murky-1.11484. For each question give a brief response that is relevant to the study on
omega-3 oil.
(a) What was the main question addressed by the statistical
study?
(b) What data were collected to answer the question?
(c) How were the data collected?
(d) What variables were measured in the data? Which variables are categorical, and which are quantitative?
(e) What were the conclusions drawn from this statistical
study? Do you believe that the conclusions are valid
based on the information provided?
(f) Do you think there were any examples of bias in the
study? If so, describe them.
(g) Did you place these data in a context that is meaningful
to you? If yes, describe your context.
(h) Are there any statements that appear to be opinions?
(i) Are there any stated or implied limitations to the study?
46. Use the following link to read an article on a connection
between Wal-Mart and weight: http://www.forbes.com/
forbes/2009/0608/024-opinions-retail-health-on-my-mind.
html. For each of the nine questions listed in Exercise 45, give
a brief response that is relevant to the study on Wal-Mart’s
weight effect.
(a) “Should the money be used for a tax cut, or should it be
used to fund new government programs?”
(b) “Should the money be used for a tax cut, or should it
be spent on programs for education, the environment,
health care, crime-fighting, and military defense?”
One of these questions drew 60% favoring a tax cut; the other,
only 22%. Which wording pulls respondents toward a tax
cut? Why?
Copyright © 2015 by Gregory D. Foley, Thomas R. Butts, Stephen W. Phelps, and Daniel A. Showalter
Advanced Quantitative Reasoning 127