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1.E: Sampling and Data (Exercises)

These are homework exercises to accompany the Textmap created for "Introductory Statistics" by OpenStax. Complementary General Chemistry question banks can be found for other Textmaps and can be accessed here. In addition to these publicly available questions, access to private problems bank for use in exams and homework is available to faculty only on an individual basis; please contact Delmar Larsen for an account with access permission.

1.2: Definitions of Statistics, Probability, and Key Terms

For each of the following eight exercises, identify: a. the population, b. the sample, c. the parameter, d. the statistic, e. the variable, and f. the data. Give examples where appropriate.

Q 1.2.1

A fitness center is interested in the mean amount of time a client exercises in the center each week.

Q. 1.2.2

Ski resorts are interested in the mean age that children take their first ski and snowboard lessons. They need this information to plan their ski classes optimally.

S 1.2.2

  1. all children who take ski or snowboard lessons
  2. a group of these children
  3. the population mean age of children who take their first snowboard lesson
  4. the sample mean age of children who take their first snowboard lesson
  5. \(X =\) the age of one child who takes his or her first ski or snowboard lesson
  6. values for \(X\), such as 3, 7, and so on

Q 1.2.3

A cardiologist is interested in the mean recovery period of her patients who have had heart attacks.

Q 1.2.4

Insurance companies are interested in the mean health costs each year of their clients, so that they can determine the costs of health insurance.

S 1.2.5

  1. the clients of the insurance companies
  2. a group of the clients
  3. the mean health costs of the clients
  4. the mean health costs of the sample
  5. \(X =\) the health costs of one client
  6. values for \(X\), such as 34, 9, 82, and so on

Q 1.2.6

A politician is interested in the proportion of voters in his district who think he is doing a good job.

Q 1.2.7

A marriage counselor is interested in the proportion of clients she counsels who stay married.

S 1.2.7

  1. all the clients of this counselor
  2. a group of clients of this marriage counselor
  3. the proportion of all her clients who stay married
  4. the proportion of the sample of the counselor’s clients who stay married
  5. \(X =\) the number of couples who stay married
  6. yes, no

Q 1.2.8

Political pollsters may be interested in the proportion of people who will vote for a particular cause.

Q 1.2.9

A marketing company is interested in the proportion of people who will buy a particular product.

S 1.2.9

  1. all people (maybe in a certain geographic area, such as the United States)
  2. a group of the people
  3. the proportion of all people who will buy the product
  4. the proportion of the sample who will buy the product
  5. \(X =\) the number of people who will buy it
  6. buy, not buy

Use the following information to answer the next three exercises: A Lake Tahoe Community College instructor is interested in the mean number of days Lake Tahoe Community College math students are absent from class during a quarter.

Q 1.2.10

What is the population she is interested in?

  1. all Lake Tahoe Community College students
  2. all Lake Tahoe Community College English students
  3. all Lake Tahoe Community College students in her classes
  4. all Lake Tahoe Community College math students

Q 1.2.11

Consider the following:

\(X =\) number of days a Lake Tahoe Community College math student is absent

In this case, \(X\) is an example of a:

  1. variable.
  2. population.
  3. statistic.
  4. data.

S 1.2.12

a

Q 1.2.12

The instructor’s sample produces a mean number of days absent of 3.5 days. This value is an example of a:

  1. parameter.
  2. data.
  3. statistic.
  4. variable.

1.3: Data, Sampling, and Variation in Data and Sampling

For the following exercises, identify the type of data that would be used to describe a response (quantitative discrete, quantitative continuous, or qualitative), and give an example of the data.

Q 1.3.1

number of tickets sold to a concert

S 1.3.1

quantitative discrete, 150

Q 1.3.2

percent of body fat

Q 1.3.3

favorite baseball team

S 1.3.3

qualitative, Oakland A’s

Q 1.3.4

time in line to buy groceries

Q 1.3.5

number of students enrolled at Evergreen Valley College

S 1.3.5

quantitative discrete, 11,234 students

Q 1.3.6

most-watched television show

Q 1.3.7

brand of toothpaste

S 1.3.7

qualitative, Crest

Q 1.3.8

distance to the closest movie theater

Q 1.3.9

age of executives in Fortune 500 companies

S 1.3.9

quantitative continuous, 47.3 years

Q 1.3.10

number of competing computer spreadsheet software packages

Use the following information to answer the next two exercises: A study was done to determine the age, number of times per week, and the duration (amount of time) of resident use of a local park in San Jose. The first house in the neighborhood around the park was selected randomly and then every 8th house in the neighborhood around the park was interviewed.

Q 1.3.11

“Number of times per week” is what type of data?

  1. qualitative
  2. quantitative discrete
  3. quantitative continuous

S 1.3.11

b

Q 1.3.12

“Duration (amount of time)” is what type of data?

  1. qualitative
  2. quantitative discrete
  3. quantitative continuous

Q 1.3.13

Airline companies are interested in the consistency of the number of babies on each flight, so that they have adequate safety equipment. Suppose an airline conducts a survey. Over Thanksgiving weekend, it surveys six flights from Boston to Salt Lake City to determine the number of babies on the flights. It determines the amount of safety equipment needed by the result of that study.

  1. Using complete sentences, list three things wrong with the way the survey was conducted.
  2. Using complete sentences, list three ways that you would improve the survey if it were to be repeated.

S 1.3.13

  1. The survey was conducted using six similar flights.
    The survey would not be a true representation of the entire population of air travelers.
    Conducting the survey on a holiday weekend will not produce representative results.
  2. Conduct the survey during different times of the year.
    Conduct the survey using flights to and from various locations.
    Conduct the survey on different days of the week.

Q 1.3.14

Suppose you want to determine the mean number of students per statistics class in your state. Describe a possible sampling method in three to five complete sentences. Make the description detailed.

Q 1.3.15

Suppose you want to determine the mean number of cans of soda drunk each month by students in their twenties at your school. Describe a possible sampling method in three to five complete sentences. Make the description detailed.

S 1.3.15

Answers will vary. Sample Answer: You could use a systematic sampling method. Stop the tenth person as they leave one of the buildings on campus at 9:50 in the morning. Then stop the tenth person as they leave a different building on campus at 1:50 in the afternoon.

Q 1.3.16

List some practical difficulties involved in getting accurate results from a telephone survey.

Q 1.3.17

List some practical difficulties involved in getting accurate results from a mailed survey.

S 1.3.17

Answers will vary. Sample Answer: Many people will not respond to mail surveys. If they do respond to the surveys, you can’t be sure who is responding. In addition, mailing lists can be incomplete.

Q 1.3.18

With your classmates, brainstorm some ways you could overcome these problems if you needed to conduct a phone or mail survey.

Q 1.3.19

The instructor takes her sample by gathering data on five randomly selected students from each Lake Tahoe Community College math class. The type of sampling she used is

  1. cluster sampling
  2. stratified sampling
  3. simple random sampling
  4. convenience sampling

S 1.3.19

b

Q 1.3.20

A study was done to determine the age, number of times per week, and the duration (amount of time) of residents using a local park in San Jose. The first house in the neighborhood around the park was selected randomly and then every eighth house in the neighborhood around the park was interviewed. The sampling method was:

  1. simple random
  2. systematic
  3. stratified
  4. cluster

Q 1.3.21

Name the sampling method used in each of the following situations:

  1. A woman in the airport is handing out questionnaires to travelers asking them to evaluate the airport’s service. She does not ask travelers who are hurrying through the airport with their hands full of luggage, but instead asks all travelers who are sitting near gates and not taking naps while they wait.
  2. A teacher wants to know if her students are doing homework, so she randomly selects rows two and five and then calls on all students in row two and all students in row five to present the solutions to homework problems to the class.
  3. The marketing manager for an electronics chain store wants information about the ages of its customers. Over the next two weeks, at each store location, 100 randomly selected customers are given questionnaires to fill out asking for information about age, as well as about other variables of interest.
  4. The librarian at a public library wants to determine what proportion of the library users are children. The librarian has a tally sheet on which she marks whether books are checked out by an adult or a child. She records this data for every fourth patron who checks out books.
  5. A political party wants to know the reaction of voters to a debate between the candidates. The day after the debate, the party’s polling staff calls 1,200 randomly selected phone numbers. If a registered voter answers the phone or is available to come to the phone, that registered voter is asked whom he or she intends to vote for and whether the debate changed his or her opinion of the candidates.

S 1.3.21

  1. convenience 
  2. cluster 
  3. stratified  
  4. systematic 
  5. simple random

Q 1.3.22

A “random survey” was conducted of 3,274 people of the “microprocessor generation” (people born since 1971, the year the microprocessor was invented). It was reported that 48% of those individuals surveyed stated that if they had $2,000 to spend, they would use it for computer equipment. Also, 66% of those surveyed considered themselves relatively savvy computer users.

  1. Do you consider the sample size large enough for a study of this type? Why or why not?
  2. Based on your “gut feeling,” do you believe the percents accurately reflect the U.S. population for those individuals born since 1971? If not, do you think the percents of the population are actually higher or lower than the sample statistics? Why?
    Additional information: The survey, reported by Intel Corporation, was filled out by individuals who visited the Los Angeles Convention Center to see the Smithsonian Institute's road show called “America’s Smithsonian.”
  3. With this additional information, do you feel that all demographic and ethnic groups were equally represented at the event? Why or why not?
  4. With the additional information, comment on how accurately you think the sample statistics reflect the population parameters.

Q 1.3.23

The Gallup-Healthways Well-Being Index is a survey that follows trends of U.S. residents on a regular basis. There are six areas of health and wellness covered in the survey: Life Evaluation, Emotional Health, Physical Health, Healthy Behavior, Work Environment, and Basic Access. Some of the questions used to measure the Index are listed below.

Identify the type of data obtained from each question used in this survey: qualitative, quantitative discrete, or quantitative continuous.

  1. Do you have any health problems that prevent you from doing any of the things people your age can normally do?
  2. During the past 30 days, for about how many days did poor health keep you from doing your usual activities?
  3. In the last seven days, on how many days did you exercise for 30 minutes or more?
  4. Do you have health insurance coverage?

S 1.3.23

  1. qualitative
  2. quantitative discrete
  3. quantitative discrete
  4. qualitative

Q 1.3.24

In advance of the 1936 Presidential Election, a magazine titled Literary Digest released the results of an opinion poll predicting that the republican candidate Alf Landon would win by a large margin. The magazine sent post cards to approximately 10,000,000 prospective voters. These prospective voters were selected from the subscription list of the magazine, from automobile registration lists, from phone lists, and from club membership lists. Approximately 2,300,000 people returned the postcards.

  1. Think about the state of the United States in 1936. Explain why a sample chosen from magazine subscription lists, automobile registration lists, phone books, and club membership lists was not representative of the population of the United States at that time.
  2. What effect does the low response rate have on the reliability of the sample?
  3. Are these problems examples of sampling error or nonsampling error?
  4. During the same year, George Gallup conducted his own poll of 30,000 prospective voters. His researchers used a method they called "quota sampling" to obtain survey answers from specific subsets of the population. Quota sampling is an example of which sampling method described in this module?

Q 1.3.25

Crime-related and demographic statistics for 47 US states in 1960 were collected from government agencies, including the FBI's Uniform Crime Report. One analysis of this data found a strong connection between education and crime indicating that higher levels of education in a community correspond to higher crime rates.

Which of the potential problems with samples discussed in [link] could explain this connection?

S 1.3.26

Causality: The fact that two variables are related does not guarantee that one variable is influencing the other. We cannot assume that crime rate impacts education level or that education level impacts crime rate.

Confounding: There are many factors that define a community other than education level and crime rate. Communities with high crime rates and high education levels may have other lurking variables that distinguish them from communities with lower crime rates and lower education levels. Because we cannot isolate these variables of interest, we cannot draw valid conclusions about the connection between education and crime. Possible lurking variables include police expenditures, unemployment levels, region, average age, and size.

Q 1.3.27

YouPolls is a website that allows anyone to create and respond to polls. One question posted April 15 asks:

“Do you feel happy paying your taxes when members of the Obama administration are allowed to ignore their tax liabilities?”1

As of April 25, 11 people responded to this question. Each participant answered “NO!”

Which of the potential problems with samples discussed in this module could explain this connection?

Q 1.3.28

A scholarly article about response rates begins with the following quote:

“Declining contact and cooperation rates in random digit dial (RDD) national telephone surveys raise serious concerns about the validity of estimates drawn from such research.”2

The Pew Research Center for People and the Press admits:

“The percentage of people we interview – out of all we try to interview – has been declining over the past decade or more.”3

  1. What are some reasons for the decline in response rate over the past decade?
  2. Explain why researchers are concerned with the impact of the declining response rate on public opinion polls.

S 1.3.28

  1. Possible reasons: increased use of caller id, decreased use of landlines, increased use of private numbers, voice mail, privacy managers, hectic nature of personal schedules, decreased willingness to be interviewed
  2. When a large number of people refuse to participate, then the sample may not have the same characteristics of the population. Perhaps the majority of people willing to participate are doing so because they feel strongly about the subject of the survey.

1.4: Frequency, Frequency Tables, and Levels of Measurement

Q 1.4.1

Fifty part-time students were asked how many courses they were taking this term. The (incomplete) results are shown below:

Part-time Student Course Loads
# of Courses Frequency Relative Frequency Cumulative Relative Frequency
1 30 0.6  
2 15    
3      
  1. Fill in the blanks in Table.
  2. What percent of students take exactly two courses?
  3. What percent of students take one or two courses?

Q 1.4.2

Sixty adults with gum disease were asked the number of times per week they used to floss before their diagnosis. The (incomplete) results are shown in Table.

Flossing Frequency for Adults with Gum Disease
# Flossing per Week Frequency Relative Frequency Cumulative Relative Freq.
0 27 0.4500  
1 18    
3     0.9333
6 3 0.0500  
7 1 0.0167  
  1. Fill in the blanks in Table.
  2. What percent of adults flossed six times per week?
  3. What percent flossed at most three times per week?

S 1.4.2

  1. # Flossing per Week Frequency Relative Frequency Cumulative Relative Frequency
    0 27 0.4500 0.4500
    1 18 0.3000 0.7500
    3 11 0.1833 0.9333
    6 3 0.0500 0.9833
    7 1 0.0167 1
  2. 5.00%
  3. 93.33%

Q 1.4.3

Nineteen immigrants to the U.S were asked how many years, to the nearest year, they have lived in the U.S. The data are as follows: 2; 5; 7; 2; 2; 10; 20; 15; 0; 7; 0; 20; 5; 12; 15; 12; 45; 10 .

Table was produced.

Frequency of Immigrant Survey Responses
Data Frequency Relative Frequency Cumulative Relative Frequency
0 2 219219 0.1053
2 3 319319 0.2632
4 1 119119 0.3158
5 3 319319 0.4737
7 2 219219 0.5789
10 2 219219 0.6842
12 2 219219 0.7895
15 1 119119 0.8421
20 1 119119 1.0000
  1. Fix the errors in Table. Also, explain how someone might have arrived at the incorrect number(s).
  2. Explain what is wrong with this statement: “47 percent of the people surveyed have lived in the U.S. for 5 years.”
  3. Fix the statement in b to make it correct.
  4. What fraction of the people surveyed have lived in the U.S. five or seven years?
  5. What fraction of the people surveyed have lived in the U.S. at most 12 years?
  6. What fraction of the people surveyed have lived in the U.S. fewer than 12 years?
  7. What fraction of the people surveyed have lived in the U.S. from five to 20 years, inclusive?

Q 1.4.4

How much time does it take to travel to work? Table shows the mean commute time by state for workers at least 16 years old who are not working at home. Find the mean travel time, and round off the answer properly.

24.0 24.3 25.9 18.9 27.5 17.9 21.8 20.9 16.7 27.3
18.2 24.7 20.0 22.6 23.9 18.0 31.4 22.3 24.0 25.5
24.7 24.6 28.1 24.9 22.6 23.6 23.4 25.7 24.8 25.5
21.2 25.7 23.1 23.0 23.9 26.0 16.3 23.1 21.4 21.5
27.0 27.0 18.6 31.7 23.3 30.1 22.9 23.3 21.7 18.6

S 1.4.4

The sum of the travel times is 1,173.1. Divide the sum by 50 to calculate the mean value: 23.462. Because each state’s travel time was measured to the nearest tenth, round this calculation to the nearest hundredth: 23.46.

Q 1.4.5

Forbes magazine published data on the best small firms in 2012. These were firms which had been publicly traded for at least a year, have a stock price of at least $5 per share, and have reported annual revenue between $5 million and $1 billion. Table shows the ages of the chief executive officers for the first 60 ranked firms.

Age Frequency Relative Frequency Cumulative Relative Frequency
40–44 3    
45–49 11    
50–54 13    
55–59 16    
60–64 10    
65–69 6    
70–74 1    
  1. What is the frequency for CEO ages between 54 and 65?
  2. What percentage of CEOs are 65 years or older?
  3. What is the relative frequency of ages under 50?
  4. What is the cumulative relative frequency for CEOs younger than 55?
  5. Which graph shows the relative frequency and which shows the cumulative relative frequency?

Graph A is a bar graph with 7 bars. The x-axis shows CEO's ages in intervals of 5 years starting with 40 - 44. The y-axis shows the relative frequency in intervals of 0.2 from 0 - 1. The highest relative frequency shown is 0.27.

Figure 1.4.1. (a)

Graph B is a bar graph with 7 bars. The x-axis shows CEO's ages in intervals of 5 years starting with 40 - 44. The y-axis shows relative frequency in intervals of 0.2 from 0 - 1. The highest relative frequency shown is 1.

Figure 1.4.1. (b)

Use the following information to answer the next two exercises: Table contains data on hurricanes that have made direct hits on the U.S. Between 1851 and 2004. A hurricane is given a strength category rating based on the minimum wind speed generated by the storm.

Frequency of Hurricane Direct Hits
Category Number of Direct Hits Relative Frequency Cumulative Frequency
  Total = 273    
1 109 0.3993 0.3993
2 72 0.2637 0.6630
3 71 0.2601  
4 18   0.9890
5 3 0.0110 1.0000

Q 1.4.6

What is the relative frequency of direct hits that were category 4 hurricanes?

  1. 0.0768
  2. 0.0659
  3. 0.2601
  4. Not enough information to calculate

S 1.4.6

b

Q 1.4.7

What is the relative frequency of direct hits that were AT MOST a category 3 storm?

  1. 0.3480
  2. 0.9231
  3. 0.2601
  4. 0.3370

1.5: Experimental Design and Ethics

Q 1.5.1

How does sleep deprivation affect your ability to drive? A recent study measured the effects on 19 professional drivers. Each driver participated in two experimental sessions: one after normal sleep and one after 27 hours of total sleep deprivation. The treatments were assigned in random order. In each session, performance was measured on a variety of tasks including a driving simulation.

Use key terms from this module to describe the design of this experiment.

S 1.5.1

Explanatory variable: amount of sleep

Response variable: performance measured in assigned tasks

Treatments: normal sleep and 27 hours of total sleep deprivation

Experimental Units: 19 professional drivers

Lurking variables: none – all drivers participated in both treatments

Random assignment: treatments were assigned in random order; this eliminated the effect of any “learning” that may take place during the first experimental session

Control/Placebo: completing the experimental session under normal sleep conditions

Blinding: researchers evaluating subjects’ performance must not know which treatment is being applied at the time

Q 1.5.2

An advertisement for Acme Investments displays the two graphs in Figure to show the value of Acme’s product in comparison with the Other Guy’s product. Describe the potentially misleading visual effect of these comparison graphs. How can this be corrected?

This is a line graph titled Acme Investments. The line graph shows a dramatic increase; neither the x-axis nor y-axis are labeled.

Figure 1.5.1. (a)

This is a line graph titled Other Guy's Investments. The line graph shows a modest increase; neither the x-axis nor y-axis are labeled.

Figure 1.5.1. (a)

As the graphs show, Acme consistently outperforms the Other Guys!

Q 1.5.3

The graph in Figure shows the number of complaints for six different airlines as reported to the US Department of Transportation in February 2013. Alaska, Pinnacle, and Airtran Airlines have far fewer complaints reported than American, Delta, and United. Can we conclude that American, Delta, and United are the worst airline carriers since they have the most complaints?

This is a bar graph with 6 different airlines on the x-axis, and number of complaints on y-axis. The graph is titled Total Passenger Complaints. Data is from an April 2013 DOT report.

Figure 1.5.2.

S 1.5.3

You cannot assume that the numbers of complaints reflect the quality of the airlines. The airlines shown with the greatest number of complaints are the ones with the most passengers. You must consider the appropriateness of methods for presenting data; in this case displaying totals is misleading.