1.1 Introduction to Statistics

1. Determine how the key terms apply to the following study. We want to know the average (mean) amount of money first-year college students spend at ABC College on school supplies (excluding books). We randomly survey 100 first-year students at the college. Three of those students spent $150, $200, and $225.

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2. Determine how the key terms apply to the following study. We want to know the average (mean) amount of money spent on school uniforms each year by families with children at Knoll Academy. We randomly survey 100 families with children in the school. Three of the families spent $65, $75, and $95.

3. Determine how the key terms apply to the following study.

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4. As part of a study designed to test the safety of automobiles, the National Transportation Safety Board collected and reviewed data about the effects of automobile crashes on test dummies. Here is the criterion they used:

Figure 1.12

Cars with dummies in the front seats were crashed into a wall at a speed of 35 miles per hour. We want to know the proportion of dummies in the driver’s seat that would have had head injuries if they had been actual drivers. We start with a simple random sample of 75 cars.^[1]

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5. An insurance company would like to determine the proportion of all medical doctors who have been involved in one or more malpractice lawsuits. The company selects 500 doctors at random from a professional directory and determines how many in the sample have been involved in malpractice lawsuits.

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6. Pharmaceutical companies often conduct surveys to determine the effectiveness of a treatment program. Suppose that a new AIDS antibody drug is currently under study. It is given to patients once the AIDS symptoms have revealed themselves. Of interest is the average (mean) length of time in months that patients live once they start the treatment. Two researchers each follow a different set of 40 patients with AIDS from the start of treatment until their deaths. The following data was collected, measuring the time (in months) between patients starting treatment and their deaths.

Researcher A: 3, 4, 11, 15, 16, 17, 22, 44, 37, 16, 14, 24, 25, 15, 26, 27, 33, 29, 35, 44, 13, 21, 22, 10, 12, 8, 40, 32, 26, 27, 31, 34, 29, 17, 8, 24, 18, 47, 33, 34

Researcher B: 3, 14, 11, 5, 16, 17, 28, 41, 31, 18, 14, 14, 26, 25, 21, 22, 31, 2, 35, 44, 23, 21, 21, 16, 12, 18, 41, 22, 16, 25, 33, 34, 29, 13, 18, 24, 23, 42, 33, 29

Determine how the key terms apply to the example for Researcher A.

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7. For each of the following exercises, identify:

1. The population
2. The sample
3. The parameter
4. The statistic
5. The variable
6. The data

Give examples where appropriate.

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

ii. 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.

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

iv. Insurance companies are interested in clients’ mean yearly health costs so that they can determine the costs of health insurance.

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

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

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

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

8. 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.

i. What is the population in which she is interested?

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

ii. 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

iii. 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

9. In a survey of 100 stocks on NASDAQ, the average percent increase for the past year was 9%.

i. The “average increase” for all NASDAQ stocks is the:

1. Population
2. Statistic
3. Parameter
4. Sample
5. Variable

ii. All of the NASDAQ stocks are the:

1. Population
2. Statistic
3. Parameter
4. Sample
5. Variable

iii. Nine percent is the:

1. Population
2. Statistic
3. Parameter
4. Sample
5. Variable

iv. The 100 NASDAQ stocks in the survey are the:

1. Population
2. Statistic
3. Parameter
4. Sample
5. Variable

v. The percent increase for one stock in the survey is the:

1. Population
2. Statistic
3. Parameter
4. Sample
5. Variable

vi. Would the data collected be qualitative, quantitative discrete, or quantitative continuous?

1.2 Data Basics

 1. The data are the colors of backpacks. You sample five students. One student has a red backpack, two students have black backpacks, one student has a green backpack, and one student has a gray backpack. What type of data is this?

2. The data are the number of books students carry in their backpacks. You sample five students. Two students carry three books, one student carries four books, one student carries two books, and one student carries one book. What type of data are the numbers of books (three, four, two, and one)?

3. The data are the weights of backpacks with books in them. You sample the same five students. The weights (in pounds) of their backpacks are 6.2, 7, 6.8, 9.1, and 4.3. Notice that backpacks carrying three books can have different weights. What type of data is this?

4. The data are the number of machines in a gym. You sample five gyms. One gym has 12 machines, one gym has 15 machines, one gym has ten machines, one gym has 22 machines, and the other gym has 20 machines. What type of data is this?

5. The data are the areas of lawns in square feet. You sample five houses. The areas of the lawns are 144 sq. feet, 160 sq. feet, 190 sq. feet, 180 sq. feet, and 210 sq. feet. What type of data is this?

6. The data are the colors of houses. You sample five houses. The colors of the houses are white, yellow, white, red, and white. What type of data is this?

7. Determine the correct data type (quantitative or qualitative) for the number of cars in a parking lot. If they are quantitative data, indicate whether they are continuous or discrete.

1.3 Data Collection and Observational Studies

1. Researchers want to investigate whether taking aspirin regularly reduces the risk of heart attack. Four hundred men between the ages of 50 and 84 are recruited as participants. The men are divided randomly into two groups; one group will take aspirin, and the other group will take a placebo. Each man takes one pill each day for three years, but he does not know whether he is taking aspirin or the placebo. At the end of the study, researchers count the number of men in each group who have had heart attacks.^[2]

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2. A researcher wants to study the effects of birth order on personality. Explain why this study could not be conducted as a randomized experiment. What is the main problem in a study that cannot be designed as a randomized experiment?

3. You are concerned about the effects of texting on driving performance. Design a study to test the response time of drivers while texting and while driving only. How many seconds does it take for a driver to respond when a leading car hits the brakes?

1. Describe the explanatory and response variables in the study.
2. What are the treatments?
3. What should you consider when selecting participants?
4. Your research partner wants to divide participants randomly into two groups: one to drive without distraction and one to text and drive simultaneously. Is this a good idea? Why or why not?
5. Identify any lurking variables that could interfere with this study.
6. How can blinding be used in this study?

4. Identify any issues with the following studies.

1. Inmates in a correctional facility are offered good behavior credit in return for participation in a study.
2. A research study is designed to investigate a new children’s allergy medication.
3. Participants in a study are told that the new medication being tested is highly promising, but they are not told that only a small portion of participants will receive the new medication. Others will receive placebo treatments and traditional treatments.

1.5 Sampling

1. Determine whether or not the following samples are representative.

1. To find the average GPA of all students in a university, use all honor students at the university as the sample.
2. To determine the most popular cereal among young people under the age of ten, stand outside a large supermarket for three hours and speak to every 20th child under age ten who enters the supermarket.
3. To find the average annual income of all adults in the United States, sample US congressmen. Create a cluster sample by considering each state as a stratum (group). By using simple random sampling, select states to be part of the cluster. Then survey every US congressman in the cluster.
4. To determine the proportion of people taking public transportation to work, survey 20 people in New York City. Conduct the survey by sitting in Central Park on a bench and interviewing every person who sits next to you.
5. To determine the average cost of a two-day stay in a hospital in Massachusetts, survey 100 hospitals across the state using simple random sampling.

2. Determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience).

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3. A high school principal polls 50 freshmen, 50 sophomores, 50 juniors, and 50 seniors regarding policy changes for after school activities. What type of sampling is used—simple random, stratified, systematic, cluster, or convenience?

4. This table displays six sets of quiz scores for an elementary-level statistics class, with each quiz worth a possible ten points. Use the random number generator to generate different types of samples from the data.

Figure 1.13

1. Create a stratified sample by column. Pick three quiz scores randomly from each column.
2. Create a cluster sample by picking two of the columns. Use the column numbers one through six.
3. Create a simple random sample of 15 quiz scores.
4. Create a systematic sample of 12 quiz scores.

5. Suppose ABC College has 10,000 part-time students (the population). We are interested in the average amount of money a part-time student spends on books in the fall term. Asking all 10,000 students is an almost impossible task.

Suppose we take two different samples.

First, we use convenience sampling and survey ten students from a first-term organic chemistry class. Many of these students are taking first-term calculus in addition to the organic chemistry class. The amount of money they spend on books is as follows:

$128 $87 $173 $116 $130 $204 $147 $189 $93 $153

The second sample is taken using a list of senior citizens who take P.E. classes, selecting every fifth senior citizen on the list until we have a total of ten senior citizens.

They spend:

$50 $40 $36 $15 $50 $100 $40 $53 $22 $22

It is unlikely that any student is in both samples.

Do you think that either of these samples is representative of (or is characteristic of) the entire 10,000 part-time student population?

Since these samples are not representative of the entire population, is it wise to use the results to describe the entire population?

Now, suppose we take a third sample. We choose ten different part-time students from the disciplines of chemistry, math, English, psychology, sociology, history, nursing, physical education, art, and early childhood development. (We assume that these are the only disciplines in which part-time students at ABC College are enrolled and that an equal number of part-time students are enrolled in each of the disciplines.) Each student is chosen using simple random sampling. Using a calculator, random numbers are generated and a student from a particular discipline is selected if he or she has a corresponding number. The students spend the following amounts:

$180 $50 $150 $85 $260 $75 $180 $200 $200 $150

Is the sample biased?

6. What type of data is “number of times per week?”

7. 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 Norfolk, Virginia. The first house in the neighborhood around the park was selected randomly, and then the resident of every eighth house in the neighborhood around the park was interviewed.

1. What was the sampling method used?
2. “Duration (amount of time)” is what type of data?
3. The colors of the houses around the park are what kind of data?
4. What is the population?

8. The following figure contains the total number of deaths worldwide as a result of earthquakes from 2000 to 2012.^[3]

Figure 1.14

Use figure above to answer the following questions.

1. What is the proportion of deaths between 2007 and 2012?
2. What percent of deaths occurred before 2001?
3. What is the percent of deaths that occurred in 2003 or after 2010?
4. What is the fraction of deaths that occurred before 2012?
5. What kind of data is the number of deaths?
6. Earthquakes are quantified according to the amount of energy they produce (e.g., 2.1, 5.0, 6.7). What type of data is that?
7. What contributed to the large number of deaths in 2010? In 2004? Explain.

9. Determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience).

1. A group of test subjects is divided into 12 groups; then four of the groups are chosen at random.
2. A market researcher polls every tenth person who walks into a store.
3. The first 50 people who walk into a sporting event are polled on their television preferences.
4. A computer generates 100 random numbers, and 100 people whose names correspond with the numbers on the list are chosen.

10. Pharmaceutical companies often conduct studies to determine the effectiveness of a treatment program. Suppose that a new AIDS antibody drug is currently under study. It is given to patients once the AIDS symptoms have revealed themselves. Of interest is the average (mean) length of time in months that patients live once starting the treatment. Two researchers each follow a different set of 40 AIDS patients from the start of treatment until their deaths. The following data was collected, measuring the time (in months) between patients starting treatment and their deaths.

Researcher A: 3, 4, 11, 15, 16, 17, 22, 44, 37, 16, 14, 24, 25, 15, 26, 27, 33, 29, 35, 44, 13, 21, 22, 10, 12, 8, 40, 32, 26, 27, 31, 34, 29, 17, 8, 24, 18, 47, 33, 34

Researcher B: 3, 14, 11, 5, 16, 17, 28, 41, 31, 18, 14, 14, 26, 25, 21, 22, 31, 2, 35, 44, 23, 21, 21, 16, 12, 18, 41, 22, 16, 25, 33, 34, 29, 13, 18, 24, 23, 42, 33, 29

1. List two reasons why the data may differ.
2. Can you tell if either researcher is correct or incorrect? Why?
3. Would you expect the data to be identical? Why or why not?
4. Suggest at least two methods the researchers might use to gather random data.
5. Suppose that the first researcher conducted his survey by randomly choosing one state in the nation and then randomly picking 40 patients from that state. What sampling method would that researcher have used?
6. Suppose that the second researcher conducted his survey by choosing 40 patients he personally knew. What sampling method would that researcher have used? What concerns would you have about this dataset based upon the data collection method?

11. Two researchers are gathering data on hours of video games played by school-aged children and young adults. They each randomly sample different groups of 150 students from the same school, collecting the following data.

Figure 1.15: Researcher A

Figure 1.16: Researcher B

1. Give a reason why the data may differ.
2. Would the sample size be large enough if the population is the students in the school?
3. Would the sample size be large enough if the population is school-aged children and young adults in the United States?
4. Researcher A concludes that most students play video games between four and six hours each week. Researcher B concludes that most students play video games between two and four hours each week. Who is correct?
5. As a way of rewarding students for participating in the survey, the researchers gave each student a gift card to a video game store. Would it affect the data if students knew about the award before the study?

12. A pair of studies was performed to measure the effectiveness of a new software program designed to help stroke patients regain their problem-solving skills. Patients were asked to use the software program twice a day, once in the morning and once in the evening. The studies observed 200 stroke patients recovering over a period of several weeks. The first study collected the data in Figure 1.17. The second study collected the data in Figure 1.18.

Figure 1.17: First study

Figure 1.18: Second study

1. Given what you know, which study is correct?
2. The first study was performed by the company that designed the software program. The second study was performed by the American Medical Association. Which study is more reliable?
3. Both groups that performed the study concluded that the software works. Is this accurate?
4. The company takes the two studies as proof that their software causes mental improvement in stroke patients. Is this a fair statement?
5. Patients who used the software were also a part of an exercise program, whereas patients who did not use the software were not. Does this change the validity of the conclusions from the second study?
6. Is a sample size of 1,000 a reliable measure for a population of 5,000?
7. Is a sample of 500 volunteers a reliable measure for a population of 2,500?
8. A question on a survey reads: “Do you prefer the delicious taste of Brand X or the taste of Brand Y?” Is this a fair question?
9. Is a sample size of two representative of a population of five?
10. Is it possible for two well-run experiments with similar sample sizes to get different data?

13. 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 that data.

1. Number of tickets sold to a concert
2. Percent of body fat
3. Favorite baseball team
4. Time in line to buy groceries
5. Number of students enrolled at Evergreen Valley College
6. Most watched television show
7. Favorite brand of toothpaste
8. Distance to the closest movie theatre
9. Age of executives in Fortune 500 companies
10. Number of competing computer spreadsheet software packages

14. 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 Norfolk. 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.

1. “Number of times per week” is what type of data?
2. “Duration (amount of time)” is what type of data?

15. 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 of six flights from Boston to Salt Lake City over Thanksgiving weekend to determine the number of babies on the flights. It determines the amount of safety equipment needed based on the results 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.

16. 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 with detailed description.

17. 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 with detailed description.

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

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

20. The instructor takes her sample by gathering data on five randomly selected students from each Lake Tahoe Community College math class. What type of sampling did she use?

21. 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. What was the sampling method?

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

1. A woman in the airport hands out questionnaires to travelers, asking them to evaluate the airport’s service. She does not stop travelers who are hurrying through the airport with their hands full of luggage, instead only asking 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 two rows and then calls on all students in those rows (row two and 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 asking for information about their age and 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 how voters are reacting 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 for whom they intend to vote and whether the debate changed their opinion of the candidates.

23. 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. In addition, 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 US 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?
3. Additional information: The survey, reported by the Intel Corporation, was filled out by individuals who visited the Los Angeles Convention Center to see the Smithsonian Institute’s road show, “America’s Smithsonian.” 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.

24. The Well-Being Index is a survey that follows trends of US 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.^[4]

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

25. 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.^[5]

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 non-sampling error?
4. During the same year, George Gallup conducted his own poll of 30,000 prospective voters. These researchers used a method they called “quota sampling” to obtain survey answers from specific subsets of the population.^[6] Quota sampling is an example of which sampling method described in this module?

26. 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.^[7]

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

• 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.

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?”^[8]

As of April 25, 11 people had responded, answering “NO!”

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

• Self-selected samples: Only people who are interested in the topic are choosing to respond.
• Sample size issues: A sample with only 11 participants will not accurately represent the opinions of a nation.
• Undue influence: The question is worded in a specific way to generate a specific response.
• Self-funded or self-interest studies: This question is generated to support one person’s claim and is designed to get the answer that the person desires.

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.”^[9]

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.”^[10]

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.

29. During the 2010-2011 academic year, 771 distance learning students at Long Beach City College responded to surveys. Highlights of the summary report are listed in Figure 1.19.

Figure 1.19

1. What percent of the students surveyed do not have a computer at home?
2. About how many students in the survey live at least 16 miles from campus?
3. If the same survey were done at Great Basin College in Elko, Nevada, do you think the percentages would be the same? Why?

30. Several online textbook retailers advertise that they have lower prices than on-campus bookstores. However, an important factor is whether the Internet retailers actually have the textbooks that students need in stock. Students need to be able to get textbooks promptly at the beginning of the college term. If the book is not available, then a student would not be able to get the textbook at all or might get a delayed delivery if the book is backordered.

A college newspaper reporter is investigating textbook availability at online retailers. He decides to investigate one textbook from each of the following seven subjects: calculus, biology, chemistry, physics, statistics, geology, and general engineering. He consults textbook industry sales data and selects the most popular nationally used textbook in each of these subjects. He visits websites for a random sample of major online textbook sellers and looks up each of these seven textbooks to see if they are available in stock for quick delivery through these retailers. Based on his investigation, he writes an article in which he draws conclusions about the overall availability of all college textbooks through online textbook retailers.

Write an analysis of his study. Is his sample representative of the population of all college textbooks? Explain why or why not. Describe some possible sources of bias in this study, and how it might affect the results of the study. Give some suggestions about what could be done to improve the study.

References

Text

The Data and Story Library, http://lib.stat.cmu.edu/DASL/Stories/CrashTestDummies.html (accessed May 1, 2013).

Gallup-Healthways Well-Being Index. http://www.well-beingindex.com/default.asp (accessed May 1, 2013).

Data from http://www.bookofodds.com/Relationships-Society/Articles/A0374-How-George-Gallup-Picked-the-President

“The Literary Digest Poll,” Virtual Laboratories in Probability and Statistics http://www.math.uah.edu/stat/data/LiteraryDigest.html (accessed May 1, 2013).

“Gallup Presidential Election Trial-Heat Trends, 1936–2008,” Gallup Politics http://www.gallup.com/poll/110548/gallup-presidential-election-trialheat-trends-19362004.aspx#4 (accessed May 1, 2013).

The Data and Story Library, http://lib.stat.cmu.edu/DASL/Datafiles/USCrime.html (accessed May 1, 2013).

LBCC Distance Learning (DL) program data in 2010-2011, http://de.lbcc.edu/reports/2010-11/future/highlights.html#focus (accessed May 1, 2013).

Data from San Jose Mercury News