OpenStax: Statistics

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1 | SAMPLING AND DATA

Chapter Objectives

By the end of this chapter, the student should be able to:

• Recognize and differentiate between key terms.
• Apply various types of sampling methods to data collection.
• Create and interpret frequency tables.

You are probably asking yourself the question, "When and where will I use statistics?" If you read any newspaper, watch television, or use the Internet, you will see statistical information. There are statistics about crime, sports, education, politics, and real estate. Typically, when you read a newspaper article or watch a television news program, you are given sample information. With this information, you may make a decision about the correctness of a statement, claim, or "fact." Statistical methods can help you make the "best educated guess."

Since you will undoubtedly be given statistical information at some point in your life, you need to know some techniques for analyzing the information thoughtfully. Think about buying a house or managing a budget. Think about your chosen profession. The fields of economics, business, psychology, education, biology, law, computer science, police science, and early childhood development require at least one course in statistics.

Included in this chapter are the basic ideas and words of probability and statistics. You will soon understand that statistics and probability work together. You will also learn how data are gathered and what "good" data can be distinguished from "bad."

1.1 | Definitions of Statistics, Probability, and Key Terms

The science of statistics deals with the collection, analysis, interpretation, and presentation of data. We see and use data in our everyday lives.

COLLABORATIVE EXERCISE

In your classroom, try this exercise. Have class members write down the average time (in hours, to the nearest half-hour) they sleep per night. Your instructor will record the data. Then create a simple graph (called a dot plot) of the data. A dot plot consists of a number line and dots (or points) positioned above the number line. For example, consider the following data:

5; 5.5; 6; 6; 6; 6.5; 6.5; 6.5; 6.5; 7; 7; 8; 8; 9

The dot plot for this data would be as follows:

Does your dot plot look the same as or different from the example? Why? If you did the same example in an English class with the same number of students, do you think the results would be the same? Why or why not?

Where do your data appear to cluster? How might you interpret the clustering?

The questions above ask you to analyze and interpret your data. With this example, you have begun your study of statistics.

In this course, you will learn how to organize and summarize data. Organizing and summarizing data is called descriptive statistics. Two ways to summarize data are by graphing and by using numbers (for example, finding an average). After you have studied probability and probability distributions, you will use formal methods for drawing conclusions from "good" data. The formal methods are called inferential statistics. Statistical inference uses probability to determine how confident we can be that our conclusions are correct.

Effective interpretation of data (inference) is based on good procedures for producing data and thoughtful examination of the data. You will encounter what will seem to be too many mathematical formulas for interpreting data. The goal of statistics is not to perform numerous calculations using the formulas, but to gain an understanding of your data. The calculations can be done using a calculator or a computer. The understanding must come from you. If you can thoroughly grasp the basics of statistics, you can be more confident in the decisions you make in life.

Probability

Probability is a mathematical tool used to study randomness. It deals with the chance (the likelihood) of an event occurring. For example, if you toss a fair coin four times, the outcomes may not be two heads and two tails. However, if you toss the same coin 4,000 times, the outcomes will be close to half heads and half tails. The expected theoretical probability of heads in any one toss is 1/2 or 0.5. Even though the outcomes of a few repetitions are uncertain, there is a regular pattern of outcomes when there are many repetitions. After reading about the English statistician Karl Pearson who tossed a coin 24,000 times with a result of 12,012 heads, one of the authors tossed a coin 2,000 times. The results were 996 heads. The fraction 996/2000 is equal to 0.498 which is very close to 0.5, the expected probability.

The theory of probability began with the study of games of chance such as poker. Predictions take the form of probabilities. To predict the likelihood of an earthquake, of rain, or whether you will get an A in this course, we use probabilities. Doctors use probability to determine the chance of a vaccination causing the disease the vaccination is supposed to prevent. A stockbroker uses probability to determine the rate of return on a client's investments. You might use probability to decide to buy a lottery ticket or not. In your study of statistics, you will use the power of mathematics through probability calculations to analyze and interpret your data.

Key Terms

In statistics, we generally want to study a population. You can think of a population as a collection of persons, things, or objects under study. To study the population, we select a sample. The idea of sampling is to select a portion (or subset) of the larger population and study that portion (the sample) to gain information about the population. Data are the result of sampling from a population.

Because it takes a lot of time and money to examine an entire population, sampling is a very practical technique. If you wished to compute the overall grade point average at your school, it would make sense to select a sample of students who attend the school. The data collected from the sample would be the students' grade point averages. In presidential elections, opinion poll samples of 1,000–2,000 people are taken. The opinion poll is supposed to represent the views of the people in the entire country. Manufacturers of canned carbonated drinks take samples to determine if a 16 ounce can contains 16 ounces of carbonated drink.

From the sample data, we can calculate a statistic. A statistic is a number that represents a property of the sample. For example, if we consider one math class to be a sample of the population of all math classes, then the average number of points earned by students in that one math class at the end of the term is an example of a statistic. The statistic is an estimate of a population parameter. A parameter is a number that is a property of the population. Since we considered all math classes to be the population, then the average number of points earned per student over all the math classes is an example of a parameter.

One of the main concerns in the field of statistics is how accurately a statistic estimates a parameter. The accuracy really depends on how well the sample represents the population. The sample must contain the characteristics of the population in order to be a representative sample. We are interested in both the sample statistic and the population parameter in inferential statistics. In a later chapter, we will use the sample statistic to test the validity of the established population parameter.

A variable, notated by capital letters such as X and Y, is a characteristic of interest for each person or thing in a population. Variables may be numerical or categorical. Numerical variables take on values with equal units such as weight in pounds and time in hours. Categorical variables place the person or thing into a category. If we let X equal the number of points earned by one math student at the end of a term, then X is a numerical variable. If we let Y be a person's party affiliation, then some examples of Y include Republican, Democrat, and Independent. Y is a categorical variable. We could do some math with values of X (calculate the average number of points earned, for example), but it makes no sense to do math with values of Y (calculating an average party affiliation makes no sense).

Data are the actual values of the variable. They may be numbers or they may be words. Datum is a single value.

Two words that come up often in statistics are mean and proportion. If you were to take three exams in your math classes and obtain scores of 86, 75, and 92, you would calculate your mean score by adding the three exam scores and dividing by three (your mean score would be 84.3 to one decimal place). If, in your math class, there are 40 students and 22 are men and 18 are women, then the proportion of men students is 22/40 and the proportion of women students is 18/40. Mean and proportion are discussed in more detail in later chapters.

NOTE

The words "mean" and "average" are often used interchangeably. The substitution of one word for the other is common practice. The technical term is "arithmetic mean," and "average" is technically a center location. However, in practice among non-statisticians, "average" is commonly accepted for "arithmetic mean."

Question 1.75

1.75

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. Identify the type of data obtained from each question used in this survey: qualitative, quantitative discrete, or quantitative continuous. a. Do you have any health problems that prevent you from doing any of the things people your age can normally do? b. During the past 30 days, for about how many days did poor health keep you from doing your usual activities? c. In the last seven days, on how many days did you exercise for 30 minutes or more? d. Do you have health insurance coverage?

Question 1.76

1.76

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. a. 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. b. What effect does the low response rate have on the reliability of the sample? c. Are these problems examples of sampling error or nonsampling error? d. 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?

Question 1.77

1.77

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 Section 1.2 could explain this connection?

Question 1.78

1.78

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?” 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?

Question 1.79

1.79

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

Question 1.80

1.80

Fifty part-time students were asked how many courses they were taking this term. The (incomplete) results are shown below: a. Fill in the blanks in Table 1.33. b. What percent of students take exactly two courses? c. What percent of students take one or two courses?

Question 1.81

1.81

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 1.34. a. Fill in the blanks in Table 1.34. b. What percent of adults flossed six times per week? c. What percent flossed at most three times per week?

Question 1.82

1.82

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; 4; 5; 10. Table 1.35 was produced. a. Fix the errors in Table 1.35. Also, explain how someone might have arrived at the incorrect number(s). b. Explain what is wrong with this statement: “47 percent of the people surveyed have lived in the U.S. for 5 years.” c. Fix the statement in b to make it correct. d. What fraction of the people surveyed have lived in the U.S. five or seven years? e. What fraction of the people surveyed have lived in the U.S. at most 12 years? f. What fraction of the people surveyed have lived in the U.S. fewer than 12 years? g. What fraction of the people surveyed have lived in the U.S. from five to 20 years, inclusive?

Question 1.83

1.83

How much time does it take to travel to work? Table 1.36 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.

Question 1.84

1.84

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 1.37 shows the ages of the chief executive officers for the first 60 ranked firms. a. What is the frequency for CEO ages between 54 and 65? b. What percentage of CEOs are 65 years or older? c. What is the relative frequency of ages under 50? d. What is the cumulative relative frequency for CEOs younger than 55? e. Which graph shows the relative frequency and which shows the cumulative relative frequency?

Question 1.85

1.85

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

A

0.0768

B

0.0659

C

0.2601

D

Not enough information to calculate

Question 1.86

1.86

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

A

0.3480

B

0.9231

C

0.2601

D

0.3370

Question 1.87

1.87

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.

Question 1.88

1.88

An advertisement for Acme Investments displays the two graphs in Figure 1.14 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?

Question 1.89

1.89

The graph in Figure 1.15 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?

Question 1.90

1.90

Seven hundred and seventy-one distance learning students at Long Beach City College responded to surveys in the 2010-11 academic year. Highlights of the summary report are listed in Table 1.39. a. What percent of the students surveyed do not have a computer at home? b. About how many students in the survey live at least 16 miles from campus? c. If the same survey were done at Great Basin College in Elko, Nevada, do you think the percentages would be the same? Why?

Question 1.91

1.91

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 back ordered. A college newspaper reporter is investigating textbook availability at online retailers. He decides to investigate one textbook for 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 that addresses the following issues: 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.

AIDS patients.

AIDS patients sampled from researcher A and researcher B.

The average length of time (in months) AIDS patients live after treatment.

The average length of time (in months) AIDS patients from the sample live after treatment.

X = the length of time (in months) AIDS patients live after treatment

the houses in the neighborhood around the park in San Antonio, Texas.

a. 0.5242

b. 0.03%

c. 6.86%

d. 823,088/823,856

e. quantitative discrete

f. quantitative continuous

g. In both years, underwater earthquakes produced massive tsunamis.

values for X, such as 3, 4, 11, and so on

Answers will vary. Sample answer: One reason may be the average age of the individuals in the two samples. Or, perhaps the drug affects men and women differently. If the ratio of men and women aren't the same in both sample groups, then the data would differ.

No, we do not have enough information to make such a claim.

Since the treatment is not the same the data might be different unless neither treatment has an effect.

Take a simple random sample from each group. One way is by assigning a number to each patient and using a random number generator to randomly select patients.

He has used a simple random sample method.

This would be convenience sampling and is not random.

The researchers are studying different groups, so there will be some variation in the data.

Yes, the sample size of 150 would be large enough to reflect a population of one school.

There are many school-aged children and young adults in the United States, and the study was done at only one school, so the sample size is not large enough to reflect the population.

Even though the specific data support each researcher’s conclusions, the different results suggest that more data need to be collected before the researchers can reach a conclusion.

Yes, people who play games more might be more likely to participate, since they would want the gift card more than a student who does not play video games. This would leave out many students who do not play games at all and skew the data.

There is not enough information given to judge if either one is correct or incorrect.

The second study is more reliable, because the company would be interested in showing results that favored a higher rate of improvement from patients using their software. The data may be skewed; however, the American Medical Association is not concerned with the success of the software and so should be objective.

The software program seems to work because the second study shows that more patients improve while using the software than not. Even though the difference is not as large as that in the first study, the results from the second study are likely more reliable and still show improvement.

No, the data suggest the two are correlated, but more studies need to be done to prove that using the software causes improvement in stroke patients.

Yes, because we cannot tell if the improvement was due to the software or the exercise; the data is confounded, and a reliable conclusion cannot be drawn. New studies should be performed.

Yes, 1,000 represents 20% of the population and should be representative, if the population of the sample is chosen at random.

No, even though the sample is large enough, the fact that the sample consists of volunteers makes it a self-selected sample, which is not reliable.

No, the question is creating undue influence by adding the word "delicious" to describe Brand X. The wording may influence responses.

No, even though the sample is a large portion of the population, two responses are not enough to justify any conclusions. Because the population is so small, it would be better to include everyone in the population to get the most accurate data.

Yes, there will most likely be a degree of variation between any two studies, even if they are set up and run the same way. Each study may be affected differently by unknown factors such as location, mood of the subjects, or time of year.

a. ordinal

b. interval

c. nominal

d. nominal

e. ratio

f. ordinal

g. nominal

h. interval

i. ratio

j. interval

k. ratio

l. ordinal

a. Inmates may not feel comfortable refusing participation, or may feel obligated to take advantage of the promised benefits. They may not feel truly free to refuse participation.

b. Parents can provide consent on behalf of their children, but children are not competent to provide consent for themselves.

c. All risks and benefits must be clearly outlined. Study participants must be informed of relevant aspects of the study in order to give appropriate consent.

a. The population is all of the clients of the fitness center.

b. A sample of the clients that use the fitness center for a given week.

c. The average amount of time that all clients exercise in one week.

d. The average amount of time that a sample of clients exercises in one week.

e. The amount of time that a client exercises in one week.

f. Examples are: 2 hours, 5 hours, and 7.5 hours

a. all children who take ski or snowboard lessons

b. a group of these children

c. the population mean age of children who take their first snowboard lesson

d. the sample mean age of children who take their first snowboard lesson

e. X = the age of one child who takes his or her first ski or snowboard lesson

f. values for X, such as 3, 7, and so on

a. the cardiologist’s patients

b. a group of the cardiologist’s patients

c. the mean recovery period of all of the cardiologist’s patients

d. the mean recovery period of the group of the cardiologist’s patients

e. X = the mean recovery period of one patient

f. values for X, such as 10 days, 14 days, 20 days, and so on

a. the clients of the insurance companies

b. a group of the clients

c. the mean health costs of the clients

d. the mean health costs of the sample

e. X = the health costs of one client

f. values for X, such as 34, 9, 82, and so on

a. all voters in the politician’s district

b. a random selection of voters in the politician’s district

c. the proportion of voters in this district who think this politician is doing a good job

d. the proportion of voters in this district who think this politician is doing a good job in the sample

e. X = the number of voters in the district who think this politician is doing a good job

f. Yes, he is doing a good job. No, he is not doing a good job.

a. all the clients of this counselor

b. a group of clients of this marriage counselor

c. the proportion of all her clients who stay married

d. the proportion of the sample of the counselor’s clients who stay married

e. X = the number of couples who stay married

f. yes, no

a. all voters (in a certain geographic area)

b. a random selection of all the voters

c. the proportion of voters who are interested in this particular cause

d. the proportion of voters who are interested in this particular cause in the sample

e. X = the number of voters who are interested in this particular cause

f. yes, no

a. all people (maybe in a certain geographic area, such as the United States)

b. a group of the people

c. the proportion of all people who will buy the product

d. the proportion of the sample who will buy the product

e. X = the number of people who will buy it

quantitative discrete, 150

quantitative continuous, 19.2%

qualitative, Oakland A’s

quantitative continuous, 7.2 minutes

quantitative discrete, 11,234 students

qualitative, Dancing with the Stars

qualitative, Crest

quantitative continuous, 8.32 miles

quantitative continuous, 47.3 years

quantitative discrete, three

a. 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.

b. 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.

Answers will vary. Sample Answer: Randomly choose 25 colleges in the state. Use all statistics classes from each of the chosen colleges in the sample. This can be done by listing all the colleges together with a two-digit number starting with 00 then 01, etc. The list of colleges can be found on Wikipedia.http://en.wikipedia.org/wiki/List_of_colleges_and_universities_in_California

Use a random number generator to pick 25 colleges.

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.

Answers will vary. Sample Answer: Not all people have a listed phone number. Many people hang up or do not respond to phone surveys.

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.

Ask everyone to include their age then take a random sample from the data. Include in the report how the survey was conducted and why the results may not be accurate.

a. Yes, in polling, samples that are from 1,200 to 1,500 observations are considered large enough and good enough if the survey is random and is well done.

b. We do not have enough information to decide if this is a random sample from the U.S. population.

c. No, this is a convenience sample taken from individuals who visited an exhibition in the Angeles Convention Center. This sample is not representative of the U.S. population.

d. It is possible that the two sample statistics, 48% and 66% are larger than the true parameters in the population at large. In any event, no conclusion about the population proportions can be inferred from this convenience sample.

a. The country was in the middle of the Great Depression and many people could not afford these “luxury” items and therefore not able to be included in the survey.

b. Samples that are too small can lead to sampling bias.

c. sampling error

d. stratified

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.

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 wording in a specific way to generate a specific response. Self-Funded or Self-Interest Studies: This question was generated to support one person’s claim and it was designed to get the answer that the person desires.

a. Possible reasons: increased use of caller id, decreased use of landlines, increased use of private numbers, voicemail, privacy managers, hectic nature of personal schedules, decreased willingness to be interviewed.

b. 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.

a.

b. 30%

c. 90%

a.

b. 5.00%

c. 93.33%

a. The Frequencies for 15 and 20 should both be two and the Relative Frequencies should both be 2/19. The mistake could be due to copying the data down wrong. The Cumulative Relative Frequency for five years should be 0.4737. The mistake is due to calculating the Relative Frequency instead of the Cumulative Relative Frequency. The Cumulative Relative Frequency for 15 years should be 0.8947

b. The 47% is the Cumulative Relative Frequency, not the Relative Frequency.

c. 47% of the people surveyed have lived in the U.S. for five years or less.

d. 5/19

e. 15/19

f. 13/19

g.  13/19

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.

a. 26 (This is the count of CEOs in the 55 to 59 and 60 to 64 categories.)

b. 12% (number of CEOs age 65 or older ÷ total number of CEOs)

c. 14/60; 0.23; 23%

d. 0.45

e. Graph A represents the cumulative relative frequency, and Graph B shows the relative frequency.

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

The graphs do not show scales of values. We do not know the period of time each graph represents; they may show data from different years. We also do not know if the vertical scales on each graph are equivalent. The scales may have been adjusted to exaggerate or minimize trends. There is no reliable information to be gleaned from these graphs, and setting them up as examples of performance is misleading.

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.

a. 4%

b. 13%

c. Not necessarily. Long beach City is the seventh largest in California the college has an enrollment of approximately 27,000 students. On the other hand, Great Basin College has its campuses in rural northeastern Nevada, and its enrollment of about 3,500 students.

Answers will vary. Sample answer: The sample is not representative of the population of all college textbooks. Two reasons why it is not representative are that he only sampled seven subjects and he only investigated one textbook in each subject. There are several possible sources of bias in the study. The seven subjects that he investigated are all in mathematics and the sciences; there are many subjects in the humanities, social sciences, and other subject areas, (for example: literature, art, history, psychology, sociology, business) that he did not investigate at all. It may be that different subject areas exhibit different patterns of textbook availability, but his sample would not detect such results.

He also looked only at the most popular textbook in each of the subjects he investigated. The availability of the most popular textbooks may differ from the availability of other textbooks in one of two ways:

• the most popular textbooks may be more readily available online, because more new copies are printed, and more students nationwide are selling back their used copies OR
• the most popular textbooks may be harder to find available online, because more student demand exhausts the supply more quickly.

In reality, many college students do not use the most popular textbook in their subject, and this study gives no useful information about the situation for those less popular textbooks.

He could improve this study by:

• expanding the selection of subjects he investigates so that it is more representative of all subjects studied by college students, and
• expanding the selection of textbooks he investigates within each subject to include a mixed representation of both the most popular and less popular textbooks.