Chapter 1 Wrap Up
Concept Check
Section Reviews
1.1 Introduction to Statistics and Key Terms
The mathematical theory of statistics is easier to learn when you know the language. This module presents important terms that will be used throughout the text.
1.2 Data Basics
Some calculations generate numbers that are artificially precise. It is not necessary to report a value to eight decimal places when the measures that generated that value were only accurate to the nearest tenth. Round off your final answer to one more decimal place than was present in the original data. This means that if you have data measured to the nearest tenth of a unit, report the final statistic to the nearest hundredth.
In addition to rounding your answers, you can measure your data using the following four levels of measurement: nominal, ordinal, interval, and ratio.
When organizing data, it is important to know how many times a value appears. How many statistics students study five hours or more for an exam? What percent of families on our block own two pets? Frequency, relative frequency, and cumulative relative frequency are measures that answer questions like these.
1.3 Data Collection and Observational Studies
In summary, making causal conclusions based on observational data can be treacherous and is not recommended. Thus, observational studies are generally only sufficient to show associations or form hypotheses that we later check using controlled experiments which we will discuss in the next section.
1.4 Designed Experiments
A poorly designed study will not produce reliable data. There are certain key components that must be included in every experiment. To eliminate lurking variables, subjects must be assigned randomly to different treatment groups. One of the groups must act as a control group, demonstrating what happens when the active treatment is not applied. Participants in the control group receive a placebo treatment that looks exactly like the active treatments but cannot influence the response variable. To preserve the integrity of the placebo, both researchers and subjects may be blinded. When a study is designed properly, the only difference between treatment groups is the one imposed by the researcher. Therefore, when groups respond differently to different treatments, the difference must be due to the influence of the explanatory variable.
“An ethics problem arises when you are considering an action that benefits you or some cause you support, hurts or reduces benefits to others, and violates some rule,” Andrew Gelman. Ethical violations in statistics are not always easy to spot. Professional associations and federal agencies post guidelines for proper conduct. It is important that you learn basic statistical procedures so that you can recognize proper data analysis.
1.5 Sampling Techniques and Ethics
Data are individual items of information that come from a population or sample. Data may be classified as qualitative (categorical), quantitative continuous, or quantitative discrete.
Because it is not practical to measure the entire population in a study, researchers use samples to represent the population. A random sample is a representative group from the population chosen by using a method that gives each individual in the population an equal chance of being included in the sample. Random sampling methods include simple random sampling, stratified sampling, cluster sampling, and systematic sampling. Convenience sampling is a nonrandom method of choosing a sample that often produces biased data.
Samples that contain different individuals result in different data. This is true even when the samples are well-chosen and representative of the population. When properly selected, larger samples model the population more closely than smaller samples. There are many different potential problems that can affect the reliability of a sample. Statistical data needs to be critically analyzed, not simply accepted.
Key Terms
Try to define the terms below on your own. Scroll over any term to check your response!
1.1 Introduction to Statistics and Key Terms
- Data analysis process
- Descriptive statistics
- Inferential statistics
- Probability
- Population
- Parameters
- Sample
- Statistic
- Individuals
- Variable
- Values
- Data
1.2 Data Basics
- Data
- Population
- Sample
- Qualitative (categorical)
- Quantitative (numerical)
- Discrete
- Continuous
- Nominal scale level
- Ordinal scale level
- Interval scale level
- Ratio scale level
- Variation
- Data analysis
1.3 Data Collection and Observational Studies
- Explanatory variable
- Response variable
- Data
- Anecdotal evidence
- Observational studies
- Designed (controlled) experiment
- Associations
- Confounding (lurking, conditional) variable
- Prospective study
- Retrospective study
- Cohort study
- Longitudinal study
- Cross-sectional study
- Case-control study
1.4 Designed Experiments
- Observational study
- Controlled (designed) experiments
- Explanatory variable
- Response variable
- Treatments
- Experimental unit
- Repeated measures
- Control group
- Placebo
- Blinding
- Double-blind
- Factors
- Levels
- Treatment combinations (interactions)
- Completely randomized
- Block design
- Matched pairs design
1.5 Sampling Techniques and Ethics
- Sample
- Simple random sample (SRS)
- Stratified sampling
- Cluster sampling
- Systematic sampling
- Sampling bias
- Sampling variability
- Convenience sampling
Extra Practice
1.1 Introduction to Statistics and Key Terms
1. Determine what the key terms refer to in the following study. We want to know the average (mean) amount of money first year college students spend at ABC College on school supplies that do not include books. We randomly surveyed 100 first year students at the college. Three of those students spent $150, $200, and $225, respectively.
2. Determine what the key terms refer to in 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, respectively.
3. Determine what the key terms refer to in the following study.
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 an automobile crash on test dummies. Here is the criterion they used:
| Speed at which Cars Crashed | Location of “driver” (i.e. dummies) |
| 35 miles/hour | Front Seat |
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]
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 the number in the sample who have been involved in a malpractice lawsuit.
6. Studies are often done by pharmaceutical companies 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 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 (in months) are collected.
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 what the key terms refer to in the example for Researcher A.
7. For each of the following 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.
a. A fitness center is interested in the mean amount of time a client exercises in the center each week.
- Solution: The population is all of the clients of the fitness center. A sample of the clients that use the fitness center for a given week. The average amount of time that all clients exercise in one week. The average amount of time that a sample of clients exercises in one week. The amount of time that a client exercises in one week. Examples are: 2 hours, 5 hours, and 7.5 hours
b. 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.
- Solution:
- all children who take ski or snowboard lessons
- a group of these children
- the population mean age of children who take their first snowboard lesson
- the sample mean age of children who take their first snowboard lesson
- X = the age of one child who takes his or her first ski or snowboard lesson
- values for X, such as 3, 7, and so on
c. A cardiologist is interested in the mean recovery period of her patients who have had heart attacks.
- Solution: the cardiologist’s patients, a group of the cardiologist’s patients, the mean recovery period of all of the cardiologist’s patients, the mean recovery period of the group of the cardiologist’s patients, X = the mean recovery period of one patient values for X, such as 10 days, 14 days, 20 days, and so on
d. Insurance companies are interested in the mean health costs each year of their clients, so that they can determine the costs of health insurance.
- Solutions:
- the clients of the insurance companies
- a group of the clients
- the mean health costs of the clients
- the mean health costs of the sample
- X = the health costs of one client
- values for X, such as 34, 9, 82, and so on
e. A politician is interested in the proportion of voters in his district who think he is doing a good job.
- Solutions: all voters in the politician’s district, a random selection of voters in the politician’s district, the proportion of voters in this district who think this politician is doing a good job, the proportion of voters in this district who think this politician is doing a good job in the sample, X = the number of voters in the district who think this politician is doing a good job, Yes, he is doing a good job. No, he is not doing a good job.
f. A marriage counselor is interested in the proportion of clients she counsels who stay married.
- Solutions:
- all the clients of this counselor
- a group of clients of this marriage counselor
- the proportion of all her clients who stay married
- the proportion of the sample of the counselor’s clients who stay married
- X = the number of couples who stay married
- yes, no
g. Political pollsters may be interested in the proportion of people who will vote for a particular cause.
- Solutions: all voters (in a certain geographic area), a random selection of all the voters, the proportion of voters who are interested in this particular cause, the proportion of voters who are interested in this particular cause in the sample, X = the number of voters who are interested in this particular cause, yes, no
h. A marketing company is interested in the proportion of people who will buy a particular product.
- Solutions:
- all people (maybe in a certain geographic area, such as the United States)
- a group of the people
- the proportion of all people who will buy the product
- the proportion of the sample who will buy the product
- X = the number of people who will buy it
- buy, not buy
a. What is the population she is interested in?
- all Lake Tahoe Community College students
- all Lake Tahoe Community College English students
- all Lake Tahoe Community College students in her classes
- all Lake Tahoe Community College math students
- Solution: d
b. 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:
- variable.
- population.
- statistic.
- data.
- Solution: a
c. The instructor’s sample produces a mean number of days absent of 3.5 days. This value is an example of a:
- parameter.
- data.
- statistic.
- variable.
- Solution: c
9. In a survey of 100 stocks on NASDAQ, the average percent increase for the past year was 9% for NASDAQ stocks.
a. The “average increase” for all NASDAQ stocks is the:
- population
- statistic
- parameter
- sample
- variable
b. All of the NASDAQ stocks are the:
- population
- statistics
- parameter
- sample
- variable
c. Nine percent is the:
- population
- statistics
- parameter
- sample
- variable
d. The 100 NASDAQ stocks in the survey are the:
- population
- statistic
- parameter
- sample
- variable
e. The percent increase for one stock in the survey is the:
- population
- statistic
- parameter
- sample
- variable
f. Would the data collected be qualitative, quantitative discrete, or quantitative continuous?
1.2 Data Basics
- Solution: qualitative (categorical) data
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)?
- Solution: quantitative discrete data
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, 4.3. Notice that backpacks carrying three books can have different weights. What type of data is this?
- Solution: quantitative continuous data
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. Indicate whether quantitative data are continuous or discrete.
1.3 Data Collection and Observational Studies + Designed Experiments
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]
2. A researcher wants to study the effects of birth order on personality.
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?
Describe the explanatory and response variables in the study.
What are the treatments?
What should you consider when selecting participants?
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?
Identify any lurking variables that could interfere with this study.
How can blinding be used in this study?
4. Identify any issues with the following studies
- Inmates in a correctional facility are offered good behavior credit in return for participation in a study.
- A research study is designed to investigate a new children’s allergy medication.
- 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 Techniques and Ethics
1. Determine whether or not the following samples are representative.
a. To find the average GPA of all students in a university, use all honor students at the university as the sample.
b. To find out the most popular cereal among young people under the age of ten, stand outside a large supermarket for three hours and speak to every twentieth child under age ten who enters the supermarket.
c. To find the average annual income of all adults in the United States, sample U.S. 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 U.S. congressman in the cluster.
d. 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.
e. 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).
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 (each quiz counts 10 points) for an elementary statistics class. Use the random number generator to generate different types of samples from the data.
| #1 | #2 | #3 | #4 | #5 | #6 |
|---|---|---|---|---|---|
| 5 | 7 | 10 | 9 | 8 | 3 |
| 10 | 5 | 9 | 8 | 7 | 6 |
| 9 | 10 | 8 | 6 | 7 | 9 |
| 9 | 10 | 10 | 9 | 8 | 9 |
| 7 | 8 | 9 | 5 | 7 | 4 |
| 9 | 9 | 9 | 10 | 8 | 7 |
| 7 | 7 | 10 | 9 | 8 | 8 |
| 8 | 8 | 9 | 10 | 8 | 8 |
| 9 | 7 | 8 | 7 | 7 | 8 |
| 8 | 8 | 10 | 9 | 8 | 7 |
a. Create a stratified sample by column. Pick three quiz scores randomly from each column.
b. Create a cluster sample by picking two of the columns. Use the column numbers: one through six.
c. Create a simple random sample of 15 quiz scores.
d. 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 and taking every fifth senior citizen on the list, for 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.
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
6. What type of data is this?
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.
The population is ______________________
8. The following figure contains the total number of deaths worldwide as a result of earthquakes from 2000 to 2012. [3]
| Year | Total Number of Deaths |
|---|---|
| 2000 | 231 |
| 2001 | 21,357 |
| 2002 | 11,685 |
| 2003 | 33,819 |
| 2004 | 228,802 |
| 2005 | 88,003 |
| 2006 | 6,605 |
| 2007 | 712 |
| 2008 | 88,011 |
| 2009 | 1,790 |
| 2010 | 320,120 |
| 2011 | 21,953 |
| 2012 | 768 |
| Total | 823,856 |
Use figure above to answer the following questions.
- What is the proportion of deaths between 2007 and 2012?
- What percent of deaths occurred before 2001?
- What is the percent of deaths that occurred in 2003 or after 2010?
- What is the fraction of deaths that happened before 2012?
- What kind of data is the number of deaths?
- Earthquakes are quantified according to the amount of energy they produce (examples are 2.1, 5.0, 6.7). What type of data is that?
- 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).
a. A group of test subjects is divided into twelve groups; then four of the groups are chosen at random.
b. A market researcher polls every tenth person who walks into a store.
c. The first 50 people who walk into a sporting event are polled on their television preferences.
d. A computer generates 100 random numbers, and 100 people whose names correspond with the numbers on the list are chosen.
10. Studies are often done by pharmaceutical companies 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 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 (in months) are collected.
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
a. Determine what the key term data refers to in the above example for Researcher A.
-
-
- values for X, such as 3, 4, 11, and so on
-
b. List two reasons why the data may differ.
c. Can you tell if one researcher is correct and the other one is incorrect? Why?
d. Would you expect the data to be identical? Why or why not?
e. Suggest at least two methods the researchers might use to gather random data.
f. 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?
g. Suppose that the second researcher conducted his survey by choosing 40 patients he knew. What sampling method would that researcher have used? What concerns would you have about this data set, 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. They collect the following data.
| Hours Played per Week | Frequency | Relative Frequency | Cumulative Relative Frequency |
|---|---|---|---|
| 0–2 | 26 | 0.17 | 0.17 |
| 2–4 | 30 | 0.20 | 0.37 |
| 4–6 | 49 | 0.33 | 0.70 |
| 6–8 | 25 | 0.17 | 0.87 |
| 8–10 | 12 | 0.08 | 0.95 |
| 10–12 | 8 | 0.05 | 1 |
| Hours Played per Week | Frequency | Relative Frequency | Cumulative Relative Frequency |
|---|---|---|---|
| 0–2 | 48 | 0.32 | 0.32 |
| 2–4 | 51 | 0.34 | 0.66 |
| 4–6 | 24 | 0.16 | 0.82 |
| 6–8 | 12 | 0.08 | 0.90 |
| 8–10 | 11 | 0.07 | 0.97 |
| 10–12 | 4 | 0.03 | 1 |
a. Give a reason why the data may differ.
-
-
- Solution: The researchers are studying different groups, so there will be some variation in the data.
-
b. Would the sample size be large enough if the population is the students in the school?
-
-
- Solution: Yes, the sample size of 150 would be large enough to reflect a population of one school.
-
c. Would the sample size be large enough if the population is school-aged children and young adults in the United States?
-
-
- Solution: 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. –>
-
d. 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?
-
-
- Solution: 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.
-
e. As part of a way to reward students for participating in the survey, the researchers gave each student a gift card to a video game store. Would this affect the data if students knew about the award before the study?
-
-
- Solution: 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.
-
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 1.18. The second study collected the data in Figure 1.19.
| Group | Showed improvement | No improvement | Deterioration |
|---|---|---|---|
| Used program | 142 | 43 | 15 |
| Did not use program | 72 | 110 | 18 |
| Group | Showed improvement | No improvement | Deterioration |
|---|---|---|---|
| Used program | 105 | 74 | 19 |
| Did not use program | 89 | 99 | 12 |
a. Given what you know, which study is correct?
-
-
- There is not enough information given to judge if either one is correct or incorrect.
-
b. 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?
-
-
- Solution: 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. –>
-
c. Both groups that performed the study concluded that the software works. Is this accurate?
-
-
- 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.
-
d. The company takes the two studies as proof that their software causes mental improvement in stroke patients. Is this a fair statement?
-
-
- Solution: 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.
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e. 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?
-
-
- 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.
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f. Is a sample size of 1,000 a reliable measure for a population of 5,000?
-
-
- Solution: Yes, 1,000 represents 20% of the population and should be representative, if the population of the sample is chosen at random.
-
g. Is a sample of 500 volunteers a reliable measure for a population of 2,500?
-
-
- 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.
-
h. 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?
-
-
- Solution: No, the question is creating undue influence by adding the word “delicious” to describe Brand X. The wording may influence responses.
-
i. Is a sample size of two representative of a population of five?
-
-
- 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.
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j. Is it possible for two experiments to be well run with similar sample sizes to get different data?
-
-
- Solution: 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.
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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 the data.
a. number of tickets sold to a concert
-
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- Solution: quantitative discrete, 150
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b. percent of body fat
-
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- Solution: quantitative continuous, 19.2%
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c. favorite baseball team
-
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- Solution: qualitative, Oakland A’s
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d. time in line to buy groceries
-
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- Solution: quantitative continuous, 7.2 minutes
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e. number of students enrolled at Evergreen Valley College
-
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- Solution: quantitative discrete, 11,234 students
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f. most-watched television show
-
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- Solution: qualitative, The Voice
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g. brand of toothpaste
-
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- Solution: qualitative, Crest
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h. distance to the closest movie theatre
-
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- Solution: quantitative continuous, 8.32 miles
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i. age of executives in Fortune 500 companies
-
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- Solution: quantitative continuous, 47.3 years
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j. number of competing computer spreadsheet software packages
-
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- Solution: quantitative discrete, three
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14. 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 Norfolk. 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.
a. “Number of times per week” is what type of data?
-
-
- Solution: quantitative discrete
-
b. “Duration (amount of time)” is what type of data?
-
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- Solution: quantitative continuous
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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. 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.
a. Using complete sentences, list three things wrong with the way the survey was conducted.
-
-
- Solution:
- 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.
- Solution:
-
b. Using complete sentences, list three ways that you would improve the survey if it were to be repeated.
-
-
- Solution:
- 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.
- Solution:
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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. Make the description detailed.
- Solution: Answers will vary. Sample Answer: Randomly choose 15 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. Use a random number generator to pick 15 colleges.
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. Make the description detailed.
- Solution: 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.
18. List some practical difficulties involved in getting accurate results from a telephone survey.
- Solution: Answers will vary. Sample Answer: Many people live in different areas than their area code. Many people hang up or do not respond to phone surveys.
19. List some practical difficulties involved in getting accurate results from a mailed survey.
- Solution: 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.
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?
- Solution: stratified sampling
- Solution: systematic
22. Name the sampling method used in each of the following situations:
a. 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.
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- Solution: convenience
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b. 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.
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- Solution: cluster
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c. 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.
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- Solution: stratified
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d. 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.
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- Solution: systematic
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e. 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.
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- Solution: simple random
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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. Also, 66% of those surveyed considered themselves relatively savvy computer users.
a. Do you consider the sample size large enough for a study of this type? Why or why not?
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- Solution: 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.
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b. 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?
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- Solution: We do not have enough information to decide if this is a random sample from the U.S. population.
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c. 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.” With this additional information, do you feel that all demographic and ethnic groups were equally represented at the event? Why or why not?
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- Solution: 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.
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d. With the additional information, comment on how accurately you think the sample statistics reflect the population parameters.
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- Solution: 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.
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24. The 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. [4]
a. Do you have any health problems that prevent you from doing any of the things people your age can normally do?
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- Solution: qualitative
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b. During the past 30 days, for about how many days did poor health keep you from doing your usual activities?
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- Solution: quantitative discrete
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c. In the last seven days, on how many days did you exercise for 30 minutes or more?
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- Solution: quantitative discrete
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d. Do you have health insurance coverage?
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- Solution: qualitative
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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.
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- Solution: 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.
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b. What effect does the low response rate have on the reliability of the sample?
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- Solution: Samples that are too small can lead to sampling bias.
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c. Are these problems examples of sampling error or nonsampling error?
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- Solution: sampling error
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d. 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. Quota sampling is an example of which sampling method described in this module? [6]
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- Solution: stratified
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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 discussed 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 responded to this question. Each participant answered “NO!”
Which of the potential problems with samples discussed in this module could explain this connection?
- Solution: 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.
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]
a. What are some reasons for the decline in response rate over the past decade?
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- 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
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b. Explain why researchers are concerned with the impact of the declining response rate on public opinion polls.
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- 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.
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29. 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 Figure 1.20.
| Have computer at home | 96% |
| Unable to come to campus for classes | 65% |
| Age 41 or over | 24% |
| Would like LBCC to offer more DL courses | 95% |
| Took DL classes due to a disability | 17% |
| Live at least 16 miles from campus | 13% |
| Took DL courses to fulfill transfer requirements | 71% |
a. What percent of the students surveyed do not have a computer at home?
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- Solution: 4%
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b. About how many students in the survey live at least 16 miles from campus?
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- Solution: 13%
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c. If the same survey were done at Great Basin College in Elko, Nevada, do you think the percentages would be the same? Why?
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- Solution: 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.
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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 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.
- 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.
References
Figures
Figure 1.9: Kindred Grey via Virginia Tech (2020). “Other Guy’s Investments.” CC BY-SA 4.0. Retrieved from https://commons.wikimedia.org/wiki/File:Other_Guy%27s_Investments.png . Adaptation of Figure 1.14 from OpenStax Introductory Statistics (2013) (CC BY 4.0). Retrieved from https://openstax.org/books/introductory-statistics/pages/1-homework#fs-idp81996208
Figure 1.10: Kindred Grey via Virginia Tech (2020). “Acme’s Investments.” CC BY-SA 4.0. Retrieved from https://commons.wikimedia.org/wiki/File:Acme%27s_Investments.png . Adaptation of Figure 1.14 from OpenStax Introductory Statistics (2013) (CC BY 4.0). Retrieved from https://openstax.org/books/introductory-statistics/pages/1-homework#fs-idp81996208
Figure 1.11: Kindred Grey (2020). “Airline Complaints 2.” CC BY-SA 4.0. Retrieved from https://commons.wikimedia.org/wiki/File:Airline_Complaints_2.png
Text
The Data and Story Library, http://lib.stat.cmu.edu/DASL/Stories/CrashTestDummies.html (accessed May 1, 2013).
“Vitamin E and Health,” Nutrition Source, Harvard School of Public Health, http://www.hsph.harvard.edu/nutritionsource/vitamin-e/ (accessed May 1, 2013).
Stan Reents. “Don’t Underestimate the Power of Suggestion,” athleteinme.com, http://www.athleteinme.com/ArticleView.aspx?id=1053 (accessed May 1, 2013).
Ankita Mehta. “Daily Dose of Aspiring Helps Reduce Heart Attacks: Study,” International Business Times, July 21, 2011. Also available online at http://www.ibtimes.com/daily-dose-aspirin-helps-reduce-heart-attacks-study-300443 (accessed May 1, 2013).
The Data and Story Library, http://lib.stat.cmu.edu/DASL/Stories/ScentsandLearning.html (accessed May 1, 2013).
M.L. Jacskon et al., “Cognitive Components of Simulated Driving Performance: Sleep Loss effect and Predictors,” Accident Analysis and Prevention Journal, Jan no. 50 (2013), http://www.ncbi.nlm.nih.gov/pubmed/22721550 (accessed May 1, 2013).
“Earthquake Information by Year,” U.S. Geological Survey. http://earthquake.usgs.gov/earthquakes/eqarchives/year/ (accessed May 1, 2013).
“Fatality Analysis Report Systems (FARS) Encyclopedia,” National Highway Traffic and Safety Administration. http://www-fars.nhtsa.dot.gov/Main/index.aspx (accessed May 1, 2013).
Meier, Paul. “The biggest public health experiment ever: the 1954 field trial of the Salk poliomyelitis vaccine.” Statistics: a guide to the unknown. San Francisco: Holden-Day (1972): 2-13.
Data from www.businessweek.com (accessed May 1, 2013).
Data from www.forbes.com (accessed May 1, 2013).
“America’s Best Small Companies,” http://www.forbes.com/best-small-companies/list/ (accessed May 1, 2013).
U.S. Department of Health and Human Services, Code of Federal Regulations Title 45 Public Welfare Department of Health and Human Services Part 46 Protection of Human Subjects revised January 15, 2009. Section 46.111:Criteria for IRB Approval of Research.
“April 2013 Air Travel Consumer Report,” U.S. Department of Transportation, April 11 (2013), http://www.dot.gov/airconsumer/april-2013-air-travel-consumer-report (accessed May 1, 2013).
Lori Alden, “Statistics can be Misleading,” econoclass.com, http://www.econoclass.com/misleadingstats.html (accessed May 1, 2013).
Maria de los A. Medina, “Ethics in Statistics,” Based on “Building an Ethics Module for Business, Science, and Engineering Students” by Jose A. Cruz-Cruz and William Frey, Connexions, http://cnx.org/content/m15555/latest/ (accessed May 1, 2013).
McClung, M. Collins, D. “Because I know it will!”: placebo effects of an ergogenic aid on athletic performance. Journal of Sport & Exercise Psychology. 2007 Jun. 29(3):382-94. Web. April 30, 2013.
Yudhijit Bhattacharjee, “The Mind of a Con Man,” Magazine, New York Times, April 26, 2013. Available online at: http://www.nytimes.com/2013/04/28/magazine/diederik-stapels-audacious-academic-fraud.html?src=dayp&_r=2& (accessed May 1, 2013).
“Flawed Science: The Fraudulent Research Practices of Social Psychologist Diederik Stapel,” Tillburg University, November 28, 2012, http://www.tilburguniversity.edu/upload/064a10cd-bce5-4385-b9ff-05b840caeae6_120695_Rapp_nov_2012_UK_web.pdf (accessed May 1, 2013).
Andrew Gelman, “Open Data and Open Methods,” Ethics and Statistics, http://www.stat.columbia.edu/~gelman/research/published/ChanceEthics1.pdf (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
- The Data and Story Library, http://lib.stat.cmu.edu/DASL/Stories/CrashTestDummies.html (accessed May 1, 2013). ↵
- Ankita Mehta. “Daily Dose of Aspiring Helps Reduce Heart Attacks: Study,” International Business Times, July 21, 2011. Also available online at http://www.ibtimes.com/daily-dose-aspirin-helps-reduce-heart-attacks-study-300443 (accessed May 1, 2013) ↵
- “Earthquake Information by Year,” U.S. Geological Survey. http://earthquake.usgs.gov/earthquakes/eqarchives/year/ (accessed May 1, 2013) ↵
- Gallup-Healthways Well-Being Index. http://www.gallup.com/poll/146822/gallup-healthways-index-questions.aspx (accessed May 1, 2013). ↵
- Dominic Lusinchi, “’President’ Landon and the 1936 Literary Digest Poll: Were Automobile and Telephone Owners to Blame?” Social Science History 36, no. 1: 23-54 (2012), https://www.jstor.org/stable/41407095 (accessed January 26, 2021). ↵
- Data from http://www.bookofodds.com/Relationships-Society/Articles/A0374-How-George-Gallup-Picked-the-President ↵
- “United States: Uniform Crime Report – State Statistics from 1960–2011.” The Disaster Center. Available online at http://www.disastercenter.com/crime/ (accessed May 2, 2013). ↵
- lastbaldeagle. 2013. On Tax Day, House to Call for Firing Federal Workers Who Owe Back Taxes. Opinion poll posted online at: http://www.youpolls.com/details.aspx?id=12328 (accessed May 1, 2013). ↵
- Scott Keeter et al., “Gauging the Impact of Growing Nonresponse on Estimates from a National RDD Telephone Survey,” Public Opinion Quarterly 70 no. 5 (2006), http://poq.oxfordjournals.org/content/70/5/759.full (http://poq.oxfordjournals.org/content/70/5/759.full) (accessed May 1, 2013). ↵
- Frequently Asked Questions, Pew Research Center for the People & the Press, http://www.people-press.org/ methodology/frequently-asked-questions/#dont-you-have-trouble-getting-people-to-answer-your-polls (accessed May 1, 2013). ↵
Process of collecting, organizing, and analyzing data
Methods of organizing, summarizing, and presenting data
The facet of statistics dealing with using a sample to generalize (or infer) about the population
The study of randomness; a number between zero and one, inclusive, that gives the likelihood that a specific event will occur
The whole group of individuals who can be studied to answer a research question
A number that is used to represent a population characteristic and can only be calculated as the result of a census
A subset of the population studied
A number calculated from a sample
The person, animal, item, thing, place, etc. that we collect information about
A characteristic of interest for each person or object in a population
Possible observations of the variable
Actual values (numbers or words) that are collected from the variables of interest
Data that describes qualities, or puts individuals into categories
Numerical data with a mathematical context
A random variable that produces discrete data
A random variable (RV) whose outcomes are measured as an uncountable, infinite, number of values
Categorical data where the the categories have no natural, intuitive, or obvious order
Categorical data where the the categories have a natural or intuitive order
Quantitative data where the difference or gap between values is meaningful
Quantitative data where the difference or gap between values is meaningful AND has a true 0 value
The level of variability or dispersion of a dataset; also commonly known as variation/variability
The independent variable in an experiment; the value controlled by researchers
The dependent variable in an experiment; the value that is measured for change at the end of an experiment
Evidence that is based on personal testimony and collected informally
Data collection where no variables are manipulated
Data collection where variables are manipulated in a controlled setting
A relationship between variables
A variable that has an effect on a study even though it is neither an explanatory variable nor a response variable
Collecting information as events unfold
Collecting or using data after events have taken place
Longitudinal study where a group of people (typically having a common factor) are studied and data is collected for a purpose
Collecting data multiple times on the same individuals, usually at fixed increments, over a period of time
Data collection on a population at one point in time (often prospective)
A study that compares a group that has a certain characteristic to a group that does not, often a retrospective study for rare conditions
Type of experiment where variables are manipulated; data is collected in a controlled setting
Different values or components of the explanatory variable applied in an experiment
Any individual or object to be measured
When an individual goes through a single treatment more than once
A group in a randomized experiment that receives no (or an inactive) treatment but is otherwise managed exactly as the other groups
An inactive treatment that has no real effect on the explanatory variable
Not telling participants which treatment they are receiving
The act of blinding both the subjects of an experiment and the researchers who work with the subjects
Variables in an experiment
Certain values of variables in an experiment
Combinations of levels of variables in an experiment
Dividing participants into treatment groups randomly
Grouping individuals based on a variable into "blocks" and then randomizing cases within each block to the treatment groups
Very similar individuals (or even the same individual) receive two different two treatments (or treatment vs. control) then the difference in results are compared
Each member of the population is equally likely to be chosen for a sample of a given sample size and each sample is equally likely to be chosen
Dividing a population into groups (strata), and then using simple random sampling to identify a proportionate number of individuals from each
A method of sampling where the population has already sorted itself into groups (clusters), randomly selecting a cluster, and using every individual in the chosen cluster as the sample
Using some sort of pattern or probability based method for choosing your sample
Bias resulting from all members of the population not being equally likely to be selected
The idea that samples from the same population can yield different results
Selecting individuals that are easily accessible and may result in biased data