Chapter 1

Step 1 — Recall the definition: The population is the entire collection of individuals or objects we want to draw a conclusion about.

Step 2 — Apply it to Researcher A's study: The goal is to learn about the recovery time for all patients who take the new drug for this respiratory virus, not just the 40 who were followed.

Answer: The population is all patients with the respiratory virus who take the new drug (every person to whom the treatment could apply).

Step 1 — Recall the definition: A sample is the subset of the population that is actually observed and measured.

Step 2 — Apply it to Researcher A's study: Researcher A did not measure every drug-taking patient — only the specific 40 patients he followed from treatment start to relief.

Answer: The sample is the 40 patients with the respiratory virus that Researcher A followed.

Step 1 — Recall the definition: A parameter is a numerical summary that describes a characteristic of the entire population.

Step 2 — Apply it to Researcher A's study: The quantity of interest is the average recovery time. Computed over the whole population, that mean is a parameter.

Answer: The parameter is the (unknown) mean number of days to symptom relief for the entire population of patients taking the drug.

Step 1 — Recall the definition: A statistic is a numerical summary computed from the sample; it estimates the corresponding parameter.

Step 2 — Apply it to Researcher A's study: The mean recovery time calculated from Researcher A's 40 patients is a statistic.

Answer: The statistic is the mean number of days to symptom relief computed from Researcher A's sample of 40 patients.

Step 1 — Recall the definition: A variable is the characteristic being measured on each member of the sample; its value changes from patient to patient.

Step 2 — Apply it to Researcher A's study: The measured characteristic is the length of time, in days, from the start of treatment until symptoms are relieved.

Answer: The variable is \(X =\) the number of days from the start of treatment until the patient's symptoms are alleviated.

Step 1 — Apply the definitions: Population = the entire group of interest; sample = the subset actually studied; parameter = a numerical fact about the population; statistic = a numerical fact computed from the sample; variable = the characteristic being measured; data = the actual values recorded.

Step 2 — Map them to this scenario (mean weekly exercise time):

a) Population: all clients of the fitness center.

b) Sample: the particular group of clients whose exercise time is actually measured.

c) Parameter: the mean weekly exercise time of all clients.

d) Statistic: the mean weekly exercise time of the sampled clients.

e) Variable: \(X =\) the amount of time (e.g., hours) a client exercises in the center per week.

f) Data: the individual recorded times, e.g., 2 hours, 5 hours, 3.5 hours, ….

Answer: a) all fitness-center clients; b) the clients actually surveyed; c) mean weekly exercise time of all clients; d) mean weekly exercise time of the sampled clients; e) weekly exercise time per client; f) the specific time values recorded (e.g., 2, 5, 3.5 hours).

Step 1 — Recall the six definitions: population (whole group), sample (subset studied), parameter (population number), statistic (sample number), variable (measured characteristic), data (the recorded values).

Step 2 — Map them to this scenario (mean age of first ski/snowboard lesson):

a) Population: all children who take ski and snowboard lessons (at the resorts of interest).

b) Sample: the particular group of children whose first-lesson ages are recorded.

c) Parameter: the mean first-lesson age of all such children.

d) Statistic: the mean first-lesson age of the sampled children.

e) Variable: \(X =\) the age at which a child takes their first ski/snowboard lesson.

f) Data: the individual ages, e.g., 4 years, 6 years, 5 years, ….

Answer: a) all children taking ski/snowboard lessons; b) the children actually surveyed; c) mean first-lesson age of all such children; d) mean first-lesson age of the sample; e) age at first lesson; f) the specific ages recorded (e.g., 4, 6, 5 years).

Step 1 — Recall the six definitions: population, sample, parameter (population number), statistic (sample number), variable (measured trait), data (recorded values).

Step 2 — Map them to this scenario (mean recovery period after heart attack):

a) Population: all of the cardiologist's patients who have had heart attacks.

b) Sample: the particular group of those patients whose recovery time is measured.

c) Parameter: the mean recovery period of all the cardiologist's heart-attack patients.

d) Statistic: the mean recovery period of the sampled patients.

e) Variable: \(X =\) the length of the recovery period (e.g., in days) for a heart-attack patient.

f) Data: the individual recovery times, e.g., 30 days, 45 days, 28 days, ….

Answer: a) all of the cardiologist's heart-attack patients; b) the patients actually measured; c) mean recovery period of all such patients; d) mean recovery period of the sample; e) recovery-period length per patient; f) the specific recovery times recorded (e.g., 30, 45, 28 days).

Step 1 — Recall the six definitions: population, sample, parameter (population number), statistic (sample number), variable (measured trait), data (recorded values).

Step 2 — Map them to this scenario (mean annual health costs of clients):

a) Population: all clients of the insurance company.

b) Sample: the particular group of clients whose annual health costs are examined.

c) Parameter: the mean annual health cost of all clients.

d) Statistic: the mean annual health cost of the sampled clients.

e) Variable: \(X =\) the annual health cost of a client (in dollars).

f) Data: the individual cost values, e.g., $2,000, $5,500, $1,200, ….

Answer: a) all clients of the insurance company; b) the clients actually sampled; c) mean annual health cost of all clients; d) mean annual health cost of the sample; e) annual health cost per client; f) the specific cost values recorded (e.g., $2,000, $5,500, $1,200).

Step 1 — Recall the six definitions: population, sample, parameter (population number), statistic (sample number), variable (measured trait), data (recorded values).

Step 2 — Map them to this scenario (proportion of voters who approve):

a) Population: all voters in the politician's district.

b) Sample: the particular group of voters who are actually surveyed.

c) Parameter: the proportion of all district voters who think the politician is doing a good job.

d) Statistic: the proportion of the sampled voters who think so.

e) Variable: \(X =\) whether a given voter thinks the politician is doing a good job (yes/no).

f) Data: the individual yes/no responses, e.g., yes, no, yes, yes, no, ….

Answer: a) all voters in the district; b) the voters actually surveyed; c) the true proportion of all district voters who approve; d) the proportion of surveyed voters who approve; e) each voter's approval (yes/no); f) the specific yes/no responses recorded.

Step 1 — Identify the population: The population is the entire group the counselor wants to draw conclusions about — here, all of the counselor's clients (those they counsel).

Step 2 — Identify the parameter of interest: The proportion of all clients who stay married. This is the population proportion, the numerical fact the counselor wants to learn.

Step 3 — Identify the variable and data: The variable is whether a given client stays married (yes/no). The data are the actual yes/no responses recorded for the clients studied.

Answer: Population = all clients the counselor counsels; parameter = the (true) proportion of those clients who stay married; variable = whether a client stays married; data = the yes/no marital-status outcomes collected.

Step 1 — Identify the population: The population is everyone whose vote the pollsters care about — all eligible voters (the people who could vote on the cause).

Step 2 — Identify the parameter of interest: The proportion of all those people who will vote for the particular cause. This is the population proportion.

Step 3 — Identify the variable and data: The variable is whether a given person will vote for the cause (yes/no). The data are the recorded yes/no voting intentions.

Answer: Population = all eligible voters; parameter = the proportion who will vote for the cause; variable = whether a person votes for the cause; data = the yes/no responses collected.

Step 1 — Identify the population: The population is all the people the marketing company is interested in — the potential buyers of the product.

Step 2 — Identify the parameter of interest: The proportion of all those people who will buy the particular product. This is the population proportion.

Step 3 — Identify the variable and data: The variable is whether a given person will buy the product (yes/no). The data are the recorded yes/no purchase responses.

Answer: Population = all potential buyers (people the company is interested in); parameter = the proportion who will buy the product; variable = whether a person buys the product; data = the yes/no purchase outcomes collected.

Step 1 — Read the question carefully: The instructor studies the mean days absent of math students, so the population must be exactly the group she wants conclusions about.

Step 2 — Eliminate options that are too broad or wrong: "All Lake Tahoe Community College students" (a) and "English students" (b) are not her group; "students in her classes" (c) is only a subset she might sample, not the full target group.

Step 3 — Select the matching group: The full target group is all Lake Tahoe Community College math students.

Answer: d) all Lake Tahoe Community College math students.

Step 1 — Recognize what \(X\) measures: \(X\) = the number of days a math student is absent. Its value changes from student to student, so it is a characteristic that varies across individuals.

Step 2 — Match to the term: A characteristic that takes different values for different members of the population is a variable. It is not the population (the students themselves), not a statistic (a single computed summary), and not the data (the actual recorded values).

Answer: a) variable.

Step 1 — Identify the source of the number: The value 3.5 days is the mean computed from the instructor's sample, not from the entire population of math students.

Step 2 — Match to the term: A numerical summary computed from a sample is a statistic. (A parameter would summarize the whole population; data are the raw values; a variable is the characteristic measured.)

Answer: c) statistic.

Step 1 — Classify the variable: "Number of times per week" is a count, so it is quantitative (numeric), not categorical.

Step 2 — Discrete or continuous? A count can only take whole-number values (0, 1, 2, …); you cannot use the park 2.5 times in a counting sense, so it is discrete.

Answer: b) quantitative discrete.

Step 1 — Identify the selection rule: Starting from one randomly chosen first house, the interviewer then takes every eighth house — a fixed interval down the list.

Step 2 — Match to a sampling method: Choosing every \(k\)th element from an ordered list (here \(k = 8\)) is the definition of systematic sampling.

Answer: b) systematic.

Step 1 — Classify the variable: "Duration (amount of time)" is measured numerically, so it is quantitative.

Step 2 — Discrete or continuous? Time can take any value in a range (e.g., 12.4 minutes, 12.43 minutes); measurements are not restricted to whole numbers, so it is continuous.

Answer: c) quantitative continuous.

Step 1 — Classify the variable: A house color (red, blue, white, …) is a label/category, not a number.

Step 2 — Confirm the type: Since the values are names rather than measurements or counts, the data are qualitative.

Answer: a) qualitative (categorical).

Step 1 — Recall the definition: The population is every individual the study aims to describe.

Step 2 — Apply it to the park study: The researchers want to characterize the people who use this San Antonio park, so the population is all of those users.

Answer: The population is all the people (residents) who use that local park in San Antonio.

Step 1 — Identify the selection rule: Twelve groups are formed, then 4 entire groups are chosen at random and everyone in those groups is used.

Step 2 — Match to a sampling method: Sampling whole pre-existing groups (clusters) rather than individuals is cluster sampling.

Answer: Cluster sampling.

Step 1 — Identify the selection rule: The researcher polls every tenth person entering the store — a fixed interval applied to an ordered stream of people.

Step 2 — Match to a sampling method: Selecting every \(k\)th element (\(k = 10\)) is systematic sampling.

Answer: Systematic sampling.

Step 1 — Identify the selection rule: The first 50 people through the door are polled simply because they are easiest to reach.

Step 2 — Match to a sampling method: Choosing whoever is most readily available, with no random mechanism, is convenience sampling.

Answer: Convenience sampling.

Step 1 — Identify the selection rule: A computer produces 100 random numbers, and the 100 people matching those numbers are selected; every person on the list has an equal chance.

Step 2 — Match to a sampling method: Equal-probability selection of individuals by a random device is simple random sampling.

Answer: Simple random sampling.

Step 1 — Set up the bins: Each row counts how many of the \(n = 40\) survival times fall in that interval (lower bound exclusive, upper bound inclusive). Relative frequency \(= \dfrac{\text{frequency}}{40}\), and cumulative relative frequency is the running total of relative frequencies.

Step 2 — Tally Researcher A (Table 1.2.8): Counting Researcher A's 40 values into each bin gives frequencies \(2, 5, 9, 5, 7, 7, 2, 3\) (sum \(= 40\)).

Step 3 — Tally Researcher B (Table 1.2.9): Counting Researcher B's 40 values gives frequencies \(3, 2, 11, 8, 6, 6, 4\) (sum \(= 40\)).

Step 4 — Check: Each frequency column sums to 40 and each final cumulative relative frequency equals 1.000, confirming the tables are complete.

Answer: The completed Researcher A and Researcher B frequency tables are shown above; frequencies are \(2,5,9,5,7,7,2,3\) for A and \(3,2,11,8,6,6,4\) for B, both summing to 40 with cumulative relative frequency reaching 1.000.

Step 1 — Recall the definition: Data are the actual recorded values of the variable for the members of the sample.

Step 2 — Apply it to Researcher A: For Researcher A, the data are the 40 measured survival lengths (3, 4, 11, 15, … , 34 days).

Answer: The data are the set of 40 recorded survival times (in days) for Researcher A's patients.

Step 1 — Think about the source of variation: The two researchers followed different sets of 40 patients, and individual patients respond differently to the drug.

Step 2 — List two reasons: (1) The two samples consist of different individuals, so natural patient-to-patient variation produces different values; (2) differences in conditions or methodology — for example, how the patients were selected, regional or demographic differences, severity of illness at start, or measurement/recording differences — can shift the results.

Answer: The data may differ because (1) each researcher sampled a different group of patients (sampling variability), and (2) factors such as how patients were chosen, their differing health conditions/severity, or measurement differences vary between the two studies.

Step 1 — Interpret "correct": Each researcher honestly recorded the survival times for their own randomly chosen sample.

Step 2 — Explain why neither is wrong: Two different random samples from the same population are expected to give different results; that difference is sampling variability, not error. Neither set of data is "correct" or "incorrect" — both are valid samples.

Answer: No. Both researchers can be reporting accurate data; the differences simply reflect natural sampling variability between two different groups of patients, so one cannot be declared right and the other wrong.

Step 1 — Consider what identical data would require: Identical data would mean both researchers happened to draw the same 40 individuals (or the same exact values), which is essentially impossible for two separate random samples.

Step 2 — Explain: Because the samples contain different patients and patients vary in how they respond, the recorded values will differ.

Answer: No, we would not expect the data to be identical. The two researchers sampled different patients, and individual responses vary, so the two data sets will naturally differ even though they describe the same population.

Step 1 — Recall what "random" requires: A random method gives every patient a known, fair chance of being selected, removing the researcher's personal choice.

Step 2 — Suggest two methods: (1) Simple random sampling — assign every eligible patient a number and use a random number generator or table to pick 40; (2) systematic sampling — order all eligible patients and select every \(k\)th one starting from a random start. (Stratified or cluster random sampling are also acceptable.)

Answer: Two workable methods are simple random sampling (number every patient and draw 40 with a random number generator) and systematic sampling (pick every \(k\)th patient from an ordered list after a random start).

Step 1 — Identify the two stages: First a state is chosen at random; then 40 patients are chosen at random from within that one state.

Step 2 — Match to a sampling method: Selecting whole geographic group(s) at random and then sampling within them is cluster sampling (the state is the cluster).

Answer: Cluster sampling.

Step 1 — Identify the selection rule: The researcher used 40 patients he already knew, choosing them because they were easy to access rather than by any random mechanism.

Step 2 — Match to a sampling method: Selecting readily available, familiar subjects is convenience sampling.

Step 3 — State the concern: A convenience sample is likely biased and not representative of the whole population — patients the researcher knows may share characteristics (location, health habits, demographics) that differ from typical patients, so conclusions drawn from this sample may not generalize.

Answer: Convenience sampling. The main concern is bias: because the patients were not chosen randomly but because the researcher knew them, the sample may not represent the broader population, making the results unreliable for general conclusions.

Step 1 — Recognize the source of variability: Two researchers each surveyed a different set of respondents, so the two data sets are independent samples drawn from the population. Random sampling means no two samples will be identical.

Step 2 — Identify a concrete reason: Differences arise from sampling variability — each sample contains different individuals whose video-game habits differ. Other contributing factors include how the question was worded, when and where each survey was administered, and ordinary measurement/reporting error.

Answer: The data differ because each researcher surveyed a different random sample of people; sampling variability (plus possible differences in question wording, timing, and respondent honesty) produces different results even when both studies are run correctly.

Step 1 — Compare sample size to population size: "The students in the school" is a relatively small, well-defined population. A survey covering a sizeable fraction of that population gives a representative picture.

Step 2 — Judge adequacy: Because the population is limited to one school, even a few hundred respondents would represent a large share of the whole, so the sample is large enough to draw reliable conclusions about that school.

Answer: Yes. For a population as small and specific as the students in one school, the sample is large enough to be reliable.

Step 1 — Reconsider the population size: "School-aged children and young adults in the United States" is an enormous and highly diverse population numbering in the tens of millions.

Step 2 — Judge adequacy: A sample taken at a single school cannot capture the geographic, economic, and cultural diversity of the whole country, and it is a tiny fraction of that population, so it is neither large enough nor representative.

Answer: No. The sample is far too small and not representative enough to support conclusions about all school-aged children and young adults in the United States.

Step 1 — Examine what each researcher is claiming: Each conclusion describes the typical play time for that researcher's own sample. The two samples are different sets of people.

Step 2 — Decide whether one must be wrong: Because the two samples differ, both conclusions can be valid descriptions of their respective samples. Neither is necessarily "correct" for the whole population — the difference reflects sampling variability, not error.

Answer: Neither is provably correct. Each conclusion accurately describes that researcher's own sample; the discrepancy is due to sampling variability, so we cannot say one researcher is right and the other wrong.

Step 1 — Consider how an incentive changes behavior: If students knew about the gift card in advance, the reward could influence who chose to participate and how they responded.

Step 2 — Identify the bias: Advance knowledge of a prize attracts self-selected participants (people interested in video games / gift cards) and may encourage rushed or exaggerated answers, biasing the data.

Answer: Yes. Knowing about the reward beforehand could bias the data through self-selection and influenced responses; a reward given only after participation, with no prior knowledge, would be far less likely to affect the results.

Step 1 — Recall that independent samples can differ: The two studies followed different groups of 200 stroke patients, so different counts are expected from sampling variability alone.

Step 2 — Decide whether one is "correct": Neither study is inherently right or wrong. Both are legitimate observations of different samples; without more information we cannot label one correct.

Answer: Neither can be declared correct. Both are valid studies of different patient groups, and the differing results reflect sampling variability rather than one study being wrong.

Step 1 — Look for a conflict of interest: The first study was run by the company that makes the software, giving it a financial stake in a favorable result. The second was run by the American Medical Association, an independent professional body.

Step 2 — Judge reliability: An independent organization with no financial interest in the outcome is less prone to bias, so its findings carry more credibility.

Answer: The second study (American Medical Association) is more reliable, because it was conducted by an independent party without a financial stake in the result, whereas the company-run study has a conflict of interest.

Step 1 — Compare the two data sets: In the first study, 142 of 200 program users improved (71%) versus 72 of 200 non-users (36%). In the second study, 105 of 200 users improved (52.5%) versus 89 of 200 non-users (44.5%).

Step 2 — Assess the shared conclusion: Both studies show users improving at a higher rate than non-users, so the qualitative claim "the software is associated with improvement" is consistent across both, even though the size of the effect differs.

Answer: It is accurate to say both studies found a higher improvement rate among software users than non-users. However, the second (independent) study shows a much smaller advantage, so the strength of the effect is uncertain — the direction of the conclusion is consistent, but the magnitude is not.

Step 1 — Separate association from causation: The studies are observational: patients were grouped by whether they used the software, not randomly assigned. Observed improvement could stem from confounding factors rather than the software itself.

Step 2 — Apply the causation rule: A correlation between software use and improvement does not by itself prove the software causes improvement. Only a properly controlled, randomized experiment can establish causation.

Answer: No. The studies are observational and show only an association, not causation. Without a randomized controlled experiment, the company cannot fairly claim the software causes mental improvement.

Step 1 — Identify the new confounding variable: Program users were also in an exercise program while non-users were not. Exercise itself can aid stroke recovery, so it is confounded with software use.

Step 2 — Reassess the conclusion from 1.2.25: Because the two groups differ in more than just software use, any improvement could be due to the exercise rather than the software. The groups are no longer comparable.

Answer: Yes, it changes the validity. Exercise is a confounding variable that differs between the groups, so the observed improvement cannot be attributed to the software alone; the conclusions from Problem 41 are weakened.

Step 1 — Compute the sampling fraction: The sample is \(1{,}000\) out of a population of \(5{,}000\): $$\frac{1{,}000}{5{,}000} = 0.20 = 20%.$$

Step 2 — Judge reliability: A randomly chosen sample of 1,000 is a large absolute size and covers 20% of the population, which is more than adequate to estimate population characteristics reliably (provided it is randomly selected).

Answer: Yes. A sample of 1,000 (20% of the population of 5,000), if randomly selected, is large enough to be a reliable measure.

Step 1 — Compute the sampling fraction: The sample is \(500\) out of a population of \(2{,}500\): $$\frac{500}{2{,}500} = 0.20 = 20%.$$

Step 2 — Account for self-selection: Twenty percent is a substantial fraction, but these are volunteers. Volunteer (self-selected) samples are prone to bias because volunteers may differ systematically from the general population.

Answer: The size (500, or 20% of 2,500) is adequate, but because the participants are self-selected volunteers, the sample may be biased and is therefore not necessarily reliable. A randomly selected sample of the same size would be preferable.

Step 1 — Read the wording critically: The question describes Brand X as having a "delicious taste" while describing Brand Y's taste neutrally. The adjective primes the respondent toward Brand X.

Step 2 — Identify the flaw: This is a leading question — its phrasing biases the answer rather than letting the respondent choose freely.

Answer: No, it is not a fair question. The word "delicious" applied only to Brand X makes it a leading (biased) question; a fair version would describe both brands neutrally, e.g., "Do you prefer the taste of Brand X or the taste of Brand Y?"

Step 1 — Compute the sampling fraction: A sample of \(2\) from a population of \(5\): $$\frac{2}{5} = 0.40 = 40%.$$

Step 2 — Judge representativeness: Although 40% is a large fraction, the absolute sample size of two is far too small to capture the variation in the population; just one unusual individual would dominate the result.

Answer: No. A sample of size two is too small to be representative of a population of five, despite being 40% of it — there is not enough data to reflect the population's variability.

Step 1 — Recall the role of randomness: Even with careful design and equal sample sizes, two experiments draw different random samples from the population.

Step 2 — Conclude: Random sampling guarantees that two independent, well-run experiments will almost always produce somewhat different data. This is expected sampling variability, not a sign of error.

Answer: Yes. Two well-run experiments with similar sample sizes can easily produce different data because they sample different individuals; this is normal sampling variability.

Step 1 — Decide qualitative vs. quantitative: Number of tickets sold is a count, so it is numerical — quantitative.

Step 2 — Decide discrete vs. continuous: Counts take only whole-number values (you cannot sell 412.5 tickets), so the data are quantitative discrete.

Answer: Quantitative discrete. Example: 412 tickets sold to a particular concert.

Step 1 — Decide qualitative vs. quantitative: Percent of body fat is a measured numerical quantity, so it is quantitative.

Step 2 — Decide discrete vs. continuous: It is measured on a scale and can take any value in a range (including fractions), so the data are quantitative continuous.

Answer: Quantitative continuous. Example: a body-fat measurement of 18.6%.

Step 1 — Decide qualitative vs. quantitative: A favorite baseball team is a name or category, not a number, so it is qualitative (categorical).

Step 2 — Confirm: There is no meaningful arithmetic on team names, which confirms the data are qualitative.

Answer: Qualitative. Example: the San Francisco Giants.

Step 1 — Decide qualitative vs. quantitative: Time waiting in line is a measured numerical quantity, so it is quantitative.

Step 2 — Decide discrete vs. continuous: Time can take any value in a range (e.g., 4.3 minutes), so the data are quantitative continuous.

Answer: Quantitative continuous. Example: 4.3 minutes spent in line.

Step 1 — Decide qualitative vs. quantitative: Number of students enrolled is a count, so it is numerical — quantitative.

Step 2 — Decide discrete vs. continuous: Counts are whole numbers (you cannot enroll a fraction of a student), so the data are quantitative discrete.

Answer: Quantitative discrete. Example: 9,800 students enrolled.

Step 1 — Decide qualitative vs. quantitative: A most-watched television show is a title/category, not a number, so it is qualitative (categorical).

Step 2 — Confirm: Show titles cannot be averaged or otherwise computed with, confirming the data are qualitative.

Answer: Qualitative. Example: the most-watched show being Jeopardy!

Step 1 — Decide qualitative vs. quantitative: A brand of toothpaste is a name or category, not a number, so it is qualitative (categorical).

Step 2 — Confirm: Brand names support no arithmetic, confirming the data are qualitative.

Answer: Qualitative. Example: the brand Colgate.

Step 1 — Decide qualitative vs. quantitative: Distance is a measured numerical quantity, so it is quantitative.

Step 2 — Decide discrete vs. continuous: Distance can take any value in a range (e.g., 2.7 miles), so the data are quantitative continuous.

Answer: Quantitative continuous. Example: 2.7 miles to the closest movie theatre.

Step 1 — Decide qualitative vs. quantitative: Age is a measured numerical quantity, so it is quantitative.

Step 2 — Decide discrete vs. continuous: Age is measured on a continuous scale (it can take any value, e.g., 52.4 years), so the data are quantitative continuous.

Answer: Quantitative continuous. Example: an executive aged 52.4 years.

Step 1 — Decide qualitative vs. quantitative: Number of competing software packages is a count, so it is numerical — quantitative.

Step 2 — Decide discrete vs. continuous: Counts take only whole-number values, so the data are quantitative discrete.

Answer: Quantitative discrete. Example: 5 competing spreadsheet packages.

Step 1 — Classify the variable: "Number of times per week" a resident uses the park is a count of visits.

Step 2 — Decide the data type: A count takes only whole-number values, so it is numerical and discrete — quantitative discrete. It is not qualitative (it is a number) and not continuous (you cannot use the park 3.5 times in the counting sense).

Answer: b) quantitative discrete.

Step 1 — Classify the variable: "Duration (amount of time)" measures how long each resident uses the park.

Step 2 — Decide the data type: Time is measured on a continuous scale and can take any value in a range (e.g., 1.75 hours), so it is quantitative continuous. It is not qualitative (it is a number) and not discrete (it is not a whole-number count).

Answer: c) quantitative continuous.

Step 1 — Examine how the survey was conducted: The airline samples only six flights, all on one route (Boston to Salt Lake City), all over a single Thanksgiving weekend, then uses that to set safety equipment for all flights.

Step 2 — (a) List three flaws: (1) The sample is far too small — six flights cannot represent the airline's entire schedule. (2) The timing is biased: Thanksgiving weekend is a peak family-travel period, so the number of babies will be unusually high and unrepresentative of typical flights. (3) The single route (Boston–Salt Lake City) is not representative of all routes, which differ in destination, length, and passenger makeup.

Step 3 — (b) List three improvements: (1) Greatly increase the sample size — survey many more flights. (2) Spread the survey across the whole year (and many days of the week) instead of one holiday weekend, so it reflects typical travel. (3) Sample a variety of routes (different origins, destinations, and flight lengths) rather than one city pair.

Answer: (a) Three flaws — sample too small (only 6 flights), biased timing (Thanksgiving overstates the number of babies), and a single unrepresentative route. (b) Three improvements — survey far more flights, spread the survey across the full year and all weekdays, and include many different routes.

Step 1 — Define the goal and population: The goal is the mean number of students per statistics class across all statistics classes in your state; the population is every statistics class offered in the state.

Step 2 — Build a frame and choose a method: Obtain a list of all colleges and universities in the state that offer statistics, and from each obtain a roster of its statistics class sections. This list of sections is the sampling frame.

Step 3 — Draw the sample and measure: Use stratified random sampling — group the schools by type (e.g., community colleges vs. universities) so each type is represented, then randomly select several statistics sections from each group. Record the enrollment of each selected class and average those counts to estimate the statewide mean class size.

Answer: A reasonable method: list all statistics sections at all schools in the state, stratify by school type, randomly select sections within each stratum, record each selected class's enrollment, and average them to estimate the mean number of students per statistics class. (Answers will vary — any clearly described, representative sampling plan is acceptable.)

Step 1 — Define the goal and population: The goal is the mean number of cans of soda drunk per month by students in their twenties at your school; the population is all students at your school who are in their twenties.

Step 2 — Build a frame and choose a method: Use the registrar's enrollment list to identify students aged 20–29. This list is the sampling frame.

Step 3 — Draw the sample and measure: Use simple random sampling — assign each eligible student a number and randomly select a sizable subset. Ask each selected student how many cans of soda they drank in the past month, then average those counts to estimate the monthly mean.

Answer: A reasonable method: get the registrar's list of students in their twenties, randomly select a representative sample, ask each how many cans of soda they drink per month, and average the responses to estimate the mean. (Answers will vary — any clearly described, representative sampling plan is acceptable.)

Step 1 — Think about who can be reached by phone: Coverage is incomplete — not everyone has a phone, some have only cell phones (and numbers may be unlisted or change), so the call list may miss whole groups, causing undercoverage bias.

Step 2 — Think about who responds: Many people do not answer unknown numbers, screen calls, or hang up, producing high nonresponse; those who do answer may differ systematically (e.g., people at home during the day), biasing results.

Step 3 — Think about answer quality: Respondents may give hurried or untruthful answers, misunderstand questions, or be influenced by the interviewer's tone, and language barriers can distort responses.

Answer: Practical difficulties include undercoverage (people without phones or with unlisted/cell-only numbers), high nonresponse (unanswered or screened calls and hang-ups), nonresponse bias (those reachable/willing differ from those not), and inaccurate answers (rushed, untruthful, or misunderstood responses, plus interviewer or language effects). (Answers will vary.)

Step 1 — Think about delivery and coverage: You need current, correct addresses; out-of-date or missing addresses mean some people never receive the survey, causing undercoverage.

Step 2 — Think about who returns it: Mailed surveys have very low response rates — many are discarded unopened — and those who bother to return them often hold stronger opinions, producing nonresponse bias.

Step 3 — Think about answer quality and timing: There is no interviewer to clarify questions, so respondents may misread or skip items; you cannot verify who actually filled it out; and replies trickle in slowly, delaying results.

Answer: Practical difficulties include undercoverage from wrong or missing addresses, very low response rates with nonresponse bias (returners differ from non-returners), no way to clarify confusing questions or verify who answered, and slow, delayed returns. (Answers will vary.)

Step 1 — Identify the obstacles to phone/mail surveys: Phone and mail surveys suffer from low response rates, unanswered or screened calls, undeliverable mail, out-of-date contact lists, and self-selection (only motivated people respond). Naming the problems first tells us what to overcome.

Step 2 — Brainstorm countermeasures: Possible strategies include offering a small incentive for completion, sending advance notices and follow-up reminders, keeping the survey short, calling at varied times of day, providing multiple response channels (web link, prepaid return envelope), training callers to build rapport, and using up-to-date, broadly representative contact lists rather than a single source.

Answer: This is an open-ended discussion problem. A good answer names concrete tactics — incentives, follow-up reminders, short well-worded questions, varied call times, prepaid return postage, and current representative contact lists — that directly counter low response rates and self-selection bias.

Step 1 — Identify the structure of the sampling: The instructor goes to each math class and selects five students at random from within every class. The classes act as naturally occurring groups.

Step 2 — Match to the definition: Selecting a random sample of individuals from within every group is stratified sampling (the strata are the math classes). Cluster sampling would instead pick whole classes and survey everyone in them; here only five per class are taken, from all classes.

Answer: b) stratified sampling

Step 1 — Identify the selection rule: The first house is chosen at random, then every eighth house thereafter is interviewed — selection follows a fixed interval through an ordered list.

Step 2 — Match to the definition: Choosing a random starting point and then taking every \(k\)th element is systematic sampling.

Answer: b) systematic

Step 1 — Classify part (a): The surveyor approaches only the travelers who are easy to reach (seated, awake, not rushing). Taking whoever is conveniently available is convenience sampling.

Step 2 — Classify part (b): Two rows are chosen at random and every student in those rows is called on. Selecting whole naturally occurring groups at random is cluster sampling.

Step 3 — Classify part (c): At each store, 100 customers are selected at random. Each store is a stratum and a random sample is drawn within every one, so this is stratified sampling.

Step 4 — Classify part (d): The librarian records data for every fourth patron. A fixed interval through the stream of patrons is systematic sampling.

Step 5 — Classify part (e): 1,200 phone numbers are chosen at random with no grouping or interval, so every voter has an equal chance — simple random sampling.

Answer: a) convenience; b) cluster; c) stratified; d) systematic; e) simple random.

Step 1 — Compute the average: Add the three quiz scores and divide by 3.

Step 2 — Decide on rounding: A standard convention is to carry one more decimal place than the original data. The scores are whole numbers (zero decimal places), so the average is reported to one decimal place.

Step 3 — Round to one decimal place: The digit after the tenths place is \(3\), so we round down, leaving the tenths digit as \(3\).

Answer: Round to one decimal place; the average is \(8.3\).

Step 1 — Recall the four levels: Nominal = labels/categories with no order; ordinal = ordered categories but uneven/unknown gaps; interval = ordered with equal gaps but no true zero; ratio = ordered with equal gaps and a meaningful zero (so ratios make sense).

Step 2 — Classify part a (brand names): Car brands are just category labels with no natural order, so this is nominal.

Step 3 — Classify part b (finishing places): 1st, 2nd, 3rd have a clear order, but the time gaps between places are not equal, so this is ordinal.

Step 4 — Classify part c (temperatures in °C): Equal-sized degree gaps make differences meaningful, but \(0\,^\circ\text{C}\) is not a true absence of temperature (so ratios are meaningless), making this interval.

Step 5 — Classify part d (weights in pounds): Equal gaps and a true zero (\(0\) lb means no weight) mean ratios are valid (\(10\) lb is twice \(5\) lb), so this is ratio.

Answer: a) nominal; b) ordinal; c) interval; d) ratio.

Step 1 — Identify what nominal data are: Nominal data are names or labels for categories (e.g., eye color, car brand). The numbers sometimes assigned to them (like jersey numbers) are just codes, not quantities.

Step 2 — Explain why arithmetic fails: Because the values carry no order and no magnitude, operations like adding or averaging produce meaningless results — the "average" of category codes does not correspond to any real category. You can only count how many fall in each category.

Step 3 — Give an example not used in this section: Blood type (A, B, AB, O) is nominal data: you can count how many people have each type, but you cannot average blood types.

Answer: Nominal data are unordered category labels, so adding or averaging them is meaningless; you can only tally counts. Example: blood type (A, B, AB, O).

Step 1 — State the shared trait: Both interval and ratio data are quantitative with equal spacing between consecutive values, so differences are meaningful for both.

Step 2 — Identify the distinguishing feature: The difference is the zero point. Interval data have an arbitrary zero (it does not mean "none"); ratio data have a true zero (it means a genuine absence of the quantity).

Step 3 — Apply to temperature (interval): \(0\,^\circ\text{C}\) does not mean "no temperature," so ratios are invalid — \(20\,^\circ\text{C}\) is not twice as hot as \(10\,^\circ\text{C}\). Temperature is interval data.

Step 4 — Apply to weight (ratio): \(0\) pounds means no weight at all, a true zero, so ratios are valid — \(20\) pounds is twice as heavy as \(10\) pounds. Weight is ratio data.

Answer: Interval data (temperature) have equal spacing but no true zero, so ratios are meaningless; ratio data (weight) have equal spacing and a true zero, so statements like "twice as much" are valid.

Step 1 — Tally each data value: Count how many times each number of pets (0–4) appears in the 20 responses.

Check the total: \(6 + 8 + 4 + 2 + 1 = 21\). That is too many — recount.

Step 2 — Recount carefully: The 20 values are 0, 1, 2, 0, 3, 1, 1, 0, 2, 1, 4, 0, 1, 2, 1, 0, 1, 3, 2, 1. Zeros: positions 1, 4, 8, 12, 16 \(\Rightarrow 5\). Ones: positions 2, 6, 7, 10, 13, 15, 17, 20 \(\Rightarrow 8\). Twos: positions 3, 9, 14, 19 \(\Rightarrow 4\). Threes: positions 5, 18 \(\Rightarrow 2\). Fours: position 11 \(\Rightarrow 1\). Total \(5 + 8 + 4 + 2 + 1 = 20\). ✓

Step 3 — Compute relative frequency (frequency \(\div 20\)) and cumulative relative frequency (running sum):

Answer: The completed frequency table is shown above; the cumulative relative frequency reaches \(1.00\), confirming all 20 responses are accounted for.

Step 1 — Locate the relevant row: "Exactly one pet" is the data value \(1\), which has frequency \(8\) in the table from Problem 74.

Step 2 — Divide by the total: Relative frequency is the frequency divided by the total number of households, \(20\).

Answer: The relative frequency of households owning exactly one pet is \(0.40\) (40%).

Step 1 — Translate "two or fewer": This means data values \(0\), \(1\), or \(2\). The cumulative relative frequency column already accumulates these — read the entry for value \(2\).

Step 2 — Read or compute the cumulative relative frequency at value \(2\):

Step 3 — Convert to a percentage:

Answer: 85% of households own two or fewer pets.

Step 1 — Identify the qualifying rows: "67.95 inches or more" covers every interval from 67.95–69.95 upward: 67.95–69.95, 69.95–71.95, 71.95–73.95, and 73.95–75.95.

Step 2 — Add their frequencies out of the 100 players:

Step 3 — Convert to a percentage:

You can check this against the cumulative column: \(1.00 - 0.63 = 0.37\), since \(0.63\) is the cumulative relative frequency up through the 65.95–67.95 row.

Answer: 37% of the heights are 67.95 inches or more.

Step 1 — Find the interval's row: The 9.01–11.03 inch interval is the fourth row of Table 1.3.5.

Step 2 — Read the cumulative relative frequency: The cumulative column sums the relative frequencies of all rows up to and including this one.

Step 3 — Interpret the value: This figure represents the proportion of all towns whose annual rainfall is below the top of this interval.

Answer: The cumulative relative frequency is \(0.72\); it means 72% of the surveyed towns receive less than 11.03 inches of rainfall per year.

Step 1 — Identify the qualifying values: "Fewer than 30 years of service" means 24, 25, 26, 27, 28, and 29 years.

Step 2 — Add their frequencies:

Step 3 — Find the total number of employees: Sum every frequency in Table 1.3.8.

Step 4 — Compute the relative frequency:

Answer: The relative frequency for fewer than 30 years of service is \(\frac{16}{70} \approx 0.229\) (about 22.9%).

Step 1 — Record the total: The 13-year death total is \(823{,}856\), which we use as the denominator throughout.

Step 2 — (a) Proportion, Years 8–13: Add Years 8 through 13: $$712 + 88{,}011 + 1{,}790 + 320{,}120 + 21{,}953 + 768 = 433{,}354.$$ Proportion \(= \dfrac{433{,}354}{823{,}856} \approx 0.526\).

Step 3 — (b) Percent before Year 2: "Before Year 2" means Year 1 only, with \(231\) deaths: \(\dfrac{231}{823{,}856} \times 100 \approx 0.028\%\).

Step 4 — (c) Percent in Year 4 or after Year 11: "After Year 11" means Years 12 and 13. Add Year 4 + Year 12 + Year 13: $$33{,}819 + 21{,}953 + 768 = 56{,}540,\qquad \dfrac{56{,}540}{823{,}856}\times 100 \approx 6.86%.$$

Step 5 — (d) Fraction before Year 13: "Before Year 13" means Years 1–12, i.e., the total minus Year 13: \(823{,}856 - 768 = 823{,}088\). Fraction \(= \dfrac{823{,}088}{823{,}856} \approx 0.999\).

Step 6 — (e) Type of data for number of deaths: A count of deaths is numeric and whole-number-valued, so it is quantitative discrete.

Step 7 — (f) Type of data for earthquake magnitude: Values like 2.1, 5.0, 6.7 are measured on a continuous scale, so magnitude is quantitative continuous.

Step 8 — (g) Explain the spikes: Both spikes reflect very large, deadly earthquakes (and the disasters they triggered). Year 11's \(320{,}120\) is consistent with a catastrophic quake followed by a tsunami, and Year 5's \(228{,}802\) likewise reflects a major quake with a tsunami; densely populated areas and poor building resilience amplify the death toll. (Exact attribution depends on the source's underlying dataset.)

Answer: (a) \(\approx 0.526\); (b) \(\approx 0.028\%\); (c) \(\approx 6.86\%\); (d) \(\approx 0.999\) (i.e., \(\tfrac{823{,}088}{823{,}856}\)); (e) quantitative discrete; (f) quantitative continuous; (g) the Year 5 and Year 11 spikes come from individual catastrophic earthquakes (and associated tsunamis) striking populated, vulnerable regions.

Step 1 — Recall the four scales: Nominal = labels/categories with no order; ordinal = ordered categories without meaningful differences; interval = ordered with meaningful differences but no true zero; ratio = ordered with meaningful differences and a true zero (ratios are meaningful).

Step 2 — Classify each item:

a) Athletic ability (Superior, Average, Above average) — ordered categories → ordinal

b) Baking temperatures in degrees — meaningful differences but no true zero (0° is not "no temperature") → interval

c) Colors of crayons — unordered labels → nominal

d) Social security numbers — identifiers, no order or magnitude → nominal

e) Incomes in dollars — true zero, ratios meaningful → ratio

f) Satisfaction coded 1/2/3 — ordered satisfaction levels → ordinal

g) Political outlook (extreme left … extreme right) — ordered categories → ordinal

h) Time of day on an analog watch — meaningful differences, no absolute zero → interval

i) Distance in miles — true zero, ratios meaningful → ratio

j) Calendar year dates (1066, 1492, …) — ordered with meaningful differences but no true zero → interval

k) Heights of women — true zero, ratios meaningful → ratio

l) Letter grades A, B, C, D, F — ordered categories → ordinal

Answer: a) ordinal; b) interval; c) nominal; d) nominal; e) ratio; f) ordinal; g) ordinal; h) interval; i) ratio; j) interval; k) ratio; l) ordinal.

Step 1 — Part (a), complete the table: There are 50 students. With 30 taking 1 course and 15 taking 2, the 3-course frequency is \(50-30-15=5\). Relative frequencies are frequency \(\div 50\): \(30/50=0.6\), \(15/50=0.3\), \(5/50=0.1\). Cumulative relative frequencies accumulate: \(0.6\), \(0.6+0.3=0.9\), \(0.9+0.1=1.0\).

Step 2 — Part (b), percent taking exactly two: \(0.3 = 30\%\).

Step 3 — Part (c), percent taking one or two: \(0.6+0.3=0.9 = 90\%\) (equivalently the cumulative relative frequency at 2 courses).

Answer: (a) 3-course frequency = 5; rel. freqs 0.6, 0.3, 0.1; cum. rel. freqs 0.6, 0.9, 1.0. (b) 30%. (c) 90%.

Step 1 — Part (a), complete the table: There are 60 adults. The 1-floss relative frequency is \(18/60=0.3000\). The frequency for 3 flosses is \(60-27-18-3-1=11\), so its relative frequency is \(11/60\approx0.1833\). Cumulative relative frequencies accumulate down the column.

The given 0.9333 for the "3" row checks out: \(0.7500+0.1833=0.9333\).

Step 2 — Part (b), percent who flossed six times: \(3/60=0.0500=5\%\).

Step 3 — Part (c), percent who flossed at most three times: This is the cumulative relative frequency at 3: \(0.9333 = 93.33\%\) (i.e. \((27+18+11)/60\)).

Answer: (a) rel. freq for 1 = 0.3000; freq for 3 = 11 (rel. freq 0.1833); cum. rel. freqs 0.4500, 0.7500, 0.9333, 0.9833, 1.0000. (b) 5%. (c) about 93.33%.

Step 1 — Part (a), find and fix the errors: Tally the 19 data values: 0 appears 2, 2 appears 3, 4 appears 1, 5 appears 3, 7 appears 2, 10 appears 2, 12 appears 2, 15 appears 2, 20 appears 2. The table lists the 15 and 20 frequencies as 1 each, but each actually occurs twice. The error likely came from counting each of those values only once instead of scanning the whole list. With the fix, the frequencies sum to 19 and the cumulative relative frequency reaches 1.0000. Corrected last two rows:

Step 2 — Part (b), what's wrong with the "47%" statement: The value 5 occurs 3 times, giving a relative frequency of \(3/19\approx0.158\), about 16% — not 47%. The 0.4737 cumulative figure means 47% lived in the U.S. five years or fewer, not exactly five years. The statement confuses cumulative relative frequency with the relative frequency of a single value.

Step 3 — Part (c), corrected statement: "About 47% of the people surveyed have lived in the U.S. for at most (5 years or fewer)." (Exactly five years applies to about 16%.)

Step 4 — Part (d), five or seven years: Frequencies \(3+2=5\), so \(\frac{5}{19}\).

Step 5 — Part (e), at most 12 years: Values \(\le 12\): \(2+3+1+3+2+2+2=15\), so \(\frac{15}{19}\).

Step 6 — Part (f), fewer than 12 years: Values \(<12\): \(15-2=13\), so \(\frac{13}{19}\).

Step 7 — Part (g), five to 20 inclusive: Values \(5\) through \(20\): \(3+2+2+2+2+2=13\), so \(\frac{13}{19}\).

Answer: (a) The frequencies for 15 and 20 should each be 2 (not 1); the 20-row cumulative becomes 1.0000. (b) It misreads the cumulative 0.4737 as applying to exactly 5 years; only about 16% (\(3/19\)) lived exactly 5 years. (c) "About 47% have lived in the U.S. at most 5 years." (d) \(\frac{5}{19}\). (e) \(\frac{15}{19}\). (f) \(\frac{13}{19}\). (g) \(\frac{13}{19}\).

Step 1 — Add the values: Sum all 50 mean commute times. Adding the table gives a total of \(1173.1\) minutes.

Step 2 — Divide by the count: The mean is $$\bar{x}=\frac{1173.1}{50}=23.462\text{ minutes}.$$

Step 3 — Round appropriately: The data are reported to one decimal place (tenths of a minute), so the mean should be rounded to the same precision: \(23.5\) minutes.

Answer: The mean travel time is about \(23.5\) minutes (\(\bar{x}=23.462\), rounded to one decimal place to match the data).

Step 1 — Part (a), frequency for ages 54 to 65: This spans the 55–59 and 60–64 class intervals (the 50–54 and 65–69 groups fall outside). Their frequencies sum to \(16+10=26\).

Step 2 — Part (b), percent 65 or older: Ages 65–69 and 70–74 give \(6+1=7\) of 60 CEOs: \(7/60\approx0.1167=11.67\%\).

Step 3 — Part (c), relative frequency of ages under 50: Classes 40–44 and 45–49 give \(3+11=14\) of 60: \(14/60\approx0.2333\).

Step 4 — Part (d), cumulative relative frequency younger than 55: Classes 40–44, 45–49, 50–54 give \(3+11+13=27\) of 60: \(27/60=0.45\).

Step 5 — Part (e), identify the graphs: The graph whose bar heights rise and then fall (matching the individual relative frequencies) shows the relative frequency; the graph whose bars only increase, leveling off at 1.0, shows the cumulative relative frequency.

Answer: (a) 26. (b) about 11.67%. (c) about 0.2333. (d) 0.45. (e) The non-decreasing graph that climbs to 1.0 is the cumulative relative frequency; the rise-then-fall graph is the relative frequency.

Step 1 — Identify the category-4 count and total: Table 1.3.15 shows 18 category-4 direct hits out of a total of 273.

Step 2 — Compute the relative frequency: $$\frac{18}{273}\approx0.0659.$$

Answer: b) 0.0659

Step 1 — Identify "at most category 3": Categories 1, 2, and 3 have \(109+72+71=252\) direct hits out of 273.

Step 2 — Compute the relative frequency: $$\frac{252}{273}\approx0.9231.$$

This equals the cumulative relative frequency through category 3.

Answer: b) 0.9231

Step 1 — Identify the role of each variable: The explanatory variable is the factor the experimenter deliberately manipulates or sets, because it is the suspected cause whose effect we want to measure.

Step 2 — Contrast with the response: The response variable is the outcome we measure on each subject, because it is the effect we expect to change when the explanatory variable changes.

Step 3 — State the relationship: In an experiment we change the explanatory variable and watch what happens to the response variable, looking for evidence that the first influences the second.

Answer: The explanatory variable is the input the researcher controls or assigns (the suspected cause); the response variable is the outcome that is measured (the suspected effect). The experiment is designed to see whether changing the explanatory variable produces a change in the response variable.

Step 1 — Find what is manipulated: The researcher decides which fertilizer each plant receives, so the explanatory variable is the type of fertilizer (Fertilizer A vs. Fertilizer B).

Step 2 — Find what is measured: Yield is the outcome recorded for each plant, so the response variable is the tomato yield (e.g., weight or number of tomatoes per plant).

Step 3 — List the treatments: The treatments are the specific levels of the explanatory variable applied to the experimental units: Fertilizer A and Fertilizer B.

Answer: Explanatory variable: type of fertilizer; response variable: tomato yield per plant; treatments: Fertilizer A and Fertilizer B.

Step 1 — Define a lurking variable: A lurking variable is a variable that is not among the explanatory or response variables being studied but that influences the response (and possibly the apparent relationship), because it can create or hide an association we wrongly attribute to the explanatory variable.

Step 2 — Explain the threat: If a lurking variable is distributed unevenly across the treatment groups, differences in the response might be caused by it rather than by the treatment, because the groups differ in more than just the treatment.

Step 3 — Explain random assignment: Randomly assigning subjects to treatment groups tends to spread lurking variables evenly across all groups, because each subject has an equal chance of landing in any group regardless of its hidden characteristics.

Answer: A lurking variable is an outside variable that affects the response but is not part of the study. Random assignment helps because, on average, it balances lurking variables across the treatment groups, so the groups are comparable and any difference in the response can be attributed to the treatment rather than to a confounding background variable.

Step 1 — Define a placebo: A placebo is a fake or inactive treatment (such as a sugar pill or saline injection) that is indistinguishable from the real treatment, given so that subjects cannot tell whether they received the active treatment.

Step 2 — Explain the placebo effect: People often respond simply to the act of being treated, a phenomenon called the placebo effect, so improvement can occur even with an inactive treatment.

Step 3 — Justify the control group: A control group that receives the placebo experiences the same psychological and procedural conditions as the treatment group, because then the only systematic difference between groups is the active ingredient.

Answer: A placebo is an inactive, look-alike treatment. Researchers give a control group a placebo so that both groups experience the placebo effect equally; this isolates the effect of the active treatment, letting them attribute any extra response in the treatment group to the treatment itself rather than to the mere act of being treated.

Step 1 — Describe a blinded experiment: In a single-blind experiment, the subjects do not know which treatment (active or placebo) they are receiving, because knowing could bias their behavior or self-reported response.

Step 2 — Describe a double-blind experiment: In a double-blind experiment, neither the subjects nor the people who interact with them and measure the response know who received which treatment, because that knowledge could bias both the subjects' responses and the researchers' assessments.

Step 3 — State the difference: The two designs differ in how many parties are kept unaware: one party (subjects only) versus two parties (subjects and the researchers/evaluators).

Answer: In a blinded (single-blind) experiment only the subjects are unaware of their treatment assignment. In a double-blind experiment both the subjects and the researchers who administer treatments or measure outcomes are unaware. Double-blinding adds protection against bias introduced by the experimenters, not just the subjects.

Step 1 — Recall the requirement of a randomized experiment: A randomized experiment requires that the experimenter assign the explanatory variable to subjects at random, because random assignment is what allows a causal conclusion.

Step 2 — Examine the explanatory variable here: The explanatory variable is season of birth (winter vs. summer). A person's birth season is fixed at birth and cannot be assigned by a researcher, because we cannot choose or change when someone is born.

Step 3 — Classify the study: Since the explanatory variable can only be observed, not manipulated or randomly assigned, this must be an observational study rather than an experiment.

Answer: Season of birth cannot be randomly assigned to subjects — it is determined by nature and fixed at birth. Because the researcher cannot manipulate or randomize the explanatory variable, the study can only be observational, so it cannot establish causation and is not a randomized experiment.

Step 1 — Recall the structure of confounding: Two variables are confounded when a third variable is associated with both, because that third variable can produce an apparent link between them even if neither causes the other.

Step 2 — Construct an example: Consider the observation that towns with more firefighters per fire tend to have more fire damage. It looks as if firefighters cause damage.

Step 3 — Identify the confounder: The confounding variable is the size or severity of the fire: larger fires both attract more firefighters and cause more damage, creating the misleading association between number of firefighters and amount of damage.

Answer: Example: ice cream sales and drowning deaths are positively associated, but neither causes the other — the confounding variable is hot weather (summer), which independently increases both ice cream sales and the number of people swimming. (Equivalently: firefighters present and fire damage, confounded by fire size.) The key point is that the confounder is linked to both variables and explains away the apparent direct relationship.

Step 1 — Spot the self-selection problem: Participants choose whether to take the vitamin, so this is not random assignment. People who opt in may differ systematically (e.g., be more health-conscious) from those who opt out, which confounds the comparison — any health difference could be due to those background traits rather than the vitamin.

Step 2 — Spot the conflict-of-interest problem: The study is funded by the vitamin company, which has a financial stake in a positive result. This creates a conflict of interest that can bias the design, analysis, or reporting toward favorable findings.

Step 3 — Confirm the two are distinct: The first is a design flaw (lack of randomization / self-selection bias); the second is a source-of-funding/bias issue. They are separate weaknesses.

Answer: Two separate problems: (1) Self-selection — because participants choose whether to take the vitamin, the groups are not randomized and likely differ in lurking variables such as overall health-consciousness, so the vitamin's effect is confounded. (2) Conflict of interest — the funding company benefits from a positive result, which can bias how the study is designed, analyzed, or reported.

Step 1 — Recall the role of an IRB: An Institutional Review Board reviews research involving human participants to ensure it is conducted ethically and legally before it begins.

Step 2 — List the three protections: The IRB ensures (a) the study's potential gains in knowledge are worth the risks to participants — a favorable risk/benefit balance; (b) participants give informed consent — they are told the risks and benefits and agree voluntarily to take part; and (c) participants' identities and data are kept confidential.

Step 3 — Explain why each matters: Each protection guards a participant's legal and ethical rights: avoiding undue harm, preserving autonomy through consent, and protecting privacy.

Answer: An IRB is responsible for ensuring that (1) the knowledge gained justifies the risk to subjects (risk/benefit assessment), (2) participants give informed consent after being told the study's risks and benefits, and (3) participants' personal information and data remain confidential.

Step 1 — Describe the practice: "Stopping as soon as the data prove your point" means watching the results accumulate and halting collection at the moment they happen to favor the desired conclusion.

Step 2 — Explain why it distorts the evidence: Random data naturally wander, so by chance the results will at some point show an apparent effect even when none exists. Choosing to stop exactly at that favorable moment cherry-picks a misleading snapshot, inflating the chance of a false positive far above the stated significance level.

Step 3 — Note that no fabrication is required: Every recorded value can be genuine, yet the selection rule for when to stop biases the analysis, because the decision to stop depends on the data turning out the way the researcher wants.

Answer: It is unethical because the stopping decision is based on the results themselves: random fluctuations guarantee that the data will eventually look favorable by chance, and stopping at that point systematically overstates the evidence and produces misleading, irreproducible conclusions. The dishonesty lies in the biased data-collection rule, not in faking any individual measurement.

Step 1 — Choose a question and define the variables: (Open-ended — answers will vary. A model response follows.) Suppose we test whether a new fertilizer increases tomato yield. The explanatory variable is the fertilizer (new vs. standard); the response variable is the tomato yield (kg per plant).

Step 2 — Describe the population and experimental units: The population is all tomato plants of the chosen variety grown under similar conditions; the experimental units are the individual tomato plants used in the study.

Step 3 — Define and assign treatments: Two treatments: the new fertilizer and the standard fertilizer (control). Plants are randomly assigned to the two groups so other factors are balanced.

Step 4 — Blinding and placebos: A "placebo" fertilizer identical in appearance ensures the researchers measuring yield do not know which plants received the new product (single/double blinding), preventing the power of suggestion from biasing measurement.

Answer: Answers will vary. A complete response names an explanatory and a response variable, identifies the population and the experimental units, specifies the treatments and a random method for assigning them, and explains how blinding and a placebo guard against the power of suggestion. (Model: fertilizer type as explanatory, yield as response, tomato plants as units, random assignment to new vs. placebo fertilizer with blinded measurement.)

Step 1 — Recall the informed-consent requirement: Participants must voluntarily agree to take part with full, undistorted knowledge of the study's nature and risks, free of coercion.

Step 2 — Evaluate each scenario:

a) Offering inmates good-behavior credit for participating is coercive — the incentive pressures a vulnerable population, undermining truly voluntary consent.

b) Studying a children's medication raises the consent-capacity issue: children cannot legally give informed consent, so consent must be obtained from a parent or legal guardian (with the child's assent where possible).

c) Telling participants the drug is "highly promising" while hiding that many will receive placebo or traditional treatment withholds material information and is misleading, so consent is not fully informed.

Answer: a) Coercion — good-behavior credit pressures a captive population, so consent is not voluntary. b) Children cannot give informed consent themselves; parental/guardian consent (and child assent) is required. c) Consent is not truly informed because participants are misled about the drug's promise and not told they may receive a placebo or traditional treatment.

Step 1 — Identify the experimental units and treatments: The 19 professional drivers are the experimental units (subjects). Each is observed under two conditions — normal sleep and 27 hours of total sleep deprivation — which are the two treatments (levels of the explanatory variable, sleep condition).

Step 2 — Identify the design type: Because every driver experiences both treatments, each subject serves as their own control: this is a matched-pairs (repeated-measures) design, a special case of a randomized block design where each subject is a block.

Step 3 — Identify the randomization and response: The order of the two treatments is assigned in random order, which controls for order/learning effects. The response variables are the performance measures, including the driving-simulation results.

Answer: This is a matched-pairs (repeated-measures) experiment: the 19 drivers are the subjects; the two treatments are normal sleep vs. 27 hours of sleep deprivation; the treatment order is randomized to control order effects; each driver acts as their own control; and the response variables are the task/driving-simulation performance measures.

Step 1 — Describe the misleading effect: Comparison graphs become misleading when the vertical (value) axis does not start at zero or uses an inconsistent/compressed scale. Truncating or stretching the axis exaggerates small differences, making Acme's product appear to outperform the "Other Guy's" by far more than the actual figures justify.

Step 2 — Explain the correction: Redraw both graphs on the same vertical scale starting at zero, with equal axis intervals and identical units. With a common, zero-based axis the true (much smaller) difference between the two products is shown honestly.

Answer: The graphs mislead by using a non-zero or unequal/compressed vertical scale that magnifies the apparent gap between the products. To correct it, plot both on the same scale beginning at zero with consistent intervals, so the real difference is shown accurately.

Step 1 — Spot the missing denominator: Complaint counts alone cannot rank airlines, because larger airlines serve far more passengers and flights and so naturally accumulate more complaints. The graph reports raw totals, not a rate.

Step 2 — Identify the fair comparison: To compare fairly we need a rate — complaints per passenger (or per flight, or per 100,000 enplanements). American, Delta, and United are large carriers, so their higher raw counts may simply reflect their size.

Answer: No. The graph shows raw complaint counts, which favor large carriers. Without adjusting for the number of passengers or flights (a complaints-per-passenger rate), we cannot conclude American, Delta, and United are the worst airlines.

Step 1 — Part (a), judge the sample size: A sample of 3,274 is reasonably large for a survey of this kind; large samples reduce sampling variability. So yes, the size itself is adequate — but size alone does not guarantee a representative sample.

Step 2 — Part (b), gut-check the percents: A "gut feeling" answer is acceptable here. Many students reason that 48% wanting to spend $2,000 on computer equipment seems high for the whole population, suggesting the reported percents are probably higher than the true population values, because the people surveyed are not typical.

Step 3 — Part (c), assess representativeness given the venue: The respondents were self-selected visitors to a Smithsonian technology road show at a convention center. Such visitors are disproportionately tech-interested, affluent, and able to travel to the event, so demographic and ethnic groups were not equally represented.

Step 4 — Part (d), comment on accuracy: Because the sample is self-selected and drawn from an unrepresentative, tech-enthusiast crowd, the statistics likely overstate computer interest and savviness in the full population. The sample statistics should not be trusted as accurate estimates of the population parameters despite the large \(n\).

Answer: (a) The size (3,274) is large enough, but size does not fix bias. (b) The percents are likely too high. (c) No — convention attendees self-selected and skew tech-interested and affluent, so groups were not equally represented. (d) The statistics probably overstate the true population parameters because of self-selection/sampling bias.

Step 1 — Recall the three data types: Qualitative (categorical) data describe a category; quantitative discrete data are counts; quantitative continuous data are measurements that can take any value in an interval (often times or amounts measured).

Step 2 — Classify (a): "Do you have any health problems that prevent you…?" is a yes/no category response — qualitative.

Step 3 — Classify (b): "About how many days did poor health keep you from your usual activities?" is a count of days — quantitative discrete.

Step 4 — Classify (c): "On how many days did you exercise 30 minutes or more?" is again a count of days — quantitative discrete.

Step 5 — Classify (d): "Do you have health insurance coverage?" is a yes/no category — qualitative.

Answer: a) qualitative; b) quantitative discrete; c) quantitative discrete; d) qualitative.

Step 1 — Part (a), why the frame was biased: In 1936, telephones, automobiles, and magazine subscriptions and club memberships were luxuries owned mainly by wealthier households. A sampling frame built from those lists systematically excluded poorer voters, so it was not representative of the U.S. electorate (this is undercoverage / sampling-frame bias).

Step 2 — Part (b), effect of low response: About 2,300,000 of 10,000,000 responded — roughly a 23% response rate. The people who chose to mail back postcards were self-selected and likely differed in opinion from non-responders, introducing nonresponse bias that makes the sample unreliable.

Step 3 — Part (c), classify the errors: A biased frame and self-selected responders are flaws in how the sample was obtained, not random chance from sampling, so these are nonsampling errors.

Step 4 — Part (d), name Gallup's method: Filling fixed quotas from specific subgroups of the population matches stratified sampling (selecting set numbers from defined strata).

Answer: (a) The lists covered mostly wealthier households, excluding poorer voters (undercoverage). (b) The ~23% response rate creates nonresponse bias, undermining reliability. (c) Nonsampling error. (d) Quota sampling corresponds to stratified sampling.

Step 1 — Read the claim: The data show a correlation — communities with more education also have higher reported crime rates. The problem asks which sampling/data pitfall could explain this.

Step 2 — Identify the likely cause: This is a case where correlation does not imply causation, driven by a confounding (lurking) variable. For example, more urban or more populous communities may have both better-funded schools (higher education) and more thorough crime reporting/higher crime, so a third variable creates the apparent link rather than education causing crime.

Answer: The connection is best explained by a confounding variable (and the correlation-is-not-causation caution): an outside factor such as urbanization or reporting practices is associated with both higher education levels and higher recorded crime, producing a misleading association.

Step 1 — Examine the question wording: The question — "Do you feel happy paying your taxes while some politicians are allowed to use loopholes and avoid paying their fair share…?" — is emotionally loaded and pushes the respondent toward a negative answer.

Step 2 — Identify the pitfall: A leading or loaded question introduces response bias (a form of nonsampling error). Because the wording prompts a particular answer, all 11 people responding "NO!" reflects the biased instrument, not genuine independent opinion. The tiny sample size of 11 also limits any conclusion.

Answer: The uniform "NO!" responses are explained by response bias from the leading, loaded wording of the question (a nonsampling error); the very small sample size compounds the problem.

Step 1 — Part (a), reasons for declining response rates: Possible reasons include caller-ID and call screening, the spread of cell phones (vs. listed landlines), distrust of telemarketers and scams, "survey fatigue," busier lifestyles, and concerns about privacy. Any well-reasoned causes are acceptable.

Step 2 — Part (b), why researchers worry: As fewer contacted people respond, the responders become more self-selected and may differ systematically from non-responders. This nonresponse bias threatens the validity of the estimates — the poll may no longer reflect the broader public, no matter how the numbers are computed.

Answer: (a) Reasons include call screening/caller ID, cell-phone-only households, distrust and privacy concerns, and survey fatigue. (b) Lower response rates raise the risk of nonresponse bias, so the responding sample may not represent the population and the poll's conclusions can become invalid.

Step 1 — Part (a), percent without a home computer: 96% have a computer at home, so the complement is \(100\%-96\%=4\%\).

Step 2 — Part (b), number living at least 16 miles away: 13% of the 771 respondents: \(0.13 \times 771 \approx 100.2\), so about 100 students.

Step 3 — Part (c), would percentages transfer to Great Basin College? Probably not. Long Beach City College is in a dense urban area, while Great Basin College in Elko, Nevada serves a rural region. Differences in internet access, commuting distances, and student demographics would likely change percentages such as "live at least 16 miles from campus" and "unable to come to campus."

Answer: (a) 4%. (b) about 100 students. (c) No — the two colleges serve different (urban vs. rural) populations, so distances, access, and demographics differ, and the percentages would likely not be the same.

Step 1 — Assess representativeness of the sample of textbooks: Martin draws conclusions about all college textbooks but examines only seven books — the single most popular title in each of seven subjects. This sample is far too small and is not representative: popular, high-demand bestsellers are exactly the books most likely to be stocked, so they don't reflect the availability of the thousands of niche, low-volume, or specialized textbooks that make up most of the population.

Step 2 — Identify sources of bias: (1) Selection bias toward best-sellers, which are kept in stock more reliably than typical or obscure titles. (2) Undercoverage — only seven subjects are covered, omitting the many fields and editions students actually need. (3) Sampling only major online retailers may overstate availability compared with the full range of sellers. These biases would make availability look better than it truly is.

Step 3 — Suggest improvements: Use a much larger sample of textbooks drawn randomly across many subjects, difficulty levels, and popularity tiers (including niche and lower-demand titles) and across multiple editions. Sample a broader, representative set of online retailers, and check availability at the actual start-of-term rush when demand peaks.

Answer: The sample is not representative: seven best-selling titles cannot speak for all college textbooks, and best-sellers are the most likely to be in stock. Main biases are selection bias toward popular books, undercoverage of subjects/editions, and using only major retailers — all of which overstate availability. Improvements: take a large random sample of textbooks spanning many subjects, popularity levels, and editions; sample a representative range of retailers; and test availability during the beginning-of-term period.