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Investigative questions

Learning goals

Formulates statistical questions that anticipate variability.
Identifies the population, variables, and intended comparison or parameter.
Distinguishes descriptive from inferential questions.
Aligns data collection with the investigative question.

Difficulty: 1 – 5

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Investigative questions 1

Which question requires a statistical investigation? a) How many seats are in the school auditorium? b) How many minutes do students at the school usually spend traveling to school? c) What time does first period begin? Explain your choice.

Hints

- Ask whether the answer could differ across observational units. - Look for a question that needs a collection of data rather than one fixed fact.

Solution

1. Questions a) and c) have single fixed answers. 2. Question b) anticipates variability among students and requires data from multiple students.

Answer

b) It requires a statistical investigation because travel times vary from student to student.

Investigative questions 2

Decide whether the question “How many goals did the home team score in yesterday’s game?” is an investigative question for a statistical study. Justify your answer.

Hints

- Ask whether repeated observations could produce varying values. - Decide whether one fixed factual record is enough to answer the question.

Solution

1. The question concerns one fixed game and has one factual answer. 2. It does not require collecting variable data from a sample to learn about a population.

Answer

No. It has one fixed factual answer and does not require a statistical investigation.

Investigative questions 3

A county parks department wants a statistical investigative question about variation in trail use among county households during June. Which question is suitable? a) How many county-trail visits did the household at 18 Cedar Lane make during June? b) What is the distribution of the number of county-trail visits made during June by households in the county? c) How many county trails were open on June 15? Explain your choice.

Hints

- Ask whether the answer is one fixed fact or a pattern that can vary across observational units. - Identify which option concerns many county households rather than one named household or one fixed date. - A statistical investigative question anticipates variability in the data.

Solution

1. Question b) concerns a variable count that can differ from household to household across a population. 2. Answering it requires collecting and analyzing data from multiple households. 3. Questions a) and c) each ask for one fixed factual value rather than a statistical pattern across a population.

Answer

b) What is the distribution of the number of county-trail visits made during June by households in the county? It is statistical because the number of visits varies among households and the question asks about that variation across a population.

Investigative questions 4

A transit planner wants to compare the reliability of Route A and Route B. Which response variable best matches that purpose: the route name, the number of passengers, or the difference in minutes between scheduled and actual arrival? Explain.

Hints

- Translate “reliability” into something observable for each trip. - Separate a group label from a measured outcome. - Reject variables that address a different operational question.

Solution

1. Reliability concerns how closely actual service matches the schedule. 2. The route name identifies the group and is not the response. 3. Passenger count measures demand rather than reliability. 4. Arrival deviation in minutes directly measures schedule reliability for each trip.

Answer

Use the difference in minutes between scheduled and actual arrival. It is a quantitative measure of reliability for each trip and can be compared between the two routes.

Investigative questions 5

A marine biologist wants to study the relationship between shell length and body mass for adult crabs in one estuary. Write an investigative question that identifies both variables and the population.

Hints

- Name the two measurements taken on each observational unit. - Use relationship wording rather than a one-variable summary. - State the location and population represented.

Solution

1. Shell length and body mass are quantitative variables measured on each crab. 2. The purpose is to investigate their relationship. 3. The population is adult crabs in the specified estuary.

Answer

One valid question is: “Among adult crabs in the estuary, what is the relationship between shell length and body mass?”

Investigative questions 6

The Cedar Valley Library System asks, “For active cardholders, what is the mean number of library items checked out during September?” What is the observational unit, and what value should be recorded for each sampled unit?

Hints

- Identify the individual object or person about which one data value is needed. - Match the recorded value to the quantity named in the investigative question.

Solution

1. The question is about active cardholders, so one active cardholder is the observational unit. 2. For each sampled cardholder, record the number of library items that cardholder checked out during September. 3. Those individual counts can be used to study the mean for the population of active cardholders.

Answer

Observational unit: one active cardholder. Value to record: that cardholder's number of library items checked out during September.

Investigative questions 7

A school district has a list of all students enrolled in its high schools. A counselor surveys a random sample of students from the list about their weekly homework hours. Which group is the population, and which is the sample?

Hints

- Which group does the district want information about? - Which students actually provide the data? - Is the sample usually smaller than the population?

Solution

1. The population is every student on the district's high-school enrollment list. 2. The sample is the students selected and surveyed.

Answer

Population: all students on the high-school enrollment list. Sample: the students chosen for the survey.

Investigative questions 8

A movie theater's sales file records the movie, seat number, showtime, and price in each row. A customer buying three seats appears in three rows. Is the observational unit a movie, a customer, or a ticket sold?

Hints

- What would one row tell you if a customer bought several seats? - Can the same movie appear in many rows? - Decide what is counted once each time a new row is entered.

Solution

1. Each row records one occupied seat sold for a particular showing. 2. A single customer can appear in several rows, and a movie can appear in many rows. 3. Therefore, the observational unit is one ticket sold.

Answer

A ticket sold.

Investigative questions 9

A principal wants to know what proportion of students brought lunch from home today. Which record would answer that question: whether the student paid in the cafeteria, the student's lunch source (home, cafeteria, elsewhere, or no lunch), or the lunch's price?

Hints

- Could two students both spend nothing in the cafeteria but eat different lunches? - What must the school know about each student to count the home-lunch group? - Does the price of a lunch tell you where it came from?

Solution

1. The principal needs to identify lunches brought from home. 2. Lunch source separates home lunches from cafeteria lunches, other lunches, and students who ate no lunch. 3. Not paying in the cafeteria does not necessarily mean bringing lunch from home.

Answer

Record where each student's lunch came from. A student who did not pay in the cafeteria might have bought lunch elsewhere or skipped lunch.

Investigative questions 10

A city library wants to estimate the mean number of books checked out during visits made by adult cardholders this year. It records the number checked out for a sample of \(180\) such visits. Identify the population, the sample, and the investigative question.

Hints

- Separate the full group the library wants to learn about from the visits actually recorded. - The requested population quantity is a mean, not an undefined idea of what is “typical.” - Phrase the investigative question about the full population rather than only the \(180\) sampled visits.

Solution

1. The population is all visits made by adult cardholders to the city library this year. 2. The sample is the \(180\) recorded visits. 3. The investigative question asks for the population mean number of books checked out per adult-cardholder visit this year.

Answer

Population: all visits made by adult cardholders to the library this year. Sample: the \(180\) recorded visits. Investigative question: “What is the mean number of books checked out during a visit made by an adult cardholder this year?”

Investigative questions 11

A package delivery service samples \(90\) packages shipped during April 2026 and records whether each arrived by its promised date. State the population and give one investigative question the data can answer.

Hints

- Which packages could have been chosen for the sample? - Notice that each package has a yes-or-no outcome. - Frame the question about all packages shipped that April, not just the sampled packages.

Solution

1. The population is all packages shipped by the delivery service during April 2026. 2. The on-time status of each sampled package can be used to study the proportion delivered on time.

Answer

Population: all packages shipped by the delivery service during April 2026. One possible question: “What proportion of packages shipped in April 2026 arrived by their promised date?”

Investigative questions 12

A school newspaper wants to know whether students prefer digital or printed yearbooks. It asks only, “Which format did Maya choose?” Explain why that question does not match the study’s purpose.

Hints

- Compare the scope of the question with the scope of the intended conclusion. - A population preference requires responses from multiple students, not one person’s choice.

Solution

1. The purpose concerns preferences across a population of students. 2. One student’s choice is a single datum and does not describe variability or proportions in the population.

Answer

The question concerns only one student, while the purpose requires data from many students to describe the population’s preferences.

Investigative questions 13

A transit agency’s goal is to learn about weekday bus delays on Route \(7\). Write an investigative question whose answer would require collecting a quantitative datum from multiple trips.

Hints

- Specify the group or period, the quantity recorded, and the purpose. - Use each trip as an observational unit and delay time as its measurement.

Solution

1. Use trips as the observational occasions and delay time as the recorded quantity. 2. One valid question is: “What is the distribution of arrival delays, in minutes, for weekday Route \(7\) trips this month?”

Answer

One valid question is: “What is the distribution of arrival delays, in minutes, for weekday Route \(7\) trips this month?”

Investigative questions 14

A researcher asks, “What is the average screen time?” What information is missing before this can become a well-defined investigative question?

Hints

- Whose screen time would the researcher measure? - Would phone, computer, and television use all count, and over what period? - Could two researchers mean different things by “average”?

Solution

1. Specify the population being studied. 2. State the time period and exactly which screen activities will count. 3. Define what “average” means, such as the mean minutes of screen time per day.

Answer

Name the population, specify the period and activities counted as screen time, and say which average is wanted—for example, the mean minutes per day.

Investigative questions 15

A community college records weekly paid-work hours for sampled full-time and part-time students. It wants an investigative question that keeps paid-work time quantitative and directly compares the two enrollment groups. Which question best matches that goal? a) Is having any paid job associated with being a full-time or part-time student? b) Is the mean weekly paid-work time different for full-time and part-time students at the college? c) What proportion of all students work more than \(20\,\text{hours}\) per week? Explain your choice.

Hints

- Keep track of both the type of response variable and the two groups that must be compared. - Ask whether an option throws away the recorded number of work hours by converting it to a category. - The matching question should use the quantitative measurements from both enrollment groups.

Solution

1. The recorded response is quantitative weekly paid-work time. 2. Choice b) compares that quantitative response between the two enrollment groups by comparing their population means. 3. Choice a) collapses work time to a yes-or-no variable, and choice c) does not make the required full-time versus part-time comparison.

Answer

b) Is the mean weekly paid-work time different for full-time and part-time students at the college? It preserves the quantitative response and directly compares the two enrollment groups.

Investigative questions 16

A museum wants to estimate the proportion of weekend visitors who used an audio guide during July. A staff member proposes the question, “Did visitors use the audio guide?” Rewrite it as a complete investigative question for the intended study.

Hints

- Specify exactly which visitors are of interest. - State the time period. - Phrase the question around the population quantity the museum wants to estimate.

Solution

1. The population is weekend museum visitors during July. 2. The variable is whether each visitor used an audio guide. 3. The goal is to estimate the population proportion who used one. 4. A complete question states all three components.

Answer

One valid question is: “What proportion of weekend visitors to the museum during July used an audio guide?”

Investigative questions 17

A dental office wants to know whether sending two appointment reminders instead of one reduces missed appointments. Write an investigative question that makes clear the explanatory variable, response variable, comparison, and population.

Hints

- Name both conditions being compared. - State the outcome that will be recorded for each appointment. - Include the group of appointments to which the conclusion should apply. - Make the direction of the proposed effect clear.

Solution

1. The explanatory variable is the number of reminders: one or two. 2. The response variable is whether a scheduled appointment is missed. 3. The question compares the population proportions of missed appointments under the two reminder conditions. 4. The population is appointments scheduled at the dental office.

Answer

One valid question is: “For appointments scheduled at this dental office, does sending two reminders instead of one reduce the proportion that are missed?”

Investigative questions 18

A researcher has existing records of neighborhood tree canopy and summer electricity use. The proposed question is, “Does increasing tree canopy cause households to use less electricity?” Rewrite the question so it matches the observational data that are actually available.

Hints

- Match the wording to whether a treatment was imposed. - Preserve the two variables in the original purpose. - Replace a causal claim with a relationship the records can support.

Solution

1. The records contain observed values; no canopy treatment was assigned. 2. The question should ask about association rather than causation. 3. It should identify the variables and the population represented by the records.

Answer

One valid revision is: “Among households represented in the neighborhood records, is greater tree-canopy coverage associated with lower summer electricity use?”

Investigative questions 19

A regional theater wants to estimate the proportion of season-ticket holders who would renew if one weekday performance were replaced by a weekend performance. Write a valid investigative question and identify the population parameter.

Hints

- Identify the full group whose behavior matters to the theater. - State the proposed schedule change in the question. - Name the population quantity, not a sample result.

Solution

1. The population is all current season-ticket holders. 2. The variable is whether each holder would renew under the proposed schedule. 3. The parameter is the population proportion who would renew. 4. The investigative question asks to estimate that proportion under the stated change.

Answer

One valid question is: “What proportion of current season-ticket holders would renew if one weekday performance were replaced by a weekend performance?” The parameter is the proportion of all current season-ticket holders who would renew under that schedule.

Investigative questions 20

A public library wants to investigate whether preferred book format, print or digital, is associated with age group among its adult cardholders. Write an investigative question suitable for a test of association.

Hints

- Name both categorical variables. - Use wording that asks whether the variables are related. - Include the population to which the conclusion should apply.

Solution

1. The variables are preferred format and age group, both categorical. 2. The analysis asks whether the variables are associated rather than independent. 3. The population is all adult cardholders of the library system.

Answer

One valid question is: “Among adult cardholders in the library system, is preferred book format associated with age group?”

Investigative questions 21

A marina wants to compare rental duration for single and tandem kayaks during the summer season. The phrase “Which kayak type is rented longer?” is too vague. Write a precise investigative question comparing population means.

Hints

- Replace “longer” with a named population summary. - Identify the two rental groups. - Include the place and time period defining the population.

Solution

1. The quantitative variable is rental duration. 2. The groups are single-kayak rentals and tandem-kayak rentals. 3. The parameter comparison is the difference in mean rental duration. 4. The population is all summer rentals at the marina.

Answer

One valid question is: “For all kayak rentals at the marina this summer, is the mean rental duration different for single kayaks and tandem kayaks?”

Investigative questions 22

A conservation group wants to estimate the mean amount of food waste produced per household per week in a county. Two data-collection plans are proposed for each sampled household: Plan A: Record whether the household discarded any food during the week (yes or no). Plan B: Weigh all food discarded by the household during the same defined week. Which plan can answer the stated goal? Explain why the other plan cannot, then write a matching investigative question.

Hints

- Match the type of population quantity in the goal to the type of value collected from each household. - Ask what a yes-or-no response could estimate and what information it loses. - The investigative question should name the same population and quantitative target as the usable data plan.

Solution

1. The target is a mean amount, so each sampled household needs a quantitative measurement of how much food waste it produced. 2. Plan B provides that quantitative amount for each household. 3. Plan A provides only a yes-or-no response, so it could estimate the proportion of households that discarded any food but not the mean amount discarded. 4. A matching investigative question asks for the population mean weekly amount of household food waste in the county.

Answer

Plan B can answer the goal because it records a quantitative amount of food waste for each sampled household. Plan A cannot estimate the mean amount; it records only whether any waste occurred. A matching investigative question is: “What is the mean amount of food waste produced per household during one week among households in the county?”

Investigative questions 23

A researcher writes, “What is the mean commute time of the \(120\) sampled employees?” Explain why this is not the main investigative question for a study intended to learn about all employees, and rewrite it.

Hints

- Distinguish the observed sample from the larger group of interest. - Decide whether the question asks for a statistic or a population parameter. - Rewrite the question around the purpose of inference.

Solution

1. The proposed question asks only for a sample statistic that can be calculated after data collection. 2. The study's purpose is to learn about a population parameter. 3. The investigative question should ask about the mean commute time of all employees represented by the sample.

Answer

The original wording asks only for the sample mean. A better investigative question is: “What is the mean commute time for all employees at the company?”

Investigative questions 24

A delivery service selects a random sample from all deliveries completed during April and asks each recipient to rate the condition of the delivered package on a scale from \(1\) to \(5\). The company wants to estimate the mean rating for all April deliveries. Identify the observational units, variable, parameter, and investigative purpose.

Hints

- Identify what receives one recorded rating. - Separate the rating itself from the population summary of all April ratings. - Use the stated month to name the full target population precisely.

Solution

1. The observational units are the individual April deliveries; each sampled delivery receives one package-condition rating. 2. The variable is the package-condition rating from \(1\) to \(5\). 3. The parameter is the mean package-condition rating for all deliveries completed during April. 4. The investigative purpose is to estimate that population mean from the random sample.

Answer

Observational units: individual deliveries completed during April. Variable: package-condition rating on the \(1\)-to-\(5\) scale. Parameter: the mean condition rating for all April deliveries. Purpose: estimate that population mean from the random sample.

Investigative questions 25

A hospital takes independent random samples of patients discharged during 2025 from Unit A and Unit B. For each sampled patient, it records whether the patient returns within \(30\,\text{days}\). Write an investigative question that concerns the two population readmission proportions rather than only the observed sample rates.

Hints

- Distinguish the sampled discharge records from the full groups the hospital wants to compare. - The response is binary, so each group has a population proportion. - Phrase the question about the two population proportions rather than the observed sample percentages.

Solution

1. The response variable is whether a discharged patient returns within \(30\,\text{days}\). 2. The two target parameters are the \(30\)-day readmission proportions for all 2025 discharges from Unit A and Unit B. 3. The investigative question should compare those population proportions rather than the two sample proportions.

Answer

One valid question is: “For patients discharged during 2025, is the \(30\)-day readmission proportion different between Unit A and Unit B?”

Investigative questions 26

An emergency dispatch center wants to study call response times. Rewrite “Are calls answered quickly?” as a statistical investigative question with a defined population and quantitative variable.

Hints

- Replace the judgment word with a measurable quantity. - Define which calls and when. - Ask a question that requires many call records rather than one fixed fact.

Solution

1. Define the population of calls by center and time period. 2. Replace “quickly” with response time measured in a stated unit. 3. Ask about a distribution or population summary that requires data from many calls.

Answer

One valid question is: “What is the distribution of the number of seconds from call arrival to answer for emergency calls received by the center during the second quarter of 2026?”

Investigative questions 27

A bike-share program wants to compare the proportion of rides ending at transit stations in 2026 with the corresponding proportion in 2025. Write a complete investigative question.

Hints

- Define one categorical outcome for each ride. - Name the two time-based populations. - Ask about the difference between population proportions.

Solution

1. The variable is whether a ride ends at a transit station. 2. The two parameters are the population proportions for rides in 2026 and rides in 2025. 3. The question asks whether those proportions differ for the bike-share system.

Answer

One valid question is: “For all rides in the bike-share system, is the proportion ending at transit stations different in 2026 from the proportion in 2025?”

Investigative questions 28

A regional orchestra selects audience visits at three performances and records the distance traveled for each visit. It wants the mean distance traveled per audience visit over the whole season. Identify the intended population, the sample, and the quantitative variable.

Hints

- The target is distance per visit, not per unique person. - Think about a person who attends two performances: how many visits is that? - Separate all visits over the season from the visits actually recorded.

Solution

1. The population is every audience visit to an orchestra performance during the season. 2. The sample is the selected audience visits at the three performances. 3. The quantitative variable is the distance traveled for each selected visit.

Answer

Population: all audience visits during the season. Sample: the selected audience visits at the three performances. Variable: distance traveled for each visit.

Investigative questions 29

A water utility wants to estimate the proportion of all residential accounts it served during 2026 that had at least one leak that year. For each sampled account, its database gives the number of leak incidents during 2026: \(0,1,2,\ldots\). What per-account variable should the analyst use to estimate the requested proportion, and what population parameter does that variable target?

Hints

- A population proportion is based on whether each observational unit satisfies a condition. - Decide how to turn a leak count such as \(0,1,2,\ldots\) into the condition named in the goal. - State the parameter for the full population of residential accounts served during 2026.

Solution

1. The target is a proportion of accounts satisfying a condition, so each sampled account should be classified according to whether it had at least one leak. 2. Convert each leak count to a binary variable: “had at least one leak in 2026” or “had no leaks in 2026.” 3. The resulting sample proportion estimates the population proportion of all residential accounts served during 2026 that had at least one leak.

Answer

Use a binary per-account variable: whether the account had at least one leak during 2026. The target parameter is the proportion of all residential accounts served during 2026 that had at least one leak.

Investigative questions 30

A computer-repair service randomly samples file-recovery jobs completed by its shops during 2026 using two backup programs. An analyst asks, “Is the mean recovery time different for the two programs?” What is missing from this investigative question? Rewrite it.

Hints

- The variable and the two-system comparison are already stated. - Which jobs do the random samples represent? - Make sure the revised question names those jobs, not only the sampled ones.

Solution

1. The proposed question identifies the response and the comparison but not the population. 2. The population is file-recovery jobs completed by the repair service's shops during 2026 using either program. 3. Add that group to the question.

Answer

The population is missing. “For file-recovery jobs completed by the repair service's shops during 2026, is the mean recovery time different between the two backup programs?”

Investigative questions 31

A transit planner asks, “For weekday route trips made by electric buses in the city's fleet, is passenger load associated with energy used per mile?” Identify the variable component, the analysis component, and the conclusion component of this investigative question.

Hints

- Separate what will be measured from how the measurements will be related. - Look for the word or phrase that indicates the intended statistical comparison. - Identify the group to which the final conclusion is meant to apply.

Solution

1. The variable component identifies passenger load and energy used per mile for each weekday route trip. 2. The analysis component asks whether the two variables are associated. 3. The conclusion component identifies the population as weekday route trips made by electric buses in the city's fleet.

Answer

Variable component: passenger load and energy used per mile for each weekday route trip. Analysis component: whether the two variables are associated. Conclusion component: the conclusion applies to weekday route trips made by electric buses in the city's fleet.

Investigative questions 32

A warehouse records package type (envelope, box, or tube) and whether each shipment arrives damaged for a random sample of shipments from one quarter. Write a complete investigative question for testing whether the two categorical variables are associated.

Hints

- Name both categorical variables exactly as they are recorded. - Use wording that asks about a relationship rather than a numerical difference. - Include the full group of shipments to which the conclusion should apply.

Solution

1. The variables are package type and damage status. 2. The analysis component is whether the variables are associated rather than independent. 3. The population is all shipments handled by the warehouse during the quarter. 4. A complete question is: “Among all shipments handled by the warehouse during the quarter, is package type associated with whether a shipment arrives damaged?”

Answer

Among all shipments handled by the warehouse during the quarter, is package type associated with whether a shipment arrives damaged?

Investigative questions 33

An engineering team records the age of each sampled municipal water meter and the absolute size of its calibration error. Write an investigative question that asks whether the two quantitative variables have a positive association for the intended population.

Hints

- Name both recorded quantitative variables. - Include the direction of the relationship requested by the study. - State the full group of meters to which the conclusion should apply.

Solution

1. The variables are meter age and absolute calibration error. 2. The analysis component is a positive association between the variables. 3. The population is all municipal water meters in the system. 4. A complete question is: “For all municipal water meters in the system, is meter age positively associated with the absolute size of calibration error?”

Answer

For all municipal water meters in the system, is meter age positively associated with the absolute size of calibration error?

Investigative questions 34

A public-health team asks, “What proportion of outdoor workers experienced breathing difficulty on smoky days?” Identify the missing location, period, and meaning of “smoky day,” and give a complete version of the question.

Hints

- Could different places and years give different answers? - How would two workers decide whether a day counts as smoky? - What rule could two researchers apply and get the same set of days?

Solution

1. The question needs a defined place and a period for both workers and reported symptoms. 2. “Smoky day” must be defined by a rule that can be applied consistently. 3. One possible definition uses the county's official smoke-advisory days during summer 2026.

Answer

Specify where and when the workers are studied and what counts as a smoky day. For example: “What proportion of outdoor workers employed in Pine County during summer 2026 reported breathing difficulty on at least one day when the county issued a smoke advisory?”

Investigative questions 35

A library samples only people who received a new library card during July 2026. The proposed investigative question is, “What proportion of all city residents prefer digital borrowing?” Explain the mismatch and rewrite the question to match the available sample.

Hints

- Identify who actually had a chance to be sampled. - Compare that group with the population named in the question. - Narrow the population wording without changing the recorded preference variable.

Solution

1. Only people who received a new library card during July 2026 can be selected, so the sample does not represent all city residents. 2. The intended population in the question is broader than the sampling frame. 3. A matching question is: “What proportion of people who received a new library card during July 2026 prefer digital borrowing?”

Answer

The sampling frame contains new cardholders from July 2026, not all city residents. A matching question is: “What proportion of people who received a new library card during July 2026 prefer digital borrowing?”

Investigative questions 36

A commuter study asks, “What is the typical travel time for employees at the industrial park?” Explain why the word “typical” does not fully determine the analysis, and rewrite the question for a specific population parameter.

Hints

- List more than one numerical summary that could describe a center. - Decide which population quantity the study will estimate. - Add a clear direction of travel and time period to the population definition.

Solution

1. The word “typical” could refer to the mean, median, mode, or another summary. 2. The investigative question should identify the parameter that the analysis will estimate. 3. One complete version is: “What is the mean one-way travel time to work for all employees at the industrial park during July 2026?”

Answer

“Typical” is ambiguous because it does not identify a specific summary. One complete version is: “What is the mean one-way travel time to work for all employees at the industrial park during July 2026?”

Investigative questions 37

A delivery service plans to compare mean delivery times for express and economy packages it ships in September 2026. An analyst asks, “Are typical delivery times different?” Explain why the question is unclear and rewrite it.

Hints

- Could two people interpret “typical” in different ways? - Which measure of center does the planned analysis use? - Which packages and month belong in the final question?

Solution

1. “Typical” could mean the mean, median, or another summary. 2. The planned comparison concerns population means, so name them directly. 3. Name the packages and month given in the plan.

Answer

“Typical” does not identify the measure being compared. A suitable question is: “For packages shipped by the service in September 2026, are mean delivery times different for express and economy shipping?”

Investigative questions 38

A clinic wants to study the mean cost of a patient visit during May 2026. A staff member asks, “How much did the clinic spend on patient care?” Explain why this does not match the goal. Rewrite it.

Hints

- What is the unit of observation: the whole clinic or each visit? - Does the word “spend” tell you whether a total or an average is wanted? - Include the given month in the revised question.

Solution

1. Total clinic spending is not the same as cost per visit. 2. Record one cost for each patient visit in May 2026 and ask about the mean across those visits.

Answer

The question asks for a total, not a mean cost per visit. A suitable question is: “What was the mean patient-care cost per visit at the clinic during May 2026?”

Investigative questions 39

A conservation team asks, “What proportion of nesting attempts were successful?” The team has not defined “successful.” Explain why an operational definition is needed and write a measurable version of the investigative question.

Hints

- Ask whether two observers would necessarily classify an outcome the same way. - Replace the vague word with an observable event. - Include the population of attempts and the breeding period.

Solution

1. Different observers could classify the same nesting attempt differently without a stated success criterion. 2. The variable must be defined so each attempt can be recorded consistently. 3. One measurable definition is producing at least one fledgling. 4. A complete question is: “What proportion of all monitored nesting attempts in the preserve during the 2026 breeding season produced at least one fledgling?”

Answer

An operational definition is needed for consistent classification. One complete question is: “What proportion of all monitored nesting attempts in the preserve during the 2026 breeding season produced at least one fledgling?”

Investigative questions 40

A company wants to learn how employees travel to work using the categories drive alone, carpool, public transit, bicycle, walk, and other. Write an investigative question that targets the full population distribution rather than only one category.

Hints

- Keep all categories of the recorded variable in the question. - Ask for how the population is divided among those categories. - Identify the complete employee population for the conclusion.

Solution

1. The variable is commute mode, a categorical variable with six categories. 2. The analysis goal is to estimate the population proportion in every category. 3. A complete question is: “Among all employees of the company, what is the distribution of usual commute mode across drive alone, carpool, public transit, bicycle, walk, and other?”

Answer

Among all employees of the company, what is the distribution of usual commute mode across drive alone, carpool, public transit, bicycle, walk, and other?

Investigative questions 41

A bus company samples weekday trips on Route 4 during September 2026. It records: <table><tr><th>Trip</th><th>Scheduled arrival</th><th>Actual arrival</th></tr><tr><td>Morning</td><td>\(8{:}00\) a.m.</td><td>\(8{:}03\) a.m.</td></tr><tr><td>Midday</td><td>\(12{:}10\) p.m.</td><td>\(12{:}10\) p.m.</td></tr><tr><td>Afternoon</td><td>\(4{:}25\) p.m.</td><td>\(4{:}32\) p.m.</td></tr></table> One analyst suggests comparing the clock times when buses arrive. Another suggests comparing minutes late, counting an early bus as 0 minutes late. Which measurement better answers whether the route runs on time? Use the records to explain, then write a question about the mean of that measurement for all weekday Route 4 trips that September.

Hints

- Would a bus arriving at noon necessarily be less punctual than one arriving at 8 a.m.? - Think about how the two recorded times for a trip relate to its punctuality. - What group of trips does the bus company want the final mean to describe?

Solution

1. Clock arrival times vary because trips are scheduled at different times of day. They do not show how late a trip was. 2. Minutes late can be compared across trips with different schedules. The three recorded trips are \(3\), \(0\), and \(7\) minutes late. 3. The investigative question should ask about mean minutes late across all weekday Route 4 trips that month, not only the three shown.

Answer

Minutes late is better: the buses have different scheduled arrival times, so their clock times do not measure punctuality. They are \(3\), \(0\), and \(7\) minutes late. One question is: “What was the mean number of minutes late for weekday Route 4 trips in September 2026?”

Investigative questions 42

A state park already records whether each visitor bought a parking pass. The park now wants to estimate the mean amount visitors spend inside the park, excluding the pass. What additional value must it collect from each sampled visitor, and why is the pass record insufficient?

Hints

- Does the existing record contain any dollar amounts? - What kind of measurement is needed to calculate a mean? - Keep the parking-pass charge out of the measurement.

Solution

1. Whether a pass was purchased is categorical and does not measure in-park spending. 2. The park must record the dollar amount each sampled visitor spends inside the park, excluding the pass. 3. Those amounts can be used to estimate a mean spending amount.

Answer

Record each sampled visitor's spending inside the park in dollars, excluding parking-pass costs. A yes-or-no pass record cannot give a mean spending amount.

Investigative questions 43

A swim club wants to know how much time members spend swimming each week. Its survey asks only whether each person swam at least once during the past week. Suppose Mia swam for \(15\) minutes and Elijah swam for \(150\) minutes that week. Why would their survey answers look the same? What population proportion could the survey estimate, and what mean could it not estimate?

Hints

- Compare what Mia and Elijah actually did with what their survey responses show. - What kind of population question can a yes-or-no response answer? - What extra information is needed to find a mean time?

Solution

1. Mia and Elijah both answer yes, even though their times are very different. 2. The yes-or-no response can estimate the proportion of members who swam at least once that week. 3. It cannot estimate mean weekly swimming minutes because no durations were recorded.

Answer

Both say yes, despite swimming for very different lengths of time. The survey could estimate the proportion who swam at least once, but not the mean minutes swum per member.

Investigative questions 44

Eero writes the investigative question, “Are the cafeteria lunches good?” Rewrite it so that it has a defined population, a measurable variable, and a clear purpose.

Hints

- Decide which students the question should describe. - Replace the word “good” with something that can be recorded. - Ask about the ratings across the group rather than one student's opinion.

Solution

1. Specify the population as students who bought a cafeteria lunch during a stated period. 2. Replace “good” with a rating on a defined scale. 3. One valid question is: “Among students who bought a cafeteria lunch this week, what is the distribution of lunch ratings on a scale from \(1\) to \(5\)?”

Answer

One valid revision is: “Among students who bought a cafeteria lunch this week, what is the distribution of lunch ratings on a scale from \(1\) to \(5\)?”

Investigative questions 45

Before collecting data, a research team asks, “What proportion of commuters in the county used public transit at least once last week?” After seeing a low sample proportion, the team changes the question to, “What proportion used public transit at least once in the past year?” What is wrong with this change?

Hints

- Compare the question that guided collection with the question asked after the results were known. - Notice that the time window determines the quantity being estimated. - Ask whether the revision was planned before collection or chosen in response to the observed result.

Solution

1. The original question defined the purpose and time period before analysis. 2. Changing the question after seeing the result makes the study answer a different question and can introduce result-driven bias.

Answer

The investigative question should not be changed in response to the observed results; the revised question concerns a different time period.

Investigative questions 46

A coach asks, “Do our runners improve?” The team has each runner's \(5\)-kilometer time from the first and final meets. Give a more precise question about the change in each runner's time.

Hints

- What two measurements are available for each runner? - Keep the measurements from the same runner together. - Once you define a change, how will its sign tell you whether a runner improved?

Solution

1. Both times belong to the same runner, so form a change for each runner. 2. Define change as final time minus first time. A negative change means the runner got faster. 3. Ask how these changes are distributed across the team.

Answer

“What is the distribution of final-meet time minus first-meet time for the team's \(5\)-kilometer runners?”

Investigative questions 47

A wildlife rehabilitation network wants to test whether the population proportion of released birds that return for additional care is greater than the corresponding proportion for released mammals. State the three essential components that a valid investigative question must include, and then write the question.

Hints

- Separate what data must be recorded from what population quantity will be compared. - Include the direction of the proposed difference. - State exactly which released animals the conclusion concerns.

Solution

1. The data component identifies animal type and whether each released animal returns for additional care. 2. The analysis component identifies the two population proportions and the direction of the comparison: birds greater than mammals. 3. The conclusion component identifies the population as animals released by the network. 4. The final question combines these components.

Answer

Components: the variables to collect, the comparison of the two population proportions with a stated direction, and the population to which the conclusion applies. One valid question is: “Among animals released by the rehabilitation network, is the proportion of birds that return for additional care greater than the proportion of mammals that return?”

Investigative questions 48

A study proposal asks, “Does noise affect concentration?” Rewrite it so that the population, explanatory variable, response variable, and planned comparison are measurable.

Hints

- Replace each vague word with something that can be recorded. - State who is being studied. - Include both conditions and the exact outcome used for comparison.

Solution

1. The population must be defined. 2. Noise must be expressed as specific conditions that can be recorded or assigned. 3. Concentration must be measured with a stated response. 4. A valid question compares the population response under the defined conditions.

Answer

One valid revision is: “For students at the school, is the mean number of correctly completed proofreading items different when students work with recorded conversation noise than when they work in silence?”

Investigative questions 49

A music program records each student's sight-reading score at the beginning and end of a semester. Write an investigative question about the population mean change that matches the paired data.

Hints

- Use the fact that both measurements come from the same student. - Define one change value for each student. - Ask about the population mean of those changes.

Solution

1. The observational unit is a student. 2. The variable for analysis is each student's end score minus beginning score. 3. The parameter is the population mean of those paired changes. 4. The question should ask for that population mean change without treating the two sets of scores as independent.

Answer

One valid question is: “For students in the music program, what is the mean change in sight-reading score from the beginning to the end of the semester?”

Investigative questions 50

A city archives only dates, locations, and repair costs for water-main breaks. An analyst proposes, “What caused each water-main break?” Explain why the question cannot be answered from the available data and write a feasible investigative question.

Hints

- List the variables that are actually available. - Remove any claim that requires unrecorded information. - Build a question that can be answered by analyzing one or more recorded variables.

Solution

1. The records do not contain measured causes or explanatory variables needed to determine cause. 2. The proposed question is not answerable from the available variables. 3. A feasible question uses dates, locations, or repair costs and asks for a distribution or comparison.

Answer

The archives do not contain evidence about causes. One feasible question is: “What is the distribution of repair costs for all water-main breaks in the city archive?”

Investigative questions 51

A university takes a simple random sample of first-year students entering this fall and randomly assigns the selected students to either a new orientation program or the current program. The response is first-semester retention. Write an investigative question that makes the intended cause-and-effect comparison and target population explicit.

Hints

- Identify the treatment that is assigned and the response that is measured. - Use the random assignment to decide whether causal wording is appropriate. - Use the sampling description to identify the population named in the question.

Solution

1. The explanatory variable is orientation-program assignment. 2. The response variable is whether a student is retained through the first semester. 3. Random assignment supports framing the comparison causally. 4. The target population is all first-year students entering the university this fall because the students were randomly sampled from that population. 5. A complete question asks whether assignment to the new program causes a different retention rate in that population.

Answer

One valid question is: “Among first-year students entering the university this fall, does assignment to the new orientation program cause a higher first-semester retention rate than assignment to the current program?”

Investigative questions 52

A coastal research group takes a random sample of county-managed beaches. At each selected beach, it records the number of plastic fragments found in a standardized survey before a cleanup policy and again one year after the policy, using the same beaches in both periods. Write an investigative question about the population mean change that reflects the paired structure.

Hints

- The same beach is measured twice, so identify the within-beach quantity that can be formed. - The target is a mean of changes across the population of county-managed beaches. - Keep the before-and-after measurements paired in the wording of the question.

Solution

1. Each sampled beach supplies a before-and-after pair. 2. Define each beach's change as the after-policy fragment count minus the before-policy count. 3. The population parameter is the mean of those changes across all county-managed beaches. 4. The investigative question should ask about the mean within-beach change rather than compare two unrelated samples.

Answer

One valid question is: “For county-managed beaches, what is the mean change in standardized plastic-fragment count from before the cleanup policy to one year after it?”

Investigative questions 53

A school counselor proposes, “Why are some students absent more often?” The available data contain commute distance, extracurricular participation, and number of absences. Rewrite the question into one feasible statistical investigation using the available variables.

Hints

- Use only variables that appear in the records. - Replace an open-ended causal “why” with a measurable relationship. - Name the student group represented by the data.

Solution

1. The available observational data cannot determine a complete cause for absences. 2. A feasible question asks about association between absences and one or more recorded variables. 3. It identifies the student population represented by the data.

Answer

One valid question is: “Among students represented in the school records, is commute distance associated with the number of absences during the school year?”

Investigative questions 54

A museum records each gallery visitor's time spent in an exhibit before and after new signs are installed. The same visitors experience both versions on different visits. Write an investigative question about the mean paired change.

Hints

- Match each person's later measurement with that person's earlier measurement. - Define a single change value per visitor. - Ask about the population mean of those paired changes.

Solution

1. Each visitor provides a before-and-after pair. 2. Define change as time after installation minus time before installation. 3. The parameter is the population mean change for visitors represented by the study. 4. The question should ask about the mean of the within-visitor changes rather than compare two independent means.

Answer

One valid question is: “For visitors represented by the study, what is the mean change in exhibit time from before to after the new signs were installed?”

Investigative questions 55

A botany lab takes a simple random sample of seedlings from its current batch of one species. Each seedling grows in an identical light box. The seedlings are randomly assigned to blue or red light, with water, soil, and temperature kept the same. Height gain is measured after \(4\) weeks. Rewrite “Is mean height gain different under blue and red light?” so that the target population and causal comparison are clear.

Hints

- What is measured after four weeks? - Which seedlings could the lab randomly select? - What does assigning the light color at random allow the researchers to investigate?

Solution

1. The population is the lab's current batch of this species. 2. Random assignment makes a cause-and-effect comparison of the two light colors a valid goal. 3. Identical boxes and other conditions avoid confounding light color with growth-chamber conditions.

Answer

“For seedlings in the lab's current batch, does blue light cause a different mean height gain over \(4\) weeks than red light?”

Investigative questions 56

A city compares satisfaction ratings from people who used a public plaza before renovation with ratings from a different sample of users after renovation. An analyst asks, “What is the mean change in satisfaction for plaza users?” Explain why that question does not match the data and rewrite it.

Hints

- Determine whether the same people were measured twice. - Ask what quantity can be formed from two independent samples. - Rewrite the comparison without implying individual before-and-after differences.

Solution

1. Mean change requires paired measurements from the same observational units before and after the renovation. 2. The study uses different users in the two periods, so individual changes cannot be calculated. 3. The available data can address a difference between two population means. 4. A matching question is: “For people who used the plaza during the study periods, is the mean satisfaction rating after renovation different from the mean rating before renovation?”

Answer

The question incorrectly asks for paired change even though different people were sampled. A suitable question is: “For people who used the plaza during the study periods, is the mean satisfaction rating after renovation different from the mean rating before renovation?”

Investigative questions 57

A food cooperative asks, “What is the mean price of a dozen eggs at local markets, and is that mean less than \(\$4.00\)?” Explain why this combines two analysis goals, and rewrite it as two separate investigative questions.

Hints

- Identify the part that asks for an unknown value. - Identify the part that compares a parameter with a fixed benchmark. - Give each statistical purpose its own complete question and population.

Solution

1. Asking for the mean price as an unknown value is an estimation goal. 2. Asking whether the mean is less than \(\$4.00\) is a directional hypothesis-testing goal. 3. The questions should be separated so each guides one analysis. 4. Suitable questions are: “What is the mean price of a dozen eggs at all local markets during the study period?” and “Is the mean price of a dozen eggs at all local markets during the study period less than \(\$4.00\)?”

Answer

The original combines estimation with a directional test. Separate questions are: 1) What is the mean price of a dozen eggs at all local markets during the study period? 2) Is the mean price of a dozen eggs at all local markets during the study period less than \(\$4.00\)?

Investigative questions 58

A transit analyst samples weekday trips on bus Routes 4, 7, and 12 during July 20–26, 2026, and asks, “Which route had the largest sample mean delay?” Explain why this is a question about the sample rather than a question about all trips. Rewrite it to compare the route populations.

Hints

- Is the question asking for a result from the sampled trips or a conclusion about other trips too? - Which days and routes define the population? - Which larger group would the revised question have to describe?

Solution

1. The largest of the three sample means can be identified from the collected trips alone. 2. The population question asks which route has the largest mean delay across all its weekday trips that week, not just the sampled trips.

Answer

The original asks about observed sample means only. A population question is: “Which of Routes 4, 7, and 12 had the greatest mean delay across all weekday trips during July 20–26, 2026?”

Investigative questions 59

A school district has data on whether each sampled student uses a reusable water bottle. An analyst proposes the question, “What proportion of schools have high reusable-bottle use?” Explain why the proposed question does not match the observational units and give a question the data can answer directly.

Hints

- Identify the unit represented by one row of data. - Compare that unit with the unit named in the proposed conclusion. - Rewrite the question using the variable that is directly recorded for each unit.

Solution

1. The observational units are students, and the recorded variable is each student's bottle use. 2. Determining whether a school has high use requires a school-level definition and enough student data within each school. 3. The current student-level data directly support a question about a student population. 4. A matching question is: “What proportion of students in the district use a reusable water bottle?”

Answer

The data are recorded for students, not schools, so the proposed school-level question requires additional aggregation and a definition of “high.” A direct question is: “What proportion of students in the district use a reusable water bottle?”

Investigative questions 60

A scooter rental service randomly samples standard commuter scooters and longer-range scooters from its fleet and records their charging times. Write two investigative questions: one about the population mean charging time for standard commuter scooters and one comparing the two population mean charging times.

Hints

- Keep charging time as the response variable for both questions. - The first question needs just one scooter model. - For the second, what must be compared across the two model populations?

Solution

1. The first question targets the mean charging time for all standard commuter scooters in the service's fleet. 2. The second compares the mean charging times for all standard commuter scooters and all longer-range scooters in that fleet. 3. Both use charging time as a quantitative response variable.

Answer

Estimation: “What is the mean charging time of standard commuter scooters in the rental service's fleet?” Comparison: “Is the mean charging time different between standard commuter and longer-range scooters in the rental service's fleet?”

Investigative questions 61

A workplace study asks, “How many messages are sent?” The available data record the number of work-related messages sent by each employee on each sampled workday. Rewrite the investigative question so the observational unit, variable, population, and parameter are clear.

Hints

- Identify what one row of the dataset represents. - State the count recorded for that unit. - Specify whether the target is a total, a mean per employee, or a mean per employee-workday.

Solution

1. The observational unit is an employee-workday. 2. The variable is the number of work-related messages sent during that employee-workday. 3. The population is all employee-workdays at the company during the study period, and the parameter is the population mean number of messages. 4. A complete question is: “What is the mean number of work-related messages sent per employee-workday at the company during the study period?”

Answer

What is the mean number of work-related messages sent per employee-workday at the company during the study period?

Investigative questions 62

A delivery company wants to compare collision risk for electric and gasoline vans, but the two fleets travel different total distances. Write an investigative question using a collision rate per mile rather than a raw collision count.

Hints

- Identify the exposure amount that differs between the two fleets. - Replace total counts with a quantity that adjusts for that exposure. - Include both fleet types, the study period, and the intended population.

Solution

1. Raw collision counts are not directly comparable when total miles traveled differ. 2. The response should be a rate with collisions as the numerator and miles traveled as the exposure. 3. A complete question is: “For the company's delivery vans during the study period, is the collision rate per mile different between electric vans and gasoline vans?”

Answer

For the company's delivery vans during the study period, is the collision rate per mile different between electric vans and gasoline vans?

Investigative questions 63

Before collecting data, a school's theater club asked, “What is the mean number of rehearsal hours per week among club members this semester?” After finding some very large values, the club instead wanted to ask, “Why do a few members rehearse much longer than the others?” Explain why the new question should not replace the original question in this study, and give one useful next step.

Hints

- Which question was chosen before the results were known? - What does a record of rehearsal hours reveal, and what does it not reveal? - Can the unusual values be described now and explained in a different study later?

Solution

1. The original question clearly names a population mean and was set before the data were examined. 2. The new question asks about causes, but recorded hours alone cannot show why some members rehearse longer. 3. Report the mean for the original question and describe the unusual values. The club can plan a separate study of possible reasons.

Answer

Keep the planned mean-hours question as the primary question. Rehearsal-hour data alone cannot tell why some values are large. Report the mean and unusual values, then collect information about possible causes in a later study.

Investigative questions 64

Noa proposes the question, “Are the new study rooms better than the old study rooms?” Explain why this is not yet a well-defined investigative question. Rewrite it as one measurable statistical question about noise.

Hints

- Identify the word whose meaning could differ from person to person. - Replace the vague judgment with a characteristic that can be recorded consistently. - Include the groups and conditions needed for a fair comparison.

Solution

1. The word “better” does not identify a specific characteristic to measure or a defined comparison. 2. A measurable question must specify the observational units, the variable, and the groups or conditions being compared. 3. One valid revision is: “How does the distribution of sound levels, in decibels, during occupied hours compare between the new and old study rooms?”

Answer

“Better” is undefined, so the proposed question does not specify what data are needed. One valid revision is: “How does the distribution of sound levels, in decibels, during occupied hours compare between the new and old study rooms?”

Investigative questions 65

A researcher asks, “What is the distribution of monthly electricity use?” Name two important boundaries that are missing, and rewrite this as a question about occupied apartments in one building.

Hints

- Which apartments provide the observations? - Could a month's electricity use change from year to year? - Include the measurement unit in the question.

Solution

1. The question does not name the apartments that make up the population. 2. It does not specify which month and year will be measured. 3. A complete question can name one building, a specific month, and kilowatt-hours.

Answer

The apartment population and the month and year are missing. One revision is: “What is the distribution of electricity use, in kilowatt-hours, among occupied apartments at Willow Court during January 2026?”

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