Free math worksheets

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

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

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

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

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.

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

- Look for a question with a clearly recordable response and a defined time period. - Reject wording that depends on personal judgment unless that judgment is operationally defined.

1. Question b) defines a measurable count, observational units, and a time period. 2. Questions a) and c) use vague terms that are not operationally defined.

b) How many visits did each sampled household make to a county trail during June?

- Identify the population mean named in the claim. - Preserve the direction of the comparison. - State the type of shipments and time period represented.

1. The variable is delivery time in days for a local shipment. 2. The parameter is the population mean delivery time for all local shipments during July 2026. 3. The directional claim is that the mean is less than \(2\,\text{days}\). 4. The question should state the population and benchmark.

One valid question is: “For all local shipments during July 2026, is the mean delivery time less than \(2\,\text{days}\)?”

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

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.

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

- Separate the full set of adult-cardholder visits from the smaller set of recorded visits. - State the question so that data from many visits are needed.

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 is: “What is the typical number of books checked out during a visit made by an adult cardholder this year?”

Population: all visits made by adult cardholders to the library this year. Sample: the \(180\) recorded visits. Investigative question: the typical number of books checked out during a visit in that population.

- Identify the complete group represented by the sample. - Phrase a question that the recorded arrival status can answer.

1. The population is all packages shipped by the company during that month. 2. One valid question is: “What proportion of packages shipped during the month arrived by the promised date?”

Population: all packages shipped during the month. One valid question: “What proportion of those packages arrived by the promised date?”

- 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.

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.

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

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

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?”

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

- Look for undefined scope and undefined measurement conditions. - Specify who is measured, over what interval, and what counts as screen time.

1. The population or group of interest must be specified. 2. The measurement period and operational definition of screen time must be specified.

The question must specify the population and how and over what time period screen time will be measured.

- Match the wording of the question to the exact comparison the college wants to make. - Check whether the population, variable, and groups are all identified. - Avoid choices that ask for a judgment or a different quantity.

1. The study compares a quantitative variable, weekly paid-work time, across two defined groups. 2. Choice b) names the population, the variable, the two groups, and the comparison of population means. 3. Choices a) and c) do not match the stated purpose.

b) It specifies the college population, the quantitative variable, the two student groups, and the comparison of their mean weekly paid-work times.

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

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.

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

- 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.

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.

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.

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

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.

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

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

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.

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?”

- Convert the operational definition of “late” into a yes-or-no variable. - State the benchmark and its direction. - Include the flights and time period to which the conclusion applies.

1. The variable is whether a domestic flight arrives at least \(15\,\text{minutes}\) late. 2. The parameter is the population proportion of the airline's domestic flights during June 2026 that are late by that definition. 3. The direction is greater than \(15\%\).

One valid question is: “For the airline's domestic flights during June 2026, is the proportion that arrive at least \(15\,\text{minutes}\) late greater than \(15\%\)?”

- Identify the yes-or-no outcome. - Name both population groups clearly. - Preserve the proposed direction of the difference.

1. The variable is whether a member uses mobile deposit. 2. The two parameters are the population proportions for the two age groups. 3. The direction is that the proportion for members younger than \(40\) is greater than the proportion for members age \(40\) or older. 4. The population is all members of the credit union.

One valid question is: “Among credit-union members, is the proportion who use mobile deposit greater for members younger than \(40\) than for members age \(40\) or older?”

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

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.

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.

- 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.

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.

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?”

- Identify what receives one recorded value. - Separate the recorded value from its population summary. - State what the company hopes to learn beyond the sample.

1. The observational units are the sampled deliveries. 2. The variable is the package-condition rating from \(1\) to \(5\) for each delivery. 3. The parameter is the mean condition rating for all deliveries in the target population. 4. The purpose is to estimate that population mean from the random sample.

Observational units: the sampled deliveries. Variable: package-condition rating on the \(1\)-to-\(5\) scale. Parameter: the mean condition rating for all target deliveries. Purpose: estimate that population mean.

- Define the yes-or-no outcome and its time window. - Name the two populations being compared. - Ask about population proportions, not the sample percentages already observed.

1. The variable is whether a discharged patient returns within \(30\) days. 2. The two parameters are the readmission proportions for all patients discharged from Units A and B during 2025. 3. The question asks whether those population proportions differ between the units.

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

- 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.

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.

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?”

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

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.

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?”

- Separate the full group the orchestra wants to understand from the visits actually measured. - Identify the value recorded for each selected visit. - Include the season in the population definition.

1. The intended population is all audience visits to orchestra performances during the season. 2. The sample is the audience visits represented by the selected people at the three performances. 3. The variable is the distance traveled for each sampled audience visit.

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

- Check whether the question states what is measured, what comparison is made, and whom the conclusion concerns. - Ask whether a reader could tell exactly which offices the study is intended to represent. - Preserve the comparison already stated while adding the missing scope.

1. The proposed question identifies the response variable and the comparison of two population means. 2. It does not identify the population to which the conclusion will apply. 3. A complete question is: “For all regional offices of the firm, is the mean time needed to restore data different between Backup System A and Backup System B?”

The missing component is the population. One complete question is: “For all regional offices of the firm, is the mean time needed to restore data different between Backup System A and Backup System B?”

- Compare the type of conclusion requested by each question. - One question asks for a decision about a fixed value; the other asks for a plausible interval. - Check that the chosen question names both the measured condition and the population.

1. Question a) asks whether the population proportion exceeds a specified value, so it is framed for a hypothesis test. 2. Question b) asks for a plausible interval for the population proportion, so it is framed for a confidence interval. 3. Question b) also states the variable and the population of broadcasts.

b) fits the goal because it asks for an interval estimate of the population proportion. Question a) instead asks for a decision about a specified benchmark.

- 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.

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.

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.

- 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.

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?”

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

- Identify the yes-or-no outcome recorded for each bag. - Translate the airport's claim into a statement about a population proportion. - Preserve the direction of the benchmark comparison.

1. The variable is whether a carry-on bag screened during July 2026 requires a manual recheck because of a false alarm. 2. The parameter is the population proportion of all carry-on bags screened at the airport during July 2026 that require such a recheck. 3. The direction is less than \(2\%\). 4. A complete question is: “For all carry-on bags screened at this airport during July 2026, is the proportion requiring a manual recheck because of a scanner false alarm less than \(2\%\)?”

For all carry-on bags screened at this airport during July 2026, is the proportion requiring a manual recheck because of a scanner false alarm less than \(2\%\)?

- 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.

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?”

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

- Ask whether the same wording could refer to different years or places. - Define the group of workers and the period during which the outcome is counted. - Make the event being measured specific enough for consistent data collection.

1. The question identifies the variable but does not specify the geographic area or time period defining the worker population and smoky days. 2. A complete population must include a location and study period. 3. One valid version is: “What proportion of outdoor workers employed in the county during summer 2026 experienced breathing difficulty on at least one day classified as smoky during that summer?”

The question needs a defined location and time period. One complete version is: “What proportion of outdoor workers employed in the county during summer 2026 experienced breathing difficulty on at least one day classified as smoky during that summer?”

- 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.

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?”

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?”

- 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.

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?”

“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?”

- Match the wording of the question to the exact parameter in the planned analysis. - Replace an informal measure-of-center word with a defined population quantity. - Include the population and both comparison groups.

1. The word “typical” does not specify which measure of center is being compared. 2. The planned analysis concerns a difference in population means, so the investigative question should state that parameter. 3. A complete version is: “For shipments sent using either method, is the mean delivery time different between Shipping Method A and Shipping Method B?”

The word “typical” is ambiguous and does not specify the means planned for analysis. A suitable question is: “For shipments sent using either method, is the mean delivery time different between Shipping Method A and Shipping Method B?”

- Identify the observational unit on which one cost value should be recorded. - Distinguish a total across the clinic from an average across visits. - Add the population of visits and the relevant study period.

1. Total clinic spending is a clinic-level aggregate and does not identify a per-visit variable. 2. The intended observational unit is a patient visit, with cost recorded for each visit. 3. A matching question is: “What is the mean patient-care cost per visit for all visits to the clinic during the study period?”

The original asks for total spending rather than a per-visit mean. A suitable question is: “What is the mean patient-care cost per visit for all visits to the clinic during the study period?”

- 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.

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?”

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?”

- 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.

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?”

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

- Make each important term precise enough to collect data. - Name who is being studied and what will be recorded. - Make the wording ask about a distribution or typical rating rather than an undefined yes-or-no judgment.

1. Specify the population as students who bought a cafeteria lunch during a stated period. 2. Replace “good” with a measurable rating scale. 3. A 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 \(5\)?”

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 \(5\)?”

- 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.

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.

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

- Use the exact measurements and time points that are available. - Make the group being studied explicit. - Define the subtraction order so the sign of each runner’s change has a clear meaning.

1. The available data provide a paired change in time for each runner. 2. Define the change as final-meet time minus first-meet time, so a negative value represents improvement. 3. A precise question is: “What is the distribution of final-meet time minus first-meet time for the team’s \(5\)-kilometer runners?”

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

- 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.

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.

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?”

- Identify the two population means being compared. - Use wording that asks for a range of plausible parameter values. - Avoid stating a greater-than, less-than, or not-equal claim.

1. The parameter is the difference between the two population mean processing times for applications submitted during 2025. 2. A confidence-interval question asks for a range of plausible values rather than asserting a direction or testing a claim. 3. The population consists of all online and mailed permit applications submitted to the agency during 2025.

One valid question is: “What is a plausible interval for the difference in mean processing time between all online and all mailed permit applications submitted to the agency during 2025?”

- 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.

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.

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

- 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.

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.

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

- 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.

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.

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?”

- 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.

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.

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?”

- 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.

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.

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?”

- Identify the population mean the group wants to estimate. - Phrase the purpose as finding a plausible interval for that parameter. - Specify one consistent quantitative measurement for every sampled household.

1. The parameter is the county population mean weekly food-waste amount. 2. An interval-estimation question asks for a plausible interval for that mean. 3. Each sampled household must provide a measured weekly amount of food waste using a consistent unit and time period.

One valid question is: “What is a plausible interval for the mean amount of food waste produced per household per week in the county?” Collect the amount of food waste produced during one defined week from each sampled household, using the same unit for all households.

- Name both program conditions. - Use the exact retention outcome and time period. - State the causal relationship the random assignment is intended to evaluate. - Limit the population to students represented by the study.

1. The explanatory variable is orientation program assignment. 2. The response is first-semester retention. 3. Random assignment makes a causal question appropriate. 4. Without a stated random-sampling plan, the population is the students participating in the experiment.

One valid question is: “Among students participating in the experiment, does assignment to the new orientation program cause a higher first-semester retention rate than assignment to the current program?”

- Match each location's later value with its own earlier value. - Define a change for each location. - Ask about the population mean of those paired changes.

1. Each beach supplies a before-and-after pair. 2. The analysis variable is the change in fragment count for each beach. 3. The parameter is the population mean change across the beaches represented. 4. The question should ask about the mean of the within-beach changes rather than compare two independent means.

One valid question is: “For beaches represented by the study, what is the mean change in plastic-fragment count from before to after the cleanup policy?”

- 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.

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.

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

- 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.

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.

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?”

- Separate the value recorded for each account from the population summary. - Use estimation language rather than a directional claim. - Keep the one-year population definition consistent.

1. The data-collection component is whether each sampled residential account had a leak during the year. 2. The analysis component is estimation of the population proportion within a range of plausible values. 3. The population is all residential accounts served during that year.

Data collection: record whether each sampled residential account had a leak during the year. Analysis: estimate, with an interval, the proportion of all residential accounts that had a leak.

- Keep the measured response and two treatment conditions from the original question. - Add the group of units to which the conclusion should apply. - Use the design description to decide whether the question may ask about causation.

1. The response variable and comparison are already stated as mean height gain under the two light treatments. 2. Because no random-sampling plan is stated, the population is the seedlings participating in the experiment. 3. Because light treatment is randomly assigned, the question may be phrased in terms of a causal effect. 4. A complete question is: “For seedlings participating in the experiment, does blue light cause a different mean height gain over \(4\) weeks than red light?”

One complete question is: “For seedlings participating in the experiment, does blue light cause a different mean height gain over \(4\) weeks than red light?”

- 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.

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?”

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?”

- 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.

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 \(\$4.00\)?”

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 \(\$4.00\)?

- Check each option for what is measured, what statistical result is sought, and whom the result concerns. - Distinguish a sample summary from a population parameter. - Look for wording that clearly signals interval estimation.

1. Question c) identifies the variable, the interval-estimation goal, and the population for the conclusion. 2. Question a) does not define a population or a specific analysis target. 3. Question b) focuses on a sample statistic rather than the population parameter and does not state an inferential goal.

c) is complete. Question a) lacks a defined population and analysis target. Question b) asks only about the sampled accounts rather than estimating the population mean.

- Decide whether the original can be answered without reasoning beyond the observed sample. - Replace sample summaries with the corresponding population quantities. - Identify the full set of trips to which the comparison should apply.

1. The original question can be answered directly from the three observed sample means. 2. It does not ask about population parameters or a conclusion beyond the sampled trips. 3. An inferential version is: “For all weekday trips during July 20–26, 2026, on Routes A, B, and C, were the mean delays different among the three routes?”

The original asks only which observed sample mean is largest. An inferential question is: “For all weekday trips during July 20–26, 2026, on Routes A, B, and C, were the mean delays different among the three routes?”

- 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.

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?”

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?”

- Keep the response variable the same in both questions. - Make the first question seek a plausible interval for one population parameter. - Make the second question seek a difference between two population parameters.

1. An estimation question for Model A is: “What is a plausible interval for the mean charging time of all Model A scooters in the fleet?” 2. A comparison question is: “For all scooters in the fleet, is the mean charging time different between Model A and Model B?” 3. The questions use the same response variable but require different analyses and conclusions.

Estimation: What is a plausible interval for the mean charging time of all Model A scooters in the fleet? Comparison: For all scooters in the fleet, is the mean charging time different between Model A and Model B?

- 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.

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?”

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

- 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.

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?”

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