Free math worksheets

- Check whether every student is included in the selection process. - Look for a plan that does not depend on being in one place or choosing to respond. - A fair selection method should not favor a particular type of student.

1. A representative random sample should give every student a known chance to be selected. 2. Plan b) includes the full roster and selects numbers randomly without relying on volunteering or location. 3. The other plans favor students who are available, respond quickly, or have high attendance.

b) Number every student and randomly select \(60\) different numbers.

- Separate the full group of interest from the people who were actually questioned. - Ask how a student had to behave in order to be included. - Consider which students had no chance to be selected.

1. The population is all middle school students in the city. 2. The sample is the first \(40\) middle school students who entered the library after school on Tuesday. 3. Students already visiting the library are likely to use library services more often than students who do not visit. 4. The sample is a convenience sample, so an inference about all middle school students may be biased.

Population: all middle school students in the city. Sample: the \(40\) students surveyed. The method may overrepresent frequent library users, so the sample may not be representative.

- Compare the hours included in the survey with the route's full operating day. - Identify riders who never have a chance to receive a card. - Consider whether travel conditions differ at excluded times.

1. The population includes riders at all times of day. 2. Only daytime riders can receive a survey card. 3. Commuters who ride before \(9\) a.m. or after \(2\) p.m. are underrepresented. 4. Their experiences may differ because crowding and schedules change by time of day.

The sample has undercoverage because riders outside \(9\) a.m. to \(2\) p.m. cannot be selected. Morning or evening commuters may be underrepresented.

- Compare the population named in the question with the names available for selection. - Random choice cannot include a person who is missing from the list. - Identify the visitor groups with zero chance of selection.

1. The population is all museum visitors. 2. The sampling list contains only annual members. 3. First-time visitors and nonmembers have no chance to be selected. 4. Random selection from an incomplete list does not make the sample representative of all visitors.

No. The list excludes nonmembers and first-time visitors, so the sample has undercoverage even though names are selected randomly.

- Identify which plan samples from the entire neighborhood. - Consider how being at the field might be connected to the opinion being measured. - Prefer the plan that gives nonusers a chance to be included.

1. Plan A favors people who already use the field in the evening and may especially value lighting. 2. Plan B selects addresses throughout the neighborhood. 3. Plan B is more likely to include both field users and nonusers, so it better represents neighborhood adults.

Plan B, because it randomly samples addresses throughout the neighborhood instead of sampling only evening field users.

- Check each generated label against the roster's valid range. - A student should appear only once in a sample without replacement. - Continue reading until five valid, distinct labels have been kept.

1. Keep \(118\) and \(407\). 2. Skip the second \(407\) because that student is already selected. 3. Keep \(052\). 4. Skip \(611\) and \(000\) because valid labels are \(001\) through \(480\). 5. Keep \(294\), skip the repeated \(052\), and keep \(371\).

\(118,\ 407,\ 052,\ 294,\ 371\). Repeated labels and labels outside \(001\) through \(480\) are skipped.

- Sampling is only one part of collecting trustworthy data. - Compare the emotional or judgmental words in the two questions. - Choose the wording that does not suggest a preferred answer.

1. Both surveys use the same random sample, so their sampling methods are equally strong. 2. Survey 1 contains loaded words that may push respondents toward opposition. 3. Survey 2 uses more neutral wording and is more likely to measure opinions accurately.

Survey 2. Its neutral wording is less likely to influence responses.

- Focus on how the very first name enters the sample. - A fixed interval alone does not make the starting position random. - The repair should give each possible starting position a chance.

1. The starting name \(10\) was chosen deliberately rather than randomly. 2. Without a random start, not every possible systematic sample has a chance to be selected. 3. Randomly select a starting number from \(1\) through \(25\), then choose every \(25\)th name.

The plan lacks a random start. Randomly choose the first position from \(1\) through \(25\), then select every \(25\)th name.

- Look for a time trend that could make early days different. - The sampling frame should include every day in the season. - Use chance to select days spread across the full period.

1. Attendance during the first part of summer may differ because of weather, school calendars, or opening promotions. 2. Consecutive early days do not represent the full season well. 3. Number the season days \(1\) through \(90\) and randomly select \(12\) different day numbers.

The first \(12\) days may not reflect attendance later in the summer. Number all \(90\) days and randomly select \(12\) distinct days.

- Ask who decides whether to enter the sample. - Think about buyers with ordinary experiences. - Compare website reviewers with the full group of buyers.

1. Posting a review is voluntary, so buyers with very strong positive or negative experiences may be more likely to respond. 2. Buyers who do not visit the website after purchasing have no chance to be included. 3. The distribution of posted opinions may differ from the opinions of all buyers.

The reviews are voluntary responses and exclude buyers who do not post on the website. Strong opinions may therefore be overrepresented.

- Distinguish the number of selections from the number of different people. - Consider what happens when one label appears twice. - State a stopping rule based on unique labels.

1. A student number can appear more than once. 2. Repeated selections do not add new students, so fewer than \(25\) distinct students may be included. 3. The program should reject repeats and continue until \(25\) different student numbers have been selected.

Repeated numbers can select the same student more than once. Reject repeats and continue until the sample contains \(25\) distinct students.

- Compare the target population with the list used for selection. - Identify which grades have zero chance to enter the sample. - Ask whether the missing group could differ on sports participation.

1. The target population is all district students. 2. The sampling frame includes only students taking physical education this semester. 3. Students in excluded grades have no chance to be selected. 4. Sports participation may be related to course enrollment or grade, so the estimate may be biased.

The sample has undercoverage because students not taking physical education this semester cannot be selected.

- Check whether every batch appears in the selection process exactly once. - Ask what thorough mixing and identical cards do to the chances of selection. - Consider why drawing without replacement matters.

1. Every batch number from \(101\) through \(500\) appears exactly once. 2. Identical, thoroughly mixed cards give each batch an equal chance to be selected. 3. Drawing without replacement prevents the same batch from being selected more than once. 4. Therefore, the method can produce a random sample of \(20\) distinct batches.

Each batch is represented once on an identical card and has an equal chance to be drawn. Because cards are not replaced, the sample contains \(20\) different batches.

- Identify the group within which the random process operates. - Compare that group with the full population named in the question. - A random sample from one subgroup is not necessarily random for the whole population.

1. The random start makes selection random within the final lunch period. 2. Students in the first two lunch periods have no chance to be selected. 3. Menu availability, waiting time, or grade assignments may differ by lunch period. 4. The sample cannot automatically represent all students.

No. The sample is random only within the final lunch period and excludes students in the other two periods.

- A random-looking action is fair only when the physical objects are comparable. - Consider whether some slips are easier to touch or grab. - Remove physical differences before drawing.

1. Different shapes and thicknesses can affect how slips mix and how easily they are drawn. 2. Students may not have equal chances of selection. 3. Use identical slips folded in the same way, mix thoroughly, and draw without looking, or use a random-number generator on a roster.

Not necessarily. Make all slips identical in size and folding before mixing, or use randomly generated roster numbers.

- Identify what all sampled people have in common. - Ask whether that shared trait could affect the opinion. - Remember that more observations do not fix a biased selection process.

1. People attending a baseball game are more likely than other residents to be interested in baseball. 2. Interest in baseball may be connected to support for a stadium. 3. The sample overrepresents baseball fans, so its large size does not remove selection bias.

Game attendees are likely to support baseball more than town residents as a whole. The sample is biased toward baseball fans.

- Ask whether chance determines which plants are selected. - Consider whether plants at one position in a row could share conditions. - Build the replacement plan from a complete list of all plants.

1. The greenhouse contains \(20\cdot 50=1000\) plants. 2. The first two plants in each row are chosen because of position, not by chance. 3. Plants at the ends of rows may receive different light, water, or foot traffic than plants elsewhere. 4. A better plan is to number all \(1000\) plants and randomly select \(40\) different numbers.

No. The first two plants in each row are selected by position and may differ from other plants. Number all plants and randomly select \(40\) distinct plant numbers.

- Focus on the list before focusing on the random choice. - Identify adults who cannot appear on that list. - Ask whether the omitted households could differ from included households.

1. Adults in households without listed landlines are absent from the sampling frame. 2. Cell-phone-only households and people with unlisted numbers have no chance to be selected. 3. Those groups may differ from listed-landline households, creating undercoverage.

The list excludes cell-phone-only and unlisted households. Random selection from an incomplete list can still be unrepresentative.

- Compare which parts of the park can enter each sample. - Consider how entrance areas may differ from remote areas. - A representative plan should not be based mainly on convenience.

1. Random selection gives every map square a chance to be included. 2. Entrance squares may have unusual litter because of heavier foot traffic and nearby trash cans. 3. Sampling across the full map is more likely to represent conditions throughout the park.

The random-square plan includes the whole park in the sampling frame and avoids favoring the unusual conditions near the entrance.

- Check whether every school contributes data to the comparison. - Focus on how trays are chosen within each school. - Think about what must be kept consistent when results from different schools are compared.

1. Every school contributes the same number of randomly selected trays, so no school is left out of the comparison. 2. Random selection within each school helps the sampled trays represent that school’s lunches. 3. For a fair comparison, staff should measure waste in the same way at every school and sample comparable kinds of days.

Strength: all six schools are represented by random samples of the same size. Condition: staff should use the same measurement method and comparable sampling times at every school.

- Compare the smallest and largest labels in the roster with the generator's endpoints. - Look for one valid label that cannot appear and one generated label that is invalid. - Make the two ranges match exactly.

1. Label \(0\) can never be generated, while \(400\) does not belong to any student. 2. One fix is to generate integers from \(0\) through \(399\). 3. Another fix is to relabel students \(1\) through \(400\) and keep the generator range \(1\) through \(400\).

The generator omits label \(0\) and can produce invalid label \(400\). Either generate \(0\)–\(399\), or relabel students \(1\)–\(400\).

- Identify how students are selected within each grade. - Compare the amount of data collected from the three grades. - Decide what part of the measurement process must be consistent.

1. Each grade is represented by its own random sample of \(20\) students. 2. Using the same sample size makes the amount of sampled data comparable across grades. 3. The researcher should ask the same question, use the same definition of a homework day, and collect data over the same time period for each grade.

The plan can support a grade-by-grade comparison because each grade has a random sample of \(20\) students. The question and data-collection period should be the same for all three grades.

- Compare the group available at the survey point with the group named in the goal. - Identify who left the process before selection occurred. - Ask whether leaving early could be related to the response.

1. The target population is everyone who started. 2. Only finishers are available at the finish line. 3. People who stopped early may have found the race especially difficult. 4. Random selection among finishers cannot represent nonfinishers.

The method excludes people who did not finish, and those runners may have experienced greater difficulty. It represents finishers, not all starters.

- Determine whether every office has a chance to enter the sample. - Ask what the beginning of the directory might overrepresent. - Base the improved plan on a complete list of offices.

1. The building has \(10\cdot24=240\) offices. 2. The first \(30\) directory listings are chosen by list position, not by chance. 3. The directory order may favor particular floors, companies, or office numbers. 4. A better plan is to label all \(240\) offices and randomly select \(30\) different labels.

The first \(30\) directory listings may favor certain parts of the building. Number all \(240\) offices and randomly select \(30\) distinct office numbers.

- Compare the number of responses with the number of different people. - Think about how repeated answers change a person's influence. - Use a rule that prevents the same member from being selected twice.

1. Selection with replacement can choose one member several times. 2. Repeated answers give those members more influence and reduce the number of distinct viewpoints. 3. Select \(12\) different members without replacement so each sampled member contributes once.

Repeated selections overrepresent some members. Select \(12\) distinct members without replacement.

- Separate the first selection from the next five selections. - Ask whether students far from the first seat can be chosen. - Make every student eligible in the replacement plan.

1. Only the first student is selected by chance. 2. The other five students are chosen because they sit nearby, so many students have little or no chance to be included. 3. Nearby students may share classroom position or other characteristics. 4. A better plan is to number every student on the class roster and randomly select \(6\) different numbers.

The five nearby students are selected by location rather than chance. Randomly select \(6\) distinct students from the complete class roster instead.

- Find the boundary of the group from which the random choices are made. - Identify trees with zero chance of selection. - Consider how edge conditions could differ from interior conditions.

1. Chance is used only within the two outer rows. 2. Interior trees have no chance to be selected. 3. Edge trees may receive different sunlight, wind, water, or pest exposure. 4. The manager must randomly sample from all orchard rows for an orchardwide inference.

The sample is random only among edge trees and excludes interior trees. Randomly select trees from throughout the orchard.

- Distinguish the sampling method from the exact composition of one resulting sample. - Ask what random selection guarantees for each student. - Remember that random samples naturally vary from one selection to another.

1. The school has \(144+168+132+156=600\) students. 2. Selecting \(48\) different students randomly from the complete roster gives every student a chance to be included. 3. A random sample can differ somewhat from the population proportions because of chance. 4. Exact proportional agreement is not required for the method to be random, although a very unusual sample may give a less accurate estimate.

No. A random sample does not have to match every population proportion exactly. The method is random because all \(600\) students are eligible and \(48\) distinct students are selected by chance.

- Identify why this particular homeroom was selected. - Ask which students have no chance to enter the sample. - Use a sampling frame that includes the entire school.

1. The homeroom is chosen because it is convenient, not by chance. 2. Students in one homeroom may share a grade level, schedule, or other characteristics that differ from the rest of the school. 3. A better plan is to select student names randomly from the complete school roster.

The sample is a convenience sample from one homeroom and may not represent the school. Randomly select students from the complete roster instead.

- Compare where sampled students are found with the behavior being estimated. - Identify the students who cannot enter the line. - Choose a sampling frame unrelated to how students travel.

1. The sampling frame contains students waiting for buses. 2. Students who walk are not in the bus-loading line and have no chance to be selected. 3. The sample will severely underrepresent or completely miss walkers. 4. The sample should be selected from the full student roster.

The plan samples only bus riders, so walkers cannot be represented. Use a random sample from the complete roster.

- Ask whether position on the pallet could affect damage. - Identify which layers have zero chance of selection. - Extend the random labels to all box positions.

1. Boxes in the top layer may experience different pressure and handling than boxes lower in the pallet. 2. Lower boxes have no chance to be selected. 3. Number box positions throughout the pallet and randomly select positions from all layers.

Top-layer boxes may have different damage rates and exclude lower layers. Randomly select box positions from every layer of the pallet.

- Random does not mean every sample has the same composition. - Think about whether chance can produce different outcomes on repeated trials. - Separate ordinary sample-to-sample variation from systematic bias.

1. Random samples do not have to match the population proportions exactly. 2. Two independent random samples can contain different subgroup counts because of chance. 3. The difference alone is not evidence that either method was nonrandom.

No. Random samples naturally vary, so different numbers of seventh graders can occur even when both methods are valid.

- Identify what determines which songs enter the sample. - Ask whether recent listening can differ from long-term listening. - Build the improved plan from the complete history.

1. The most recent \(30\) songs are selected by time order, not by chance. 2. A user may have listened to one genre more often recently than across the full history. 3. The sample may therefore overrepresent recent listening patterns. 4. A better plan is to number all song occurrences in the history and randomly select \(30\) different positions.

The \(30\) most recent songs may not represent the full listening history. Randomly select \(30\) distinct song positions from the complete history instead.

- Identify the behavior required to enter the sample. - List common ways families leave school without using that location. - Consider whether transportation choice could affect the opinion.

1. Families whose children ride buses do not use the pickup lane. 2. Families whose children walk, bike, attend after-school programs, or are picked up elsewhere may also be missed. 3. These transportation choices may be connected to traffic opinions.

Bus-rider families and families whose children walk or bike may be underrepresented. Families using after-school care may also be missed.

- Compare the people on the list with the current membership. - Identify both the extra group and the missing group. - Correct the sampling list before applying a random method.

1. Former members can be selected even though they are not part of the current membership. 2. Recent members have no chance to be selected because their names are missing. 3. Random selection from an inaccurate list does not represent the current population. 4. The center should update the spreadsheet so it contains every current member and no former members before drawing the sample.

Former members may be selected incorrectly, while recent members are excluded. The center should update the list to include exactly the current members before sampling.

- Identify which valid IDs the program can never choose. - Random selection is valid only within the set made eligible. - Make the program’s selection list match the full roster.

1. Students whose IDs end in \(0\) have no chance to be selected. 2. The program therefore excludes part of the school before making random choices. 3. A valid plan must allow every student ID on the roster to be selected. 4. The program should sample from the complete set of student IDs and reject only repeated selections if \(75\) distinct students are needed.

The program excludes students whose IDs end in \(0\), so not every student is eligible. Use the complete roster of IDs when making the random selections.

- Check whether every school contributes data. - Focus on how students are selected within each school. - Think about what could be lost by combining all schools into one number.

1. Every school is represented by the same number of randomly selected students. 2. Random selection within each school helps the sample represent that school. 3. Schools may differ in schedules, programs, or student experiences. 4. Reporting results separately preserves those differences instead of hiding them in one combined result.

Strength: every school has a random sample of \(10\) students, so the schools can be compared. The district should report school results separately because student experiences may differ from one school to another.

- Match the eligible times to the population that can be represented. - Identify the shift with zero chance of selection. - Limit the conclusion to the part of the factory included in the sampling frame.

1. The random times can represent production during the day and evening shifts. 2. Night-shift production has no chance to be observed. 3. The sample can support an inference about the included shifts. 4. It cannot support a factorywide inference unless the night shift is also sampled.

The sample can describe day- and evening-shift production, but not production across all three shifts.

- Separate the original random selection from the way replacement students were chosen. - Ask whether volunteering gives every remaining student the same chance to enter the sample. - Improve the plan by keeping the replacement process random.

1. The original selection was random because every student on the roster had a chance to be selected. 2. Replacing nonresponders with volunteers changes the sampling method because students with strong interest or more free time may be more likely to volunteer. 3. The final sample may therefore overrepresent students who chose to participate. 4. A better plan is to randomly select replacement student ID numbers from the students who were not chosen originally.

The volunteer replacements can introduce bias because volunteers may differ from the students who did not respond. The school should randomly select replacement students from the remaining roster instead.

- Think about who decides whether to participate. - Ask whether every student is equally likely to see the invitation. - Build the improved plan from a complete list of the population.

1. Participation is voluntary, so students with strong opinions may be more likely to respond. 2. Students who do not see the link or lack access at that time cannot respond. 3. A better plan is to select student ID numbers randomly from the complete school roster and survey those students.

The poll may have voluntary-response bias and may miss students who do not see the link. A better plan is to randomly select students from the complete roster.

- Distinguish between choosing a group randomly and choosing people from throughout the population. - Ask whether residents of one building could share characteristics. - Consider how the sample could include more parts of the complex.

1. The building is chosen randomly, so chance is used. 2. All sampled households come from one building. 3. Pet rules, apartment sizes, or resident characteristics may differ among buildings. 4. A sample drawn from several buildings would better represent the entire complex.

The plan uses chance, but one building may differ from the others. Surveying households selected randomly across all \(12\) buildings would be more representative.

- A random selection is only as complete as the list it uses. - Separate the selected sample from the group that actually replies. - Think about households that could be absent at either stage.

1. The voter list may omit households with no registered voters. 2. Some selected households may not return the survey, creating nonresponse. 3. Officials should check whether omitted and nonresponding households differ from responding households on recreation needs.

They should check coverage of households without registered voters and the amount and pattern of nonresponse. Either issue could make the responding sample unrepresentative.

- Look at which major groups are guaranteed to appear. - Then examine how many smaller groups within each grade are represented. - A plan can have both a useful feature and a remaining source of variation.

1. Selecting one homeroom from each grade ensures that all three grades are represented. 2. Chance is used within each grade when the homeroom is selected. 3. One homeroom may differ from other homerooms in that grade, so three whole classes may show more variation than a sample of students spread across many homerooms.

Strength: every grade is included through random selection. Limitation: each grade is represented by only one homeroom, which may not reflect the other homerooms.

- The sampling frame should include every word occurrence in the novel. - Give each occurrence one unique position number. - Select distinct position numbers by chance rather than choosing pages or locations first.

1. Use an electronic copy to place all word occurrences in one complete sequence. 2. Number the word occurrences from \(1\) through the total number of words in the novel. 3. Use a random number generator to select \(30\) different valid numbers. 4. Include the word occurrence at each selected position in the sample.

Number every word occurrence in the novel, then use a random number generator to select \(30\) different position numbers. This gives every word occurrence an equal chance to be selected.

- Ask why those five locations were chosen. - Consider whether the nearby environment could affect the measured characteristic. - Use the complete shoreline, not ease of access, as the sampling frame.

1. Parking-lot access determines which spots are sampled, so the method is a convenience sample. 2. Conditions near the parking lot may differ because of runoff, shade, or foot traffic. 3. A better plan is to divide the shoreline map into equal segments, number the segments, and randomly select segments to inspect.

The sample may overrepresent conditions near the parking lot. Divide the full shoreline into numbered segments and randomly select segments from the entire pond edge.

- Compare each grade's size with the district total. - Use the same part-to-whole relationship in the sample of \(100\). - Check that the three sample counts add to the requested size.

1. Grade \(6\) is \(\frac{300}{1200}=\frac14\) of the district, so select \(\frac14\cdot100=25\). 2. Grade \(7\) is \(\frac{420}{1200}=0.35\) of the district, so select \(0.35\cdot100=35\). 3. Grade \(8\) is \(\frac{480}{1200}=0.40\) of the district, so select \(0.40\cdot100=40\). 4. The counts total \(25+35+40=100\).

Grade \(6\): \(25\) students Grade \(7\): \(35\) students Grade \(8\): \(40\) students

- Keep track of the group selected and the smaller group that supplied data. - Ask whether choosing not to reply could be related to the survey topic. - Think of a way to learn whether missing responses matter.

1. Random selection applies to the \(150\) chosen households. 2. The analyzed group contains only the \(42\) respondents. 3. Respondents may differ from nonrespondents, so the final data can have nonresponse bias. 4. A useful next step is to contact a random group of nonrespondents or compare known characteristics of respondents and nonrespondents.

The conclusion is not justified because nonresponse may make the \(42\) respondents different from the original random sample. The researcher should follow up with nonrespondents or check whether the two groups differ.

- Examine whether the replacement student is chosen by chance. - Ask whether some roster positions are more likely to become replacements. - Keep the replacement process consistent with the original selection method.

1. The original \(50\) labels are random. 2. The “next person” replacements are determined by alphabetical position, not chance. 3. Students following frequently absent students can be overrepresented. 4. A better rule is to randomly select replacement labels from the remaining roster or follow up with the originally selected students.

The replacements are not random. Randomly select replacements from the remaining roster, or collect responses later from the originally selected students.

- Separate the random selection of ticket numbers from the way the survey is delivered. - Identify which selected people have no opportunity to respond. - Keep all randomly selected ticket holders eligible for data collection.

1. The original selection gives every ticket number a chance to be chosen. 2. The survey reaches only selected ticket holders with email addresses on file. 3. Selected ticket holders without recorded email addresses cannot respond, so the responding group may differ from all ticket holders. 4. The theater should contact every selected ticket holder using an available method, such as email, mail, or a phone number, rather than excluding selected people without email addresses.

The final responses may have undercoverage because selected ticket holders without email addresses on file cannot participate. The theater should provide a way to contact every selected ticket holder.

- Identify what is being counted and what the population consists of. - Ask whether one person can enter the sample more than once for the same reason. - Choose a sampling unit that gives each person one roster entry.

1. One student can appear in several club, team, and activity photographs. 2. Students with many activities can be counted repeatedly, while other students may appear only once. 3. Photographs therefore do not give students equal chances to be represented. 4. The sample should select unique students from a student roster instead.

Students can appear in different numbers of photographs, so active students are overrepresented. Randomly sample unique students from the roster instead.

- Compare the selection process before comparing the numbers sampled. - Ask whether membership in each sample could be connected to the opinion. - A larger sample reduces random variation but does not remove selection bias.

1. Group A has a larger sample, but participation and pep-rally attendance can be related to school attitudes. 2. Group B uses a smaller sample selected randomly from the full roster. 3. Representativeness depends strongly on how the sample is selected, so Group B is more trustworthy.

Group B. Its random roster sample is more likely to represent the school, even though it is smaller. A large biased sample can still give a biased result.

- Find each travel group's fraction of the school. - Apply each fraction to the sample size. - Use the total of the three selections as a check.

1. Bus riders are \(\frac{196}{560}=0.35\) of the school, so select \(0.35\cdot40=14\). 2. Car riders are \(\frac{224}{560}=0.40\), so select \(0.40\cdot40=16\). 3. Walkers or bikers are \(\frac{140}{560}=0.25\), so select \(0.25\cdot40=10\). 4. The sample sizes total \(14+16+10=40\).

Bus: \(14\) Car: \(16\) Walk or bike: \(10\)

- Think of a characteristic shared by neighbors on one street. - Ask whether that characteristic might be connected to internet access. - Spread the sample across more of the town.

1. A single street may contain mostly one type of housing or one income range. 2. Internet access may be connected to those street-level characteristics. 3. The plan can be improved by randomly selecting households from the townwide address list or by selecting several streets across different areas.

If the chosen street has unusually similar housing or income, it may not represent the town. Randomly sample households townwide or use several randomly selected streets.

- Separate problems with selecting people from problems with recording their answers. - Check whether every possible situation has an answer choice. - Consider students with more than one valid response.

1. The sample selection may be random, but the response choices omit students in other clubs and students in no club. 2. Some students belong to more than one club, yet the wording asks for only one. 3. A better survey allows multiple selections and includes “another club” and “no club.”

The question has incomplete response choices and does not handle multiple memberships. Allow students to select all that apply, including “another club” and “no club.”

- Identify who can appear in the search results. - Ask whether one person can contribute more than one data item. - Consider which opinions are most likely to be posted publicly.

1. Only students who use the platform and post publicly with the hashtag can be included. 2. Students with strong opinions may post more often, and one student may create several posts. 3. The posts are not a random sample of students and may overrepresent active users and strong opinions.

The sample excludes students who do not post publicly with the hashtag and can count frequent posters multiple times. It is not representative of all students.

- Name the exact region where chance was used. - Ask whether conditions in that region match the rest of the lake. - Expand the set of eligible locations to match the desired population.

1. The random points represent fish accessible within the shallow cove. 2. Fish size may vary with water depth, temperature, or habitat. 3. For a lakewide inference, the sampling frame must include locations throughout the lake, with random points chosen across different areas.

The data can describe fish in the sampled cove, not necessarily the whole lake. Randomly choose locations throughout the lake, including different habitats and depths.

- Determine who has an occupied seat and who does not. - Ask whether absence could be connected to the topic being studied. - Choose a list that contains every student in the target population.

1. Randomly choosing occupied seats represents students who attended the assembly. 2. Absent students have no occupied seat and cannot be selected. 3. Attendance at the assembly may be related to opinions about attendance policies. 4. The sample should be drawn from the complete student roster.

The sample excludes students absent from the assembly. Randomly select students from the complete roster instead.

- Compare the included packages with all packages. - Ask which deliveries are most likely to take extra time. - Predict the direction in which excluding them changes the average.

1. The target population is all packages. 2. The sampling frame includes only first-attempt deliveries. 3. Packages requiring another attempt usually take longer. 4. Excluding them is likely to make the estimated average delivery time too low.

The estimate is likely too low because slower packages requiring additional attempts cannot enter the sample.

- Separate the selection of households from the selection of people within households. - Ask whether every eligible person at a selected address has the same chance. - Use chance at both selection stages.

1. Addresses are selected randomly, but the person within each household is chosen by age. 2. Younger household members are always favored, and older members can be excluded whenever someone younger lives with them. 3. The within-household respondent should also be selected by a random rule.

The youngest-person rule favors younger residents. After selecting addresses, randomly select one eligible household member at each address.

- Distinguish who is selected from who is able to respond. - Ask whether the survey format creates unequal barriers. - Modify data collection so selected households can participate.

1. The household selection is random. 2. Some selected households may be unable or less willing to complete an English-only questionnaire. 3. Their nonresponse can be connected to language and possibly to the survey topic. 4. Offering the survey in commonly used languages would reduce this source of nonresponse bias.

An English-only survey may lose responses from selected households that use other languages, creating nonresponse bias. Provide translated versions or interpretation support.

- A good sampling label should connect to exactly one member of the population. - Check for people with no label and labels attached to more than one person. - Use the list designed to identify students rather than storage spaces.

1. Students without lockers have no chance to be selected. 2. Students sharing a locker are connected to one label, so the plan needs an extra nonrandom rule to choose between them. 3. Locker labels do not create one unique, equally likely entry per student. 4. Student roster numbers are a better sampling frame.

Students without lockers are excluded, and shared lockers do not correspond to one unique student. Use the student roster instead.

- Ask whether shelf position might carry meaning. - A repeated fixed position can align with a repeated arrangement rule. - Replace the fixed choice with independent random choices.

1. The third position may be connected to how employees arrange featured, new, or low-priced books. 2. Using the same position can create a repeated placement pattern. 3. Randomly select one valid position independently on each shelf, or randomly select book inventory numbers from the complete catalog.

The third position may have a special placement pattern. Randomly choose a position on each shelf or sample random inventory numbers from the full catalog.

- Identify the exact flow of people from which selections are made. - Ask why different fans might use different entrances. - Expand the sampling frame to cover the whole stadium.

1. Selection is systematic and random only among fans using Gate A. 2. Parking lots, transit stops, or seating sections may direct different travelers to different gates. 3. Fans entering through Gates B, C, and D have no chance to be selected. 4. Staff should sample at all entrances or use tickets from the full attendance list.

The method represents Gate A users, not necessarily all fans. Travel method may be related to entrance, so all gates should be included.