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Random sampling methods

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5546697

A middle school has \(720\) seventh-grade students. The principal randomly selects \(60\) of them to answer a survey. What is the population, and what is the sample?

Hints

- The population is the full group the study is about. - The sample is the group actually selected and observed.

Solution

1. The population is the entire group the principal wants information about: all \(720\) seventh-grade students. 2. The sample is the smaller group actually selected for the survey: the \(60\) students.

Answer

Population: all \(720\) seventh-grade students Sample: the \(60\) randomly selected students

5415117

A city library wants to know how often all middle school students use its services. A librarian surveys the first \(40\) students who enter the library after school on Tuesday. Identify the population, identify the sample, and explain why this sampling method may not represent the population.

Hints

- 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.

Solution

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.

Answer

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.

5415127

A school wants a random sample of \(60\) students from its \(900\)-student roster. Which plan best supports a valid inference about the whole school? a) Ask students to volunteer, then randomly choose \(60\) names from the volunteers. b) Number every student on the full roster from \(1\) to \(900\), then randomly select \(60\) different numbers. c) Randomly choose one lunch period, then survey \(60\) students from that lunch period. d) Ask each homeroom teacher to nominate students until \(60\) names have been collected.

Hints

- A random step is not enough if the people entering that random step already come from a restricted group. - Check which plan begins with all \(900\) students as possible selections. - Distinguish a random sample of the whole school from a random sample of only volunteers or one subgroup.

Solution

1. A valid random sample for the whole school needs a selection process that covers the whole school, not just volunteers, one lunch period, or teacher nominees. 2. Plan b) starts with the complete roster and uses random selection from that full sampling frame. 3. Plans a), c), and d) can systematically leave out or favor certain students even though a) and c) include a random step.

Answer

b) Number every student on the full roster and randomly select \(60\) different numbers.

5415147

A transit agency wants to learn how satisfied riders are with a bus route. It places survey cards only on buses that run between \(9\) a.m. and \(2\) p.m. Explain the likely sampling problem and name a group of riders that may be underrepresented.

Hints

- 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.

Solution

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.

Answer

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

5415177

A museum wants feedback from all visitors, including first-time visitors. It randomly selects \(80\) names from its annual-membership list. Is the sample representative of the stated population? Explain.

Hints

- 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.

Solution

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.

Answer

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

5415197

A parks department wants to estimate support for adding lights to a neighborhood field. Which method is more likely to produce a representative sample, and why? Plan A: Ask \(50\) people attending an evening soccer game at the field. Plan B: Use a map of neighborhood addresses, randomly select \(50\) addresses, and ask one adult at each selected address.

Hints

- 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.

Solution

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.

Answer

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

5415207

Students are numbered \(001\) through \(480\). A random-number generator produces \(118,\ 407,\ 407,\ 052,\ 611,\ 294,\ 000,\ 052,\ 371\). The school needs five different students. Reading from left to right, which student numbers should be selected? Explain how invalid and repeated results are handled.

Hints

- 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.

Solution

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 \(371\).

Answer

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

54152211

Two surveys use the same random sample of residents. Survey 1 asks, “Do you support the costly plan to replace perfectly usable playground equipment?” Survey 2 asks, “Do you support the plan to replace playground equipment?” Which survey is more likely to measure residents' opinions accurately? Explain.

Hints

- 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.

Solution

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.

Answer

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

5415257

A school roster lists all sixth graders first, then all seventh graders, then all eighth graders. A student selects every \(25\)th name, starting with name \(10\), to survey opinions about a school dance. What is missing from this plan, and how can it be fixed?

Hints

- 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.

Solution

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.

Answer

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

5415267

A district wants a sample of \(100\) middle school students. To preserve the grade proportions, it has already decided to sample \(25\) Grade \(6\) students, \(35\) Grade \(7\) students, and \(40\) Grade \(8\) students. Which selection method best supports an inference about all district middle school students? a) Within each grade, randomly select the required number from a complete district roster for that grade. b) Within each grade, take the required number of students whose last names come first alphabetically. c) Within each grade, ask for volunteers and then randomly select the required number from those volunteers. d) Within each grade, randomly select the required number only from students who take first-period physical education.

Hints

- The grade counts have already been chosen, so the remaining issue is how students enter the sample. - Ask whether every student in each grade can be selected. - A proportional sample can still be biased if selection within a grade is not representative.

Solution

1. The grade quotas make the sample composition proportional, but proportional quotas alone do not make the sample representative. 2. Plan a) uses a complete sampling frame within every grade and randomly selects the required number from each frame. 3. Plan b) uses a fixed positional rule, plan c) restricts selection to volunteers, and plan d) undercovers students who are not in first-period physical education. 4. Therefore, plan a) is the method that best supports inference about the district population.

Answer

a) Randomly select the required number from a complete roster within each grade.

5415287

A community pool wants to estimate its typical daily attendance during a \(90\)-day summer season. It records attendance on the first \(12\) days of the season. Describe why this sample may be biased and give a random method for choosing \(12\) days.

Hints

- 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.

Solution

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.

Answer

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

5415297

A company wants to know how satisfied all buyers are with a new backpack. It analyzes only reviews posted on the company's website. Give two reasons the reviews may not form a representative sample of buyers.

Hints

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

Solution

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.

Answer

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

5415307

To select \(25\) students from a roster, a program repeatedly chooses a random student number but allows the same number to be selected more than once. The program stops after \(25\) selections. Why might this produce fewer than \(25\) students, and how should the rule change?

Hints

- 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.

Solution

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.

Answer

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

5415337

A district wants to estimate the percentage of students who participate in after-school sports. It surveys a random sample from students currently enrolled in physical education classes, even though some grades do not take physical education this semester. What is the main sampling concern?

Hints

- 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.

Solution

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.

Answer

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

5415367

A school plans a \(40\)-student sample with \(14\) bus riders, \(16\) students who arrive by car, and \(10\) students who walk or bike. The bus riders and car riders are randomly selected from complete lists for those groups. For the walk-or-bike group, the school uses the first \(10\) students who arrive at the bike rack one morning. Does this plan support an inference about all students' travel modes? If not, identify the problem and give a specific repair.

Hints

- Check the selection process separately inside each travel-mode group. - Ask whether every student who walks or bikes has a comparable chance to be chosen. - Changing only the defective group is enough to repair the stratified plan.

Solution

1. The bus and car groups use random selection from complete group lists. 2. The walk-or-bike group is selected by arrival order at one location and time, so students who arrive later or use a different entrance may be systematically excluded. 3. Because one stratum is selected by convenience rather than randomly, the overall plan is not a valid stratified random sample. 4. A repair is to make a complete list of all students who walk or bike and randomly select \(10\) students from that list.

Answer

No. The walk-or-bike group is a convenience sample. Use a complete list of students who walk or bike and randomly select \(10\) from that list.

5415377

A factory makes parts in batches numbered \(101\) through \(500\). An inspector writes each batch number once on identical cards, mixes the cards thoroughly, and draws \(20\) cards without putting any card back. Explain why this method can produce a valid random sample of \(20\) different batches.

Hints

- 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.

Solution

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.

Answer

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.

5415387

A cafeteria serves three lunch periods. To estimate satisfaction with lunch, staff survey every \(10\)th student entering only the final lunch period, using a random start. Does the random start make the sample representative of all students? Explain.

Hints

- 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.

Solution

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.

Answer

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

5415407

A teacher writes each student's name on a slip of paper and mixes the slips in a box. Some slips are folded twice, and others are left flat. The teacher draws \(12\) slips without looking. Is this necessarily a fair random sample? Explain how to improve the method.

Hints

- 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.

Solution

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.

Answer

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

5415417

A local newspaper wants to estimate the percentage of town residents who support building a new baseball stadium. Reporters survey \(250\) people as they enter a minor-league baseball game. Why is this sample likely biased, even though \(250\) is a fairly large number?

Hints

- 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.

Solution

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.

Answer

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

5415427

A greenhouse has \(20\) rows with \(50\) plants in each row. A researcher samples the first two plants in every row, for a total of \(40\) plants. Is this a random sample of all plants in the greenhouse? Explain and give a better sampling plan.

Hints

- 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.

Solution

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.

Answer

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.

5415437

A survey company uses only listed landline phone numbers to sample adults in a county. It calls numbers at random from that list. Explain why random calling does not guarantee a representative county sample.

Hints

- 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.

Solution

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.

Answer

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

5415447

A cleanup team wants to estimate the kinds of litter in a large park. A map divides the park into \(120\) equal squares. The team randomly selects \(15\) squares and records every piece of litter in them. Explain why this is a stronger plan than inspecting the \(15\) squares nearest the entrance.

Hints

- 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.

Solution

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.

Answer

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

5415467

A district wants to compare typical food waste among six schools. Staff randomly select \(30\) lunch trays from each school. Explain one strength of this plan for comparing the schools and one condition needed for the comparison to be fair.

Hints

- 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.

Solution

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.

Answer

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.

5415487

Students on a roster are labeled \(0\) through \(399\). A program is set to generate integers from \(1\) through \(400\). Explain the mismatch and give two valid fixes.

Hints

- 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.

Solution

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 \(400\).

Answer

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

5415497

A school has \(120\) sixth graders, \(180\) seventh graders, and \(300\) eighth graders. A researcher randomly selects \(20\) students from each grade to compare the average number of homework minutes among the three grades. Explain why this plan can support a comparison among grades and identify one feature the researcher should keep the same for all three samples.

Hints

- 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.

Solution

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.

Answer

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.

5415507

An organizer wants to estimate how difficult a \(5\text{K}\) race felt to everyone who started. At the finish line, volunteers randomly select \(50\) finishers to survey. Why can this be biased even though the finishers are selected randomly?

Hints

- 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.

Solution

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.

Answer

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

5415527

A \(10\)-story office building has \(24\) offices on each floor. A researcher wants a random sample of \(30\) offices but uses only the first \(30\) offices listed in the lobby directory. Explain why this sample may not represent the building and describe a better random plan.

Hints

- 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.

Solution

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.

Answer

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

5415547

A small club has \(18\) members. To create a sample of \(12\) survey responses, a computer randomly selects members with replacement, so some members answer more than once. Explain how repeated responses affect the sample and give a better plan for surveying members.

Hints

- 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.

Solution

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.

Answer

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

5415557

A teacher wants a random sample of \(6\) students. She selects one student from the front row and then chooses the five students sitting closest to that student. Explain why this is not a random sample of the class and give a better plan.

Hints

- 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.

Solution

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.

Answer

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

5415567

An orchard manager checks tree health by randomly selecting \(25\) trees from the two outer rows because those rows are easiest to reach. Why is the word “randomly” not enough to justify an inference about the whole orchard?

Hints

- 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.

Solution

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.

Answer

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

5415587

A school has \(144\) students with winter birthdays, \(168\) with spring birthdays, \(132\) with summer birthdays, and \(156\) with fall birthdays. The school randomly selects \(48\) different students from the complete roster. A student claims the sample is invalid unless the four seasons appear in exactly the same proportions as in the school. Is the claim correct? Explain.

Hints

- 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.

Solution

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.

Answer

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.

5415607

A school wants to survey students about their favorite music. A teacher chooses the homeroom that is free during first period and surveys every student in that class. Explain why this sample may not represent the whole school and give a better random plan.

Hints

- 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.

Solution

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.

Answer

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

5415617

A school newspaper selects every \(8\)th student from the bus-loading line to estimate the percentage of all students who walk to school. Explain why the sampling plan cannot answer the question well.

Hints

- 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.

Solution

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.

Answer

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

5415627

A warehouse inspector randomly chooses \(20\) boxes from the top layer of a large pallet to estimate damage among all boxes on the pallet. Why might this sample be biased, and how can the inspector use box positions more fairly?

Hints

- 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.

Solution

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.

Answer

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

5415657

Two students independently select random samples of \(25\) names from the same complete roster. Their samples contain different numbers of seventh graders. Does this difference show that one selection method was not random? Explain.

Hints

- 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.

Solution

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.

Answer

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

5415667

A music app wants to estimate the percentage of songs in each genre in a user’s complete listening history. It analyzes only the \(30\) most recently played songs. Explain why this sample may be biased and give a better random plan.

Hints

- 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.

Solution

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.

Answer

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

5415687

A school surveys adults in the afternoon pickup lane to estimate how all families feel about traffic around the school. Name two family groups that may be underrepresented.

Hints

- 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.

Solution

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.

Answer

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

5415697

A community center selects names at random from an old membership spreadsheet. The file still contains former members and does not include people who joined during the past three months. Explain how both problems can make the sample unrepresentative and state what should be done before sampling.

Hints

- 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.

Solution

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.

Answer

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.

5415707

A school uses a computer program to choose \(75\) student IDs at random, but the program is set to ignore every ID ending in \(0\). Explain why the resulting sample is not a valid random sample of the whole school and give a correction.

Hints

- 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.

Solution

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.

Answer

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.

5415727

A district wants to compare student experiences among its \(25\) schools. It randomly selects \(10\) students from each school. Explain one strength of this plan for comparing schools and one reason the district should report each school’s results separately.

Hints

- 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.

Solution

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.

Answer

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.

5415737

A factory runs three shifts. An inspector randomly selects \(40\) production times from the day and evening shifts but excludes the night shift because no inspector is available then. What conclusion is the sample able to support, and what conclusion is not supported?

Hints

- 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.

Solution

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.

Answer

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

5415137

A school randomly selects \(60\) student ID numbers from its complete \(720\)-student roster. Fifteen selected students do not respond, so the school replaces them with students who volunteer after seeing an announcement. Explain why the final sample may not represent the whole school and describe a better way to handle the missing responses.

Hints

- 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. - Think about whether randomly choosing replacements would fully remove a problem caused by nonresponse.

Solution

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 first step is to follow up with the students who were originally selected. If replacements are necessary, choose them randomly from the remaining roster, while recognizing that nonresponse can still cause bias.

Answer

The volunteer replacements can introduce bias because volunteers may differ from the students who did not respond. The school should first follow up with the originally selected students; if replacements are necessary, select them randomly and recognize that nonresponse may still affect representativeness.

5415157

A student council asks, “Should the school cancel homework on Fridays?” Students may answer an online poll if they notice the link. Describe two reasons the results may not represent the entire student body, and give one better sampling plan.

Hints

- 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.

Solution

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.

Answer

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.

5415167

An apartment manager wants to estimate the average number of pets per household in a \(12\)-building complex. She randomly selects one building and surveys every household in that building. Explain why the selection is random but may still fail to represent the whole complex.

Hints

- 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.

Solution

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.

Answer

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

5415187

A county mails a recreation survey to \(200\) addresses selected randomly from a voter-registration list. The county wants information about all households. What should officials check before treating the results as representative? Give two concerns.

Hints

- 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.

Solution

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.

Answer

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

5415217

A middle school has grades \(6\), \(7\), and \(8\). To study lunch satisfaction, the principal randomly selects one homeroom from each grade and surveys every student in those three homerooms. State one strength and one limitation of this plan.

Hints

- 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.

Solution

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.

Answer

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

5415237

A student wants to estimate the average length of words in a \(240\)-page novel. Describe a random method for selecting \(30\) words that does not favor words near the beginning or end of pages.

Hints

- 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.

Solution

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.

Answer

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.

5415247

A science team wants to estimate algae coverage along the entire edge of a pond. The team samples only the five spots closest to the parking lot. Explain the bias and describe a better random plan that can be carried out from a map.

Hints

- 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.

Solution

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.

Answer

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.

5415277

A random sample of \(150\) households receives a mailed survey, but only \(42\) households respond. The researcher reports that the \(42\) responses are automatically representative because the original \(150\) households were selected randomly. Explain why the conclusion is not justified and name one useful next step.

Hints

- 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.

Solution

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.

Answer

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.

5415317

A researcher randomly selects \(50\) students from a roster. If a selected student is absent, the researcher replaces that student with the next person on the alphabetical list. Explain why the replacement rule weakens the random sample. Give a better rule.

Hints

- 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.

Solution

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.

Answer

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

5415327

A theater sold \(630\) tickets online and at the box office. An employee randomly selects \(42\) ticket numbers from the complete sales list but sends the survey only to selected ticket holders whose email addresses are on file. Explain why the final responses may not represent all ticket holders, even though the ticket numbers were selected randomly. Give a better collection plan.

Hints

- 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.

Solution

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.

Answer

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.

5415347

A student randomly selects \(40\) pages from a yearbook and counts every student photograph on those pages to estimate the percentage of students in clubs. Why is “student photograph” not an appropriate sampling unit for an inference about students?

Hints

- 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.

Solution

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.

Answer

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

5415357

Two groups estimate support for a new school schedule. Group A surveys \(400\) students who volunteer at a pep rally. Group B randomly selects \(80\) students from the complete school roster. Which result is more trustworthy for the whole school? Explain why sample size alone does not decide.

Hints

- 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.

Solution

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.

Answer

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.

5415397

A town randomly selects one residential street and surveys every household on that street about internet access. Give one situation in which this plan could be seriously unrepresentative, and suggest an improvement.

Hints

- 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.

Solution

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.

Answer

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.

54154511

A random sample of students is asked, “Which school club do you belong to?” The only answer choices are art, music, robotics, and athletics. Explain why the results may still be misleading, and give a better set of response choices.

Hints

- 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.

Solution

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.”

Answer

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.”

5415477

A principal searches posts containing the school's hashtag to estimate how all students feel about a new policy. Explain two ways this sample can differ from the student population.

Hints

- 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.

Solution

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.

Answer

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.

5415517

Biologists want to estimate the average length of fish in a lake. They use a net at five randomly chosen points, but all five points are inside one shallow cove. What population can the data reasonably describe, and what change is needed for a lakewide inference?

Hints

- 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.

Solution

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.

Answer

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.

5415537

After a school assembly, researchers randomly choose \(80\) occupied seats and survey the students in them about school attendance policies. About \(15\%\) of students were absent from the assembly. Explain why the seat selection may not represent the entire school.

Hints

- 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.

Solution

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.

Answer

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

5415577

A delivery company estimates its average delivery time using a random sample of packages that were delivered on the first attempt. Packages requiring a second attempt are excluded. Explain the likely bias.

Hints

- 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.

Solution

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.

Answer

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

5415597

To survey household opinions, a researcher randomly selects \(100\) addresses and asks the youngest person at each address to answer. Why does random selection of addresses not make the respondents a random sample of household members?

Hints

- 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.

Solution

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.

Answer

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

5415637

A city randomly selects \(300\) households for a survey, but the questionnaire is available only in English. The city has many households where another language is used most often. Explain how the final responses can become unrepresentative.

Hints

- 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.

Solution

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.

Answer

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

5415647

A school randomly selects locker numbers to survey students, but \(40\) students do not have lockers and \(18\) pairs of students share lockers. Give two reasons locker numbers are a poor substitute for student numbers.

Hints

- 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.

Solution

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.

Answer

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

5415677

A bookstore arranges shelves by genre. An employee selects the third book from every shelf to estimate the average price of all books. Explain one risk of using the same position on every shelf and propose a random alternative.

Hints

- 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.

Solution

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.

Answer

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.

5415717

A stadium has four entrances. To estimate how fans travel to games, staff survey every \(20\)th fan entering through Gate A after a random start. Why might the sample differ from all stadium fans?

Hints

- 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.

Solution

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.

Answer

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

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