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Random sampling methods

Learning goals

Distinguishes random sampling from convenience or voluntary-response methods.
Describes how to choose a simple random sample.
Explains how random sampling supports generalization to a population.
Identifies bias from convenience or voluntary-response sampling.
Identifies observational units and variables in a data set.

Difficulty: 1 – 5

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Random sampling methods 1

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

Random sampling methods 2

A town wants to know how many of its seventh graders ride a bike to school. A survey asks \(24\) seventh graders from one school. What is the population, and what is the sample?

Hints

- Which group does the town want to learn about? - Which students actually provided information? - The sample does not need to include everyone in the population.

Solution

1. The question is about every seventh grader in the town, so that is the population. 2. The sample is the \(24\) seventh graders who were asked.

Answer

Population: all seventh graders in the town. Sample: the \(24\) seventh graders surveyed.

Random sampling methods 3

A track coach selects \(6\) runners and records two sprint times for each runner. Are there \(6\) runners or \(12\) runners in the sample? Explain.

Hints

- What is being sampled: runners or sprint times? - Does timing the same person twice make that person two different runners?

Solution

1. Each runner is counted once as a person in the sample. 2. Recording two times per runner gives \(12\) measurements, but only \(6\) sampled runners.

Answer

There are \(6\) runners in the sample. The \(12\) sprint times are measurements, not \(12\) different runners.

Random sampling methods 4

For a class survey, a teacher writes every student's name on an identical card, mixes the cards in a bag, and draws \(5\) without looking. Does every student have a chance to be selected? Explain.

Hints

- Is any student missing from the bag? - Could a card's appearance make it easier to pick than another card?

Solution

1. Every student has one card in the bag. 2. The cards are identical and mixed, so each student has a chance to be drawn. 3. This is a random way to select students.

Answer

Yes. Every student has one identical card, and the cards are mixed before the draw, so the selection is random.

Random sampling methods 5

A student council posts a lunch survey online. Students enter the sample only if they choose to fill it out. Is this a random sample or a voluntary-response sample?

Hints

- Who makes the decision to enter the survey? - Is anyone's name chosen by a lottery or another chance method?

Solution

1. Students decide for themselves whether to take the survey. 2. No chance-based method chooses who answers, so it is a voluntary-response sample.

Answer

It is a voluntary-response sample because each student decides whether to participate.

Random sampling methods 6

A school wants to learn what all students think of the cafeteria. It asks only students enrolled in art classes this semester. Which students have no chance to be included in this survey?

Hints

- Who can receive the survey? - Think about students outside the classes where the survey is given.

Solution

1. Only students in art classes are asked. 2. Students who are not taking an art class this semester cannot enter the sample.

Answer

Students who are not enrolled in art classes this semester have no chance to be included.

Random sampling methods 7

A city library wants to know how often all middle school students use its services. A librarian surveys the first \(40\) middle school 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.

Random sampling methods 8

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.

Random sampling methods 9

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.

Random sampling methods 10

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.

Random sampling methods 11

A parks department wants to estimate support among neighborhood adults for adding lights to a field. Which plan is more likely to give a representative sample, and why? Plan A: Ask \(50\) people attending an evening soccer game at the field. Plan B: Randomly select \(50\) neighborhood addresses from a complete map, then randomly choose one adult who lives at each selected address.

Hints

- Could attending an evening game affect someone's opinion of lights? - Which plan gives adults who never use the field a chance to respond? - Does choosing a street address tell you which person to ask?

Solution

1. Plan A favors people who already use the field in the evening and may especially value lighting. 2. Plan B includes addresses throughout the neighborhood and chooses an adult by chance at each selected address. 3. Plan B can include both field users and nonusers, so it is more likely to represent neighborhood opinions, although selecting one adult per address does not give every adult an equal chance when households differ in size.

Answer

Plan B. It samples throughout the neighborhood instead of taking only people at a soccer game. It is not perfect when households have different numbers of adults.

Random sampling methods 12

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.

Random sampling methods 13

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

Hints

- Separate the rule for spacing selected names from the rule for choosing where to begin. - Ask whether each possible starting position had a chance to be selected. - What would have to change so that different starting positions could be selected?

Solution

1. The starting name 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.

Random sampling methods 14

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.

Random sampling methods 15

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. - How could chance be used to select from all \(90\) days, not just the beginning?

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. This gives each day an opportunity to be selected, though it does not guarantee an even spread across the season.

Answer

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

Random sampling methods 16

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.

Random sampling methods 17

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.

Random sampling methods 18

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.

Random sampling methods 19

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.

Random sampling methods 20

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.

Random sampling methods 21

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.

Random sampling methods 22

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.

Random sampling methods 23

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.

Random sampling methods 24

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.

Random sampling methods 25

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.

Random sampling methods 26

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.

Random sampling methods 27

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 \(1\)–\(400\).

Random sampling methods 28

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.

Random sampling methods 29

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.

Random sampling methods 30

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.

Random sampling methods 31

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.

Random sampling methods 32

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.

Random sampling methods 33

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.

Random sampling methods 34

A school randomly selects \(48\) different students from its complete roster. Aino says the sample is valid only if students with winter, spring, summer, and fall birthdays appear in exactly the same proportions as in the whole school. Is Aino correct? Explain.

Hints

- Focus on how students were chosen, not on the exact mix in one result. - Could two random selections of the same size have different birthday counts?

Solution

1. Randomly selecting students from the full roster gives every student a chance to be included. 2. A random sample can have different birthday-season proportions from the school because of chance. 3. Exact matching is not required for the selection method to be random.

Answer

No. Random selection does not guarantee that the sample has exactly the same birthday-season proportions as the school.

Random sampling methods 35

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.

Random sampling methods 36

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.

Random sampling methods 37

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.

Random sampling methods 38

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.

Random sampling methods 39

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.

Random sampling methods 40

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.

Random sampling methods 41

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.

Random sampling methods 42

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.

Random sampling methods 43

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.

Random sampling methods 44

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.

Random sampling methods 45

A school has \(240\) students on one complete numbered roster. The principal wants to select \(12\) different students using a random-number generator. Describe how to select the sample. What should the principal do if a number appears twice?

Hints

- What list contains every student who could be selected? - How can you connect a generated number to a student? - Would drawing the same student twice give you two different people?

Solution

1. Use the roster numbers already assigned to students, from \(1\) through \(240\). 2. Generate random whole numbers in that range and match them with students. 3. Ignore repeated numbers and continue until \(12\) different students are chosen.

Answer

Randomly generate student numbers from \(1\) through \(240\). Keep each number only once, and continue until \(12\) different students have been selected.

Random sampling methods 46

A school numbers \(60\) students from \(1\) to \(60\). A teacher wants to select every tenth student. Sofia suggests always starting with student \(1\). Amir suggests drawing a starting number from \(1\) through \(10\) using identical slips, then taking every tenth student. Which plan gives a student at position \(7\) a chance to be chosen? Explain why. If Amir draws \(4\), which six students are selected?

Hints

- Which ending digits can appear when the start is always \(1\)? - How does drawing a start change which students can be included? - Once the start is known, what fixed step joins successive positions?

Solution

1. Sofia's fixed start selects only positions ending in \(1\), so position \(7\) has no chance. 2. Amir's random start can be \(7\), so position \(7\) has a chance to be chosen. 3. If the draw gives \(4\), add \(10\) repeatedly to get \(4,14,24,34,44,54\).

Answer

Amir's plan. Position \(7\) can be chosen if the starting slip is \(7\); Sofia's plan cannot select it. For a start of \(4\): \(4,14,24,34,44,54\).

Random sampling methods 47

A middle school selects \(15\) students from its full roster by drawing names. The school sends the questions through an app, but \(4\) selected students cannot open the app. How can the school collect answers from those \(4\) students without changing who was selected?

Hints

- Did the problem happen when names were drawn or when questions were delivered? - Could the school reach the same selected students another way?

Solution

1. The original \(15\) students were selected by drawing names from the full roster. 2. App access is a problem with collecting responses, not with choosing names. 3. Offer the same questions to the \(4\) selected students on paper or in person.

Answer

Give the same questions on paper or in person to the \(4\) students who cannot use the app, while keeping the original \(15\) selected students.

Random sampling methods 48

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.

Random sampling methods 49

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.

Random sampling methods 50

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.

Random sampling methods 51

A county uses voter records to make a list of household addresses. It removes repeated addresses, then randomly chooses \(200\) different addresses and mails a recreation survey. The county wants information about all households. What are two reasons the survey results might still fail to represent all households?

Hints

- Which households could be missing from a list based on registered voters? - Does selecting an address guarantee a completed survey? - Think about the difference between the addresses chosen and the households that reply.

Solution

1. Households with no registered voters may be missing from the address list, so they cannot be selected. 2. Some selected households may not reply to the mailed survey. 3. If households left off the list or households that do not reply have different recreation needs, the results may be biased.

Answer

The list may leave out households with no registered voters, and selected households may not return the survey.

Random sampling methods 52

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.

Random sampling methods 53

Ayu 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

- Define a sampling frame that gives every word occurrence a chance to be selected. - Avoid a procedure tied to where a word appears on a page. - Make the selections by chance without selecting the same occurrence twice.

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.

Random sampling methods 54

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.

Random sampling methods 55

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.

Random sampling methods 56

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.

Random sampling methods 57

A theater has a list of \(630\) different ticket buyers, with each buyer listed once even if they bought several tickets. An employee randomly selects \(42\) buyers but sends a survey only to selected buyers whose email addresses are on file. Explain why the responses may not represent all buyers, even though buyers were selected randomly. Give a better collection plan.

Hints

- Separate the random choice of buyers from the way the survey is delivered. - Identify which selected buyers have no way to receive the survey. - Consider how to reach people who do not have email on file.

Solution

1. Selecting from a list of unique buyers gives every buyer a chance to be chosen. 2. The survey reaches only selected buyers with email addresses on file. 3. Selected buyers without recorded email addresses cannot respond, so the responding group may differ from all buyers. 4. The theater should contact each selected buyer by an available method instead of excluding people without email addresses.

Answer

Selected buyers without an email address on file are left out. Contact every selected buyer through an available method, such as email, mail, or phone.

Random sampling methods 58

Ishaan randomly selects \(40\) pages from a yearbook and uses every student photograph on those pages as a sampling unit. For each photograph, Ishaan records whether the pictured student belongs to at least one club, then uses those records to estimate the percentage of students in clubs. Why is “student photograph” not an appropriate sampling unit for an inference about students?

Hints

- Compare the unit being sampled with the population unit the inference is about. - Ask whether one student can contribute more than one sampled unit. - Think about a sampling frame in which each student can appear only once.

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.

Random sampling methods 59

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.

Random sampling methods 60

A school wants to estimate what percentage of students travel to school by bus, car, or walking or biking. It plans to survey \(40\) students: \(14\) bus riders, \(16\) car riders, and \(10\) students who walk or bike. The bus and car riders are randomly chosen from complete group lists, but the walk-or-bike group consists of the first \(10\) students at the bike rack one morning. Give two reasons this plan cannot reliably estimate how students travel to school. Describe a better sampling plan.

Hints

- Were the numbers of bus, car, and walk-or-bike students determined by the actual school population or chosen in advance? - Think about walkers who never go near the bike rack. - What would happen if every student had a chance to enter the sample before their travel method was known?

Solution

1. The numbers in the three travel groups were fixed before sampling, so their proportions in the sample cannot reveal the actual proportions in the school. 2. Selecting the walk-or-bike group at one bike rack by arrival order can also miss many students who walk or bike. 3. Instead, randomly choose \(40\) students from the complete school roster without setting travel-group quotas, then ask each selected student how they travel.

Answer

Fixed group counts prevent the sample from showing the school's real travel-mode proportions. The bike-rack group is also a convenience sample. Randomly select \(40\) students from the full roster, without fixed group counts, and ask how they travel.

Random sampling methods 61

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.

Random sampling methods 62

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.

Random sampling methods 63

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.

Random sampling methods 64

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.

Random sampling methods 65

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.

Random sampling methods 66

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

Hints

- Separate the selection of household addresses from the selection of respondents. - Does being the oldest adult in a household give someone any chance to answer? - What could make the within-household choice fairer?

Solution

1. Addresses are selected randomly, but the respondent within each household is chosen by age. 2. Younger adults are always favored, and older adults cannot respond when someone younger lives with them. 3. The researcher should randomly choose one adult within each selected household instead, while noting that adults in larger households would still have a smaller chance of selection.

Answer

The youngest-adult rule favors younger people. Randomly choose an adult from each selected household instead, though household size may still affect each adult's chance.

Random sampling methods 67

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.

Random sampling methods 68

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.

Random sampling methods 69

A bookstore arranges its 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 match a repeated arrangement rule. - What list would give all books a chance to be selected, regardless of their shelf size?

Solution

1. The third position may be connected to how employees arrange featured, new, or low-priced books. 2. Using the same position on every shelf can repeat the same placement bias. 3. Give each book in the complete catalog a unique inventory number and randomly select book numbers from that list.

Answer

The third position may have a special placement pattern. Randomly select unique book inventory numbers from the complete catalog instead.

Random sampling methods 70

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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