Free math worksheets

- Compare the stated goal with what can happen after a selected number is returned. - Ask whether a sampled unit can appear more than once. - State the selection rule that prevents duplicates.

1. Returning each selected number allows the same employee to be chosen more than once. 2. The goal requires distinct employees, so repeated selections are not acceptable. 3. Sampling should be done without replacement.

The procedure samples with replacement, so one employee could appear multiple times. The researcher should sample without replacement so each selected employee can be chosen only once.

- Consider what happens to a unit after it is measured. - Ask whether the same physical unit could be selected again. - Match the sampling rule to the requirement for distinct units.

1. A destructively tested bulb cannot be returned to the production lot for another selection. 2. The sample should therefore be taken without replacement. 3. Without replacement also ensures that the sample consists of distinct bulbs rather than allowing one unit to be selected repeatedly.

Sample without replacement. Each tested bulb is destroyed and cannot be selected again, and the resulting sample should contain distinct bulbs from the lot.

- Ask whether the selected unit returns to the population. - Determine which units are available before the second draw. - Recalculate probabilities using the units that remain.

1. With replacement, Unit A is returned before the second draw, so its probability is \(\frac{1}{5}\). 2. Without replacement, Unit A is not returned, so its probability of appearing on the second draw is \(0\). 3. The remaining four units each have conditional probability \(\frac{1}{4}\) on the second draw without replacement.

With replacement, Unit A has probability \(\frac{1}{5}\) of being selected again. Without replacement, its probability is \(0\).

- Compare the target population with the available records. - Ask whether every observational unit can be included. - Decide whether estimation from a subset is necessary.

1. Complete records are available for every artwork in the target population. 2. The target population is limited to sales during one month. 3. Using all records is a census and gives the exact population mean for that month. 4. Sampling is unnecessary unless cost or data quality creates a reason not to use all records.

Use a census of all artworks sold during the month. The full data are available, so the gallery can calculate the exact mean sale price for that population.

- Do not decide only from the fact that both plans use chance. - Ask whether every possible group of \(8\) students could be the final sample. - Compare the set of possible samples under the two plans.

1. In an SRS of \(8\), every possible group of \(8\) students must have the same chance to be selected. 2. In Plan A, drawing \(8\) well-mixed slips without replacement gives every group of \(8\) students the same chance. 3. In Plan B, only two fixed groups of \(8\) students can ever be selected; most groups of \(8\) have probability \(0\). 4. Therefore, Plan A produces an SRS of \(8\), but Plan B does not.

Plan A produces an SRS of \(8\). Plan B does not, because most possible groups of \(8\) students can never be selected.

- Notice whether entire groups are selected or individuals are selected within every group. - Ask what characteristic is used to divide the population. - Connect the sample size from each group to that group's share of the population.

1. The population is divided into nonoverlapping groups based on grade level. 2. A random sample is selected within every group. 3. This is a stratified random sample with proportional allocation. Selecting sample sizes in proportion to grade enrollment makes the sample's grade distribution match the population's grade distribution.

A stratified random sample with proportional allocation. Grade levels are the strata, students are randomly selected within each stratum, and each grade's sample size is proportional to its enrollment.

- Determine whether individuals or whole groups are randomly selected. - Check what happens to every member of a selected group. - Name the groups used in the sampling plan.

1. Entire blocks are selected at random. 2. Every occupied household in each selected block is surveyed. 3. This is a cluster random sample, with neighborhood blocks serving as clusters.

A cluster random sample. The neighborhood blocks are clusters, and all occupied households in the randomly selected clusters are surveyed.

- Look for a repeated selection pattern. - Identify how the first selected unit is chosen. - Distinguish a fixed rule from selecting nearby or available units.

1. A random starting position is chosen. 2. Tickets are then selected at a fixed periodic interval. 3. This is a systematic random sample. The random start and the predetermined interval distinguish it from taking whichever tickets are easiest to reach.

A systematic random sample. Its random start and fixed interval of every fortieth ticket provide the random mechanism.

- Find each housing group's share of the full student population. - Apply each share to the desired sample size. - Check that the three allocations add to the total sample size.

1. The total population size is \(4000+2500+1500=8000\). 2. Commuters: \(\frac{4000}{8000}\cdot160=80\). 3. Residence halls: \(\frac{2500}{8000}\cdot160=50\). 4. Nearby apartments: \(\frac{1500}{8000}\cdot160=30\).

Select \(80\) commuter students, \(50\) residence-hall students, and \(30\) students living in nearby apartments.

- Relate the population size to the requested sample size. - The first selected position must be chosen randomly within one interval. - Apply the same spacing after the random start.

1. The interval is \(\frac{2400}{60}=40\). 2. Randomly choose a starting position from \(1\) through \(40\). 3. Select that pallet and every fortieth pallet after it until \(60\) pallets are included.

Use an interval of \(40\). Choose a random start from the first \(40\) pallets, then select every fortieth pallet.

- Identify what is listed and randomly selected. - Separate the units being sampled from the people counted within each unit. - Check whether every set of the requested size can be selected.

1. The complete population consists of the thirty festival dates. 2. Eight distinct dates are selected at random from the full list. 3. This is a simple random sample of dates. 4. The observational units are festival dates, not individual attendees.

It is a simple random sample of eight festival dates. The observational units are the festival dates, with attendance recorded for each selected date.

- Look for how the first unit is selected. - Identify the repeated spacing rule. - Connect the orderly sequence to practical data collection.

1. The plan uses a random starting point and a fixed interval. 2. It is a systematic random sample. 3. It is efficient because the inspector can sample directly from the moving production sequence without constructing and contacting a scattered list of selected containers.

It is a systematic random sample. The method is efficient because selections are evenly spaced along the production line after one random start.

- Identify what is randomly selected first. - Ask whether the units inside each selected group are then all included. - The clusters must form groups to which each recorded boarding event belongs unambiguously.

1. The population is partitioned by bus route for the stated period, and each boarding event belongs to one route. 2. Entire routes are randomly selected. 3. Every boarding event in each selected route during the period is included. 4. This is cluster random sampling, with bus routes as the clusters.

It is a cluster random sample. The bus routes are the clusters, and all passenger boarding events on the selected routes during the study period are included.

- Determine whether every category contributes sampled individuals. - Identify whether individuals or whole categories are selected. - Connect the design to representation of distinct visitor types.

1. The population is divided into nonoverlapping visitor categories. 2. Random samples are selected within every category. 3. This is stratified random sampling. 4. It ensures that smaller but distinct visitor types are included rather than left to chance in one overall sample.

It is a stratified random sample. Sampling within each visitor category guarantees representation of day visitors, campers, and permit holders.

- Ask whether all districts or only selected districts contribute data. - Determine whether every teacher or only sampled teachers are used within a district. - Name the method based on both group selection and within-group selection.

1. Randomly selecting some districts and surveying all eligible teachers in them is cluster random sampling. 2. Randomly sampling teachers within every district is stratified random sampling with districts as strata. 3. The distinction is whether only selected groups or all groups contribute individuals.

The first plan is cluster random sampling. The second plan is stratified random sampling.

- Find each group's fraction of the total workforce. - Apply each fraction to the sample size. - Check that the allocations add to the requested total.

1. The total number of employees is \(1200+800=2000\). 2. Hourly employees: \(\frac{1200}{2000}\cdot100=60\). 3. Salaried employees: \(\frac{800}{2000}\cdot100=40\).

Select \(60\) hourly employees and \(40\) salaried employees, using random selection within each group.

- Divide the population size by the desired sample size. - Choose the initial position from one complete interval. - Keep the spacing fixed after the first selection.

1. The interval is \(\frac{1800}{90}=20\). 2. Randomly choose one starting number from \(1\) through \(20\). 3. Select that permit and every twentieth permit afterward.

Use an interval of \(20\). Choose a random start from \(1\) through \(20\), then select every twentieth permit.

- Review what must happen inside each stratum. - Check whether every store group has a chance to appear in the final sample. - Decide what population the existing sample could represent without adding another selection step.

1. A stratified random sample of the full population requires random selection within every stratum. 2. The medium-sales stratum has no chance to contribute stores to the sample. 3. Therefore, the procedure is not a stratified random sample of all stores and cannot represent the full store population without additional sampling from the omitted stratum.

The claim is incorrect. A stratified random sample must include random selections from every stratum. Because medium-sales stores are omitted, the procedure is not a stratified random sample of all stores.

- Recall the two defining features of a systematic random sample. - Check whether the starting location is determined by chance. - Keep the interval but change only the feature that is not random.

1. The fixed interval of ten is present, but the starting position is not random. 2. Always starting after the depot can favor locations tied to a particular part of the route. 3. The supervisor should randomly select one of the first ten stops as the starting point and then include every tenth stop around the route.

The procedure lacks a random start. Randomly choose one of the first ten stops, then select every tenth stop from that point.

- Think of the sample as occupying a fixed number of positions from the full roster. - Use symmetry: no listed volunteer is favored over another. - Compare the number selected with the total number eligible.

1. An SRS of size \(8\) gives each roster member the same inclusion probability. 2. The probability that a specified volunteer is included is \(\frac{8}{50}=0.16\). 3. The value is the same for every volunteer because the random-selection procedure treats every roster position symmetrically.

The inclusion probability is \(\frac{8}{50}=0.16\), or \(16\%\). Every volunteer has this same probability because the sample is selected as an SRS from the full roster.

- Identify what is chosen directly by the random process. - Check whether every member of a chosen group is included. - Name the group that contains the individual observational units.

1. Complete groups of consecutive days are selected rather than individual days from throughout the season. 2. Each week is a cluster. 3. Randomly selecting weeks and including every day in the selected weeks is cluster random sampling.

This is cluster random sampling. The weeks are the clusters, and all seven days in each selected week are included.

- Identify the role of the first random choice and the repeated interval. - Determine what event stops the sampling process. - Ask which unknown quantity controls how many selections occur.

1. The procedure uses a random start and a fixed interval of \(20\), so it is systematic random sampling. 2. The inspector continues until the shift ends. 3. Because the total number of packages processed during the shift is not known in advance and may vary, the number of sampled packages also varies.

This is systematic random sampling. The final sample size depends on how many packages are processed after the random start before the shift ends, so it is not fixed in advance.

- Compare the database's largest valid ID with the generator's largest possible output. - Identify any customer whose selection probability is zero. - Correct the endpoint and preserve the intended distinct-sample rule.

1. Customer ID \(5000\) has probability zero of being selected. 2. The other IDs have positive selection probabilities, so customers are not treated equally. 3. Any sample produced by the procedure excludes ID \(5000\) and therefore is not an SRS from the full database. 4. Change the random integer range to include both endpoints \(1\) and \(5000\), and use a proper without-replacement rule for a fixed-size distinct sample.

The procedure excludes customer \(5000\), so it cannot produce an SRS from all customers. Use a random integer range of \(1\) through \(5000\) and sample without replacement when distinct customers are required.

- Compare each three-digit group with the valid label range. - Keep track of labels that have already been accepted. - Continue reading until five distinct valid labels have been found.

1. Discard \(403\) because it is outside the label range; select \(027\) and \(275\). 2. Discard the second \(027\) because sampling is without replacement, and discard \(000\) because it is not a label. 3. Select \(196\), discard \(812\) as out of range, and select \(054\) and \(221\). 4. The sample is \(027, 275, 196, 054, 221\).

The selected labels are \(027, 275, 196, 054, 221\). Discard \(403\) and \(812\) as out of range, the repeated \(027\) because sampling is without replacement, and \(000\) because it is not a valid label.

- Ask which grouping is connected to the variable being estimated. - Consider whether people within each proposed group would tend to have similar responses. - Think about which grouping guarantees representation from meaningful parts of the workforce.

1. A useful stratification variable forms groups that are internally more similar for the response and meaningfully different from one another. 2. Work location is related to commute time, while the last digit of employee ID has no expected relationship to commute time. 3. Stratifying by work location can ensure representation from every location and can reduce sampling variability.

Stratify by work location because it is related to commute time. This ensures that every location is represented and can produce a more precise estimate than stratifying by employee ID digit.

- Trace the diagram from the full hospital network to the people who are ultimately sampled. - Identify what type of unit is selected first. - Then identify what type of unit is selected from within each of those selected units.

1. The first random selection chooses hospitals from the network. 2. Within each selected hospital, a second random selection chooses nurses. 3. Because random selection occurs in successive stages, the method is multistage random sampling. 4. The first-stage sampling units are hospitals, and the second-stage sampling units are individual nurses within the selected hospitals.

The method is multistage random sampling. The first-stage sampling units are hospitals, and the second-stage sampling units are individual nurses within the selected hospitals.

- Use the full definition, including both equal chances for samples and fixed sample size. - Check whether each plan always returns the requested number of students. - Distinguish a random process from a simple random sample of a stated size.

1. A simple random sample of size twenty must contain exactly twenty students. 2. Under Plan A, every set of twenty distinct student numbers has the same chance of selection. 3. Plan B does not guarantee a sample size of twenty. 4. Therefore, Plan A produces an SRS of size twenty.

Plan A. It selects exactly twenty distinct students and gives every possible group of twenty the same chance of being chosen. Plan B is random but is not an SRS of fixed size twenty.

- Compare whether the groups are internally similar or internally diverse. - Consider whether all individuals in selected groups can be surveyed efficiently. - Choose the method whose group structure matches the population description.

1. Each class section is intended to resemble the heterogeneous school population. 2. Selecting entire sections is operationally efficient. 3. A cluster random sample of class sections fits these conditions better than stratifying by grade and sampling scattered students within every grade.

Use a cluster random sample by class section. Randomly choose several sections and survey every student in them because each section roughly mirrors the school's heterogeneity and entire sections are easy to reach.

- Ask whether every group or only selected groups contribute data. - Determine whether all units or only sampled units are taken from a group. - Do not classify the plan solely because it uses groups.

1. Every destination region contributes selected packages. 2. Packages are randomly sampled within each region rather than all packages being taken from selected regions. 3. Therefore, regions are strata and the method is stratified random sampling. 4. In cluster sampling, only some regions would be selected and all packages in them would be included.

It is a stratified random sample, not a cluster sample. The company samples individual packages within every region; cluster sampling would randomly select only some regions and include every package in those selected regions.

- Compare the generator's full output range with the member-label range. - Decide what to do with an output that does not name a member. - Ensure no member can appear twice in the final sample. - Continue until the required number of distinct valid labels has been obtained.

1. Generate three-digit integers from \(000\) through \(999\). 2. Ignore \(000\) and any value from \(851\) through \(999\) because those values do not label a member. 3. Keep each valid label from \(001\) through \(850\) only the first time it appears. 4. Continue until \(40\) distinct valid labels have been kept, then select the corresponding members.

Generate random three-digit integers from \(000\) through \(999\). Keep valid labels \(001\) through \(850\), ignore invalid labels and repeated valid labels, and continue until \(40\) distinct members have been selected.

- Determine whether every division contributes sampled individuals. - Identify the study goal: overall estimation or comparison among groups. - Explain why a small group might need more representation than proportional allocation gives.

1. Employees are divided into nonoverlapping groups by job division. 2. A simple random sample is selected within every division. 3. This is stratified random sampling. 4. Equal allocation provides enough observations from the small division for a meaningful comparison across divisions.

It is a stratified random sample. Taking the same number from each division deliberately gives the small division adequate representation for comparing division-level opinions.

- Identify what is randomly selected in each plan. - Notice whether all members of selected groups are included. - Compare efficiency with how much diversity each sample may capture.

1. Selecting individuals from the combined roster is a simple random sample. 2. Selecting whole dorm floors and surveying everyone on them is a cluster random sample. 3. The SRS may be more geographically scattered but directly randomizes individuals. 4. The cluster sample is easier to administer but may be less precise if residents within a floor are similar.

The first plan is an SRS; the second is a cluster random sample. Cluster sampling is easier to carry out, but an SRS may better spread selections across the university and avoid strong within-floor similarity.

- Identify the two defining steps of the selection plan. - Ask whether the list order is connected to the measured variable. - Focus especially on any repeating pattern that matches the interval.

1. The plan uses a random start and fixed interval, so it is systematic random sampling. 2. The order is acceptable if surname position is not related to the variable being studied in a way that creates a repeating pattern at the sampling interval. 3. Without such a relationship, the method can spread selections across the roster.

It is a systematic random sample. The alphabetical order is not a concern if surname order has no relevant periodic relationship with the survey variable.

- Decide whether the groups are internally similar or miniature versions of the county. - Identify whether every group needs representation. - Specify where random selection occurs.

1. Divide all residential properties into urban, suburban, and rural strata. 2. Select a simple random sample of properties within each stratum. 3. These groups are internally more similar in property characteristics and differ from one another. 4. Sampling within every group supports separate and overall estimates; selecting only some groups as clusters would omit major area types.

Use a stratified random sample: classify every residential property as urban, suburban, or rural, then randomly sample properties within each category. The area types are appropriate strata because properties within a type are relatively similar and all types must be represented.

- Identify natural groups that reduce travel costs. - Decide whether each group should resemble the full population. - State what is randomly selected and who is surveyed afterward.

1. Villages are natural geographic clusters and each is intended to contain varied households. 2. Randomly select several villages from the full list. 3. Survey every eligible household in each selected village. 4. The plan reduces travel while retaining random selection of clusters.

Randomly select villages as clusters and survey all eligible households in each selected village. This is practical because travel is concentrated in a few locations, provided villages reasonably reflect the region's household diversity.

- Focus on the selection mechanism rather than the display order. - Ask whether every possible group of the requested size can be chosen. - Contrast direct random selection with fixed-interval selection.

1. The selection uses random choice from the entire roster. 2. Every possible sample of the stated size can still have the same chance of selection. 3. Sorting does not matter for an SRS when selection is not based on position patterns. 4. The order would matter more for a systematic sample if it interacted with the interval.

No. The sorted order does not prevent an SRS because account numbers are selected randomly from the entire roster, not by a position-based pattern.

- Decide whether units are sampled from every group or whether entire groups are selected. - Identify the group whose members are all included once that group is chosen. - Consider what characteristics a useful group should share with the full population.

1. The shelving units are clusters because entire selected shelves are inspected. 2. Randomly selecting shelves and including every file on them is cluster random sampling. 3. The method is most appropriate when each shelf contains a mixture that resembles the full population of files; shelves dominated by particular years or departments could make the selected clusters unrepresentative.

This is a cluster random sample. Each shelving unit is a cluster, and all files in the selected clusters are inspected. The method works best when individual shelves are heterogeneous and resemble the full collection rather than being separated by year or department.

- Determine whether the calendar is divided into groups before dates are selected. - Ask whether sampling occurs within every group or only within a few selected groups. - Consider why time of year could be related to the measured response.

1. The months form nonoverlapping strata, and random dates are selected within every stratum. 2. The method is stratified random sampling. 3. Sampling within every month guarantees representation across the year and can reduce variation caused by seasonal boating patterns.

This is stratified random sampling with month as the stratifying variable. It guarantees that every month is represented, which is useful when harbor traffic changes by season.

- Check whether every valid label is handled by the same rule. - Decide what rejecting out-of-range labels changes about the relative chances of valid labels. - Think about why repeated labels must be skipped when the sample requires distinct boats.

1. Every valid boat number is treated the same by the random digit table. 2. Invalid labels are ignored, so they do not favor any registered boat. 3. Repeated valid labels are skipped, which makes the sample contain \(40\) distinct boats. 4. Every possible set of \(40\) boats has the same chance of being selected, so the result is an SRS of size \(40\).

Yes. The random digit table treats all valid boat numbers equally, and rejecting invalid labels and duplicates produces \(40\) distinct selections without favoring any boat. The procedure gives a simple random sample of \(40\) boats.

- Count how many separate random-selection steps occur. - Identify the units selected at each step. - Compare the final step with what would happen if every member of a selected group were included.

1. Counties are selected randomly at the first stage. 2. Rental households are then randomly selected within the chosen counties at the second stage. 3. This is a multistage random sample. 4. In a one-stage cluster sample, every rental household in each selected county would be included rather than taking a second random sample within the selected counties.

The study uses multistage random sampling: counties are sampled first, then households are sampled within those counties. A one-stage cluster sample would include all rental households in every selected county.

- Determine whether every group contributes to the sample. - Check whether only a few complete groups or individuals within all groups are selected. - Use that distinction to classify the role of the batches.

1. The population is divided into nonoverlapping production batches. 2. A random item is selected within every batch. 3. This is stratified random sampling with batch as the stratifying variable. 4. The batches are not sampled as clusters because every batch contributes an item; a cluster sample would randomly select only some batches and inspect all or many items within those selected batches.

This is stratified random sampling. Each batch is a stratum because a random selection is made within every batch. In cluster sampling, only some batches would be selected as whole groups.

- Check whether one person can belong to more than one proposed group. - Check whether every person belongs to at least one group. - Split the two yes-or-no characteristics into all possible combinations.

1. Strata must be nonoverlapping and together cover the full population. 2. The proposed categories overlap because some customers satisfy both conditions. 3. They also fail to cover customers who satisfy neither condition. 4. The researcher can create four nonoverlapping strata: both conditions, newsletter only, purchase only, and neither, then randomly sample within each stratum.

The proposed strata overlap and do not cover every customer. Use four mutually exclusive, exhaustive groups: both, newsletter only, purchase only, and neither, then sample randomly within each group.

- Think about the set of all possible two-person samples. - Identify which pairs the procedure makes impossible. - Compare the procedure with selecting both names from the same full list.

1. In an SRS of two employees, every possible pair from the full roster must have the same chance of selection. 2. The procedure can select only pairs whose members are in the same department. 3. Pairs from different departments have probability zero, so possible pairs do not have equal chances. 4. Both employees should instead be selected without replacement from the full roster.

No. Cross-department pairs cannot be selected, so not every possible pair has the same chance. Select two distinct employees randomly from the full roster without replacement.

- Think about what sorting independent random numbers does to the roster order. - Ask whether any farm is favored for an early position. - Extend the argument from individual positions to every possible set of \(30\) farms.

1. Independent continuous random numbers create a random ordering of the farms, with ties having probability zero. 2. Every ordering of the roster is equally likely. 3. Therefore, every set of \(30\) farms is equally likely to occupy the first \(30\) positions. 4. The selected farms form an SRS of size \(30\).

The random numbers create an equally likely random ordering of the roster. Every possible set of \(30\) farms has the same chance to appear in the first \(30\) positions, so the result is an SRS.

- Identify whether selections are made separately inside population groups. - Compare the sampled fraction within each group. - Recall that an SRS from the full population treats every individual symmetrically.

1. Random samples are taken within both population groups, so the method is stratified random sampling. 2. A Group X member has inclusion probability \(\frac{45}{90}=0.5\). 3. A Group Y member has inclusion probability \(\frac{45}{910}\approx 0.0495\). 4. The individual inclusion probabilities are unequal, so the combined sample is not an SRS of \(90\) people from the full population.

The design is disproportionate stratified random sampling. Group X inclusion probability: \(0.5\). Group Y inclusion probability: \(\frac{45}{910}\approx 0.0495\). Because the probabilities differ, the sample is not an SRS of \(90\) people.

- Track what happens to the final sample size when a duplicate occurs. - Compare the procedure with the fixed size required by an SRS of \(20\). - Modify the selection rule so every selected student is distinct and the size is fixed.

1. Sampling with replacement allows the same student to be drawn multiple times. 2. Discarding duplicates without making additional draws can leave fewer than \(20\) distinct students. 3. Therefore, the procedure does not guarantee a sample of size \(20\) and cannot be an SRS of \(20\) distinct students. 4. The registrar should sample \(20\) students without replacement or continue drawing with replacement until \(20\) distinct students have been selected.

The claim is false. Duplicate draws can make the final sample contain fewer than \(20\) distinct students. Sample without replacement or continue drawing with replacement until \(20\) distinct students are selected.

- Determine whether every time group contributes an observation. - Identify the random selection made inside each group. - Consider what an unrestricted sample might fail to represent.

1. The one-hour periods are nonoverlapping strata. 2. One item is randomly selected within every stratum, so the method is stratified random sampling. 3. The design guarantees representation from every hour and can capture time-of-day changes that an SRS of \(24\) items might miss by chance.

This is stratified random sampling with hour as the stratifying variable. It guarantees coverage of all \(24\) hours and is useful when process behavior changes over the day.

- Consider how similar homes within one block may be. - Ask whether one selected group can represent variation among all groups. - Improve the plan by increasing the number of independently selected groups.

1. Homes within one block may be unusually similar in housing type, income, shade, or neighborhood policy. 2. With only one selected cluster, block-specific characteristics cannot be separated from citywide patterns. 3. A better design randomly selects many blocks spread across the city and records every home within each selected block. 4. Multiple clusters provide replication at the cluster level and better represent variation among blocks.

One block may be atypical, so a one-cluster sample can be dominated by that block's characteristics. Randomly select several blocks and survey every home in each selected block.

- Identify the first division of the population. - Classify the selection rule used separately inside each division. - Name the overall method and the within-group method.

1. The population is first divided into two nonoverlapping strata based on shift. 2. A systematic random sample with a random start and fixed interval of \(50\) is taken within each stratum. 3. The overall design is stratified random sampling, using systematic random sampling inside the day and night strata. 4. This guarantees representation from both shifts.

The design is stratified random sampling by shift, with a systematic random sample taken within each stratum.

- Treat the small division and the remaining employees as separate population groups. - Determine the sampled fraction in each group. - Classify the overall design based on random selection within the groups.

1. The population is divided into remote-work and other-division strata. 2. The remote-work stratum is surveyed by census, so each member has inclusion probability \(1\). 3. The other stratum is sampled randomly, so each member has inclusion probability \(\frac{160}{1960}=\frac{4}{49}\approx 0.0816\). 4. The overall design is disproportionate stratified sampling.

This is disproportionate stratified sampling. Remote-work employees have inclusion probability \(1\). Other employees have inclusion probability \(\frac{160}{1960}=\frac{4}{49}\approx 0.0816\).

- Follow one school through both stages of selection. - Compare a school’s chance within a small district with its chance within a large district. - Recall what must be equal for every possible school sample in an SRS.

1. Conditional on its district being selected, a school in the \(4\)-school district has selection probability \(\frac{1}{4}\), while a school in the \(20\)-school district has probability \(\frac{1}{20}\). 2. Because districts have the same first-stage selection chance, schools in smaller districts have larger overall inclusion probabilities. 3. Therefore, the final set is a multistage random sample but not an SRS of all public high schools.

No. Schools in smaller districts are more likely to be selected; for example, the within-district probabilities are \(\frac{1}{4}\) and \(\frac{1}{20}\). The design is multistage, not an SRS of schools.

- Compare each group’s share of the sample with its share of the population. - Give each group’s sample mean the weight of that group in the county. - Check that the weights account for the entire population.

1. The sample gives rural and urban households equal representation even though they make up different proportions of the county. 2. The population weights are \(\frac{100}{1000}=0.10\) for rural households and \(\frac{900}{1000}=0.90\) for urban households. 3. The weighted estimate is \(0.10\cdot 42+0.90\cdot 30=31.2\,\text{miles}\).

An unweighted average overrepresents rural households. The appropriately weighted countywide estimate is \(31.2\,\text{miles}\) per week.

- Identify the random start and fixed interval. - Compare the interval with any pattern in the ordered list. - Ask whether repeated positions in that pattern differ in the response variable.

1. A random start followed by a fixed interval makes this a systematic random sample. 2. The interval matches the repeating airline pattern. 3. The sample may repeatedly select the same airline or the same position in the pattern. 4. Because airlines may have different delay patterns, one resulting sample may be badly unrepresentative and the estimator may have high sampling variability.

It is a systematic random sample, but the interval of twelve aligns with the list's periodic structure. The sample may repeatedly select one airline or one position in the pattern, producing an unrepresentative estimate with high sampling variability.

- Identify whether whole groups or individual residents are selected first. - Compare the selected counties with where most residents in the state live. - Distinguish an unlucky random outcome from a failure to use random selection.

1. Entire counties are selected and every resident in each selected county is surveyed, so this is a cluster random sample. 2. Counties can differ substantially in characteristics related to the survey response. 3. With only two selected clusters, the realized sample can have high sampling variability. 4. Because both selected counties are rural while most state residents live in urban counties, this particular random sample represents the statewide resident population poorly even though the cluster selection itself was random.

It is a cluster random sample. Selecting only two counties can produce high sampling variability, and in this realization both selected clusters are rural even though most state residents live in urban counties, so the sample is poorly representative of the statewide resident population.

- Use the definition based on possible samples, not only individual chances. - Identify whether the plan forces a particular group composition. - Name the method that samples within every group.

1. The plan fixes the number selected from every grade. 2. Samples with different grade compositions have probability zero. 3. Therefore, not every possible sample of the total size has the same chance of selection. 4. The method is stratified random sampling, not an SRS of the entire school.

No. It is a stratified random sample because exactly five students are selected within each grade. It is not an SRS of all students since samples with other grade compositions cannot be chosen.

- Identify how the roster is divided before selections are made. - Determine whether random selections occur within every group. - Consider why a small population group might need more observations than proportional allocation would provide.

1. The population is divided into two nonoverlapping groups based on disability status. 2. A random sample is taken within each group, so the method is stratified random sampling. 3. Selecting equal numbers gives the smaller group enough observations for a meaningful comparison between groups. 4. The allocation is disproportionate, so an overall campus estimate would need to account for the groups' actual population proportions.

This is a stratified random sample with disability status as the stratifying variable. Equal allocation gives adequate representation to the smaller group for comparison, although the sample is not proportional and cannot be combined naively for an overall estimate.

- Identify exactly what is counted in the every-twentieth rule. - Ask whether each person contributes the same number of chances for selection. - Distinguish a trip or event from the individual who generates it.

1. The procedure is systematic random sampling of boarding events because every twentieth boarding is selected after a random start. 2. A rider who boards many times has more opportunities to be selected than a rider who boards once. 3. The same rider could also appear more than once. 4. Therefore, the procedure is not a systematic random sample of distinct riders unless the agency adds a method that identifies and samples each rider only once from a rider-level frame.

It is a systematic random sample of boardings, not of distinct riders. Frequent riders have more chances to be selected and may appear multiple times, so a rider-level sampling frame is needed for a sample of distinct riders.

- Identify which objects are selected directly by the random mechanism. - Check whether individuals can appear in arbitrary combinations. - Separate equal inclusion chances from the stronger definition of an SRS.

1. The buildings, not individual residents, are selected at random. 2. All residents in selected buildings are included, so the design is cluster random sampling. 3. It is not an SRS of residents because possible resident samples are constrained to whole-building groups and do not all have the same chance of selection. 4. Unequal building sizes also make the final sample size variable, although each resident's inclusion probability can still equal the probability that the resident's building is selected.

The sample is a cluster random sample, not an SRS of residents. Whole buildings are selected and all residents within them are surveyed, so only samples made of complete building clusters can occur.

- Identify the unit selected at each random stage. - Ask what would happen if the interviewer simply chose the first available adult. - Compare an adult's selection chance in a one-adult household with the chance in a four-adult household.

1. Households are randomly selected at the first stage. 2. One adult is randomly selected within each chosen household at the second stage. 3. The procedure is multistage random sampling. 4. The second random step prevents the interviewer from favoring the adult who is easiest to contact and gives eligible adults within the same household equal chances of selection. 5. Adults do not necessarily have equal overall inclusion probabilities: an adult in a household with fewer eligible adults has a greater chance of being selected than an adult in a larger household. Adult-level estimates may therefore require weighting by household eligibility count.

This is multistage random sampling: households are sampled first, then one adult is sampled within each selected household. The second random step avoids convenience selection within households. However, adults in smaller eligible households have higher overall inclusion probabilities, so adult-level estimates may require weighting.

- Compare the number and structure of samples the procedure can produce with all possible \(100\)-person sets. - Identify a plausible set of employees that the procedure could never select. - Separate equal chances for individuals from equal chances for complete samples.

1. The random start gives each employee an equal chance to be included in some \(100\)-person consecutive block. 2. However, only \(1000\) consecutive circular blocks can be selected. 3. Most possible sets of \(100\) employees, especially sets scattered across hire dates, have probability zero. 4. Therefore, the sample is not an SRS of \(100\) even though individual inclusion probabilities are equal.

The claim is false. A random start allows only consecutive \(100\)-person blocks, so most possible samples of \(100\) employees cannot occur. Equal individual inclusion probabilities do not make the design an SRS.

- Check whether two books can receive the same random value. - Examine what happens when a tie affects the last available place. - Ask whether the tie rule favors any fixed identifying feature.

1. Ties are possible because only \(100\) random integers are used for \(600\) books. 2. Resolving a cutoff tie by catalog number gives some tied books a systematic advantage, so not every group of \(30\) books has the same chance of selection. 3. The procedure can be fixed by randomly breaking every cutoff tie or by generating a random ordering of all \(600\) books and selecting the first \(30\).

No. The deterministic catalog-number tie-break means the result is not an exact SRS. Randomly break cutoff ties, or randomly order all \(600\) books and select the first \(30\).

- Identify the unit that was randomly selected and the unit that supplied responses. - Consider shared influences within each selected group. - Think about whether correlated observations provide as much information as independent ones.

1. Students were sampled in clusters, not independently across the district. Responses within a classroom may be correlated. 2. Students in the same classroom may share the same teacher, assignments, reminders, or school resources, making their reading-program participation more similar. 3. Treating correlated observations as independent can underestimate the true sampling variability, so the reported margin of error may be too small.

a) The students are clustered within only \(8\) classrooms, so their responses may not be independent. b) Shared classroom experiences can make responses similar. c) The usual calculation may underestimate the margin of error.

- Write the selection process as a clinic step followed by a patient-within-clinic step. - Compare the second-step chance in a small clinic with the chance in a large clinic. - Adjust the first-stage clinic chances so they offset clinic size.

1. Every clinic has the same first-stage selection probability. 2. Within a selected clinic, a patient has probability equal to one divided by that clinic's patient count of being chosen. 3. Patients in small clinics therefore have greater overall inclusion probabilities than patients in large clinics. 4. One correction is to select clinics with probability proportional to clinic size and then select the same number of patients within each selected clinic, or to account for unequal probabilities with appropriate weights.

Equal-probability clinic selection followed by one patient per clinic favors patients in small clinics. Select clinics with probability proportional to patient count and then sample equally within selected clinics, or use sampling weights that reflect the unequal probabilities.