Sampling methods practice problems

Ask what device selected the individuals and at what level it acted. If no chance device was used, the design is convenience or voluntary response.

(a) A random number generator draws individuals directly from a list of all 1,450 students, so every group of 100 students is equally likely. This is a simple random sample.

(b) A random start followed by a fixed interval of every 25th bottle is a systematic random sample. Chance sets only the starting point.

(c) The population is split into groups of similar individuals and an SRS is taken inside every group. The sample takes some voters from every one of the four age groups, so this is a stratified random sample.

(d) Whole routes are chosen at random and every student on a chosen route is surveyed, with no one sampled from the other 54 routes. This is a cluster random sample.

(e) The first 30 shoppers at one store are simply the easiest people to reach, and no chance device is involved. This is a convenience sample.

(f) The 900 readers selected themselves by choosing to mail the form in. This is a voluntary response sample.

Keep (c) and (d) apart with one line: stratified takes some individuals from every group, cluster takes every individual from some groups.

(a) Taking every $k$th name from a list after a random start is a systematic random sample. The interval is $k = \frac{N}{n} = \frac{1200}{60} = 20$, so the auditor picks a random integer from 1 to 20 as the start and then takes every 20th name.

(b) With a start of 7, the selected numbers are $7$, $7 + 20 = 27$, and $7 + 40 = 47$. The 60th selection is $7 + 20(60 - 1) = 7 + 1180 = 1187$.

(b) Check the count: from 7 to 1187 in steps of 20 gives $\frac{1187 - 7}{20} + 1 = 59 + 1 = 60$ employees, as required.

(c) There are 20 possible starts, each equally likely, and each employee belongs to exactly one of the 20 resulting samples. So any given employee is selected with probability $\frac{1}{20} = 0.05$, which matches $\frac{60}{1200} = 0.05$.

(d) No. An SRS requires every group of 60 employees to be equally likely, and here only 20 different samples are possible. The group made of employees 1 through 60 has probability 0, so equal chance for each individual is not enough to make the design an SRS.

One risk worth naming: if the payroll list repeats with period 20, for example if every 20th name is a shift supervisor, the sample over-represents or misses that group every time the method is used.

(a) Find the sampling fraction first: $f = \frac{200}{4000} = 0.05$, so the sample takes 5% of each stratum.

(a) Apply it to each level. Elementary: $1800 \times 0.05 = 90$. Middle: $1200 \times 0.05 = 60$. High: $1000 \times 0.05 = 50$. Check the total: $90 + 60 + 50 = 200$.

(b) Number the elementary students 1 to 1800 and use a random number generator to produce distinct integers in that range, ignoring repeats, until 90 different students are chosen. Repeat separately for the 1,200 middle school students, taking 60, and for the 1,000 high school students, taking 50. Combine the three groups into the sample of 200.

(c) Homework time plausibly differs a lot between elementary, middle, and high school students but less within a level. Stratifying guarantees all three levels appear in the right proportion and, when strata differ like this, it produces an estimate that varies less from sample to sample than an SRS of the same size.

(d) The sample was drawn at random from a frame covering all 4,000 district students, so the estimate describes the mean homework time of all 4,000 students in this district. It says nothing about students in other districts, who had no chance of selection.

(a) Nobody was selected by a chance device. The 4,200 people chose to answer an open poll, so this is a voluntary response sample.

(b) The percentage saying yes is $\frac{2856}{4200} = 0.68$, or 68%, and the share who answered is $\frac{4200}{50000} = 0.084$, or 8.4%. Only about 1 in 12 of the people who saw the post answered it.

(b) Call that 8.4% a participation rate, not a response rate. A response rate counts the people who answered out of the people a study selected, which is why Design B in a later problem has one and this poll does not. Nobody here was selected by anyone, so there was nothing to respond to, and the 8.4% measures who volunteered rather than who was reached.

(c) The main flaw is voluntary response bias: people who feel strongly enough about the stadium to stop and answer are not typical of city residents, and on a team's own fan page the people who feel strongly are overwhelmingly supporters. The estimate is therefore biased upward, so 68% overstates support among all city residents.

(c) There is undercoverage as well. Residents who do not follow the team on social media had no chance to answer, and they are exactly the people least likely to favor spending public money on a stadium, which pushes the estimate in the same upward direction.

(d) At best the 68% describes the 4,200 people who chose to answer. It is not a valid estimate for the roughly 50,000 who saw the post, and it is certainly not an estimate for all city residents.

(e) No. Bias is a lean in the method, so it repeats in the same direction no matter how many people answer. A poll of 42,000 self-selected fans would give a narrower but equally wrong estimate. The fix is a different method: a random sample drawn from a frame that covers all city residents.

(a) Whole floors are chosen at random and everyone on a chosen floor is surveyed, so the floors are clusters and this is a cluster random sample. The sample holds about $8 \times 30 = 240$ residents.

(a) Note the two matching fractions: the office sampled $\frac{8}{240} = \frac{1}{30}$ of the floors and about $\frac{240}{7200} = \frac{1}{30}$ of the residents, which is what taking whole clusters at random should produce.

(b) A stratified design takes an SRS inside every group, so every floor would contribute someone. Here 232 of the 240 floors contribute nobody at all, and the 8 selected floors contribute everybody. Strata are also built to be alike within a group, while clusters are meant to look like the whole population on their own.

(c) The chance mechanism selected floors from a frame of dorm floors, so the result generalizes to the 7,200 students living in university housing. Undergraduates who live off campus or commute had zero chance of selection, so extending the result to all 12,000 is undercoverage bias.

(c) State the direction as well: students who live in dorms eat in dining halls far more than commuters do, so their views on dining hall hours are unrepresentative and the estimate leans toward the interests of on-campus residents. To reach all 12,000, the office needs a frame listing every undergraduate, for example stratifying by housing status and taking an SRS within each stratum.

(a) Design A: $\frac{540}{600} = 0.90$, a response rate of 90%. Design B: $\frac{168}{800} = 0.21$, a response rate of 21%.

(b) Design A suffers undercoverage bias. Adults in households with no landline, which today means most younger adults, have no chance of being selected at all. Landline households skew older, and older adults are more likely to have a primary care doctor, so Design A likely overstates the true proportion.

(b) Design B suffers nonresponse bias. Fully $800 - 168 = 632$ of the selected households, or 79%, did not answer. People who take the time to return a health survey are plausibly those already connected to regular care, so Design B likely overstates the proportion as well.

(c) Compare where each flaw lives. Design A's flaw is in the sampling frame: the landline directory permanently excludes part of the population, and no amount of extra effort with that frame reaches cell-only households. Design B's frame is the city address list, which does cover the population; only the response step failed.

(c) So Design B is easier to repair, through follow-up mailings, a second contact, or in-person visits to a random subset of the non-responders. Design A needs a new frame, such as a list that includes cell phone numbers, before it can estimate anything about all city adults.

Note that the 90% response rate does not make Design A the better study. A high response rate within a frame that misses part of the population still leaves an estimate that is systematically wrong.

(a) Group A: $\frac{219}{300} = 0.73$, or 73% saying yes. Group B: $\frac{129}{300} = 0.43$, or 43%. The gap is $0.73 - 0.43 = 0.30$, or 30 percentage points.

(b) The voters came from an SRS and were split into the two groups at random, so the two groups are alike in composition and the flaw is not in who was asked. A 30 percentage point gap between two random halves of the same sample is far too large to explain as ordinary sampling variability. The flaw is in the measurement, and its name is response bias from question wording.

(b) Notice what the random split buys here. The two wordings were randomly assigned to two halves of one sample, which makes this a small randomized experiment sitting inside the survey, so the wording itself can be named as the cause of the 30 point gap. Compare this with problem 8, where random selection alone supports no causal claim at all.

(c) Version A calls the increase "small" and pairs it with a concrete, appealing benefit in the same sentence, which pushes the estimate upward: 73% overstates real support. Version B leads with "raising taxes" and gives a vaguer benefit, which pushes the estimate downward: 43% understates it.

(d) No. Each wording carries its own systematic lean, so asking 3,000 voters per version would tighten each estimate around its own biased value rather than move either toward the truth. A larger sample shrinks variability, never bias.

(e) Neither number is a trustworthy estimate of support. The town should ask again with neutral wording that states the actual tax rate and the service in the same sentence, without the word "small" and without leading with the word taxes, and report the estimate from that survey instead.

(a) The teacher surveyed the students already in front of her, with no chance device selecting them, so this is a convenience sample. The response rate is $\frac{96}{120} = 0.80$, or 80%.

(b) Keep one target in view for both answers: the question asks which way each flaw pushes 6.4 hours as an estimate of the mean for all 1,600 students, so both directions have to be stated against that same number.

(b) First, undercoverage from the convenience design. Every student who does not take her course had no chance of selection, and because the course is a junior and senior elective, that leaves out nearly every freshman and sophomore in the school. Upperclassmen carry heavier course loads and more outside commitments, so they plausibly sleep less than the school as a whole, which pushes 6.4 hours downward as an estimate of the school mean.

(b) Second, nonresponse bias. The $120 - 96 = 24$ students who did not return the form are 20% of the group, and the busiest students, who are also the ones sleeping least, are the most likely to skip a voluntary form. Dropping the shortest sleepers lifts the average of whoever is left, so nonresponse pushes 6.4 hours upward as an estimate of the school mean. A third defensible answer is response bias, since students self-report sleep and tend to round the number up.

(b) The two leans run in opposite directions, so the net error in 6.4 hours cannot be signed, let alone subtracted out. That is the honest reading of a broken design: it is repaired by sampling again, not by adjusting the number it produced.

(c) The sampling fraction is $f = \frac{160}{1600} = 0.10$, so take 10% of each grade: $480 \times 0.10 = 48$, $420 \times 0.10 = 42$, $380 \times 0.10 = 38$, and $320 \times 0.10 = 32$. Check the total: $48 + 42 + 38 + 32 = 160$.

(c) Selection: number the 480 students in the first grade level from 1 to 480, use a random number generator to produce integers in that range, discard repeats, and stop at 48 distinct students. Do the same within each of the other three grades to get 42, 38, and 32 students, then combine the four groups. Follow up with any student who does not respond rather than replacing that student with someone easier to reach, since replacing them rebuilds the nonresponse problem.

(d) Scope: the 160 students were selected at random from a frame covering all 1,600 students in the school, so 6.9 hours estimates the mean sleep for all students at this school. It does not describe students at other schools, who had no chance of being selected.

(d) Causation: no. Random selection supports generalizing to the school, and nothing more. A cause and effect conclusion requires random assignment of treatments in an experiment, and this is an observational study with no treatment assigned. Students who sleep more may also differ in workload, jobs, or activities, and any of those could explain a difference in test scores.