1.1 A student council surveys 100 students by taking random samples of 25 freshmen, 25 sophomores, 25 juniors, and 25 seniors.

A) Simple random sample
B) Stratified random sample
C) Cluster random sample
D) Systematic random sample
E) None of the above

• B). This is a stratified random sample because the population was divided into strata (class levels), and random samples were taken from each.

1.2 A quality control worker at a factory selects the first 5 items she sees as her sample for the day.

A) Simple random sample
B) Stratified random sample
C) Cluster random sample
D) Systematic random sample
E) None of the above

• E). This is a convenience sample.

1.3 A principal orders t-shirts and wants to check some of them to make sure they were printed properly. She randomly selects 3 of the 10 boxes of shirts and checks every shirt in those 3 boxes.

A) Simple random sample
B) Stratified random sample
C) Cluster random sample
D) Systematic random sample
E) None of the above

• C). The shirts were divided into clusters (boxes) and the principal sampled every shirt from some of the clusters.

1.4 A researcher recruits participants from his social media followers who are willing to complete a questionnaire.

A) Quota sample
B) Purposive sample
C) Convenience sample
D) Snowball sample
E) None of the above

• C). This is a subtype of convenience sample: voluntary (self-selection) sample.

1.5 A teacher puts her students’ names in a box and pick without looking to get 2 students to answer the question at class

A) Simple random sample
B) Convenience sample
C) Voluntary sample
D) None of the above

• A). No grouping, and all students in the class have an equal chance of being selected.

1.6 A conference organizer gathers feedback from every participant on the conference.

Simple random sample
A) Stratified random sample
B) Cluster random sample
C) Systematic random sample
D) Purposive sample
E) Convenience sample
F) Voluntary sample
G) Quota sample
H) None of the above

• H). This is not a sample, but the population itself, as everyone is selected.

1.7 A restaurant manager implements a discount program by randomly selecting one of the first 100 clients to receive a special discount. Subsequently, every 100th client is offered the same discount.

A) Simple random sample
B) Stratified random sample
C) Cluster random sample
D) Systematic random sample
E) None of the above

• D). This is an example of systematic random sampling. The manager uses a fixed interval (every 100th client) after randomly selecting a starting point within the first 100 clients.

1.8 A middle school principal wants to know how students experience their school life in this semester, so he selected some students at random in each school grade.

A) Simple random sample
B) Stratified random sample
C) Cluster random sample
D) Systematic random sample
E) None of the above

• B). This is a stratified random sample because the population was divided into strata (school grades), and a random sample was taken from each group (students per grade).

1.9 Which sampling strategy achieves the intended design?

A college professor wants to survey a sample of students taking her large lecture course. There are about 150 students in this course, and 10 of those students are graduate students. She wants to take a systematic random sample of approximately 30 students.

A) Randomly select 15 men and 15 women from the class for the survey.
B) Randomly select 2 graduate students and 28 other students for the survey.
C) Randomly select one of the first 5 students to arrive to class, and every 5th student thereafter to take the survey.
D) Randomly select one of the first 15 students to arrive to class, and every 15th student thereafter to take the survey.
E) Assign each student a number and use a computer to randomly select 30 students for the survey.

• C). This is a systematic sample since the students arrive in some order, she selects the \(k^{th}\) student, and then selects every \(k^{th}\) student thereafter. Also, \(\frac{150}{5}=30\) students.

1.10 Why use a simple random sample instead of a systematic random sample in the following example?

A large bakery produces cakes in an assembly line. Sergio noticed a potential defect that seems to be showing in every other cake. He decides that in the next production run, he’ll take a random sample of cakes to get a better idea of how many cakes might have this defect.

A) If the defect is in every other cake, a systematic random sample could miss it completely depending on what Sergio chooses for k.
B) A simple random sample can save time since it is less involved than a systematic random sample.
C) A simple random sample guarantees that Sergio will get some cakes from each hour of production.

• A). Suppose he starts with the \(8^{th}\) cake and \(k=10\) He would pick cakes 8, 18, 28, 38…. If the defect is in every other cake starting with the cake 1, then none of the cakes in Sergio’s sample would have the defect.

1.11 Why use a cluster random sample instead of a simple random sample? (Choose all that apply)

An airline offers a certain flight once per day that usually contains about 250 passengers. The flight offers seats in first class (most expensive), business class, and economy class (least expensive). The airline wants to survey 500 passengers of this flight about their overall satisfaction. The passengers will be selected using a cluster random sample where each flight is a cluster.

A) A cluster sample chooses some passengers from each flight, so the airline will be more likely to get a representative sample.
B) Contacting all of the passengers on a few flights will probably be more efficient than contacting passengers spread out across many flights.
C) A simple random sample might result in a disproportionate number of passengers from one or more class, but a cluster sample is less likely to suffer this issue. 

• B and C.

1.12 Which sampling strategy achieves the intended design?

A researcher aims to study school performance among high school students in the USA. They define a sample size of 1000 respondents. To achieve this, the researcher randomly selects 10 states out of 50. From each selected state, they randomly choose 5 high schools. Within each school, they survey 2 entire classes picked at random.

• Multistage cluster sampling.

  Stage-1: Random selection of states: ensures geographical diversity by selecting 10 states randomly, reducing bias in regional representation.

  Stage-2: Random selection of high schools: narrows the sample by randomly selecting 5 schools within each chosen state, which ensures variation within states.

  Stage-3: Random selection of classes within schools: involves randomly choosing 2 entire classes per school, ensuring diverse student representation within each school.

  In this design, clusters (states/schools/classes) are the primary focus of selection, rather than individual students directly.

1.13 Which of the following best describes how the chosen sampling is applied in this context?

A researcher aims to study the healthcare challenges faced by sex workers in a metropolitan city. Given the sensitive nature of the topic and the difficulty of accessing this population, they decide to use snowball sampling.

A) The researcher randomly selects sex workers from a city directory and invites them to participate.
B) The researcher begins with a few participants from NGOs and asks them to refer other sex workers willing to participate.
C) The researcher divides the city into zones and surveys sex workers found in specific areas.
D) The researcher conducts a public survey and asks respondents if they know any sex workers who can be contacted.

• B). Snowball sampling starts by identifying an initial small pool of participants who are accessible. In this case, the researcher connects with sex workers through NGOs or community organizations that already have established trust and relationships with the target population.

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