Random Sampling: Identify and Justify the Method — Worked Solutions

A is without replacement: the selected slip is not returned, so that apartment cannot be drawn again. B is with replacement: the slip returns before later draws, so the apartment may be selected multiple times.

A is an SRS because every two-student pair has the same chance. B permits only two pairs and leaves other pairs impossible, so it is not an SRS. Mix ten equal numbered slips and draw two without replacement for an equivalent SRS.

The first is stratified random sampling: non-overlapping grade strata group students by a shared attribute, an SRS is taken from each, and results are combined. The second is cluster random sampling: whole homerooms are randomly selected and all their students measured. Ideally mixed homerooms reflect school variety and are similar to one another.

This is systematic random sampling. It combines a random starting point with a fixed interval of 10 between selections. A repeating list pattern at interval 10 could make every selected household share a characteristic, so suitability depends on the order.

Use the complete 600-member roster and a random-number generator to select 60 distinct labels uniformly without replacement. Every possible set of 60 members must have the same chance, not merely every member an equal individual chance. Without replacement avoids asking one member multiple times.

Use family, individual, and student categories as non-overlapping strata. One allocation is 20 random members from each category; proportionate 20,25,15 is also valid. Select an SRS within every stratum and combine the chosen members into one sample. Membership type is the shared within-stratum attribute.

Select an SRS of three of the 30 neighborhood groups, then measure all 20 members in each selected group. The groups are desirable clusters because each reflects varied membership types and the groups are similar to one another. Do not randomly select only some members inside these clusters and still call it this one-stage method.

Choose the starting position uniformly from 1 through 10, then take that position and every tenth position through the list. This yields 60 selections. Both the random start and interval 10 are part of the method.

Stratifying by category ensures all categories supply observations and can ensure enough student-member responses for comparisons; an equal allocation prioritizes such comparisons. Cluster sampling might be preferable if interviewers must visit neighborhoods and measuring every selected group reduces travel. The choice depends on comparison needs and collection cost, not a universal ranking.

Return the selected label to the selection pool before the next draw. A member could be selected more than once. That is different from the distinct-label without-replacement procedures above.