Introducing Statistics: Questions, Samples, and Context — Worked Solutions

The population is all 8,000 annual museum members, so N=8,000. The sample is the 160 responding selected members, so n=160. Selection determines which members supply data; it does not change which members the museum wants to understand.

One member’s recorded “10 a.m.” preference is a datum. All 160 recorded preferences form the data set. The datum describes the opening-time preference of that particular member.

The team can collect measurements from selected trees to answer a question about all 40,000 trees in the region. The population is large and reaching every tree is difficult and costly; a sample makes data collection feasible. The sampling design still matters for extending the description to the population.

The new threshold was chosen after inspecting the responses, so it was not the defined original purpose. Report the answer to the original more-than-half question. Label the one-third idea as a follow-up question and, if it will guide a decision, plan a new investigation with that purpose fixed in advance.

One valid question is “What proportion of the 900 current Urbanopolis community-garden plot holders would use an on-site storage locker at least once a week next spring?” Collect one yes/no response under that fixed weekly-use condition and analyze the sample proportion saying yes. A different clearly specified binary locker-use question is valid, such as any intended use next spring, if its population, yes/no condition, time period and decision purpose are explicit. An unrestricted numerical frequency or desired-size question would not satisfy this task’s binary-response condition.

N=900 and n=90 if all 90 selected holders respond. One holder’s yes/no response to the exact question constructed in part (a) is a possible datum; all 90 responses under that same rule form the complete data set. The sample is those measured selected holders, not every holder who would like a locker.

The council obtains information from a subset to answer a question about the larger population. Contacting all 900 requires more staff time, and some holders may be difficult to reach before the decision date. These are reasons to sample; they do not by themselves establish that the selected holders represent the population.

54/90=0.60, so 60% of the 90 sampled plot holders answer yes to the exact intended-use condition and time period in the learner’s question. For the example weekly-use question, 60% say they would use a locker at least once a week next spring; a valid any-use question instead requires that matching interpretation. The calculation alone does not establish that exactly 60% of all 900 holders will meet the chosen condition; selection, nonresponse and intentions versus later behavior matter.

Ask each selected holder the same clear binary question constructed in part (a), retaining that question’s intended-use condition and time period. Record each yes/no response under that rule and divide the number of yes responses by the number of obtained responses. Keep the question fixed for the original report. Record an unexpected pattern as a reason for a clearly labeled follow-up investigation rather than quietly changing the original question after seeing results. Equivalent consistent collection plans can earn credit.