Question 12·Hard·Evaluate Statistical Claims: Observational Studies and Experiments
A sports scientist investigated the relationship between the amount of strength-training runners perform each week and their finishing times in a city 10-kilometer race. From the list of all 2,300 runners registered for the race, the scientist randomly selected 150 runners to interview immediately after they finished. The table shows the data for the 112 runners in the sample who completed the interview.
| Weekly strength-training | Finish time > 50 min | Finish time ≤ 50 min | Total |
|---|---|---|---|
| Fewer than 3 hours | 31 | 17 | 48 |
| 3 hours or more | 26 | 38 | 64 |
| Total | 57 | 55 | 112 |
Which of the following is the largest population to which the conclusions of this study can be appropriately generalized?
For questions about which population you can generalize to, first locate the description of how the sample was chosen and ask: "From what big group were these people (or items) randomly selected?" That big group is your candidate population. Then check each answer choice: eliminate options that describe only the sample itself, subgroups defined by outcomes (like only those with a certain score or time), or groups that were never part of the original selection process. Finally, pick the largest group that still matches the population from which the sample was randomly drawn.
Hints
Clarify what the question is asking
Focus on the words "largest population" and "appropriately generalized". This is about which big group the study’s results can represent, not about which numbers appear in the table.
Look for the group used in the random selection
Find the sentence that describes how the runners were randomly selected. From which group were these runners chosen at random?
Match answer choices to that group
Among the answer choices, identify which describes the entire group from which the random sample was drawn, rather than a smaller subset or a different kind of subgroup (like only fast finishers).
Think about "largest" carefully
Several answer choices may sound reasonable, but the question wants the largest population that still fits the idea of being represented by the sample. Eliminate any choice that is just the sample itself or a subgroup defined after the fact (such as by finish time).
Step-by-step Explanation
Identify sample and population in a study
In statistics, the sample is the group you actually collect data from (here, the runners who were interviewed). The population is the larger group you want to say something about using that sample.
The question asks: What is the largest population to which the conclusions can be appropriately generalized? That means: what is the biggest group that the sample can reasonably represent?
Find exactly who was randomly selected
The prompt says: "From the list of all 2,300 runners registered for the race, the scientist randomly selected 150 runners to interview."
So the random selection was from the list of all 2,300 registered runners. This is crucial: random selection tells us which group our sample can represent, assuming no serious bias.
Connect random selection to generalization
We can appropriately generalize conclusions only to groups from which the sample was randomly drawn.
Now compare the groups mentioned in the problem:
- The 112 runners are just the ones who completed the interview. They are the final sample used, but they are not the population.
- The 150 runners selected are the intended sample; they themselves came from a larger group.
- The group of runners with finish time ≤ 50 minutes is a subgroup defined by performance, not the group we sampled from.
- The 2,300 registered runners are the full list from which the random selection was made.
The largest group that still matches "from which the sample was randomly selected" will be the correct choice.
Choose the largest appropriate population
Because the 150 selected runners (and thus the 112 who responded) were randomly chosen from the full list of 2,300 registered runners, the study’s conclusions can be generalized to that full group, but not beyond it.
Therefore, the largest population to which the conclusions can be appropriately generalized is all 2,300 runners registered for the city 10-kilometer race.