The empirical law of large numbers and the hospital problem: systematic investigation of the impact of multiple task and person characteristics

Sensitivity to Sample Size in the Context of the Empirical Law of Large Numbers: Comparing the Effectiveness of Three Approaches to Support Early Secondary School Students

“…Regarding task variants focusing on a 'static-contrast' vs a 'dynamic' view, prior research has reported problems mostly relating to the 'static-contrast' view of the eLLN (e.g., Batanero et al 1996;Rasfeld 2004), while a 'dynamic' view appears to be less problematic. In contrast to the three aforementioned aspects, which appear to be important for the difficulty of corresponding items based on prior research, Weixler et al (2019) produced evidence that the aspect sample (e.g., childbirths) vs. test series (e.g., coin tosses) plays only a subordinate role regarding item difficulty.…”

“…Fifth, it can be differentiated (Weixler et al 2019) between formulations of the eLLN that focus on (one or multiple) 'samples' from a population or that focus on (one or multiple) 'test series' (see Fig. 2 for examples).…”

“…However, besides these common aspects, multiple different formulations of the hospital problem have been used and it has, for example, also been transformed into other contexts, e.g., coin tosses. This has led to partially conflicting evidence (see e.g., Lem 2015;Weixler et al 2019), as the different possible foci of the eLLN have not been systematically considered. Regarding differences in solution rates, tasks one can characterize as 'one' seem to lead to higher solution rates than tasks one can characterize as 'multiple' (e.g., Evans and Dusoir 1977;Sedlmeier and Gigerenzer 1997) and tasks one can characterize as 'accuracy' seem to lead to higher solution rates than tasks one can characterize as 'tail' (e.g., Well et al 1990).…”

“…Moreover, little research has focused on the systematic analysis and comparison of different educational approaches that may help students increase their sensitivity to sample size in the context of the eLLN and to better cope with corresponding tasks-covering different foci of the eLLN-and demands in general. In particular, a study by Weixler et al (2019), which disen-K tangled partially conflicting evidence regarding the eLLN by using a systematic approach to investigating different task variants of the hospital problem, suggests a new approach based on so-called "maximum-deviation tasks." These tasks follow an extreme-value heuristic or "Extremalprinzip" (e.g., Grieser 2017) and are similar to boundary examples or boundary problems (see also A.…”

Tversky and Kahneman’s Cognitive Illusions: Who Can Solve Them, and Why?

“…According to Weixler et al (2019), performances range between 0% (Fischbein and Schnarch, 1997) and 85% (Evans and Dusoir, 1977), the authors attributing the wide range of solution rates to the fact that the tasks used usually varied in one or more features and that the groups of people investigated were different. In disentangling the effects of concrete task and participant characteristics (see below; e.g., grades: Roth et al, 2015;gender: Watson, 2000; see also section "Person-Related and Task-Related Characteristics Associated With the Ability to Solve Cognitive Illusions"), Weixler et al (2019) found that, for example, problem-solving is facilitated in particular when the deviation from the expected relative frequency is maximal (cf. Lem, 2015), the ratio between the large and the small sample is large (cf.…”