Predicted Variability and Working Memory 1 Running head: PREDICTED VARIABILITY AND WORKING MEMORY Statistical Reasoning and Working Memory

In this study, the relationships between statistical reasoning and WM is examined in terms of a variability prediction task. The study consists of two parts. The first part is the variability prediction task which replicates the settings of the experiment of Kareev, Arnon, and Horwitz-Zeliger (2002), but with extra trials added. In this experiment, the participants first have a chance to observe a population of items and are asked to predict the number of the target items in a 7-item sample drawn from the population of 28 items, when the items remain present or absent during prediction making. Given that the expected target number is smaller, the larger the predicted number, the more variable the prediction will be. The second part is for measuring a participant’s WM capacity. In order to get a more thorough measurement of WM capacity, 7 tasks designed for testing the functions of the central executive of WM are administered. The structural equation modeling is conducted to link the data of these two parts. The generated model shows that among the executive functions, only the update function can predict the performance of the prediction process. This result supports the original hypothesis of Kareev et al. (2002) and highlights the link between WM and ”bounded rationality”. Predicted Variability and Working Memory 3 Statistical Reasoning and Working Memory We have plenty of experiences of making statistical reasoning to the events, from predicting whether or not it will rain tomorrow to predicting how much longer the economic recession will last. However, people seem not to apply the knowledge of statistics, for instance, to the prediction of the event frequency and probability (see Tversky & Kahneman, 1973). Recently some psychologists have argued that people do hold and apply the correct knowledge of statistics when making statistical reasoning, but rely on the distorted sample information for prediction making (Fiedler & Juslin, 2006). Their rationale is that people rely on the sample temporarily maintained in WM (Working Memory) to make predictions and the prediction bias correlates with the extent of the representativeness of the sample to the population. Since WM, extended from STM (Short-Term Memory) in concept, is thought to be a cognitive platform for temporary storage of information and for implementation of the mental processes (e.g., predicting the mean of a population) relevant to the stored information (see Baddeley, 1986, 2000; Baddeley & Hitch, 1974; Cowan, 2005), one’s performance in statistical reasoning should be influenced by the capacity of WM, presumably. Psychologists have acknowledged for a long time that the capacity of WM is limited, that is, one can only process a few amount, not every piece, of information at every moment. This phenomenon can be easily observed from the decline of the recall accuracy while another mental process is performing. This in turn becomes a standard method to measure the capacity of a person’s WM. For instance, the sentence span task developed by Daneman and Carpenter (1980) is meant to test to what extent that people can retain the last word in every sentence while being asked to comprehend the sentences at the same time. Similarly, the operation span reflects the number of consonants one can correctly recall while being asked to verify the arithmetic problems during learning the consonants Predicted Variability and Working Memory 4 (Turner & Engle, 1989). Of course, there are more WM capacity measures developed in the past decades. In addition, there are clear individual/age differences on the WM capacity that makes itself as a good index of cognitive ability and the WM capacity has been evident to correlate positively with reading comprehension (Baddeley & Hitch, 1974), the ability to solve the syntactic ambiguity (King & Just, 1991), decision making (Hinson, Jameson, & Whitney, 2003), and reasoning intelligence (Sub, Oberauer, Wittmann, Wilhelm, & Schulze, 2002). In light of the correlations between the WM capacity and many cognitive tasks, Kareev et al. (2002) tried to examine if the WM capacity influences the perception of variability of a population. These authors asked the participants to first have a look at an opaque box, in which 28 items (i.e., balls of two colors or cylinders of different heights) are contained and then predict the makeup of a sample consisting of 7 items drawn from the box. In the out-of-sight condition, the participants first watched the box with the lid open for 1 minutes and then made their predictions about the number of target items 1 in the sample with the lid closed. In the full-view condition, the lid was open throughout the prediction making. The main concern of these authors was the estimated population variability of the participants of different sizes of WM capacity. Since the variability of a sample of size n, namely variance, is smaller by a factor of (n− 1)/n than that in the population (Hays, 1963), the estimated variance will be downward attenuated more when n gets smaller. These authors hypothesized that the sample on which people relied to estimate the population parameters was the sampled items temporarily maintained in their WM. Thus, the low span people should make a prediction to the number of the target item less variable than that done by the high span people, for a smaller WM capacity can afford maintaining less items simultaneously. Predicting directly the population variance was hard to people, so the predicted variability was shown in terms of the participants’ estimate of the number of the target item. For Predicted Variability and Working Memory 5 instance, in the ball condition (10 red balls vs. 18 blue balls), the mean number of the blue balls predicted by the participants would be smaller, when the prediction was made relying on a larger sample. This is because that the red balls get a larger chance to occur in a larger sample which in turn attenuate the perceived proportion of the blue balls. In order to examine the influence of WM, these authors adopted the digit span2 as the index of WM capacity for the participants. The results, basically, supported their hypothesis that the high span participants did predict a larger variability. However, a few potential worries in their study are hard to neglect. First, although the difference on the predicted variability between the high and low span groups is significant3, the effect size computed from their data is not satisfactorily large (partial η2 = .09). Second, the split-group design adopted by these authors might make the group difference more extreme which might mislead our inference to a significant influence of WM capacity on the variability prediction. Third, the digit span task is more relevant to STM (Short-Term memory) in terms of the definition of WM. The prediction of the target number presumably relies on not only the storage of the seen items but also the process of number estimating (i.e., counting or computing). In addition, the digit span is reported to have a low correlation with the individual differences on many cognitive aptitude tests (e.g., Cowan et al., 2005; Daneman & Merikle, 1996; Mukunda & Hall, 1992). At last, the the number of valid participants in their experiment was 454 only, which is relatively small in order to get a reliable result on most statistic tests. Each of these concerns might be a cause to lower the validity of their inference that the predicted variability is influenced by WM capacity. Therefore, we have reason to re-examine their results not only for getting a more reliable result but also for seeking the evidence of whether the statistical reasoning relies on the WM capacity. To circumvent the above potential flaws, this study recruits more participants (e.g., more than 100), adopts more sensitive WM tasks, and analyzes the data using more subtle design. Predicted Variability and Working Memory 6