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The emphasis given to probability and statistics in the K-12 mathematics curriculum has brought attention to the various approaches to probability and statistics concepts, as well as how to teach these concepts. Teachers from fourth, fifth, and sixth grades from a small suburban Catholic school engaged their students (n=87) in a study to compare learning traditional probability concepts to learning traditional and subjective probability concepts. The control group (n=44) received instruction in traditional probability, while the experimental group (n=43) received instruction in traditional and subjective probability. A Multivariate Analysis of Variance and a Bayesian t-test were used to analyze pretest and posttest scores from the Making Decisions about Chance Questionnaire (MDCQ). Researcher observational notes, teacher journal entries, student activity worksheet explanations, pre- and post-test answers, and student interviews were coded for themes. All groups showed significant improvement on the post-MDCQ (p < .01). There was a disordinal interaction between the combined fifth- and sixth-grade experimental group (n=28) and the control group (n=28), however the mean difference in performance on the pre-MDCQ and post-MDCQ was not significant (p=.096). A Bayesian t-test indicated that there is reasonable evidence to believe that the mean of the experimental group exceeded the mean of the control group. Qualitative data showed that while students have beliefs about probabilistic situations based on their past experiences and prior knowledge, and often use this information to make probability judgments, they find traditional probability problems easier than subjective probability. Further research with different grade levels, larger sample sizes or different activities would develop learning theory in this area and may provide insight about probability judgments previously labeled as misconceptions by researchers.

Doctor of Philosophy / Department of Agricultural Economics / Ted C. Schroeder / Risk is an inevitable part of agricultural production and all producers face various forms of risk. Output price has been shown to be the major contributor to the risk in cattle feeding, yet few choose to manage this risk. This study used subjective price expectations and price distributions of survey participants to determine how producer's expectations compare with that of the market. In addition, demographic information gathered from survey participants allowed for further examination as to how these factors effect price outlook and variability. Data used for this study were gathered through survey responses from Kansas State University Extension meeting and workshop participants and other meetings targeted to livestock producers. First, data were aggregated and analyzed at a group level. Only two of the twelve price forecast were significantly lower than the futures settlement price. On the other hand, all but one of the aggregated group volatility expectations was different. Typically nearby contract price risk expectation was underestimated and distant contract price risk expectation was overestimated. Individual respondent's discreet stated price and price distribution information was fitted to a continuous distribution and an implied mean and standard deviation were determined. These were compared to market price and price risk data. Respondent's expectation of price was significantly lower than the market for distant months for five of the six groups. Individual volatilities resulting from each fitted distribution were significantly lower from the volatility measure resulting from Black's model. Demographic data were estimated to show the impact of this information on overall error of price forecast and price risk expectations. Those living outside the Northeast and Northern Plains tended to have larger error in their expectation of price volatility. Larger backgrounding operations reported lower price variance error and selling more fed cattle each year increased price risk expectation error. Lastly, prior use of risk management tools for the most part did not have an impact on error in either price expectation or price volatility expectation.

A particular field in research on judgment and decision making (JDM) is concerned with realism of confidence in one’s knowledge. An interesting finding is the so-called format dependence effect, which implies that assessment of the same probability distribution generates different conclusions about over- or underconfidence depending on the assessment format. In particular, expressing a belief about some unknown continuous quantity (e.g., a stock value) in the form of an intuitive confidence interval is severely prone to overconfidence as compared to expressing the belief as an assessment of a probability judgment. This thesis gives a tentative account of this finding in terms of a Naïve Sampling Model, which assumes that people accurately describe their available information stored in memory, but they are naïve in the sense that they treat sample properties as proper estimators of population properties (Study 1). The effect of this naivety is directly investigated empirically in Study 2. A prediction that short-term memory is a constraining factor for sample size in judgment, suggesting that experience per se does not eliminate overconfidence is investigated and verified in Study 3. Age-related increments in overconfidence were observed with intuitive confidence interval but not for probability judgment (Study 4). This thesis suggests that no cognitive processing bias (e.g., Tversky &amp; Kahneman, 1974) over and above naivety is needed to understand and explain the overconfidence “bias” with intuitive confidence interval and hence the format dependence effect.

O propósito desta dissertação é apresentar o uso da opinião de especialistas e outras questões relevantes acerca do assunto na avaliação das incertezas em estudos de análise de risco e confiabilidade, com apresentação de um estudo de caso prático. Em estudos de confiabilidade umas das principais preocupações está na determinação das frequências de ocorrência dos eventos e seu comportamento ao longo do tempo. Muitas vezes, os dados de frequência estão obsoletos, não estão disponíveis ou mesmo, não são suficientes para se avaliar a probabilidade de ocorrência de eventos. Nestes casos, a elicitação da opinião de especialista surge como uma alternativa a suplementar estas ausências de dados possibilitando assim uma melhor análise das incertezas. Baseado na condição da subjetividade, a elicitação dos especialistas tem como objetivo quantificar as incertezas a partir da experiência prévia e estado atual de conhecimento. Combinado com métodos matemáticos, a elicitação possibilita o gerenciamento de conflitos de informações de forma a atingir o consenso e possibilitar uma análise subjetiva dos problemas. / The purpose of this dissertationis to present the use of expert opinion and other relevant issues on the subjective assessment of uncertainties in risk analysis and reliability studies, presenting a practical case study. In reliability studies a major concern is to determine the frequencies of occurrence of events and their behavior through time. Often, the available data are not representative enough to evaluate the event probability or it is obsolete for use. In these cases, the elicitation of expert opinion is an alternative to supplement these data absences, Thus enabling a better uncertainties analysis. Based on the subjectivity condition, the elicitation of experts aims to quantify the uncertainty considering the previous experiences and current state of knowledge. Combined with mathematical elicitation methods, it enables the manegement of information conflicts in order to reach consensus and makes possible a subjective analysis of problems.

<p>A particular field in research on judgment and decision making (JDM) is concerned with realism of confidence in one’s knowledge. An interesting finding is the so-called format dependence effect, which implies that assessment of the same probability distribution generates different conclusions about over- or underconfidence depending on the assessment format. In particular, expressing a belief about some unknown continuous quantity (e.g., a stock value) in the form of an intuitive confidence interval is severely prone to overconfidence as compared to expressing the belief as an assessment of a probability judgment. This thesis gives a tentative account of this finding in terms of a Naïve Sampling Model, which assumes that people accurately describe their available information stored in memory, but they are naïve in the sense that they treat sample properties as proper estimators of population properties (Study 1). The effect of this naivety is directly investigated empirically in Study 2. A prediction that short-term memory is a constraining factor for sample size in judgment, suggesting that experience per se does not eliminate overconfidence is investigated and verified in Study 3. Age-related increments in overconfidence were observed with intuitive confidence interval but not for probability judgment (Study 4). This thesis suggests that no cognitive processing bias (e.g., Tversky & Kahneman, 1974) over and above naivety is needed to understand and explain the overconfidence “bias” with intuitive confidence interval and hence the format dependence effect.</p>

<p>According to the heuristics and biases approach, the representativeness heuristic (RH) is one of the heuristics available for assessing subjective probabilities (A. Tversky & D. Kahneman, 1974). A subjective probability assessed by the RH is determined by how representative the target object is of the target category. Several aspects of the RH are argued to cause systematic biases, for example: (<i>i</i>) When the RH is used, the category is represented by one single prototypical exemplar. This feature is argued to cause biases such as misperception of chance and insensitivity to sample size. (<i>ii</i>) The RH assesses the inverse rather than the conditional probability. This feature is argued to cause biases such as the conjunction fallacy and base-rate neglect.</p><p>The present thesis focuses on the cognitive aspects of the RH. Three studies were conducted. Overall, data indicated that the RH does not play a major role when subjective probabilities are assessed. Study I indicated that subjective probabilities are not typically determined by how representative the target object is of the target category. Study II indicated that the category is not represented by one single prototypical exemplar when subjective probabilities are assessed. Study III indicated that conjunction fallacies are not caused by the RH. </p><p>The results presented in Studies I-III cast serious doubts on the claim that subjective probabilities are routinely assessed using the RH. Rather, Studies I-II suggested that subjective probabilities are based on exemplar memory and Study III suggested that the conjunction fallacy is caused by people combining component probabilities in a an inappropriate way. In the General Discussion, it is suggested that people use a weighted average rule when combining component probabilities into conjunction probabilities. A simulation showing the ecological relevance of the weighted average rule is presented.</p>

According to the heuristics and biases approach, the representativeness heuristic (RH) is one of the heuristics available for assessing subjective probabilities (A. Tversky &amp; D. Kahneman, 1974). A subjective probability assessed by the RH is determined by how representative the target object is of the target category. Several aspects of the RH are argued to cause systematic biases, for example: (i) When the RH is used, the category is represented by one single prototypical exemplar. This feature is argued to cause biases such as misperception of chance and insensitivity to sample size. (ii) The RH assesses the inverse rather than the conditional probability. This feature is argued to cause biases such as the conjunction fallacy and base-rate neglect. The present thesis focuses on the cognitive aspects of the RH. Three studies were conducted. Overall, data indicated that the RH does not play a major role when subjective probabilities are assessed. Study I indicated that subjective probabilities are not typically determined by how representative the target object is of the target category. Study II indicated that the category is not represented by one single prototypical exemplar when subjective probabilities are assessed. Study III indicated that conjunction fallacies are not caused by the RH. The results presented in Studies I-III cast serious doubts on the claim that subjective probabilities are routinely assessed using the RH. Rather, Studies I-II suggested that subjective probabilities are based on exemplar memory and Study III suggested that the conjunction fallacy is caused by people combining component probabilities in a an inappropriate way. In the General Discussion, it is suggested that people use a weighted average rule when combining component probabilities into conjunction probabilities. A simulation showing the ecological relevance of the weighted average rule is presented.

O propósito desta dissertação é apresentar o uso da opinião de especialistas e outras questões relevantes acerca do assunto na avaliação das incertezas em estudos de análise de risco e confiabilidade, com apresentação de um estudo de caso prático. Em estudos de confiabilidade umas das principais preocupações está na determinação das frequências de ocorrência dos eventos e seu comportamento ao longo do tempo. Muitas vezes, os dados de frequência estão obsoletos, não estão disponíveis ou mesmo, não são suficientes para se avaliar a probabilidade de ocorrência de eventos. Nestes casos, a elicitação da opinião de especialista surge como uma alternativa a suplementar estas ausências de dados possibilitando assim uma melhor análise das incertezas. Baseado na condição da subjetividade, a elicitação dos especialistas tem como objetivo quantificar as incertezas a partir da experiência prévia e estado atual de conhecimento. Combinado com métodos matemáticos, a elicitação possibilita o gerenciamento de conflitos de informações de forma a atingir o consenso e possibilitar uma análise subjetiva dos problemas. / The purpose of this dissertationis to present the use of expert opinion and other relevant issues on the subjective assessment of uncertainties in risk analysis and reliability studies, presenting a practical case study. In reliability studies a major concern is to determine the frequencies of occurrence of events and their behavior through time. Often, the available data are not representative enough to evaluate the event probability or it is obsolete for use. In these cases, the elicitation of expert opinion is an alternative to supplement these data absences, Thus enabling a better uncertainties analysis. Based on the subjectivity condition, the elicitation of experts aims to quantify the uncertainty considering the previous experiences and current state of knowledge. Combined with mathematical elicitation methods, it enables the manegement of information conflicts in order to reach consensus and makes possible a subjective analysis of problems.

The law of small numbers bias is a representativeness heuristic that often leads individuals to draw extensive conclusions from small samples while underestimating the generalizability in larger ones. This study investigated whether individuals overestimate perceived discrimination in small employment samples and underestimate it in large ones. A pre-registered scenario-based experiment was conducted, where participants (N = 874) estimated probability of discrimination versus chance in skewed hiring distributions. We manipulated employment sample size (filling four vs. 100 positions) and ethnic majority (hiring more immigrants or Swedes) using a 2x2 between-subject design. A tendency for people to overestimate perceived discrimination by underestimating the impact of chance in small employment samples was revealed. Conversely, in large employment samples, people tended to underestimate perceived discrimination by overestimating the impact of chance. Hence, results aligned with the law of small numbers. Furthermore, participants were more inclined to attribute an event as discriminatory when organizations hired more Swedes than immigrants, reflecting (accurate) prototypes of discrimination. This study's implications are discussed concerning the repercussions of underestimating and overestimating perceived discrimination in hiring situations. Future research suggestions are also provided.