Inferring uncertainty from interval estimates: Effects of alpha level and numeracy

Interval estimates are commonly used to descriptively communicate the degree of uncertainty in numerical values. Conventionally, low alpha levels (e.g., .05) ensure a high probability of capturing the target value between interval endpoints. Here, we test whether alpha levels and individual differences in numeracy influence distributional inferences. In the reported experiment, participants received prediction intervals for fictitious towns’ annual rainfall totals (assuming approximately normal distributions). Then, participants estimated probabilities that future totals would be captured within varying margins about the mean, indicating the approximate shapes of their inferred probability distributions. Results showed that low alpha levels (vs. moderate levels; e.g., .25) more frequently led to inferences of over-dispersed approximately normal distributions or approximately uniform distributions, reducing estimate accuracy. Highly numerate participants made more accurate estimates overall, but were more prone to inferring approximately uniform distributions. These findings have important implications for presenting interval estimates to various audiences

Inferring uncertainty from interval estimates: Effects of alpha level and numeracy

Interval estimates are commonly used to descriptively communicate the degree of uncertainty in numerical values. Conventionally, low alpha levels (e.g., .05) ensure a high probability of capturing the target value between interval endpoints. Here, we test whether alpha levels and individual differences in numeracy influence distributional inferences. In the reported experiment, participants received prediction intervals for fictitious towns' annual rainfall totals (assuming approximately normal distributions). Then, participants estimated probabilities that future totals would be captured within varying margins about the mean, indicating the approximate shapes of their inferred probability distributions. Results showed that low alpha levels (vs. moderate levels; e.g., .25) more frequently led to inferences of over-dispersed approximately normal distributions or approximately uniform distributions, reducing estimate accuracy. Highly numerate participants made more accurate estimates overall, but were more prone to inferring approximately uniform distributions. These findings have important implications for presenting interval estimates to various audiences

Logistic Mixed Model of PVT Correctness (vs. Error).

<p><i>Note.</i> B = unstandardized regression coefficient; β<i>_x</i> = <i>x</i>-standardized regression coefficient.</p

Percentage Correct PVT Responses, by Year, Mathematics Achievement Status and Item Difficulty.

<p>Percentage Correct PVT Responses, by Year, Mathematics Achievement Status and Item Difficulty.</p

Linear Regression Models of PVT Improvement (Grade 8 Sum Correct – Grade 5 Sum Correct).

<p><i>Note.</i></p>†<p><i>y-</i>standardized only (categorical predictors);</p><p>other βs fully standardized (both <i>x-</i>standardized and <i>y-</i>standardized).</p

Partial Proportional Odds Model of Calibration Scores.

<p><i>Note.</i></p>†<p>Proportional odds assumption retained;</p><p>β<i>_x</i> = <i>x-</i>standardized regression coefficient.</p

Uncanny Sums and Products May Prompt “Wise Choices”: Semantic Misalignment and Numerical Judgments

Automatized arithmetic can interfere with numerical judgments, and semantic misalignment may diminish this interference. We gave 92 adults two numerical priming tasks that involved semantic misalignment. We found that misalignment either facilitated or reversed arithmetic interference effects, depending on misalignment type. On our number matching task, digit pairs (as primes for sums) appeared with nouns that were either categorically aligned and concrete (e.g., pigs, goats), categorically misaligned and concrete (e.g., eels, webs), or categorically misaligned concrete and intangible (e.g., goats, tactics). Next, participants were asked whether a target digit matched either member of the previously presented digit pair. Participants were slower to reject sum vs. neutral targets on aligned/concrete and misaligned/concrete trials, but unexpectedly slower to reject neutral versus sum targets on misaligned/concrete-intangible trials. Our sentence verification task also elicited unexpected facilitation effects. Participants read a cue sentence that contained two digits, then evaluated whether a subsequent target statement was true or false. When target statements included the product of the two preceding digits, this inhibited accepting correct targets and facilitated rejecting incorrect targets, although only when semantic context did not support arithmetic. These novel findings identify a potentially facilitative role of arithmetic in semantically misaligned contexts and highlight the complex role of contextual factors in numerical processing