A palynologist uses a regression model to estimate ancient rainfall \( R \) from pollen ratios: \( R = 2.5P - 15 \), where \( P \) is a standardized pollen index. If the correlation between \( P \) and \( R \) improves, reducing the error margin by 20%, and originally the standard error was 3.75 mm, what is the new standard error? Also, if the original 95% confidence interval was ±7.5 mm, what is the new width?

["Improving Rainfall Estimates in Paleoclimate: How Regression Refinement Reduces Uncertainty", "Palynology—the study of pollen and spores—plays a vital role in reconstructing past climates. By analyzing standardized pollen ratios, scientists infer historical environmental conditions, such as ancient rainfall levels. A recent advancement in statistical modeling has enhanced the precision of these reconstructions. Using a regression model of the form:", "[\nR = 2.5P - 15\n]", "where ( R ) is estimated rainfall (in mm) and ( P ) is a standardized pollen index, researchers now achieve improved accuracy. This article explores how enhanced correlation reduces uncertainty, focusing on two key statistical measures: the standard error and the confidence interval.", "---", "### From Correlation to Precision: The Role of Model Improvement", "In the original model, the correlation between ( P ) and ( R ) was sufficient but allowed for a standard error of 3.75 mm. However, when researchers refined their regression using richer datasets and better preprocessing, the relationship became stronger—meaning ( P ) now explains more variance in ( R ). This improvement correlates directly with reduced estimation error.", "The standard error quantifies the typical deviation of individual rainfall estimates from the regression line. A 20% reduction in error margin means the new standard error is calculated as:", "[\n\ ext{New Standard Error} = \ ext{Original Standard Error} \ imes (1 - 0.20) = 3.75 \ imes 0.80 = 3.00 , \ ext{mm}\n]", "Thus, the refinement successfully lowers uncertainty, bringing measurements closer to the true paleoprecipitation values.", "---", "### Confidence Intervals: Reflecting Reduced Uncertainty", "The original 95% confidence interval for ( R ) was ±7.5 mm, reflecting weekness in the estimate. Since standard error directly determines confidence margin (typically 1.96 × standard error at 95% confidence), reducing the standard error also narrows the interval. The original width was:", "[\n2 \ imes 3.75 = 7.5 , \ ext{mm}\n]", "With the improved standard error of 3.00 mm, the new 95% confidence interval spans:", "[\n2 \ imes 3.00 = 6.00 , \ ext{mm}\n]", "This narrower interval signifies greater confidence in the reconstructed rainfall, reinforcing the reliability of pollen-based paleoclimate models.", "---", "### Conclusion", "Advances in statistical modeling—enabled by stronger correlations and refined data—have significantly improved ancient rainfall estimates derived from pollen ratios. In this case, a 20% reduction in standard error lowered uncertainty from ±3.75 mm to ±3.00 mm, and the 95% confidence interval shrank from ±7.5 mm to ±6.00 mm. These developments not only enhance scientific precision but also strengthen the role of palynology in understanding climate change through deep time.", "For researchers and climate historians, such methodological progress marks a crucial step toward more accurate and trustworthy paleoenvironmental reconstructions."]