\boxed{13}Question: At a science fair, a judge evaluates 12 projects, each scored on creativity and technical skill. The average creativity score is 84 with a standard deviation of 6, and the average technical score is 78 with a standard deviation of 5. If the correlation coefficient between creativity and technical scores is 0.6, what is the predicted technical score for a project with a creativity score of 90?

["Predicting Technical Scores Using a Correlation: A Science Fair Example", "When evaluating student projects at a science fair, judges rely on quantitative metrics to fairly assess performance. Consider a scenario where a science fair judge evaluates 12 student projects, each scored on creativity and technical skill. Understanding how these two metrics relate can help predict outcomes—in this case, estimating a project’s technical score based on its creativity score. This article explores how to calculate the predicted technical score using correlation statistics.", "### Understanding the Data and Correlation", "Given:\n- Average creativity score: 84\n- Standard deviation (creativity): 6\n- Average technical score: 78\n- Standard deviation (technical): 5\n- Correlation coefficient (creativity vs. technical): 0.6\n- A project has a creativity score of 90", "The correlation coefficient of 0.6 indicates a moderate positive relationship—as creativity increases, technical scores tend to increase, though with some variation.", "### Using the Linear Regression Formula", "To predict the technical score ((Y)) given creativity ((X)), we use linear regression. The formula for the regression line is:", "[\nY = a + bX\n]", "Where:\n- (b = r \cdot \dfrac{\sigma_Y}{\sigma_X}) is the slope\n- (a = \bar{Y} - b\bar{X}) is the intercept", "Substitute the values:\n- (r = 0.6)\n- (\sigma_Y = 5), (\sigma_X = 6)\n- (\bar{X} = 84), (\bar{Y} = 78)", "Calculate the slope:\n[\nb = 0.6 \cdot \dfrac{5}{6} = 0.5\n]", "Now calculate the intercept:\n[\na = 78 - (0.5 \cdot 84) = 78 - 42 = 36\n]", "Thus, the regression equation is:\n[\nY = 36 + 0.5X\n]", "### Predicting the Technical Score for Creativity 90", "Plug (X = 90) into the equation:\n[\nY = 36 + 0.5 \cdot 90 = 36 + 45 = 81\n]", "### Conclusion", "Based on the correlation and given data, the predicted technical score for a project with a creativity score of 90 is 81. This means a strong positive relationship exists between creativity and technical achievement in this sample—projects scoring well in creativity tend to also perform strongly technically.", "Such predictive models help judges identify high-potential projects, even when exact scores aren’t available. By understanding the statistical link, evaluation becomes more objective and data-driven—key attributes of successful science fair assessments.", "---", "Keywords: science fair correlation, predict technical score, creativity correlation technical score, statistical prediction science project, correlation coefficient regression example, average scores correlation, student project evaluation"]