1. **State the problem:** We have ACT and SAT scores for 8 students and want to analyze the relationship between these scores using a scatter plot, find the line of best fit, and make predictions. 2. **Coordinates used for the scatter plot:** (18, 800), (25, 1100), (30, 1300), (21, 1000), (20, 900), (27, 1200), (16, 600), (22, 800). 3. **Correlation:** The scatter plot shows a positive, strong correlation because as ACT scores increase, SAT scores also increase consistently. 4. **Line of best fit formula:** We use the linear regression formula $$y = mx + b$$ where $y$ is SAT score, $x$ is ACT score, $m$ is slope, and $b$ is y-intercept. 5. **Calculate slope $m$:** Calculate means: $$\bar{x} = \frac{18+25+30+21+20+27+16+22}{8} = \frac{179}{8} = 22.375$$ $$\bar{y} = \frac{800+1100+1300+1000+900+1200+600+800}{8} = \frac{7700}{8} = 962.5$$ Calculate numerator and denominator for slope: $$\sum (x_i - \bar{x})(y_i - \bar{y}) = (18-22.375)(800-962.5) + ... + (22-22.375)(800-962.5) = 10287.5$$ $$\sum (x_i - \bar{x})^2 = (18-22.375)^2 + ... + (22-22.375)^2 = 132.875$$ Slope: $$m = \frac{10287.5}{132.875} \approx 77.43$$ 6. **Calculate intercept $b$:** $$b = \bar{y} - m\bar{x} = 962.5 - 77.43 \times 22.375 \approx 962.5 - 1732.5 = -770$$ 7. **Line of best fit:** $$y = 77.43x - 770$$ 8. **Predict ACT score for SAT = 1000:** Solve for $x$: $$1000 = 77.43x - 770$$ $$77.43x = 1770$$ $$x = \frac{1770}{77.43} \approx 22.86$$ 9. **Predict SAT score for ACT = 32:** $$y = 77.43 \times 32 - 770 = 2477.76 - 770 = 1707.76$$ **Final answers:** - Line of best fit: $$y = 77.43x - 770$$ - Predicted ACT for SAT 1000: approximately 22.86 - Predicted SAT for ACT 32: approximately 1707.76