1. **Problem Statement:** We have data for the price of an economy class airline ticket based on weeks remaining before departure. We want to create a scatter plot, find a regression line, predict the price at 10 weeks, and estimate the correlation coefficient. 2. **Data:** - Weeks before departure: $8, 7, 6, 5, 4, 3$ - Prices: $154, 154, 193, 240, 315, 489$ 3. **Step 1: Create a scatter plot** Plot points $(8,154), (7,154), (6,193), (5,240), (4,315), (3,489)$ on a coordinate plane with weeks on the x-axis and price on the y-axis. 4. **Step 2: Find the regression line** We use the least squares regression line formula: $$y = mx + b$$ where $m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}$ and $b = \frac{\sum y - m \sum x}{n}$. Calculate sums: $$\sum x = 8+7+6+5+4+3 = 33$$ $$\sum y = 154+154+193+240+315+489 = 1545$$ $$\sum xy = 8\times154 + 7\times154 + 6\times193 + 5\times240 + 4\times315 + 3\times489 = 1232 + 1078 + 1158 + 1200 + 1260 + 1467 = 7395$$ $$\sum x^2 = 8^2 + 7^2 + 6^2 + 5^2 + 4^2 + 3^2 = 64 + 49 + 36 + 25 + 16 + 9 = 199$$ $$n=6$$ Calculate slope $m$: $$m = \frac{6 \times 7395 - 33 \times 1545}{6 \times 199 - 33^2} = \frac{44370 - 50985}{1194 - 1089} = \frac{-5615}{105} = -53.476$$ Calculate intercept $b$: $$b = \frac{1545 - (-53.476) \times 33}{6} = \frac{1545 + 1764.7}{6} = \frac{3309.7}{6} = 551.62$$ So regression line: $$y = -53.476x + 551.62$$ 5. **Step 3: Predict price at 10 weeks before departure** Substitute $x=10$: $$y = -53.476 \times 10 + 551.62 = -534.76 + 551.62 = 16.86$$ This prediction is very low compared to given data and likely unreasonable because the data shows prices increasing as weeks decrease, so extrapolating beyond the data range can be unreliable. 6. **Step 4: Estimate correlation coefficient $r$** Formula: $$r = \frac{n\sum xy - \sum x \sum y}{\sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)}}$$ Calculate $\sum y^2$: $$154^2 + 154^2 + 193^2 + 240^2 + 315^2 + 489^2 = 23716 + 23716 + 37249 + 57600 + 99225 + 239121 = 460627$$ Calculate denominator: $$\sqrt{(6 \times 199 - 33^2)(6 \times 460627 - 1545^2)} = \sqrt{(1194 - 1089)(2763762 - 2387025)} = \sqrt{105 \times 376737} = \sqrt{39557385} = 6289.5$$ Calculate numerator: $$6 \times 7395 - 33 \times 1545 = 44370 - 50985 = -5615$$ So $$r = \frac{-5615}{6289.5} = -0.893$$ This indicates a strong negative correlation, consistent with price decreasing as weeks increase. --- **Final answers:** - Regression line: $$y = -53.476x + 551.62$$ - Predicted price at 10 weeks: $$16.86$$ (likely unreasonable) - Estimated correlation coefficient: $$-0.893$$