1. **Problem Statement:** We have a scatter plot showing the relationship between hours spent studying ($x$) and money earned ($y$) for 10 students. We need to (a) comment on the relationship and (b) predict the money earned if a student studies 10 hours. 2. **Step a: Comment on the relationship** - The points are: $(18, 23), (20, 21), (23, 20), (25, 19), (25, 21), (27, 18), (32, 16), (38, 17), (40, 16), (41, 23)$. - Observing the scatter plot, as $x$ (hours studying) increases, $y$ (money earned) generally decreases, indicating a negative correlation. - However, some points like $(25, 21)$ and $(41, 23)$ deviate from this trend, suggesting some variability. - Overall, there is a weak negative linear relationship between hours spent studying and money earned. 3. **Step b: Predict money earned for $x=10$ hours** - To predict, we find the line of best fit (linear regression) using the formula: $$y = mx + c$$ - Calculate the slope $m$ and intercept $c$ using the formulas: $$m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}$$ $$c = \frac{\sum y - m \sum x}{n}$$ 4. **Calculate sums:** - $n=10$ - $\sum x = 18+20+23+25+25+27+32+38+40+41 = 289$ - $\sum y = 23+21+20+19+21+18+16+17+16+23 = 194$ - $\sum xy = 18\times23 + 20\times21 + 23\times20 + 25\times19 + 25\times21 + 27\times18 + 32\times16 + 38\times17 + 40\times16 + 41\times23 = 4143$ - $\sum x^2 = 18^2 + 20^2 + 23^2 + 25^2 + 25^2 + 27^2 + 32^2 + 38^2 + 40^2 + 41^2 = 8763$ 5. **Calculate slope $m$:** $$m = \frac{10 \times 4143 - 289 \times 194}{10 \times 8763 - 289^2} = \frac{41430 - 56066}{87630 - 83521} = \frac{-14636}{4109} \approx -3.56$$ 6. **Calculate intercept $c$:** $$c = \frac{194 - (-3.56) \times 289}{10} = \frac{194 + 1028.84}{10} = \frac{1222.84}{10} = 122.28$$ 7. **Regression line:** $$y = -3.56x + 122.28$$ 8. **Predict $y$ for $x=10$:** $$y = -3.56 \times 10 + 122.28 = -35.6 + 122.28 = 86.68$$ **Interpretation:** The predicted money earned for a student studying 10 hours is approximately 86.68 (units of money). **Note:** This prediction is an extrapolation outside the original data range and may not be reliable.