1. **Problem Statement:** We have two problems involving paired data and linear regression. **Problem 2:** Given casino size and revenue data, find the linear regression equation and predict revenue for a casino size of 200 thousand square feet. **Problem 3:** Given time and height data of a soccer ball, find the linear correlation coefficient $r$, interpret it, and discuss potential mistakes without a scatterplot. --- 2. **Linear Regression Equation (Problem 2a):** The linear regression equation is given by: $$y = mx + b$$ where $m$ is the slope and $b$ is the y-intercept. To find $m$ and $b$, use: $$m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}$$ $$b = \frac{\sum y - m \sum x}{n}$$ *Note:* Since the paired data from the preceding exercise is not provided here, we cannot compute exact values. However, once $m$ and $b$ are found, the equation is complete. --- 3. **Prediction for Casino Size 200 (Problem 2b):** Use the regression equation: $$y = m(200) + b$$ This gives the predicted revenue. *Is it likely to be accurate?* Prediction accuracy depends on whether 200 thousand square feet is within the range of the original data (interpolation) or outside (extrapolation). Extrapolation is less reliable. --- 4. **Linear Correlation Coefficient $r$ (Problem 3a):** The formula for $r$ is: $$r = \frac{n\sum xy - \sum x \sum y}{\sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)}}$$ Given: Time $x$: 0.0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8 Height $y$: 0.0, 1.7, 3.1, 3.9, 4.5, 4.7, 4.6, 4.1, 3.3, 2.1 Calculate sums: $$\sum x = 9.0$$ $$\sum y = 28.9$$ $$\sum x^2 = 11.4$$ $$\sum y^2 = 108.15$$ $$\sum xy = 24.68$$ $$n = 10$$ Calculate numerator: $$10 \times 24.68 - 9.0 \times 28.9 = 246.8 - 260.1 = -13.3$$ Calculate denominator: $$\sqrt{(10 \times 11.4 - 9.0^2)(10 \times 108.15 - 28.9^2)} = \sqrt{(114 - 81)(1081.5 - 835.21)} = \sqrt{33 \times 246.29} = \sqrt{8127.57} \approx 90.18$$ Therefore: $$r = \frac{-13.3}{90.18} \approx -0.1475$$ --- 5. **Interpretation of $r$ (Problem 3b):** An $r$ value of approximately $-0.15$ indicates a very weak negative linear correlation between time and height. --- 6. **Potential Mistake Without Scatterplot (Problem 3c):** Without a scatterplot, one might incorrectly assume a linear relationship exists. The data shows a parabolic pattern (height increases then decreases), so linear regression is not appropriate. This could lead to misleading conclusions. --- **Final answers:** - Problem 2a: Linear regression equation $y = mx + b$ (values depend on data). - Problem 2b: Predicted revenue at $x=200$ is $y = m(200) + b$; accuracy depends on data range. - Problem 3a: $r \approx -0.15$. - Problem 3b: Very weak negative linear correlation. - Problem 3c: Mistake is assuming linearity without scatterplot.