1. **Problem:** Determine if there is sufficient evidence of a linear correlation between casino size and revenue, and whether increasing size can increase revenue. 2. **Given Data:** Size (thousands sq ft): $160, 227, 140, 144, 161, 147, 141$ Revenue (millions): $189, 157, 140, 127, 123, 106, 101$ 3. **Step 1: Calculate the linear 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 sums: $\sum x = 1120$, $\sum y = 943$, $\sum xy = 153,964$, $\sum x^2 = 181,066$, $\sum y^2 = 139,927$, $n=7$ Calculate numerator: $$7 \times 153,964 - 1120 \times 943 = 1,077,748 - 1,056,160 = 21,588$$ Calculate denominator: $$\sqrt{(7 \times 181,066 - 1120^2)(7 \times 139,927 - 943^2)} = \sqrt{(1,267,462 - 1,254,400)(979,489 - 889,249)} = \sqrt{13,062 \times 90,240} \approx \sqrt{1,178,000,000} \approx 34,320$$ Calculate $r$: $$r = \frac{21,588}{34,320} \approx 0.629$$ Interpretation: $r=0.629$ indicates a moderate positive linear correlation. 4. **Step 2: Linear regression equation $y = mx + b$** Slope $m$: $$m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} = \frac{21,588}{13,062} \approx 1.653$$ Intercept $b$: $$b = \frac{\sum y - m \sum x}{n} = \frac{943 - 1.653 \times 1120}{7} = \frac{943 - 1,851.36}{7} = \frac{-908.36}{7} \approx -129.77$$ Regression equation: $$\hat{y} = 1.653x - 129.77$$ 5. **Step 3: Predict revenue for size 200** $$\hat{y} = 1.653 \times 200 - 129.77 = 330.6 - 129.77 = 200.83$$ 6. **Step 4: Accuracy of prediction** Since $r$ is moderate, prediction has some uncertainty. Also, 200 is within the data range, so prediction is reasonable but not guaranteed. --- 7. **Problem 2a:** Find linear regression equation (already done above): $$\hat{y} = 1.653x - 129.77$$ 8. **Problem 2b:** Predict revenue for size 200 (done above): $$200.83$$ million dollars approximately. 9. **Problem 3a:** Calculate linear correlation coefficient $r$ for time and height. Data: Time $t$: $0.0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8$ Height $h$: $0.0, 1.7, 3.1, 3.9, 4.5, 4.7, 4.6, 4.1, 3.3, 2.1$ Calculate sums: $n=10$ $\sum t = 9.0$ $\sum h = 31.0$ $\sum th = 22.68$ $\sum t^2 = 11.4$ $\sum h^2 = 117.88$ Calculate numerator: $$10 \times 22.68 - 9.0 \times 31.0 = 226.8 - 279 = -52.2$$ Calculate denominator: $$\sqrt{(10 \times 11.4 - 9.0^2)(10 \times 117.88 - 31.0^2)} = \sqrt{(114 - 81)(1178.8 - 961)} = \sqrt{33 \times 217.8} = \sqrt{7187.4} \approx 84.77$$ Calculate $r$: $$r = \frac{-52.2}{84.77} \approx -0.616$$ 10. **Problem 3b:** Interpretation $r \approx -0.616$ indicates a moderate negative linear correlation between time and height. 11. **Problem 3c:** Mistake without scatterplot Without a scatterplot, one might incorrectly assume a linear relationship when the data actually follows a parabolic trajectory (height first increases then decreases), so linear correlation is misleading. --- **Final answers:** - Casino size and revenue have moderate positive correlation ($r=0.629$). - Regression equation: $\hat{y} = 1.653x - 129.77$. - Predicted revenue for size 200: $200.83$ million. - Time and height have moderate negative correlation ($r=-0.616$). - Scatterplot is essential to avoid misinterpretation of nonlinear data.