1. **State the problem:** We have data on the time to climb (in minutes) and the height of rock formations (in feet). We want to analyze the relationship between these two variables by drawing a scatter plot and finding the line of best fit. 2. **Formula for line of best fit:** The line of best fit is a linear equation of the form $$y = mx + b$$ where $y$ is the height, $x$ is the time to climb, $m$ is the slope, and $b$ is the y-intercept. 3. **Calculate slope ($m$):** The slope is given by $$m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}$$ where $n$ is the number of data points. 4. **Calculate intercept ($b$):** The intercept is $$b = \frac{\sum y - m \sum x}{n}$$ 5. **Compute sums:** - $n = 20$ - $\sum x = 30 + 45 + 115 + 87 + 160 + 60 + 135 + 117 + 55 + 145 + 95 + 150 + 67 + 120 + 100 + 88 + 138 + 75 + 167 + 120 = 1929$ - $\sum y = 85 + 120 + 298 + 194 + 434 + 241 + 351 + 339 + 180 + 400 + 248 + 370 + 155 + 296 + 271 + 255 + 315 + 215 + 405 + 317 = 5739$ - $\sum x^2 = 30^2 + 45^2 + ... + 120^2 = 217,865$ - $\sum xy = 30\times85 + 45\times120 + ... + 120\times317 = 628,865$ 6. **Calculate slope:** $$m = \frac{20 \times 628,865 - 1929 \times 5739}{20 \times 217,865 - 1929^2} = \frac{12,577,300 - 11,066,031}{4,357,300 - 3,721,641} = \frac{1,511,269}{635,659} \approx 2.38$$ 7. **Calculate intercept:** $$b = \frac{5739 - 2.38 \times 1929}{20} = \frac{5739 - 4590}{20} = \frac{1149}{20} = 57.45$$ 8. **Line of best fit:** $$y = 2.38x + 57.45$$ 9. **Interpretation:** For every additional minute spent climbing, the height increases by approximately 2.38 feet. The base height when time is zero is about 57.45 feet. 10. **Scatter plot:** Plot the points $(x, y)$ from the data and draw the line $y = 2.38x + 57.45$ to visualize the trend. This analysis helps Jeremy estimate climb height based on time for planning the trip.