- Observe the data trend: as exercise minutes increase, sleep hours increase, suggesting a positive slope.
- Perform a linear regression to calculate the slope \(m \approx 0.058\) and intercept \(b \approx 4.138\).
- Match the resulting model, \(y = 0.058x + 4.138\), against the provided options.
- The best linear model is \(\boxed{y = 0.058x + 4.138}\).
Explanation
- Analyzing the Data and Trends
To find the best linear model for the relationship between exercise minutes (\(x\)) and sleep hours (\(y\)), we first analyze the given data points: \((30, 6.25), (37, 6), (40, 6.5), (45, 7), (52, 6.5), (55, 7.25), (61, 7.5), (65, 8), (71, 8.65), (76, 8.5)\). We observe that as the number of minutes spent exercising (\(x\)) increases, the number of hours slept (\(y\)) generally increases, which indicates a positive correlation. Therefore, the slope (\(m\)) of the best-fitting line must be positive. This immediately allows us to rule out any models with a negative slope, such as \(y = -0.058x - 4.138\) and \(y = -0.058x + 4.138\).
- Calculating the Line of Best Fit
Next, we perform a linear regression to find the exact line of best fit in the form \(y = mx + b\). Using the calculated values from our data analysis, we find the slope and the y-intercept. The slope is approximately \(m \approx 0.0578\), and the y-intercept is approximately \(b \approx 4.1378\). Rounding these to three decimal places as seen in the provided options, we obtain a slope of \(0.058\) and a y-intercept of \(4.138\). Substituting these values into our linear equation gives \(y = 0.058x + 4.138\).
- Selecting the Best Model
Finally, we compare our calculated model \(y = 0.058x + 4.138\) with the given options. Our model matches one of the choices perfectly. The positive slope \(0.058\) confirms the positive correlation observed in the data, and the y-intercept \(4.138\) provides the best-fitting starting value for the sleep duration when the exercise is zero. Therefore, the best model is the third option. The final answer is \(\boxed{y = 0.058x + 4.138}\).
Examples
This type of linear modeling is widely used in health sciences and data analytics. For instance, a sports scientist might use this method to model the relationship between an athlete's training intensity and their recovery time. By establishing a linear equation like \(y = mx + b\), they can make informed predictions about how much extra sleep an athlete might need based on their training load, helping to prevent overtraining and optimize performance. Understanding how to derive these models from observed data is a foundational skill in statistics and practical problem solving.