"[DATA] An Unhealthy Commute [See Problem 11 in Section 12.3] The following data represent commute times (in minutes) and a score on a well-being survey.

b. Determine the slope of the least-squares regression line between commute time and well-being score."

Verified step by step guidance

1. Identify the variables: Let \(x\) represent the commute time (in minutes) and \(y\) represent the Gallup-Healthways Well-being Index Composite Score.
2. Calculate the means of \(x\) and \(y\): Compute \(\bar{x} = \frac{\sum x_i}{n}\) and \(\bar{y} = \frac{\sum y_i}{n}\), where \(n\) is the number of data points.
3. Calculate the slope \(b\) of the least-squares regression line using the formula: \(b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}\)
4. To find \(\sum (x_i - \bar{x})(y_i - \bar{y})\), subtract the mean from each \(x_i\) and \(y_i\), multiply the differences for each pair, then sum all these products.
5. To find \(\sum (x_i - \bar{x})^2\), subtract the mean from each \(x_i\), square the result, and sum all these squared differences. Then divide the sum from step 4 by the sum from this step to get the slope.

Key Concepts

Least-Squares Regression Line

The least-squares regression line is a straight line that best fits the data points by minimizing the sum of the squared vertical distances (residuals) between observed values and the line. It models the relationship between an explanatory variable (x) and a response variable (y), allowing predictions of y based on x.

Slope of the Regression Line

The slope represents the average change in the response variable (y) for each one-unit increase in the explanatory variable (x). It indicates the direction and strength of the relationship: a positive slope means y increases as x increases, while a negative slope means y decreases as x increases.

Calculation of the Slope Using Data

To calculate the slope, use the formula b = r(s_y / s_x), where r is the correlation coefficient, s_y is the standard deviation of y, and s_x is the standard deviation of x. Alternatively, the slope can be found by dividing the change in y by the change in x between two points on the line.