Overview
This lecture explained the relationship between slope, y-intercept, and correlation coefficient in regression lines, and provided templates for interpreting slope and y-intercept in context.
Slope, Trend, and Correlation
- The slope (b) of a regression line shows whether the trend is positive (upward) or negative (downward).
- The sign of the slope matches the sign of the correlation coefficient (r); both describe the direction of association.
- The numeric values of slope and r are different, but their signs are always the same.
Interpreting Slope
- The slope is the coefficient next to x in the regression equation.
- Slope means the average change in the y-value for a one-unit increase in the x-value.
- To interpret slope: Identify x variable (in units), y variable (in units), and use "increase" or "decrease" with the slope value and correct units.
- Example: For every 1 inch increase in height, shoe size increases by an average of 0.570 size.
Interpreting Y-intercept
- The y-intercept (a) is the predicted average value of y when x is zero.
- To interpret, state: "When x is zero, the predicted y value is the y-intercept."
- Before interpreting, check if x=0 and the resulting y value make sense in the context.
- If x=0 or the y value at x=0 does not make sense, do not interpret the y-intercept.
Key Terms & Definitions
- Slope (b) — The rate of change of y for a one-unit increase in x in a regression line.
- Y-intercept (a) — The expected value of y when x equals zero.
- Correlation coefficient (r) — A number between -1 and 1 indicating the strength and direction of the linear relationship between x and y.
Action Items / Next Steps
- Practice interpreting slope and y-intercept with given regression equations.
- Always check if x=0 and corresponding y values are meaningful before interpreting the y-intercept.