Final answer:
A residual plot with no pattern and points evenly distributed about the x-axis suggests a line of best fit is appropriate. Linear or curved patterns in residuals indicate that a linear model may not be the best fit and alternatives should be considered.
Step-by-step explanation:
When examining the shape of the residual plot from a line of best fit, we look for specific patterns to determine if the linear model is appropriate. If the residual plot shows that the points have no pattern and are evenly distributed about the x-axis, this suggests the linear model is a good fit for the data (options a and b). However, if the residuals display a linear or curved pattern, this indicates that a linear model may not be the best representation of the data (options c and d).
A linear pattern in the residuals may suggest that the relationship between the variables is linear but the slope of the line of best fit is incorrect. A curved pattern points to a non-linear relationship which may be better modeled with a different type of function. Therefore, the residuals should ideally have a random scattering around the horizontal axis with no apparent pattern for a linear model to be considered appropriate.
In relation to linear regression, understanding the scatter plot is crucial. When the residuals show no pattern and are again randomly distributed, the x and y variables are likely good candidates for linear regression. If the pattern of points suggests a relationship that isn't linear, an alternative method would better fit the data.