Final answer:
A quadratic model is appropriate for a set of data when the second differences are constant, signaling a parabolic relationship. It is not suitable when the first differences are constant or the second differences are zero, as these indicate linear relationships.
Step-by-step explanation:
A quadratic model is appropriate to use for a set of data when the second differences are constant. This is a characteristic of quadratic relationships because the graph of a quadratic equation is a parabola, which has a curved shape rather than a straight line. When the first differences are constant, this suggests a linear relationship. It's essential to analyze the scatter plot of the data to look for this curvature in the distribution of data points. A line is suitable when the data points lie roughly along a straight line without much deviation, indicating a linear relationship. Lastly, when the second differences are zero, it typically implies that the data points lie perfectly on a line, which directly suggests a first-order polynomial or linear model, not a quadratic one.
Using the least-squares regression line is common, as it minimizes the sum of squared residuals, providing the 'best fit' for linear models. However, when the scatter plot indicates a curved pattern, statisticians should consider fitting a curve, such as a quadratic model, instead of a straight line. This is because a quadratic model can better accommodate the curvature and provide more accurate predictions for the values of y given different values of x.