Which statement about a cubic trend is true?

Which statement about a cubic trend is true?

• A. It cannot be detected with four or more ordered conditions
• B. It is indicated by three changes in direction
• C. It is indicated by two changes in direction
• D. It requires at least five conditions

Answer: (C)

A cubic trend is described by a third-degree pattern, so its slope is a quadratic function of the order position. A quadratic can cross zero at most twice, which means the direction of the trend can change at most twice as you move through the ordered conditions. In other words, you can see the sequence go up then down, then up again (or the opposite)—two turning points. That matches the statement that a cubic trend is indicated by two changes in direction. The other options overstate or misstate the pattern: three changes in direction would require more than a cubic can provide, and detecting a cubic trend doesn’t require five conditions—four ordered points are enough to estimate a cubic trend.

A cubic trend is described by a third-degree pattern, so its slope is a quadratic function of the order position. A quadratic can cross zero at most twice, which means the direction of the trend can change at most twice as you move through the ordered conditions. In other words, you can see the sequence go up then down, then up again (or the opposite)—two turning points. That matches the statement that a cubic trend is indicated by two changes in direction. The other options overstate or misstate the pattern: three changes in direction would require more than a cubic can provide, and detecting a cubic trend doesn’t require five conditions—four ordered points are enough to estimate a cubic trend.