Where the relationship between the dependent and and independent variables is quadratic, represented by equation - Yi= β0 + β1X1i+β2X1i2 + errori, with the plot for X vs Y resembling a bottom-opening parabola, would β1 be greater or less than zero, and would β2 be greater or less than zero?

Answer 1

In a quadratic relationship, the plot of X vs Y resembles a bottom-opening parabola. In this scenario, Yi = β0 + β1X1i + β2X1i^2 + errori.

β1 represents the coefficient for the linear term (X1i), and β2 represents the coefficient for the quadratic term (X1i^2).

In a bottom-opening parabola, the slope of the line initially increases and then decreases. This indicates that as X1 increases, Y increases first, reaches a maximum point, and then starts decreasing.

Therefore, in this case:
- β1 would be greater than zero because it represents the linear term, indicating that as X1 increases, Y also tends to increase before reaching the maximum point.
- β2 would be less than zero because it represents the quadratic term. A negative β2 coefficient implies that the parabola's concavity is downward, in line with the shape of a bottom-opening parabola.

To summarize:
- β1 > 0 (greater than zero)
- β2 < 0 (less than zero)