Below is a partial multiple regression computer output based on a quadratic regression model to predict student enrollment at a local university. The dependent variable is the annual enrollment given in thousands of students, the independent variable X is the increase in tuition stated in thousands of dollars per year, and X2is the square of tuition increase given in squared thousands of dollars per year.
Interpret β0 (the y intercept) and β1(the β coefficient for the X variable). Does the parabola open upward or downward? Why?
Source SS df
Model 29.44 2
Error 59.66 15
Variable Coefficient (Sb)
Intercept 8.01 1.45
x -1.35 0.55
x2 0.46 0.43
Recall that the equation of a quadratic regression is
y = b0 + b1x + b2x^2 + e
Substituting in the estimated coefficients gives
y = 8.01 - 1.35x + 0.46x^2 + e
This parabola opens upwards as the coefficient of x^2 is positive (remember that the coefficient of the term with the square term decides whether the parabola opens up or down).
The quadratic regression models are examined. The parabola to open upward or downward are given.