Suppose the estimated quadratic model Yt = 500 + 20 t - t2 is the best-fitting trend of sales of XYZ Inc. using data for the past twenty years (t = 1, 2,.., 20).
Which statement is incorrect?
A. The trend was higher in year 10 than in year 20.
B. Latest year sales are no better than in year zero.
C. The turning point would be in period 10.
D. Sales are increasing by about 20 units per year.
Need help Please
To determine which statement is incorrect, we need to evaluate each option based on the given quadratic model for sales (Yt = 500 + 20t - t^2) over the past twenty years (t = 1, 2,.., 20).
A. The trend was higher in year 10 than in year 20.
To determine whether the trend was higher in year 10 than in year 20, we can evaluate the sales values for these years using the quadratic model.
For year 10 (t = 10):
Y10 = 500 + 20(10) - (10)^2 = 500 + 200 - 100 = 600
For year 20 (t = 20):
Y20 = 500 + 20(20) - (20)^2 = 500 + 400 - 400 = 500
Therefore, the trend was actually higher in year 10 (600) compared to year 20 (500). So, statement A is correct.
B. Latest year sales are no better than in year zero.
To determine whether the latest year sales are better than in year zero, we need to compare the sales values for these years.
For year zero (t = 0):
Y0 = 500 + 20(0) - (0)^2 = 500
For the latest year (t = 20):
Y20 = 500 + 20(20) - (20)^2 = 500 + 400 - 400 = 500
The sales values for both year zero and the latest year are the same, so statement B is correct.
C. The turning point would be in period 10.
The turning point of a quadratic function occurs at the vertex, which is the point where the function changes from increasing to decreasing or vice versa. In a quadratic function of the form Yt = a + bt + ct^2, the turning point is at t = -b/(2c).
For the given quadratic model, b = 20 and c = -1, so the turning point is at t = -20/(2*(-1)) = 10. Hence, statement C is correct.
D. Sales are increasing by about 20 units per year.
In the given quadratic model Yt = 500 + 20t - t^2, the coefficient of the linear term, 20, represents the rate of change in sales over time. Since it is positive, it indicates an increase in sales. Therefore, statement D is correct.
Based on the analysis, all the given statements A, B, C, and D are correct.