a) Yes, there is sufficient statistical evidence of an upward trend in shoe sales. This can be determined by looking at the parameter and T-ratio for the T variable. The parameter is positive (2100) and the T-ratio (6.18) is significant as the P-value is 0.0001.
b) To determine if there is a statistically significant seasonal pattern in sales, one would typically conduct an F-test. However, in this case, the parameters for each seasonal variable (D1, D2, D3) are individually significant. Thus, it is highly likely that the combined effect of these variables would also be significant. As for the seasonal pattern exhibited by the data, the sales are highest in the summer quarter (D3) and lowest in the autumn quarter (not explicitly shown in the given data).
c) To forecast sales of Rubax shoes for 2005(III) and 2006(II), you would need to plug the appropriate values into the estimated forecast equation. The equation is Q1 = a + bt + c1D1 + c2D2 + c3D3. Since the data only goes up to 2004(IV), it is not explicitly stated what the values of D1, D2, and D3 would be for 2005(III) and 2006(II). Thus, without knowing the values of these variables, it is not possible to provide an accurate forecast using the estimated equation.
d) To improve the forecast equation, there are several variables that could be considered. Some possible variables to include would be price (of Rubax shoes), price of competitors, advertising expenses (by Rubax), advertising expenses by competitors, population of young adults, number of retailers, and number of shoe styles produced. By including additional relevant variables, the forecast equation will have a more comprehensive set of predictors, resulting in a potentially more accurate forecast.