26 The quarterly sales data number of copies sold for a coll

26. The quarterly sales data (number of copies sold) for a college textbook over the past three years follow. Year 2 Year 3 Year 1 Quarter 1690 1800 1850 1100 940 900 2625 2900 2930 2615 2500 2360 a. Construct a time series plot. What type of pattern exists in the data? b. Use a regression model with dummy variables as follows to develop an equation to account for seasonal effects in the data. Qtr 1- 1 if Quarter 1,0 otherwise, Qtr2 1 if Quarter 2, 0 otherwise; Qtr3 1 if Quarter 3, 0 otherwise c. Compute the quarterly forecasts for next year. d. Lett 1 to refer to the observation in quarter 1 of year 1; t 2 to refer to the obser- vation in quarter 2 of year 1 and t 12 to refer to the observation in quarter 4 of year 3. Using the dummy variables defined in part (b and also using t, develop an equation to account for seasonal effects and any linear trend in the time series. Based upon the seasonal effects in the data and linear trend, compute the quarterly forecasts for next year.

Solution

a.

The Cyclical Pattern is exist in the data because the time series plot shows an alternating sequence of points
below and above the trend line lasting more than one year.

b. The regression model, is

Y=b₀+b₁ Qtr1+b₂ Qtr2+b₃ Qtr3+b₄ S

561.9595\\

c.

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.989744
R Square	0.979592
Adjusted R Square	0.971939
Standard Error	124.9667
Observations	12
ANOVA
	df	SS	MS	F	Significance F
Regression	3	5996942	1998981	128.003	4.23E-07
Residual	8	124933.3	15616.67
Total	11	6121875
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	2491.667	72.14954	34.53476	5.4E-10	2325.29	2658.044
Qtr1	-711.667	102.0349	-6.97474	0.000116	-946.959	-476.374
Qtr2	-1511.67	102.0349	-14.8152	4.24E-07	-1746.96	-1276.37
Qtr3	326.6667	102.0349	3.20152	0.012584	91.37388	561.9595\\