In: Statistics and Probability
Year | Quarter | Sales |
1 | 1 | 10 |
1 | 2 | 31 |
1 | 3 | 43 |
1 | 4 | 16 |
2 | 1 | 11 |
2 | 2 | 33 |
2 | 3 | 45 |
2 | 4 | 17 |
3 | 1 | 13 |
3 | 2 | 34 |
3 | 3 | 48 |
3 | 4 | 19 |
4 | 1 | 15 |
4 | 2 | 37 |
4 | 3 | 51 |
4 | 4 | 21 |
a. Construct a time series plot. What type of pattern exists in the data?
b. Use the following dummy variables to develop an estimated regression equation to account for seasonal effects and any linear trend in the data: Qtr1 = 1 if Quarter 1, 0 otherwise; Qtr2 = 1 if Quarter 2, 0 otherwise; Qtr3 = 1 if Quarter 3, 0 otherwise. Compute the forecast of bike sales for quarter 1 of next year.
c. Compute the forecast of bike sales for quarter 2 of next year.
d. Compute the forecast of bike sales for quarter 3 of next year.
e. Compute the forecast of bike sales for quarter 4 of next year.
a. Construct a time series plot. What type of pattern exists in the data?
There is a horizontal seasonal pattern in the data.
b. Use the following dummy variables to develop an estimated regression equation to account for seasonal effects and any linear trend in the data: Qtr1 = 1 if Quarter 1, 0 otherwise; Qtr2 = 1 if Quarter 2, 0 otherwise; Qtr3 = 1 if Quarter 3, 0 otherwise. Compute the forecast of bike sales for quarter 1 of next year.
The estimated regression equation is:
y = 13.25 -4.5*Qtr 1 + 16.5*Qtr 2 + 29*Qtr 3 + 0.5*t
The forecast of bike sales for quarter 1 of next year is 17.25.
c. Compute the forecast of bike sales for quarter 2 of next year.
The forecast of bike sales for quarter 2 of next year is 38.75.
d. Compute the forecast of bike sales for quarter 3 of next year.
The forecast of bike sales for quarter 3 of next year is 51.75.
e. Compute the forecast of bike sales for quarter 4 of next year.
The forecast of bike sales for quarter 4 of next year is 23.25.
The data is:
Year | Quarter | y | Qtr 1 | Qtr 2 | Qtr 3 | t |
1 | 1 | 10 | 1 | 0 | 0 | 1 |
1 | 2 | 31 | 0 | 1 | 0 | 2 |
1 | 3 | 43 | 0 | 0 | 1 | 3 |
1 | 4 | 16 | 0 | 0 | 0 | 4 |
2 | 1 | 11 | 1 | 0 | 0 | 5 |
2 | 2 | 33 | 0 | 1 | 0 | 6 |
2 | 3 | 45 | 0 | 0 | 1 | 7 |
2 | 4 | 17 | 0 | 0 | 0 | 8 |
3 | 1 | 13 | 1 | 0 | 0 | 9 |
3 | 2 | 34 | 0 | 1 | 0 | 10 |
3 | 3 | 48 | 0 | 0 | 1 | 11 |
3 | 4 | 19 | 0 | 0 | 0 | 12 |
4 | 1 | 15 | 1 | 0 | 0 | 13 |
4 | 2 | 37 | 0 | 1 | 0 | 14 |
4 | 3 | 51 | 0 | 0 | 1 | 15 |
4 | 4 | 21 | 0 | 0 | 0 | 16 |
The output is:
R² | 0.998 | |||||
Adjusted R² | 0.998 | |||||
R | 0.999 | |||||
Std. Error | 0.674 | |||||
n | 16 | |||||
k | 4 | |||||
Dep. Var. | y | |||||
ANOVA table | ||||||
Source | SS | df | MS | F | p-value | |
Regression | 2,990.0000 | 4 | 747.5000 | 1644.50 | 3.44E-15 | |
Residual | 5.0000 | 11 | 0.4545 | |||
Total | 2,995.0000 | 15 | ||||
Regression output | confidence interval | |||||
variables | coefficients | std. error | t (df=11) | p-value | 95% lower | 95% upper |
Intercept | 13.2500 | |||||
Qtr 1 | -4.5000 | 0.4900 | -9.184 | 1.72E-06 | -5.5784 | -3.4216 |
Qtr 2 | 16.5000 | 0.4827 | 34.186 | 1.61E-12 | 15.4377 | 17.5623 |
Qtr 3 | 29.0000 | 0.4782 | 60.642 | 3.03E-15 | 27.9474 | 30.0526 |
t | 0.5000 | 0.0377 | 13.266 | 4.12E-08 | 0.4170 | 0.5830 |
Predicted values for: y | ||||||
Qtr 1 | Qtr 2 | Qtr 3 | t | Predicted | ||
1 | 0 | 0 | 17 | 17.250 | ||
0 | 1 | 0 | 18 | 38.750 | ||
0 | 0 | 1 | 19 | 51.750 | ||
0 | 0 | 0 | 20 | 23.250 |
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