1. **Problem Statement:** We have monthly sales data (in 000 units) for seven months: Feb(19), Mar(18), Apr(15), May(20), Jun(18), Jul(22), Aug(20). We need to forecast sales for September and October using three methods: linear trend, four-month moving average, and weighted average. 2. **Linear Trend Equation:** We assign time periods $t=1$ for Feb, $t=2$ for Mar, ..., $t=7$ for Aug. Sales data points: $(1,19), (2,18), (3,15), (4,20), (5,18), (6,22), (7,20)$. The linear trend equation is $y = a + bt$ where: $$b = \frac{n\sum{(t y)} - \sum{t} \sum{y}}{n\sum{t^2} - (\sum{t})^2}$$ $$a = \frac{\sum{y} - b \sum{t}}{n}$$ Calculate sums: $\sum{t} = 1+2+3+4+5+6+7 = 28$ $\sum{y} = 19+18+15+20+18+22+20 = 132$ $\sum{t^2} = 1^2+2^2+3^2+4^2+5^2+6^2+7^2 = 140$ $\sum{t y} = 1\times19 + 2\times18 + 3\times15 + 4\times20 + 5\times18 + 6\times22 + 7\times20 = 1\times19 + 36 + 45 + 80 + 90 + 132 + 140 = 542$ Number of points $n=7$. Calculate slope $b$: $$b = \frac{7 \times 542 - 28 \times 132}{7 \times 140 - 28^2} = \frac{3794 - 3696}{980 - 784} = \frac{98}{196} = 0.5$$ Calculate intercept $a$: $$a = \frac{132 - 0.5 \times 28}{7} = \frac{132 - 14}{7} = \frac{118}{7} \approx 16.857$$ So, the trend equation is: $$y = 16.857 + 0.5 t$$ Forecast for September ($t=8$): $$y_8 = 16.857 + 0.5 \times 8 = 16.857 + 4 = 20.857$$ Forecast for October ($t=9$): $$y_9 = 16.857 + 0.5 \times 9 = 16.857 + 4.5 = 21.357$$ 3. **Four-Month Moving Average:** Average sales of the previous 4 months to forecast next month. Forecast September (average of May, Jun, Jul, Aug): $$\frac{20 + 18 + 22 + 20}{4} = \frac{80}{4} = 20$$ Forecast October (average of Jun, Jul, Aug, Sep forecast): $$\frac{18 + 22 + 20 + 20}{4} = \frac{80}{4} = 20$$ 4. **Weighted Average:** Using weights 0.6 for August, 0.3 for July, 0.1 for June. Forecast September: $$0.6 \times 20 + 0.3 \times 22 + 0.1 \times 18 = 12 + 6.6 + 1.8 = 20.4$$ Forecast October (using September forecast 20.4, August 20, July 22 with same weights): $$0.6 \times 20.4 + 0.3 \times 20 + 0.1 \times 22 = 12.24 + 6 + 2.2 = 20.44$$ **Final forecasts:** - Linear trend: September $\approx 20.86$, October $\approx 21.36$ - Four-month moving average: September $= 20$, October $= 20$ - Weighted average: September $= 20.4$, October $= 20.44$