Question 3 of 7: Sales Forecast for a Tablet Computer, With a Missing Month

Question 3: Sales Forecast for a Tablet Computer, With a Missing Month (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Seven known months of actual sales, February through September, with April's figure lost when the sales report was misplaced; no other demand drivers (promotions, launch dates, pricing) are supplied.

Find. A justified point forecast for October sales, and a discussion of the forecast's reliability and how it could be improved.

Approach (part a). Handle the missing April value first (it must not silently become a zero or be guessed as a "typical" month), then fit a linear trend using only the seven genuine observations and test it with $R^2$ before trusting it, comparing against a naive and a short moving-average forecast to pick the best-justified method.

1. Handle the missing month. April is not simply absent from the record — it is a genuinely lost data point, and the two forecasting methods below respond to that differently. A trend regression can be fit directly on the seven known (month, sales) pairs without needing April at all (regression does not require evenly spaced, gap-free data). A short trailing moving average likewise only needs the most recent months (July–September), so April's absence has zero effect on either the point forecast or its justification. For context only, linear interpolation between March (300) and May (740) gives an implied April $\approx\boxed{520\ \text{units}}$, consistent with the general upward run of the series — useful as a sanity check, but not used directly in either forecast below.
2. Check for a linear trend. Regressing sales $y$ against month index $t$ ($t=1$ for February $\ldots\ t=8$ for September, $t=3$/April omitted) over the seven known points gives slope $\approx86.4$ units/month, intercept $\approx341.2$, with coefficient of determination $$R^2\approx\boxed{0.640}.$$ This is a materially stronger fit than a coin-flip trend line — about 64% of the month-to-month variation is explained by a steady climb from the 450/300-unit spring months toward the 950–1,000-unit summer months — so, unlike a weak-trend series, a trend-based forecast here has real support in the data. Projecting to October ($t=9$): $$F_{Oct}^{trend}=86.41(9)+341.2\approx\boxed{1{,}119\ \text{units}}.$$
3. Compare against naive and moving-average. Naive (last actual value, September) gives $F_{Oct}=\boxed{800}$; this ignores the sustained June–August plateau around 950–1,000 units and would understate the trend the regression just confirmed. A 3-month moving average (Jul, Aug, Sep) smooths recent volatility: $$F_{Oct}=\frac{950+1000+800}{3}=\boxed{916.7\ \text{units}}.$$
4. Select and justify. With $R^2\approx0.64$ — roughly three times stronger than a borderline trend fit — the linear trend is the best-supported of the three candidates and is adopted as the primary forecast, $\boxed{F_{Oct}\approx1{,}119\ \text{units}}$, bracketed below by the moving average (917, a more conservative estimate that assumes the recent plateau holds rather than continuing to climb) and the naive estimate (800, a floor case if September's dip persists). Because the trend forecast extrapolates beyond every value actually observed (max observed $=1{,}000$ in June/August), it is reported with an explicit caveat in part (b) rather than as a bare point estimate.

(b) Discussion and improvement. The trend forecast is the best-supported of the three candidates, but it still rests on only seven data points and extrapolates beyond the highest month actually observed (1,000 units) — a real risk if the summer plateau (June–August) was driven by a one-time promotion or a new-model launch rather than a sustained ramp, in which case October could just as easily revert toward the 800–950 range. The lost April figure is itself a process problem, not just a forecasting inconvenience: a sales-report-tracking gap that recurs will keep degrading every future forecast's data quality. To improve the forecast I would: (1) fix the reporting process (a shared, backed-up sales log rather than a single misplaceable report) so no future month goes missing; (2) collect at least 24 months of history to test for a genuine annual seasonal pattern (back-to-school and holiday-season cycles are common for consumer electronics, and this series' summer climb could be exactly that); (3) bring in causal information the retailer already has — any manufacturer launch calendar, planned promotions, and competitor pricing — since a launch-driven plateau is a known event, not random noise, and should be modelled explicitly; and (4) track forecast error (MAD or MSE) month over month once more history exists, and switch to exponential smoothing with a moderate smoothing constant so the model adapts to a genuine trend shift without over-reacting to any single month.