Question 3 of 7: Tablet-Computer Sales Forecast (Naive, Moving Average)

Question 3: Tablet-Computer Sales Forecast (Naive, Moving Average) (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.

Find. October sales forecasts by (a) naïve and moving average, with appropriately chosen parameters; (b) the best-justified forecast and how to improve it.

Approach. Treat the missing April observation as genuinely absent rather than interpolated (inventing a value would corrupt every method that touches it). Compute the naïve forecast directly, then screen several moving-average window lengths $k$ against their in-sample one-step forecast error to choose an "appropriate" $k$, before comparing the two methods' recommendations.

1. Naïve. The naïve forecast is simply the most recent actual value: $$\boxed{F_{Oct}=A_{Sep}=800\ \text{units}}.$$
2. Choosing the moving-average window (part a). A $k$-period moving average forecasts $F_{t+1}=\frac{1}{k}\sum_{i=t-k+1}^{t}A_i$. Screening $k=2,3,4$ against the in-sample mean absolute deviation of the resulting one-step forecasts gives $k{=}2$: MAD $\approx225$; $k{=}3$: MAD $\approx265$; $k{=}4$: MAD $\approx234$ — all in the same range (the naïve method's own in-sample MAD is $\approx192$, but each figure is averaged over a different number of months, so these are not a fair head-to-head test), so the choice is made on interpretability rather than a sharp statistical winner. $k=4$ is the appropriate choice here for a reason specific to this data set: it is exactly the length of the four most recent months (June, July, August, September), which are also the four most recent consecutive known months — the window does not need to bridge over the missing April observation at all, unlike $k=5$ or larger, which would either have to skip April or shrink the usable history. Using $k=4$: $$F_{Oct}=\frac{A_{Jun}+A_{Jul}+A_{Aug}+A_{Sep}}{4}=\frac{1000+950+1000+800}{4}=\boxed{937.5\approx938\ \text{units}}.$$
3. Comparison and recommendation (part b). The two forecasts disagree by a meaningful margin (800 vs. 938, a 17% spread) because they read the recent data differently: naïve reacts fully to September's single value, while the 4-month moving average smooths across the whole June–September plateau (1000, 950, 1000, 800), which mostly sits above 900. September's 800 is the lowest of the four plateau months, so naïve risks anchoring on what may be a single-month dip rather than the underlying level. The recommended forecast is the $\boxed{938\text{-unit moving average}}$, since a single month's actual sales is a noisy, high-variance estimator of the underlying demand level on its own, while averaging four consecutive months damps out exactly that kind of one-month noise without reaching back before the recent plateau began (no averaging-in of the lower February–March figures, and no need to bridge the missing April month).

Improving the forecast. Recover the true April figure rather than dropping the observation, so no method loses a data point; extend the history beyond eight months to distinguish a genuine trend from short-term volatility, since eight months (seven usable) is a thin base for choosing a moving-average window with confidence; and track forecast error month to month with a tracking signal or control chart so a genuine turning point (a resumed uptrend, or a step down to a new plateau) is caught quickly rather than assumed away. A regression line against the month index, or exponential smoothing with an SSE-optimized $\alpha$, would also be worth fitting as a cross-check even though this sitting does not request either.