Solutions: Seasonal Adjustment

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   Seasonal averages: Q1 = (85+72)÷2 = 78.5; Q2 = (120+132)÷2 = 126; Q3 = (60+65)÷2 = 62.5; Q4 = (95+103)÷2 = 99.

   Overall mean = (78.5+126+62.5+99)÷4 = 366÷4 = 91.5

   SI_Q1 = 78.5÷91.5 = 0.858; SI_Q2 = 126÷91.5 = 1.377; SI_Q3 = 62.5÷91.5 = 0.683; SI_Q4 = 99÷91.5 = 1.082

   Sum = 0.858 + 1.377 + 0.683 + 1.082 = 4.000 ✓

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   (a) SI_Q3 = 0.72 means Q3 sales are typically 28% below the annual quarterly average. Q3 is a below-average season for this business.

   (b) Deseasonalised Q3 sales = $84,000 ÷ 0.72 = $116,667 (to the nearest dollar).

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   SI_Q2 = 1.35 means Q2 sales are typically 35% above the annual quarterly average. In other words, Q2 is a strong season — its sales are 35% higher than what would be expected if sales were evenly spread across all four quarters.

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   For quarterly data, all four seasonal indices must sum to 4.

   SI_Q3 = 4 − (0.85 + 1.10 + 0.95) = 4 − 2.90 = 1.10

   This method is valid because the defining property of seasonal indices is that they must sum to the number of seasons (4). Given three indices, the fourth is determined.

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   (a) SI = monthly average ÷ overall mean (145). Results (to 2 d.p.):

   Jan: 190÷145 = 1.31; Feb: 175÷145 = 1.21; Mar: 150÷145 = 1.03; Apr: 130÷145 = 0.90; May: 115÷145 = 0.79; Jun: 110÷145 = 0.76; Jul: 120÷145 = 0.83; Aug: 130÷145 = 0.90; Sep: 140÷145 = 0.97; Oct: 155÷145 = 1.07; Nov: 175÷145 = 1.21; Dec: 145÷145 = 1.00

   (b) Months with SI > 1: January (1.31), February (1.21), March (1.03), October (1.07), November (1.21). Reason: QLD schools use high air-conditioning in the hot months (Jan–Mar, Oct–Nov), driving electricity consumption above the annual average during summer and late spring. December is exactly at the mean.

   (c) Deseasonalised June = 118 ÷ 0.76 = 155.3 MWh. The overall average is 145 MWh. The deseasonalised value of 155.3 MWh is above the overall average, suggesting that once the seasonal suppression of June is removed, the school’s underlying consumption level for that June is slightly higher than normal.

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   (a) Seasonal averages: Q1 = (95+100+108)÷3 = 101; Q2 = (185+198+210)÷3 = 197.67; Q3 = (210+225+238)÷3 = 224.33; Q4 = (130+142+154)÷3 = 142.

   Overall mean = (101+197.67+224.33+142)÷4 = 665÷4 = 166.25

   SI_Q1 = 101÷166.25 = 0.61; SI_Q2 = 197.67÷166.25 = 1.19; SI_Q3 = 224.33÷166.25 = 1.35; SI_Q4 = 142÷166.25 = 0.85. Sum ≈ 4.00 ✓

   (b) Deseasonalised = Actual ÷ SI:

   Y1: Q1=95÷0.61=155.7; Q2=185÷1.19=155.5; Q3=210÷1.35=155.6; Q4=130÷0.85=152.9

   Y2: Q1=100÷0.61=163.9; Q2=198÷1.19=166.4; Q3=225÷1.35=166.7; Q4=142÷0.85=167.1

   Y3: Q1=108÷0.61=177.0; Q2=210÷1.19=176.5; Q3=238÷1.35=176.3; Q4=154÷0.85=181.2

   (c) Deseasonalised Q1: 155.7 (Y1), 163.9 (Y2), 177.0 (Y3). These increase each year by approximately $10,000–13,000, revealing a clear upward trend in underlying performance. Without deseasonalising, the raw Q1 values of $95k, $100k, $108k also show growth but are compressed by the seasonal suppression of Q1 (SI = 0.61).

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   (a) Deseasonalised January = 280,000 ÷ 1.40 = 200,000. Deseasonalised July = 175,000 ÷ 0.88 = 198,864 (approx. 198,900).

   (b) After removing seasonal effects, the two months have almost identical underlying performance (200,000 vs 198,900). The difference is negligible. Neither month has a meaningfully stronger underlying performance than the other.

   (c) The manager’s reasoning is statistically flawed. The raw comparison (280,000 vs 175,000) is misleading because January naturally has a much higher seasonal index (1.40 vs 0.88). After deseasonalising, July’s performance is virtually identical to January’s. Marketing decisions should be based on deseasonalised performance, not raw figures affected by seasonal patterns.

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   (a) Sum of indices = 1.22 + 0.96 + 0.78 + 1.04 = 4.00 ✓ The indices are consistent.

   (b) Expected Q3 revenue = overall mean × SI_Q3 = $250,000 × 0.78 = $195,000. This is what Q3 revenue would be expected to be, solely due to the seasonal effect.

   (c) Deseasonalised Q3 = $185,000 ÷ 0.78 = $237,179. This means that once the below-average seasonal nature of Q3 is removed, the underlying revenue level is $237,179 — comfortably above the $250,000 quarterly average? No — $237,179 is below $250,000. Interpretation: After deseasonalising, Q3 revenue of $185,000 corresponds to an underlying trend level of $237,179, which is still somewhat below the long-run average of $250,000. This Q3 was below average even after accounting for its typically weak season.

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   (a) SI_Q1 = Actual ÷ Deseasonalised = 55.0 ÷ 42.3 = 1.30

   Check with Year 2: 58.2 ÷ 44.8 = 1.299 ≈ 1.30 ✓. Year 3: 61.8 ÷ 47.5 = 1.301 ≈ 1.30 ✓. The seasonal index is consistent across all three years.

   (b) Deseasonalised Q1 values: 42.3 (Y1), 44.8 (Y2), 47.5 (Y3). These increase by 2.5 (Y1→Y2) and 2.7 (Y2→Y3). The average annual increase is approximately 2.6 thousand visitors per year. The underlying trend is upward, gaining about 2,600 visitors per Q1 each year.

   (c) Estimated deseasonalised Q1 Y4 = 47.5 + 2.6 = 50.1 thousand (using the average annual increase).

   Estimated actual Q1 Y4 arrivals = 50.1 × 1.30 = 65.1 thousand.

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    (a) Seasonally adjusted revenue grows by $3,200 per month. From October (Month 10) to December (Month 12) is 2 months.

    Seasonally adjusted December = $82,000 + 2 × $3,200 = $82,000 + $6,400 = $88,400.

    (b) Forecast actual December = Seasonally adjusted × SI_Dec = $88,400 × 1.85 = $163,540.

    (c) Residual = Actual − Forecast = $166,000 − $163,540 = +$2,460.

    Interpretation: Actual December revenue exceeded the forecast by $2,460 (about 1.5% above the model’s prediction). The model performed well. The small positive residual could reflect an unusually strong Christmas trading period, a one-off promotional event, or minor irregular variation. It does not indicate any model failure.