The least-squares regression line for the deseasonalised sales data of a product in one shop is given by Deseasonalised Sales = 128 + 1.56 × month number where January 2021 represents month number 1, February 2021 month number 2 and so on. The seasonal index for the month of March is 0.85. The actual sales for March 2022 is closest to

Question

BlackTom AI · Accepted Answer

For a multiplicative seasonal model, actual sales = deseasonalised sales × seasonal index.

1. Relevant knowledge points and concepts
- The regression line gives predicted deseasonalised sales (trend + cycle) as Deseasonalised Sales = 128 + 1.56 × month number.
- Multiplicative seasonal decomposition means deseasonalised = actual / seasonal index, so actual = deseasonalised × seasonal index.
- Month numbering: January 2021 = 1, so month numbers increase by 1 each calendar month.

2. Complete reasoning process step by step
- Determine the month number for March 2022.
  - January–December 2021 correspond to months 1–12.
  - January 2022 is month 13, February 2022 is month 14, March 2022 is month 15.
- Use the regression to compute the predicted deseasonalised sales for month 15.
  - Deseasonalised Sales = 128 + 1.56 × 15.

3. Calculations with intermediate steps
- Multiply slope by month number: 1.56 × 15 = (1.56 × 10) + (1.56 × 5) = 15.6 + 7.8 = 23.4.
- Add the intercept: 128 + 23.4 = 151.4. This is the predicted deseasonalised sales for March 2022.
- Convert to actual sales using the seasonal index (March index = 0.85):
  - Actual sales = 151.4 × 0.85.
  - 151.4 × 0.85 = 151.4 × (1 − 0.15) = 151.4 − 22.71 = 128.69.

4. Why the correct answer is correct
- The regression output is explicitly for deseasonalised data, so we must apply the seasonal index multiplicatively to recover actual sales. The arithmetic yields $128.69, matching option B.

5. Why the other options are incorrect
- A ($130.02): does not match the correct multiplication; likely a rounding or arithmetic error.
- C ($151.40): equals the deseasonalised forecast, not the actual (seasonally adjusted) value.
- D ($127.36): differs from the correct product 151.4 × 0.85; likely from using an incorrect month number or index.

6. Conclusion
The computed actual sales for March 2022 is $128.69, which confirms option B.

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