1. **Understand the data and problem:** We have quarterly sales (in thousands) for 3 years (2022-2024): \n| Year | Q1 | Q2 | Q3 | Q4 | |------|----|----|----|----| | 2022 | 42 | 45 | 48 | 51 | | 2023 | 44 | 47 | 52 | 56 | | 2024 | 49 | 54 | 59 | 63 | \nWe need to compute 4-quarter moving averages (MA), center them to get centered moving averages (CMA), estimate trend and comment, and find seasonal variations using the ratio-to-moving-average method. \n2. **Compute 4-quarter moving averages (MA):** The 4-quarter MA at quarter t is the average of sales at quarters $t, t+1, t+2, t+3$. Since we have 12 data points (4 quarters x 3 years), valid MAs cover quarters 1 to 9. \nQuarter numbering from start (Q1 2022 = 1 to Q4 2024 = 12): Values: 42, 45, 48, 51, 44, 47, 52, 56, 49, 54, 59, 63 \nCalculate MAs: $MA_1 = \frac{42 + 45 + 48 + 51}{4} = \frac{186}{4} = 46.5$ \n$MA_2 = \frac{45 + 48 + 51 + 44}{4} = \frac{188}{4} = 47$ \n$MA_3 = \frac{48 + 51 + 44 + 47}{4} = \frac{190}{4} = 47.5$ \n$MA_4 = \frac{51 + 44 + 47 + 52}{4} = \frac{194}{4} = 48.5$ \n$MA_5 = \frac{44 + 47 + 52 + 56}{4} = \frac{199}{4} = 49.75$ \n$MA_6 = \frac{47 + 52 + 56 + 49}{4} = \frac{204}{4} = 51$ \n$MA_7 = \frac{52 + 56 + 49 + 54}{4} = \frac{211}{4} = 52.75$ \n$MA_8 = \frac{56 + 49 + 54 + 59}{4} = \frac{218}{4} = 54.5$ \n$MA_9 = \frac{49 + 54 + 59 + 63}{4} = \frac{225}{4} = 56.25$ \n3. **Center the moving averages to find CMA:** Centered MA is the average of two consecutive MAs, assigned between quarters. \n$CMA_1 = \frac{MA_1 + MA_2}{2} = \frac{46.5 + 47}{2} = 46.75$ \n$CMA_2 = \frac{MA_2 + MA_3}{2} = \frac{47 + 47.5}{2} = 47.25$ \n$CMA_3 = \frac{MA_3 + MA_4}{2} = \frac{47.5 + 48.5}{2} = 48$ \n$CMA_4 = \frac{MA_4 + MA_5}{2} = \frac{48.5 + 49.75}{2} = 49.125$ \n$CMA_5 = \frac{MA_5 + MA_6}{2} = \frac{49.75 + 51}{2} = 50.375$ \n$CMA_6 = \frac{MA_6 + MA_7}{2} = \frac{51 + 52.75}{2} = 51.875$ \ $CMA_7 = \frac{MA_7 + MA_8}{2} = \frac{52.75 + 54.5}{2} = 53.625$ \n$CMA_8 = \frac{MA_8 + MA_9}{2} = \frac{54.5 + 56.25}{2} = 55.375$ \nThe CMAs correspond to quarters 2 to 9 (Q2 2022 to Q1 2024). \n4. **Estimate the trend from CMAs:** Observe increasing CMAs from 46.75 to 55.375 over quarters 2 to 9. \nThis indicates an upward trend in sales over time. \n5. **Calculate seasonal variation using ratio-to-moving-average method:** Seasonal variation $= \frac{\text{actual sales}}{\text{CMA}}$ for each quarter where CMA is available. \nMatch quarters: \n- Quarter 2 (Q2 2022): Sales = 45, CMA = 46.75 \n- Quarter 3 (Q3 2022): Sales = 48, CMA = 47.25 \n- Quarter 4 (Q4 2022): Sales = 51, CMA = 48 \n- Quarter 5 (Q1 2023): Sales = 44, CMA = 49.125 \n- Quarter 6 (Q2 2023): Sales = 47, CMA = 50.375 \n- Quarter 7 (Q3 2023): Sales = 52, CMA = 51.875 \n- Quarter 8 (Q4 2023): Sales = 56, CMA = 53.625 \n- Quarter 9 (Q1 2024): Sales = 49, CMA = 55.375 \nCalculate ratios (rounded to four decimals): \nQ2 2022: $\frac{45}{46.75} = 0.9624$ \nQ3 2022: $\frac{48}{47.25} = 1.0158$ \nQ4 2022: $\frac{51}{48} = 1.0625$ \nQ1 2023: $\frac{44}{49.125} = 0.8959$ \nQ2 2023: $\frac{47}{50.375} = 0.9335$ \nQ3 2023: $\frac{52}{51.875} = 1.0024$ \nQ4 2023: $\frac{56}{53.625} = 1.0444$ \nQ1 2024: $\frac{49}{55.375} = 0.8842$ \n6. **Average seasonal indices per quarter:** Group ratios by quarters where they appear: \n- Q1 (Quarters 5 and 9): $(0.8959 + 0.8842)/2 = 0.8901$ \n- Q2 (Quarters 2 and 6): $(0.9624 + 0.9335)/2 = 0.9479$ \ - Q3 (Quarters 3 and 7): $(1.0158 + 1.0024)/2 = 1.0091$ \ - Q4 (Quarters 4 and 8): $(1.0625 + 1.0444)/2 = 1.0535$ \n**Interpretation:** - Sales in Q3 and Q4 tend to be above trend (seasonality >1). - Sales in Q1 and Q2 tend to be below trend. - The sales show an overall increasing trend. \n**Final summary:** - 4-quarter moving averages smooth data and show trend. - Centered moving averages give refined trend estimates. - Trend is upward. - Seasonal factors indicate stronger sales in Q3 and Q4.