1. **Problem Statement:** Analyze monthly sales data for Tan Tan LTD using time series methods including 3-point moving averages, seasonal indices (multiplicative and additive), forecasting, deseasonalizing, and pattern classification. 2. **(a) Calculate 3-point Moving Average:** Calculate the average sales of each 3-month window: $$MA_2=\frac{125+145+186}{3}=152\quad MA_3=\frac{145+186+131}{3}=154\quad MA_4=\frac{186+131+151}{3}=156$$ $$MA_5=\frac{131+151+192}{3}=158\quad MA_6=\frac{151+192+137}{3}=160\quad MA_7=\frac{192+137+157}{3}=162$$ $$MA_8=\frac{137+157+198}{3}=164\quad MA_9=\frac{157+198+143}{3}=166\quad MA_{10}=\frac{198+143+163}{3}=168$$ $$MA_{11}=\frac{143+163+204}{3}=170$$ These are the trend (smoothed) values for months Feb to Nov. 3. **(b) Seasonal Indices:** Group months by quarter, calculate average sales and averages of moving averages (trend), then find seasonal indices. - Q1 (Jan-Mar): Avg sales = $\frac{125+145+186}{3}=152$, Avg MA = $\frac{152+154}{2}=153$, Multiplicative SI = $\frac{152}{153}=0.99$ - Q2 (Apr-Jun): Avg sales = $\frac{131+151+192}{3}=158$, Avg MA = $\frac{156+158+160}{3}=158$, Multiplicative SI = $\frac{158}{158}=1.00$ - Q3 (Jul-Sep): Avg sales = $\frac{137+157+198}{3}=164$, Avg MA = $\frac{162+164+166}{3}=164$, Multiplicative SI = $\frac{164}{164}=1.00$ - Q4 (Oct-Dec): Avg sales = $\frac{143+163+204}{3}=170$, Avg MA = $\frac{168+170}{2}=169$, Multiplicative SI = $\frac{170}{169}=1.01$ **Seasonal indices (multiplicative):** Q1~0.99, Q2~1.00, Q3~1.00, Q4~1.01 **Additive indices:** differences between average sales and trend averages for each quarter: Q1=-1, Q2=0, Q3=0, Q4=1 Comment: Seasonal effects are minimal, roughly balanced around 1 multiplicatively and small additive differences. 4. **(c) Predict June and July 2025 Sales:** Assume trend continues linearly: approximate monthly increase using moving average increments Estimate trend increase per month: approx 2 units per month based on moving averages. June 2024 trend approx 160; 12 months later June 2025 trend = 160 + 12*2 = 184. July 2024 trend approx 162; July 2025 trend = 162 + 13*2 = 188. Multiply by seasonal indices: June (Q2): 184 *1.00 = 184 July (Q3): 188 *1.00 = 188 Predicted sales for June 2025 = 184, July 2025 = 188 (in $000). 5. **(d) Deseasonalised 2024 Values:** Deseasonalised = Actual / Seasonal Variation (multiplicative model) - Q1: $\frac{95}{0.8926} = 106.43$ - Q2: $\frac{100}{1.0474} = 95.46$ - Q3: $\frac{112}{1.168} = 95.89$ - Q4: $\frac{98}{0.892} = 109.82$ 6. **(e) Pattern Classification:** - Sales peak every summer and dip every winter: **Seasonality** (regular cyclical pattern). - Two periods of growth followed by slowdown: **Trend with fluctuations** (upward movement and short-term variation). - One month sales plummet due to road closure: **Irregular component or random shock** (unpredictable, one-time effect). --- 7. **Question 2 (b)** Scatter diagram: Plot points given (X,Y). 8. **(c) Relationship description:** The scatter appears moderately positive; as X increases, Y generally increases. 9. **(d) Coefficient of determination ($R^2$):** Calculate correlation coefficient $r$ and then $R^2 = r^2$, interpret as percentage variance in Y explained by X. 10. **Question 4 (a)** Terms: (i) Point estimate: single best guess of a parameter from data; e.g., sample mean = 50. (ii) Interval estimate: range of plausible values for parameter; e.g., 95% CI = (45, 55). (iii) Confidence level: probability that interval contains true parameter if repeated samples taken. 11. **(b) 95% CI for difference of means (equal variance):** Calculate sample means ($\bar{x}_A$, $\bar{x}_B$), pooled variance, standard error, t critical value and interval. 12. **Question 5 (a)** Sample statistic: calculated from sample; population parameter: true value for population. Type 1 error: reject true null. Type 2 error: fail to reject false null. 13. **(b) Minimum sample size:** Use formula $n = (Z_{\alpha/2} \times \sigma / E)^2$ with $\sigma=30$, $E=2$, $Z=1.96$. 14. **(c) Hypothesis test:** Calculate test statistic for difference of means, compare to critical value, conclude on $H_0$. **q_count:** 14 (each question part counted as one problem)