Let's break down the problem and solve it in steps.

a. Predict the sales for January to June and July to December in year 2006.

The trend line equation is given as:

$y = 255 + 1.2x$

where $y$ represents the sales in RM thousand, and $x$ represents the time period (measured in semi-annual intervals).

For each period, we need to first calculate the predicted sales using the trend equation and then adjust these predictions using the seasonal variation index.

1. Determine the value of $x$ for the year 2006:
   • In January 2001 (start of the time period), $x = 1$.
   • Since $x$ increases by 1 for each 6-month period, by January to June 2006, $x$ will have increased by 10 semi-annual periods (from January 2001).

So, for January to June 2006: $x = 12 \, \text{(since there are 12 semi-annual periods from Jan 2001 to Jan 2006)}$

For July to December 2006: $x = 13$

2. Predict sales for January to June 2006: Using the trend equation $y = 255 + 1.2x$: $y_{\text{Jan-Jun}} = 255 + 1.2 \times 12 = 255 + 14.4 = 269.4$ After adjusting for the seasonal variation index of +45 RM thousand: $\text{Predicted sales for Jan-Jun 2006} = 269.4 + 45 = 314.4 \, (\text{in RM thousand}) \approx 314 \, (\text{nearest thousand})$

3. Predict sales for July to December 2006: $y_{\text{Jul-Dec}} = 255 + 1.2 \times 13 = 255 + 15.6 = 270.6$ After adjusting for the seasonal variation index of -45 RM thousand: $\text{Predicted sales for Jul-Dec 2006} = 270.6 - 45 = 225.6 \, (\text{in RM thousand}) \approx 226 \, (\text{nearest thousand})$

Summary for 2006:

• January to June 2006: 314 RM thousand
• July to December 2006: 226 RM thousand

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b. Predict the sales for January to June and July to December in year 2005.

1. Determine the value of $x$ for the year 2005: For January to June 2005: $x = 10$ For July to December 2005: $x = 11$

2. Predict sales for January to June 2005: $y_{\text{Jan-Jun}} = 255 + 1.2 \times 10 = 255 + 12 = 267$ After adjusting for the seasonal variation index of +115 RM thousand: $\text{Predicted sales for Jan-Jun 2005} = 267 + 115 = 382 \, (\text{in RM thousand})$

3. Predict sales for July to December 2005: $y_{\text{Jul-Dec}} = 255 + 1.2 \times 11 = 255 + 13.2 = 268.2$ After adjusting for the seasonal variation index of +85 RM thousand: $\text{Predicted sales for Jul-Dec 2005} = 268.2 + 85 = 353.2 \, (\text{in RM thousand}) \approx 353 \, (\text{nearest thousand})$

Summary for 2005:

• January to June 2005: 382 RM thousand
• July to December 2005: 353 RM thousand

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Final Results:

• 2006 Predictions:

  • January to June: 314 RM thousand
  • July to December: 226 RM thousand

• 2005 Predictions:

  • January to June: 382 RM thousand
  • July to December: 353 RM thousand

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Would you like more details or explanations on any part of the solution?

Here are five follow-up questions to deepen your understanding:

1. How is the seasonal variation index affecting the sales predictions?
2. What would happen if the seasonal index was zero for both periods?
3. How can the trend equation be modified for non-linear sales growth?
4. How do we calculate $x$ values for other future or past periods?
5. How can we interpret the slope of 1.2 in terms of the business's growth?

Tip: Always account for both trend and seasonal variations when forecasting sales for better accuracy.