Question

Determine the equation of the straight-line trend.

Hint:

When calculating a straight-line trend using least squares: 
- Assign coded time values (\( t \)) to simplify calculations. 
- Use the formulas \( a = \frac{\sum y}{N} \) and \( b = \frac{\sum (t \cdot y)}{\sum t^2} \).

Answer 1

Step 1: Assign coded time variables (\( t \)) for convenience:
Let \( t = -2, -1, 0, 1, 2, 3 \) for the years 1996 to 2001, respectively.

Step 2: Tabulate the data:
 

Step 3: Use the least squares formula for straight-line trend:

\[ y = a + bt, \]

where:

\[ a = \frac{\sum y}{N}, \quad b = \frac{\sum (t \cdot y)}{\sum t^2}. \]

Substitute the values:

\[ a = \frac{\sum y}{N} = \frac{35.6}{6} = 5.93, \quad b = \frac{\sum (t \cdot y)}{\sum t^2} = \frac{22.4}{19} = 1.18. \]

Step 4: Write the equation:

\[ y = 5.93 + 1.18t. \]