Consider the following data Year (x)20102011201220132014Profit (Rs. i…

The correct answer is Option (3) → \(y=x-1999\)

There are 5 years (an odd number), so time is measured from the middle year 2012: \(X = x - 2012\). Then \(\sum X = 0\), and the normal equations of the least-squares line \(y = a + bX\) reduce to

    \( \displaystyle a = \frac{\sum y}{n}, \quad b = \frac{\sum Xy}{\sum X^2} \)

With X = −2, −1, 0, 1, 2 and y = 10, 12, 14, 16, 13:

    \(\sum y = 65\)
    \(\sum X^2 = 4 + 1 + 0 + 1 + 4 = 10\)
    \(\sum Xy = -20 - 12 + 0 + 16 + 26 = 10\)

    \( \displaystyle a = \frac{65}{5} = 13, \quad b = \frac{10}{10} = 1 \)

So the trend line is

    \(y = 13 + X\)

    \(y = 13 + (x - 2012)\)

    \(y = x - 1999\)