Based on the data available for the production (\(y_i\) in thousand tons) of a cloth factory for 7 years (\(x_i\)) using the method of least squares, the straight line trend is given by \(y = a + bx\) with \(\sum y_i=608, \sum x_i=0, \sum x_iy_i= 116, \sum {x_i}^2= 28\). Then, the increase in production per year is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 4,143 tons
The 7 years are coded so that \(\sum x_i = 0\) (x = −3, −2, …, 3), so one unit of x is one year. With \(\sum x_i = 0\) the normal equations of the least squares line give
\( \displaystyle a = \frac{\sum y_i}{n},\quad b = \frac{\sum x_iy_i}{\sum x_i^2} \)
The slope b is the increase in production per year:
\( \displaystyle b = \frac{116}{28} = 4.143 \text{ thousand tons} \)
\( b \approx 4{,}143 \text{ tons per year} \)