Question

Given: \(y = a + b(x - 2022)\) is the least-squares straight-line trend.

Year (x)	2020	2021	2022	2023	2024
Profit (Rs. '000) (y)	2	3	4	5	2

• A. 15
• B. 5
• C. 16
• D. 2/3

Hint:

When fitting a linear trend, always check if you can simplify the time variable. If the number of years is odd, choosing the middle year as the origin (as done here with `X = x - 2022`) makes \(\sum X = 0\). This greatly simplifies the calculations for the least squares parameters 'a' and 'b'.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The problem asks to find the parameters 'a' and 'b' for a linear trend line fitted using the method of least squares. The equation is given in a coded form, which simplifies calculations by shifting the origin of the time variable 'x'.
Step 2: Key Formula or Approach:
The trend line is \(y = a + bX\), where \(X = x - 2022\). The normal equations for finding 'a' and 'b' are: \[ \sum y = na + b \sum X \] \[ \sum Xy = a \sum X + b \sum X^2 \] Since the time variable is coded such that \(\sum X = 0\), the formulas simplify to: \[ a = \frac{\sum y}{n} \] \[ b = \frac{\sum Xy}{\sum X^2} \] Step 3: Detailed Explanation:
Let's create a calculation table with the coded time variable \(X = x - 2022\).
\begin{tabular}{|c|c|c|c|c|} \hline Year (x) & Profit (y) & X = x - 2022 & Xy & \(X^2\)
\hline 2020 & 2 & -2 & -4 & 4
\hline 2021 & 3 & -1 & -3 & 1
\hline 2022 & 4 & 0 & 0 & 0
\hline 2023 & 5 & 1 & 5 & 1
\hline 2024 & 2 & 2 & 4 & 4
\hline Sum & \(\sum y = 16\) & \(\sum X = 0\) & \(\sum Xy = 2\) & \(\sum X^2 = 10\)
\hline \end{tabular}
The number of data points is n = 5.
Now, we can calculate 'a' and 'b' using the simplified formulas: \[ a = \frac{\sum y}{n} = \frac{16}{5} = 3.2 \] \[ b = \frac{\sum Xy}{\sum X^2} = \frac{2}{10} = 0.2 \] The question asks for the value of the ratio \(\frac{a}{b}\). \[ \frac{a}{b} = \frac{3.2}{0.2} = \frac{32}{2} = 16 \] Step 4: Final Answer:
The value of \(\frac{a}{b}\) is 16.