Question

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

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: Problem Definition: The objective is to determine the coefficients 'a' and 'b' for a linear regression model fitted using the method of least squares. The provided equation utilizes a coded time variable to simplify computations. Step 2: Methodology: The linear trend line is represented as \(y = a + bX\), where \(X = x - 2022\). The standard normal equations are: \[ \sum y = na + b \sum X \] \[ \sum Xy = a \sum X + b \sum X^2 \] Given that the coded time variable satisfies \(\sum X = 0\), the equations simplify to: \[ a = \frac{\sum y}{n} \] \[ b = \frac{\sum Xy}{\sum X^2} \] Step 3: Calculation and Derivation: A tabular calculation is performed using 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 total number of data points is n = 5. 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 problem requires the computation of the ratio \(\frac{a}{b}\). \[ \frac{a}{b} = \frac{3.2}{0.2} = \frac{32}{2} = 16 \] Step 4: Conclusion: The calculated value of the ratio \(\frac{a}{b}\) is 16.