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: Problem Definition: The objective is to determine the coefficients 'a' and 'b' for a linear trend line, \(y = a + bX\), fitted using the method of least squares. The time variable 'x' is transformed to a coded variable \(X = x - 2022\) to simplify computations.
Step 2: Applicable Formulas: The standard normal equations for a linear regression are given by:
\[ \sum y = na + b \sum X \]\[ \sum Xy = a \sum X + b \sum X^2 \]
Due to the coding \(X = x - 2022\), the sum of the coded time variable is zero (\(\sum X = 0\)). This simplifies the formulas to:
\[ a = \frac{\sum y}{n} \]
\[ b = \frac{\sum Xy}{\sum X^2} \]
Step 3: Calculation and Derivation: A table is constructed to compute the necessary sums. The coded variable \(X\) is derived by subtracting 2022 from each year. The table includes columns for Year (x), Profit (y), Coded Time (X), the product Xy, and the square of X (\(X^2\)).

Year (x)	Profit (y)	X = x - 2022	Xy	\(X^2\)
2020	2	-2	-4	4
2021	3	-1	-3	1
2022	4	0	0	0
2023	5	1	5	1
2024	2	2	4	4
Sum	\(\sum y = 16\)	\(\sum X = 0\)	\(\sum Xy = 2\)	\(\sum X^2 = 10\)

The total number of data points, \(n\), is 5.

Applying the simplified formulas:
\[ a = \frac{16}{5} = 3.2 \]
\[ b = \frac{2}{10} = 0.2 \]
The problem requires the calculation of the ratio \(\frac{a}{b}\).
\[ \frac{a}{b} = \frac{3.2}{0.2} = \frac{32}{2} = 16 \]
Step 4: Conclusion: The ratio \(\frac{a}{b}\) is determined to be 16.