The following data shows the percentage of rural and urban Indian hou…

The correct answer is Option (2) → (A), (C) and (D) only

Let \(t = x - 2022\), so t = −2, −1, 0, 1, 2. Then \(\sum t = 0\) and the normal equations for \(y = a + bt\) reduce to

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

with n = 5 and \(\sum t^2 = 4 + 1 + 0 + 1 + 4 = 10\).

Rural (y = 2, 3, 3, 4, 5):

    \( \sum y = 17,\quad a = 3.4 \)
\( \sum ty = -4 - 3 + 0 + 4 + 10 = 7,\quad b = 0.7 \)

Urban (y = 6, 7, 9, 15, 20):

    \( \sum y = 57,\quad a = 11.4 \)
\( \sum ty = -12 - 7 + 0 + 15 + 40 = 36,\quad b = 3.6 \)

For 2025, t = 3.

(A) A straight line trend by the method of the least squares for rural Indians is \(y_t = 3.4 + 0.7\,(x - 2022)\) — Correct. \(a = \frac{17}{5} = 3.4\) and \(b = \frac{7}{10} = 0.7\).

(B) A straight line trend by the method of the least squares for urban Indians is \(y_t = 11.4 + 4.3\,(x - 2022)\) — Incorrect. The slope is \(\frac{36}{10} = 3.6\), not 4.3. The trend is \(y_t = 11.4 + 3.6\,(x - 2022)\).

(C) The forecast for the year 2025 for urban group using the trend equation is 22.2%. — Correct.

    \( 11.4 + 3.6 \times 3 = 22.2 \)

(D) The forecast for the year 2025 for rural group using the trend equation is 5.5%. — Correct.

    \( 3.4 + 0.7 \times 3 = 5.5 \)