GATE 2025 ST – Question 24

Regression Analysis · Linear Models and Estimation · 1 mark · Multiple choice

For data (x_i,y_i) with Σx_i²>0, β̂ minimizes Σ(y_i−βx_i)². Put v_i=y_i−x_i,u_i=2x_i, and let γ̂ minimize Σ(v_i−γu_i)². If β̂=10, find γ̂.

Show answer and explanation

Correct answer: (A) 4.5

Explanation

β̂=Σx_iy_i/Σx_i². The transformed slope is Σu_iv_i/Σu_i²=[2Σx_i(y_i−x_i)]/[4Σx_i²]=(β̂−1)/2. Thus **γ̂=4.5, A**.

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