Based on 10 data points (π‘₯1, 𝑦1 ),(π‘₯2, 𝑦2 ) (π‘₯10, 𝑦10) on a variable (𝑋, π‘Œ), the simple regression lines of π‘Œ on 𝑋 and 𝑋 on π‘Œ are obtained as 2𝑦 βˆ’ π‘₯ = 8 and 𝑦 βˆ’ π‘₯ = βˆ’3, respectively. Let π‘₯Μ…= 1 10 βˆ‘ π‘₯𝑖 10 𝑖=1 and 𝑦̅ = 1 10 βˆ‘ 𝑦𝑖 10 𝑖=1 . Then, which of the following statements is/are TRUE?
Question:

Based on 10 data points (π‘₯1, 𝑦1 ), (π‘₯2, 𝑦2 ), … , (π‘₯10, 𝑦10) on a variable (𝑋, π‘Œ), the simple regression lines of π‘Œ on 𝑋 and 𝑋 on π‘Œ are obtained as 2𝑦 βˆ’ π‘₯ = 8 and π‘¦βˆ’π‘₯ =βˆ’3, respectively. Let π‘₯Μ…=110βˆ‘i=110π‘₯𝑖\frac{ 1}{ 10} βˆ‘^{10}_{i=1} π‘₯_𝑖 and 𝑦̅ = 110βˆ‘i=110y𝑖\frac{ 1}{ 10} βˆ‘^{10}_{i=1} y_𝑖. Then, which of the following statements is/are TRUE?

Updated On: Oct 1, 2024
  • βˆ‘i=110xi=140βˆ‘^{10}_{i=1}x_i=140
  • βˆ‘i=110yi=110βˆ‘^{10}_{i=1}y_i=110
  • βˆ‘i=110(xiβˆ’xyi)(βˆ‘i=110(xiβˆ’x)2)(βˆ‘i=110(yiβˆ’y)2)=βˆ’12\frac{βˆ‘^{10}_{i=1}(x_i-xy_i)}{\sqrt({βˆ‘^{10}_{i=1}(x_i-x)^2)(βˆ‘^{10}_{i=1}(y_i-y)^2)}}=-\frac{1}{\sqrt2}
  • βˆ‘i=110(xiβˆ’x)2βˆ‘i=110(yiβˆ’y)2\frac{βˆ‘^{10}_{i=1}(x_i-x)^2}{βˆ‘^{10}_{i=1}(y_i-y)^2}=2
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The Correct Option is A, B, D

Solution and Explanation

The correct options are: A, B and D
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