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Statistics Question on Limit Theorems

Based on 10 data points (𝑥1, 𝑦1 ), (𝑥2, 𝑦2 ), … , (𝑥10, 𝑦10) on a variable (𝑋, 𝑌), the simple regression lines of 𝑌 on 𝑋 and 𝑋 on 𝑌 are obtained as 2𝑦 − 𝑥 = 8 and 𝑦−𝑥 =−3, respectively. Let 𝑥̅=110∑i=110𝑥𝑖\frac{ 1}{ 10} ∑^{10}_{i=1} 𝑥_𝑖 and 𝑦̅ = 110∑i=110y𝑖\frac{ 1}{ 10} ∑^{10}_{i=1} y_𝑖. Then, which of the following statements is/are TRUE?

A

∑i=110xi=140∑^{10}_{i=1}x_i=140

B

∑i=110yi=110∑^{10}_{i=1}y_i=110

C

∑i=110(xi−xyi)(∑i=110(xi−x)2)(∑i=110(yi−y)2)=−12\frac{∑^{10}_{i=1}(x_i-xy_i)}{\sqrt({∑^{10}_{i=1}(x_i-x)^2)(∑^{10}_{i=1}(y_i-y)^2)}}=-\frac{1}{\sqrt2}

D

∑i=110(xi−x)2∑i=110(yi−y)2\frac{∑^{10}_{i=1}(x_i-x)^2}{∑^{10}_{i=1}(y_i-y)^2}=2

Answer

∑i=110xi=140∑^{10}_{i=1}x_i=140

Explanation

Solution

The correct options are: A, B and D