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Question:

Let x₁, x₂, ..., xₙ be n observations, and let ¯x be their arithmetic mean and σ² be their variance. Statement 1: Variance of 2x₁, 2x₂, ..., 2xₙ is 4σ². Statement 2: Arithmetic mean of 2x₁, 2x₂, ..., 2xₙ is 4¯x. Select the correct statement.

Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.

Statement 1 is false, Statement 2 is true,

Statement 1 is true, Statement 2 is true, Statament 2 is a correct explanation for Statement 1

Statement 1 is true, Statement 2 is false.

Solution:

σ2=∑x21n−(∑xin)2Variance of2x1.2xn=∑(2xi)2n−(∑2xin)2=4σ2.Statement 1 is true.A.M. of2xi..2xn=2x1+2x22xnn=2¯¯¯xStatement 2 is false.Hence, option 'D' is correct.