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Let x1, x2,..., xn be n observations, and let x̄ be their arithmetic mean and σ² be the variance.
Statement-1: Variance of 2x1, 2x2,..., 2xn is 4σ².
Statement-2: Arithmetic mean 2x1, 2x2,..., 2xn is 4x̄.
- Statement-1 is false, Statement-2 is true.
- Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
- Statement-1 is true, Statement-2 is false.
Correct answer: Statement-1 is true, Statement-2 is false.
Solution
Statement-1 is correct because scaling the observations by a factor of 2 results in the variance being multiplied by the square of that factor, which is 4. However, Statement-2 is incorrect since scaling the mean by 2 results in the mean being doubled, not quadrupled.
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