JEE Main · 2012 · 29 Apr 2012 · Offline · Statistics

Let

x_{1}, x_{2}, \dots\dots, x_{n}

be n observations, and let

\overline{x}

be their arithematic mean and

σ^{2}

be their variance.

Statement 1: Variance of

2 x_{1}, 2 x_{2}, \dots\dots, 2 x_{n}

4 σ^{2}

.
Statement 2: Arithmetic mean of

2 x_{1}, 2 x_{2}, \dots\dots, 2 x_{n}

4 \overline{x}

(A)Statement 1 is false, statement 2 is true
(B)Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1
(C)Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1
(D)Statement 1 is true, statement 2 is false✓ Correct

Answer: (D)

Solution

Multiplying every observation by

2

multiplies the variance by

2^{2}

, so the variance of

2 x_{1}, \dots, 2 x_{n}

4 σ^{2}

and the first statement is true. The mean, however, scales by the same single factor as the data, so the mean of

2 x_{1}, \dots, 2 x_{n}

2 \overset{x}{ˉ}

, not

4 \overset{x}{ˉ}

; the second statement is therefore false. Hence the first statement is true while the second is false.

JEE Main · 2012 · 29 Apr 2012 · Offline · Single correct option