Q74 / 160 · AP EAMCET 2024 (20 May Shift 1) · Mathematics

$\bar{x}$ and $\bar{y}$ are the arithmetic means of the runs of two batsmen A and B in 10 innings respectively and $\sigma_{\mathrm{A}}, \sigma_{\mathrm{B}}$ are the standard deviations of their runs in them. If batsman A is more consistent than B , then he is also a higher run scorer only when

  • Correct answer
    A
    $0 \lt \frac{\sigma_A}{\sigma_B} \lt \frac{\bar{x}}{\bar{y}} ; \frac{\bar{x}}{\bar{y}}\gt1$
  • B
    $\frac{\bar{x}}{\bar{y}}\gt\frac{\sigma_A}{\sigma_B}\gt1$
  • C
    $\frac{\bar{x}}{\bar{y}} \lt \frac{\sigma_A}{\sigma_B}\gt1$
  • D
    $\frac{\bar{x}}{\bar{y}}\gt1 ; 1 \leq \frac{\bar{x}}{\bar{y}} \lt \frac{\sigma_A}{\sigma_B}$

Solution

Batsman A is more consistent than B $\Rightarrow$ Coefficient of variation of $\mathrm{A} \lt $ coefficient of variation B $\Rightarrow \frac{\sigma_A}{\bar{x}} \lt \frac{\sigma_B}{\bar{y}} \Rightarrow 0 \lt \frac{\sigma_A}{\sigma_B} \lt \frac{\bar{x}}{\bar{y}}$ A scores higher runs $\Rightarrow \Sigma x_{i A}\gt\Sigma x_{i B}$ $\Rightarrow \bar{x}\gt\bar{y} \Rightarrow \frac{\bar{x}}{\bar{y}}\gt1$

Hence, the correct answer is option: 'A'.

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