Question

Let \( \overline{X} \) and \( \overline{Y} \) be the arithmetic means of the runs of two batsmen A and B in 10 innings respectively, and \( \sigma_A, \sigma_B \) 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

• A. \( 0 < \frac{\sigma_A}{\sigma_B} < \frac{\overline{X}}{\overline{Y}} < 1 \)
• B. \( \frac{\overline{X}}{\overline{Y}} \geq \frac{\sigma_A}{\sigma_B} \)
• C. \( \frac{\overline{X}}{\overline{Y}} < \frac{\sigma_A}{\sigma_B} \)
• D. \( \frac{\overline{X}}{\overline{Y}} > 1; 1 \leq \frac{\overline{X}}{\overline{Y}} \leq \frac{\sigma_A}{\sigma_B} \)

Hint:

To determine a player's consistency, compare the coefficient of variation (CV). A lower CV means higher consistency. The key condition to check for both consistency and higher runs is ensuring that \( \frac{\sigma_A}{\sigma_B} \) remains lower than \( \frac{\overline{X}}{\overline{Y}} \).

Answer 1

The Correct Option is A

We are given the mean and standard deviations of two batsmen A and B. The consistency of a batsman is measured using the coefficient of variation (CV), which is given by: \[ \text{CV} = \frac{\sigma}{\text{Mean}} \] Step 1: Define the Consistency Condition
Batsman A is more consistent than batsman B if: \[ \frac{\sigma_A}{\overline{X}}<\frac{\sigma_B}{\overline{Y}} \] Rearranging this inequality: \[ \frac{\sigma_A}{\sigma_B}<\frac{\overline{X}}{\overline{Y}} \] Step 2: Condition for Higher Runs
For A to be a higher scorer than B, we must also ensure that: \[ \frac{\overline{X}}{\overline{Y}}<1 \] which means that A’s mean score should be relatively high compared to B's. Combining these two conditions: \[ 0<\frac{\sigma_A}{\sigma_B}<\frac{\overline{X}}{\overline{Y}}<1 \] Conclusion:
Thus, the correct condition for batsman A to be both more consistent and a higher scorer than B is: \[ 0<\frac{\sigma_A}{\sigma_B}<\frac{\overline{X}}{\overline{Y}}<1 \] which matches option (1).

Answer 2

Understanding Consistency and Scoring Comparison of Batsmen
We are given the mean and standard deviations of two batsmen, A and B. To compare them, we use the Coefficient of Variation (CV), which measures consistency as:

\[ \text{CV} = \frac{\sigma}{\text{Mean}} \]

Step 1: Condition for Greater Consistency
Batsman A is more consistent than Batsman B if A has a lower coefficient of variation. This translates to:

\[ \frac{\sigma_A}{\overline{X}} < \frac{\sigma_B}{\overline{Y}} \]

Rewriting the inequality for comparison:

\[ \frac{\sigma_A}{\sigma_B} < \frac{\overline{X}}{\overline{Y}} \]

Step 2: Condition for Higher Average Score
To ensure that Batsman A also scores more on average than Batsman B:

\[ \frac{\overline{X}}{\overline{Y}} < 1 \]

Step 3: Combine Both Conditions
Combining both consistency and higher scoring conditions, we get:

\[ 0 < \frac{\sigma_A}{\sigma_B} < \frac{\overline{X}}{\overline{Y}} < 1 \]

Conclusion:
Hence, for Batsman A to be both more consistent and a higher scorer than Batsman B, the correct condition is:

\[ \boxed{0 < \frac{\sigma_A}{\sigma_B} < \frac{\overline{X}}{\overline{Y}} < 1} \]

Correct Option: (1)