ANSWER :

[tex] \\ [/tex]

• ➤ Batsman X is better scorer than Batsman Y in the test series.

• ➤ Batsman Y is more consistent than Batsman X in the test series.

______________________________________________________

SOLUTION :

[tex] \\ \\ [/tex]

❑ Given :-

• Runs scored by Batsman X and Batsman Y in different innings.

❑ To Determine :-

• Which of the two Batsman X and Y is a better scorer ?

• Who between the two Batsman X and Y is more consistent ?

___________________________________________

▣ Note :-

• ➫ In order to determine which of the two batsman is a better scorer, we have to compare Average (Mean) runs scored by the two batsmen in different innings.

• ➫ In order to determine consistency of batting between the two batsman, we have to compare Coefficient of Variation (C.V.) of the runs scored by the batsmen in the test series.

___________________________________________

❍ Calculation of Average and C.V. of Batsman X :-

[tex] \\ [/tex]

From the table, we have,

• ΣX = 500

• N = 10

Hence,

• [tex]\bigstar \: \: \tt{ \bar{X} = \dfrac{ \sum \: X }{N}}[/tex]

[tex]\implies\: \: \tt{ \bar{X} = \dfrac{500}{10}}[/tex]

[tex]\therefore\: \: \tt{ \bar{X} = \underline{\: 50 \:}}[/tex]

• Average of the runs scored by Batsman X in different innings is 50.

Again,

• Σx² = 17498

• N = 10

So,

• [tex]\rhd\: \: \sf{ \sigma_x \: = \sqrt{ \dfrac{ \sum x {}^{2} }{N}}}[/tex]

[tex]: \to \: \: \sf{\sigma_x \: = \sqrt{ \dfrac{17498}{10}}}[/tex]

[tex]: \to \: \: \sf{\sigma_x \: = \sqrt{1749.8}}[/tex]

[tex]: \to \: \: \sf{\sigma_x \: = 41.83}[/tex]

Hence,

• [tex]\bigstar \: \: \tt{C.V. \: _x = \dfrac{\sigma_x}{ \bar{ X}} \times 100\%}[/tex]

[tex] \implies \: \: \tt{C.V. \: _x = \dfrac{41.83}{50} \times 100\%}[/tex]

[tex]\therefore \: \: \tt{C.V. \: _x = \underline{\: 83.66 \: \%}}[/tex]

• Coefficient of Variation of the runs scored by the Batsman X in the test series is 83.66%.

______________________________________________________

❍ Calculation of Average and C.V. of Batsman Y :-

[tex] \\ [/tex]

From the table, we have,

• ΣY = 330

• N = 10

Hence,

• [tex]\bigstar \: \: \tt{ \bar{Y} = \dfrac{ \sum \: Y }{N}}[/tex]

[tex]\implies\: \: \tt{ \bar{Y} = \dfrac{330}{10}}[/tex]

[tex]\therefore\: \: \tt{ \bar{Y} = \underline{\: 33 \:}}[/tex]

• Average of the runs scored by Batsman Y in different innings is 33.

Again,

• Σy² = 5462

• N = 10

So,

• [tex]\rhd\: \: \sf{ \sigma_y \: = \sqrt{ \dfrac{ \sum y {}^{2} }{N}}}[/tex]

[tex]: \to \: \: \sf{\sigma_y \: = \sqrt{ \dfrac{5462}{10}}}[/tex]

[tex]: \to \: \: \sf{\sigma_y \: = \sqrt{546.2}}[/tex]

[tex]: \to \: \: \sf{\sigma_y \: = 23.37}[/tex]

Hence,

• [tex]\bigstar \: \: \tt{C.V. \: _y = \dfrac{\sigma_y}{ \bar{ Y}} \times 100\%}[/tex]

[tex] \implies \: \: \tt{C.V. \: _y = \dfrac{23.37}{33} \times 100\%}[/tex]

[tex]\therefore \: \: \tt{C.V. \: _y = \underline{\: 70.82 \: \%}}[/tex]

• Coefficient of Variation of the runs scored by the Batsman Y in the test series is 70.82%.

______________________________________________________

⦿ Determination :-

[tex] \\ [/tex]

Here,

• Average of runs scored by Batsman X = 50

• Average of runs scored by Batsman Y = 33

► As, the average score of Batsman X is more than that of Batsman Y; therefore, Batsman X is a better scorer than Batsman Y in the test series.

And,

• Coefficient of Variation of runs scored by Batsman X = 83.66%

• Coefficient of Variation of runs scored by Batsman Y = 70.82%

► As, the coefficient of variation of Batsman X is greater than that of Batsman Y; therefore, Batsman Y is relatively more consistent batsman than Batsman X in the test series.