Question

In 2007, the batting average of MLB players have μ=0.283μ=0.283 and σ=0.027σ=0.027. In 2013, the batting averages have μ=0.271μ=0.271 and σ=0.029σ=0.029. In 2007, Ronnie Belliard had a batting average of 0.2900.290. In 2013, Manny Machado had a batting average of 0.2830.283. Which of these players performed better, in relation to the year they played?

a. Manny Machado performed better than Ronnie Belliard since the z-score of Machado’s batting average is higher than that of Belliard’s batting average.

b. Ronnie Belliard performed better than Manny Machado since the z-score of Belliard’s batting average is higher than that of Machado’s batting average.

c. Neither performed better in the relation of the year they played since both have the same z-score.

d. Ronnie Belliard performed better than Manny Machado since Belliard’s batting average is higher than Machado’s batting average while the comparison does not take account the years they have played.

Answer 1

Q.Which of these players performed better, in relation to the year they played?

a. Manny Machado performed better than Ronnie Belliard since the z-score of Machado’s batting average is higher than that of Belliard’s batting average.

Explanation: Calculation of Z score as per below formula:

	2007		2013
	Mean (Mu)	Std.Dev(Sigma)	Mean (Mu)	Std.Dev(Sigma)
MLB players	0.283	0.027	0.271	0.029
Ronnie Belliard	0.290
Manny Machado			0.283
Ronnie Z-score==>	=(0.29-0.283)/0.027	Manny Z-score==>	=(0.283-0.271)/0.029
	0.259		0.414

Theoretical note:

The value of the z-score indicates how many standard deviations the score is away from the mean. If a z-score is equal to 0, it is on the mean.

A positive z-score indicates the score is higher than the mean average. For example, if a z-score is equal to +1, it is 1 standard deviation above the mean.

A negative z-score reveals the score is below the mean average. For example, if a z-score is equal to -2, it is 2 standard deviations below the mean.

For Example: Normal variable with Mean=1010 is converted in to Z score on the right side.