Chapter 2.2, Problem 52E

(a)

To determine

Percentage of the players with 100 plate appearances and batting averages of 0.363 and more.

Answer to Problem 52E

0.15% of the Major League Basketball players with 100 plate appearances had batting averages of 0.363 and higher.

Explanation of Solution

Given information:

Mean, $\mu =0.261$

Standard deviation, $\sigma =0.034$

According to 68 − 95 − 99.7 rule:

68% of the data of a normal distribution lies with 1 standard deviation from the mean.

95% of the data of a normal distribution lies with 2 standard deviation from the mean.

99.7% of the data of a normal distribution lies with 1 standard deviation from the mean.

Then

The general Normal density graph is represented as:

  

Note that

0.363 lies $3\sigma$ above the mean.

  $\mu +3\sigma =0.261+3\left(0.034\right)=0.363$

According to 68 − 95 − 99.7 rule:

99.7% of the data values lie within $3\sigma$ of the mean.

Although,

Data values in total are 100%.

Then

  $100\%-99.7\%=0.30\%$

0.30% of the data values lie more than $3\sigma$ from the mean.

We also know that

Normal distribution is symmetric about the mean.

That implies

0.15% of the data values are more than $3\sigma$ below the mean.

And

0.15% of the data values are more than $3\sigma$ above the mean.

Therefore,

0.15% of the Major League Basketball players with 100 plate appearances had batting averages of 0.363 and higher.

(b)

To determine

Percentile of the player in the distribution with batting average of 0.227.

Answer to Problem 52E

The player with batting average of 0.227 is at 16^th percentile.

Explanation of Solution

Given information:

Mean, $\mu =0.261$

Standard deviation, $\sigma =0.034$

According to 68 − 95 − 99.7 rule:

68% of the data of a normal distribution lies with 1 standard deviation from the mean.

95% of the data of a normal distribution lies with 2 standard deviation from the mean.

99.7% of the data of a normal distribution lies with 1 standard deviation from the mean.

Then

The general Normal density graph is represented as:

  

Note that

0.227 lies $\sigma$ (1 standard deviation) below the mean.

  $\mu -\sigma =0.261-0.034=0.227$

According to 68 − 95 − 99.7 rule:

68% of the data values lie within $\sigma$ (1 standard deviation) of the mean.

Although,

Data values in total are 100%.

Then

  $100\%-68\%=32\%$

32% of the data values lie more than $\sigma$ (1 standard deviation) from the mean.

We also know that

Normal distribution is symmetric about the mean.

That implies

16% of the data values are more than $\sigma$ (1 standard deviation) above the mean.

And

16% of the data values are more than $\sigma$ (1 standard deviation) below the mean.

The data value represented by the x^th percentile includes x% of the data values below it.

That implies

16% of the players have batting average of 0.227 or less.

Thus,

The player with batting average of 0.227 is at 16^th percentile.