91Ó°ÊÓ

Mean,$\mathrm{μ}=0.261$

Standard deviation,$\mathrm{σ}=0.034$

Graph of normal probability- We can claim that the distribution is roughly Normal if the Normal probability plot has a linear structure.

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 $3\mathrm{σ}$above the mean.

$\mathrm{μ}+3\mathrm{σ}=0.261+3(0.034)=0.363$

According to $68−95−99.7$rule:

$99.7\%$of the data values lie within $3\mathrm{σ}$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 $3\mathrm{σ}$from the mean.

We also know that

Around the mean, the normal distribution is symmetric.

That implies

$0.15\%$of the data values are more than $3\mathrm{σ}$below the mean.

And

$0.15\%$of the data values are more than $3\mathrm{σ}$above the mean.

Therefore,

With $100$plate appearances, $0.15$percent of Major League Basketball players had batting averages of $0.363$or better.

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 $\mathrm{σ}$($1$standard deviation) below the mean.

$\mathrm{μ}−\mathrm{σ}=0.261−0.034=0.227$

According to $68−95−99.7$rule:

$68\%$of the data values lie within $\mathrm{σ}$($1$standard deviation) of the mean.

Although,

Data values in total are $100\%$

Then

$100\%−68\%=32\%$

$32\%$of the data values lie more than $\mathrm{σ}$($1$standard deviation) from the mean.

We also know that

Around the mean, the normal distribution is symmetric.

That implies

$16\%$of the data values are more than $\mathrm{σ}$($1$standard deviation) above the mean.

And

$16\%$of the data values are more than $\mathrm{σ}$($1$standard deviation) below the mean.

The data value represented by the $Xth$percentile includes $x\%$of the data values below it.

That implies

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

Thus,

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