Question

Batter up! In baseball, a player’s batting average is the proportion of times the player gets a hit out of his total number of times at-bat. The distribution of batting averages in a recent season for Major League Baseball players with at least 100 plate appearances can be modeled by a Normal distribution with mean $\mu =0.261$ and standard deviation $\sigma =0.034$Sketch the Normal density curve. Label the mean and the points that are $1,2,$ and $3$ standard deviations from the mean.

Short answer

Mean,$\mu =0.261$

Standard deviation,$\sigma =0.034$

Step-by-step solution

Given information

Mean,$\mu =0.261$

Standard deviation,$\sigma =0.034$

Calculation

In a normal distribution, $68$percent of the data lies within $1$ standard deviation of the mean.

In a normal distribution, $95\%$of the data lies within $2$standard deviations of the mean.

A normal distribution has $99.7\%$of its data inside $1$ standard deviation of the mean.

Then

The general Normal density graph is represented as:

Now,

On both sides of the mean, find the standard values.

From left to the mean,

$3$standard deviation:

$\mu -3\sigma =0.261-3(0.034)=0.159$

$2$standard deviation:

$\mu -2\sigma =0.261-2(0.034)=0.193$

$1$standard deviation:

$\mu -\sigma =0.261-0.034=0.227$

From right to the mean,

$1$standard deviation:

$\mu +\sigma =0.261+0.034=0.295$

$2$standard deviations:

$\mu +2\sigma =0.261+2(0.034)=0.329$

$3$standard deviation:

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

To show the Normal density curve, write the estimated values in the figure below: