Question

in baseball, a players batting average is the proportion of times the player gets a hit out of the 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 $sigma = 0.034$. what percent of players have batting averages between 0.250 and 0.300? round your answer to 4 decimal places and then convert to a percentage.

Explanation:

Step1: Calculate z - scores

The z - score formula is $z=\frac{x-\mu}{\sigma}$.
For $x = 0.250$, $z_1=\frac{0.250 - 0.261}{0.034}=\frac{- 0.011}{0.034}\approx - 0.3235$.
For $x = 0.300$, $z_2=\frac{0.300 - 0.261}{0.034}=\frac{0.039}{0.034}\approx1.1471$.

Step2: Use the standard normal table

We want to find $P(-0.3235From the standard - normal table, $P(Z < 1.1471)\approx0.8749$ and $P(Z < - 0.3235)\approx0.3735$.
Then $P(-0.3235$P(-0.3235

Step3: Convert to percentage

To convert the proportion to a percentage, we multiply by 100.
$0.5014\times100 = 50.14\%$.

Answer:

50.14%