in baseball, a players batting average is the proportion of times the player gets a hit out of the total…

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.

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.

Answer

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 $P(-0.3235<Z<1.1471)$. From the standard - normal table, $P(Z < 1.1471)\approx0.8749$ and $P(Z < - 0.3235)\approx0.3735$. Then $P(-0.3235<Z<1.1471)=P(Z < 1.1471)-P(Z < - 0.3235)$. $P(-0.3235<Z<1.1471)=0.8749 - 0.3735=0.5014$.

Step3: Convert to percentage

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

Answer:

50.14%