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 $sigma = 0.034$. a player with a batting average below 0.200 is at risk of sitting on the bench during important games. about what percent of players are at risk? round to 4 decimal places then convert to a percent.

Explanation:

Step1: Calculate the z - score

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $x = 0.200$, $\mu=0.261$, and $\sigma = 0.034$. So, $z=\frac{0.200 - 0.261}{0.034}=\frac{- 0.061}{0.034}\approx - 1.7941$.

Step2: Find the proportion from the standard normal table

We want to find $P(X<0.200)$, which is equivalent to $P(Z < - 1.7941)$ in the standard - normal distribution. Looking up the value in the standard normal table (or using a calculator with a normal - distribution function), $P(Z < - 1.7941)\approx0.0364$.

Step3: Convert to percentage

To convert the proportion to a percentage, we multiply by 100. So the percentage is $0.0364\times100 = 3.64\%$.

Answer:

3.64%