Question

question 1 of 10
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 z - score formula 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 $P(X\lt0.200)$, which is equivalent to $P(Z\lt - 1.7941)$ in the standard - normal distribution. Looking up the value in the standard normal table (or using a calculator with a normal - distribution function), we find that $P(Z\lt - 1.7941)\approx0.0365$.

Step3: Convert to percentage

To convert the proportion to a percentage, we multiply by 100. So the percentage is $0.0365\times100 = 3.6500\%$.

Answer:

3.6500%