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 to 4 decimal places and then convert to a percentage.)

Explanation:

Step1: Calculate the z - scores

The formula for the z - score 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: Find the probabilities

We use the standard normal distribution table (or a calculator with a normal - distribution function).
The probability \(P(Z\lt z_1)\) where \(z_1=-0.3235\) is \(P(Z\lt - 0.3235)\approx0.3735\).
The probability \(P(Z\lt z_2)\) where \(z_2 = 1.1471\) is \(P(Z\lt1.1471)\approx0.8749\).

Step3: Calculate the probability between the two z - scores

\(P(-0.3235\lt Z\lt1.1471)=P(Z\lt1.1471)-P(Z\lt - 0.3235)\).
\(P(-0.3235\lt Z\lt1.1471)=0.8749 - 0.3735=0.5014\).

Answer:

\(50.14\%\)