Question

Batter up! Refer to Exercise $48$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?

Short answer

Around 3.67% of the players have the risk of sitting on the bench during important games.

Step-by-step solution

Given information

Batting average value,$x=0.200$

Mean,$\mu =0.261$

Standard deviation,$\sigma =0.034$

Concept

The formula used:$z=\frac{x-\mu }{\sigma }$

Calculation

Calculate the $Z-$ score,

$z=\frac{x-\mu }{\sigma }=\frac{0.200-0.261}{0.034}\approx -1.79x$

To find the equivalent probability, use the normal probability table in the appendix.

See the usual normal probability table for $P(z<-1.79)$and the row that starts with$-1.7$ and the column that starts with.09.

$$\begin{gathered} P(x<0.200)=P(z<-1.79) \\ =1-P(z>-1.79) \\ =1-0.9633 \\ =0.0367 \\ =3.67\% \end{gathered}$$

Therefore,

Around $3.67\%$ of the players have a batting average below $0.200$ and are at risk of sitting on the bench during important games.