Question

Baseball salaries Did Ryan Madson’s , who was paid $\$1,400,000$, have a high salary or a low salary compared with the rest of the team? Justify your answer by calculating and interpreting Madson’s percentile and z-score.

Short answer

Here $48.28\%$of player's salary on the World Champion $2008$ Philadelphia Phillies baseball team is less than M's salary and percentile was $50\%$

Step-by-step solution

Step 1. Given

Step 2. Concept

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

Step 3. Calculation

M's annual compensation was $\$1,400,000$in $2008$. M's income is at the percentile since there are $14$salaries that are smaller than his:

$$\begin{gathered} Percentile=\frac{14}{29} \\ \approx 0.4828 \\ =48.28\% \end{gathered}$$

This means that M's salary is $48.28\%$less than that of the World Champion $2008$Philadelphia Phillies baseball team. It's worth noting that, because M's income was the median, we may argue that his percentile would be $50\%$

The following is the standardized

$z=\frac{x-\mu }{\sigma }$

salary score for M:

$$\begin{gathered} =\frac{1,400,000-3,388,617}{3,767,484} \\ \approx -0.53 \end{gathered}$$

Because M's percentile was $50\%$, he had a typical income relative to the rest of the team, but because his z-score was negative, suggesting a salary below the mean team salary, several players made significantly more than he did.

Therefore $48.28\%$of players' salary on the World Champion 2008 Philadelphia Phillies baseball team is less than M's salary and the percentile was $50\%$