Question

A sampling of a group of student athlete's weights are normally distributed. If Abe'sweight of 181 lbs. has a z-score of 3 and Carlie's weight of 116 lbs. has a z-score of -2,what is the average weight of this group?

Answer 1

$\begin{gathered} \text{The Z-score is given by the equation:} \\ Z=\frac{x-\mu}{\sigma} \\ \mu\colon\text{ mean} \\ \sigma\colon s\tan dard\text{ deviation} \\ x\colon\text{ observed value} \end{gathered}$$\begin{gathered} \text{For Abe:} \\ 3=\frac{181-\mu}{\sigma} \\ 3\sigma=181-\mu \\ \sigma=\frac{181-\mu}{3}\ldots A) \end{gathered}$$\begin{gathered} \text{For Carlie's} \\ -2=\frac{116-\mu}{\sigma} \\ -2\sigma=116-\mu \\ \sigma=\frac{116-\mu}{-2}\ldots B) \end{gathered}$$\begin{gathered} \text{Equating equation A and B} \\ \frac{181-\mu}{3}=\frac{116-\mu}{-2} \\ -2(181-\mu)=3(116-\mu) \\ -362+2\mu=348-3\mu \\ 2\mu+3\mu=348+362 \\ 5\mu=710 \\ \mu=\frac{710}{5} \\ \mu=142 \end{gathered}$

Hence, the average weight of the group is 142 lbs