Question

When electing an office bearer in a company the votes are normally distributed with mean μ and standard deviation σ. More than 85 people voting is 5%. Less than 27 people voting is 15%. Find the mean μ and standard deviation σ.

Answer 1

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

$P(x>85)=1-P(z<z_1=\frac{85-\mu}{\sigma})=0.05$

$P(x<27)=P(z<z_2=\frac{27-\mu}{\sigma})=0.15$

From z-table:

$z_1=1.645$

$z_2=-2.17$

$\frac{85-\mu}{\sigma}=1.645$

$\frac{27-\mu}{\sigma}=-2.17$

$\mu=85-1.645\sigma$

$27-85+1.645\sigma=-2.17\sigma$

$3.81\sigma=58$

$\sigma=58/3.81=15.22$

$\mu=85-1.645\cdot 15.22=60$