Answer: The score p81 that separates the bottom 81% from the top 19% is 80.12
Step by step:
First, calculate the "z" value. Z is a normally distributed random variable with 0 mean and standard deviation 1. The score value corresponding to the desired percentile p81 can be determined from a z value as follows:
[tex]z=\frac{s-\mu}{\sigma}\\z_{p81}=\frac{p_{81}-70}{11.5}\\\implies p_{81}=z_{p81}\cdot 11.5+70[/tex]
We use a z-table (check online) to find the z value for the 81-st percentile. I found [tex]z_{p81}=0.88[/tex] and so we use that value to calculate the score for the percentile:
[tex]p_{81}=0.88\cdot 11.5+70=80.12[/tex]
The score p81 that separates the bottom 81% from the top 19% is 80.12
