Question

To qualify for a police academy, applicants are given a test of physical fitness. The scores are normally distributed with a mean of 65 and a standard deviation of 14. If only the top 20% are selected, find the cutoff score.

Answer 1

mean of 65

standard deviation of 14

top 20% are selected

We need to use the Z-Score table to find the value of Z that represents 80%

In this case, the closest value would be 0.84 as can be seen in the picture below...

Now, we can use this equation...

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

where,

Z=0.84,

x is the cutoff score,

μ is the mean, and

σ is the standard deviation

Now, we replace those values and solve for x, as it follows,

$0.84=\frac{x-65}{14}$

$\begin{gathered} \frac{x-65}{14}=0.84 \\ \frac{14\left(x-65\right)}{14}=0.84\cdot\: 14 \\ x-65=11.76 \\ x-65+65=11.76+65 \\ x=76.76 \end{gathered}$

Therefore, the cutoff score is 76.76