Question

A recent survey shows that people are retiring at the mean age of 60 and standard deviation 2.5. What proportion of people are retiring over the age of 55? What is the median retirement age? What is the percentile of the retirement age of 65? What retiring age corresponds to the 16th percentile?

Answer 1

Answer:

a) 60 years

b) 97.72th percentile

c) 52.65 years

Explanation:

Given that:

mean (μ) = 60 years and standard deviation (σ) = 2.5. The z score is given by the equation:

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

a) the median correspond to 50th percentile.

The 50th percentile from the z table corresponds to a z score of 0. Using the z score equation:

$z=\frac{x-\mu}{\sigma}\\0=\frac{x-60}{2.5}\\ x-60 = 0\\x=60$

The median retirement age is 60 years

b) When age is 65 years.

$z=\frac{x-\mu}{\sigma}=\frac{65-60}{2.5} =2$

From the normal distribution table: P(x < 65) = P(z <  2) = 0.9772 = 97.72th percentile

c) The 16th percentile from the z table corresponds to a z score of -2.94. Using the z score equation:

$z=\frac{x-\mu}{\sigma}\\-2.94=\frac{x-60}{2.5}\\ x-60 = -7.35\\x=60-7.35=52.65$

An age of 52.65 corresponds to 16th percentile