Question

# A normal population has mean \mu = 20 and standard deviation \sigma = 4. What proportion of the population is less than 18?

Random variables
A normal population has mean $$\displaystyle\mu={20}$$ and standard deviation $$\displaystyle\sigma={4}$$.
What proportion of the population is less than 18?

2020-10-26

Step 1
From the provided information,
Mean $$\displaystyle{\left(\mu\right)}={20}$$
Standard deviation $$\displaystyle{\left(\sigma\right)}={4}$$
$$\displaystyle{X}\sim{N}{\left({20},{4}\right)}$$
Step 2
The required proportion of the population which is less than 18 can be obtained as:
$$\displaystyle{P}{\left({X}{<}{18}\right)}={P}{\left({\frac{{{x}-\mu}}{{\sigma}}}{<}{\frac{{{18}-{20}}}{{{4}}}}\right)}$$
$$\displaystyle={P}{\left({Z}{<}-{0.5}\right)}={0.3085}$$ (Using standard normal table)
Thus, the required proportion is 0.3085.