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

Question
Random variables
asked 2020-10-25
A normal population has mean \(\displaystyle\mu={20}\) and standard deviation \(\displaystyle\sigma={4}\).
What proportion of the population is less than 18?

Answers (1)

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:
\(\displaystyle{P}{\left({X}{<}{18}\right)}={P}{\left({\frac{{{x}-\mu}}{{\sigma}}}{<}{\frac{{{18}-{20}}}{{{4}}}}\right)}\)</span>
\(\displaystyle={P}{\left({Z}{<}-{0.5}\right)}={0.3085}\)</span> (Using standard normal table)
Thus, the required proportion is 0.3085.
0

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