Question

A random variable is normally distributed with a mean of 20 and a standard deviation of 4. If an observation is randomly selected from the​ distribution,
a. What value will be exceeded 25​% of the​ time?
b. What value will be exceeded ​85% of the​ time?
c. Determine two values of which the smaller has 10​% of the values below it and the larger has 10​% of the values above it.
d. What value will 15​% of the observations be​ below?

Answer 1

We remind that for the normally distributed random variable $X$ with a mean of 20 and a standard deviation of 4 one has (see e.g., https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm): $P(a\leq X\leq b)=\frac{1}{4\sqrt{2\pi}}\int_a^be^{-\frac12(\frac{x-20}{4})^2}dx$ . For computations we used the following program in Anaconda (https://www.anaconda.com/):

_{from scipy import integrate}

_{import numpy as np}

_import math

_{func = lambda x:(1/(4*math.sqrt(2)*math.sqrt(math.pi)))*math.exp(-1/2*((x-20)/4)*((x-20)/4))}

_{e = integrate.quad(func, a,b)}

Instead of a and b we inserted values that we are interested in.

(a) The aim is to find $\alpha$ : $P(\alpha<X)=0.25$. We substitute different values of $\alpha$ in:

$\frac{1}{4\sqrt{2\pi}}\int_{\alpha}^{+\infty}e^{-\frac12(\frac{x-20}{4})^2}dx$ and get: $\alpha\approx22.70$ .

(b) We look for $\alpha$ such that: $P(\alpha<X)=0.85$. By substituting different choices of $\alpha$ we get: $\alpha\approx15.85$ .

(c) The aim is to find such $\alpha$ that: $P(\alpha<X)=0.9$ . We get: $\alpha\approx14.87$ . Now we look for $\beta$ satisfying $P(X<\beta)=0.9$. We find that $\beta\approx25.13$.

(d) The aim is to find $\alpha$ : $P(X<\alpha)=0.15$ . We get: $\alpha\approx15.85$.