Answer to Question #159763 in Statistics and Probability for nalinda linda

Question

Amira knows that the time it takes her to commute to work is approximately normally distributed with a mean of 45 minutes and a standard deviation of 3 minutes. What time must she leave home in the morning so that she is 95% sure of arriving at work by 9.00am? 

Answer 1

Assume that random variable $X$ denotes a time that Amira needs to arive at work.

We need to find such a minimum number $\alpha$ that $P(X\leq\alpha)=0.95$. We remind that the density fo the normal distribution with parameters $\mu=45$ and $\sigma=3$ is: $p(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac12(\frac{x-\mu}{\sigma})^2}=\frac{1}{3\sqrt{2\pi}}e^{-\frac12(\frac{x-45}{3})^2}$ . Thus, $P(X\leq\alpha)=\int_{-\infty}^{\alpha}\frac{1}{3\sqrt{2\pi}}e^{-\frac12(\frac{x-45}{3})^2}dx$. The latter yields $0.95$ in case $\alpha=49.94$. We used the following code in Anaconda to get the result:

_{from scipy import integrate}

_{import numpy as np}

_import math

_{func = lambda x:(1/(3*math.sqrt(2)*math.sqrt(math.pi)))*math.exp(-1/2*((x-45)/3)*((x-45)/3))}

_{e = integrate.quad(func, -np.inf, 49.94)}

_print(e)

We subsituted different values in the command $integrate.quad$ to get the result

Thus, it is enough to leave at 8.10 (more precisely, at 8.10 and 6 seconds) to arrive at work till 9.00am.