Question

if a person tests positive for Tuberculosis and starts treatment immediately it takes on avarage 6 months ( 183 days with a standard deviation of 14 days) of treatment until they are completely TB free .what is the probability that a person will be healthy and TB free in less than 160 days

Answer 1

Let $X=$ the length of treatment in days: $X\sim N(\mu, \sigma^2 )$

Then $Z=\dfrac{X-\mu}{\sigma}\sim N(0, 1)$

Given $\mu=183\ days, \sigma=14\ days$

$$P(X<160)=P(Z<\dfrac{160-183}{14})\approx$$

$$\approx P(Z<-1.642857)\approx0.050206$$

The probability that a person will be healthy and TB free in less than 160 days is approximately $5.0206\ \%.$