normal distribution and/ or probability distribution

Question

If a person tests positive for Tuberculosis (TB) and starts treatment immediately, it takes on average 6 months (183 days, with a standard deviation of 14 days) of treatment until they are completely TB free. To receive the positive TB test individuals may go for the tuberculin skin test which takes on average 2.5 days to receive these test results with a standard deviation of 0.5 days, after which they are immediately given TB treatment. 

10. If we are interested in the total time until a person is TB free from the time of the TB test, what is the mean of the distribution that describes this time from TB test until a TB-free status? 

11. If we are interested in the total time until a person is TB free from the time of the TB test, what is the variance of the distribution that describes this time from TB test until a TB-free status? 

12. If we are interested in the total time until a person is TB free from the time of the TB test – what is the probability that it takes more than 200 days until someone is completely TB free? 

Please round all numeric answers to three 

Answer 1

TB test until results = 2.5 days mean 0.5 days SD

TB treatment until cured = 183 days mean 14 days SD

combining independent random variables;

mean of sum = sum of means

variance of sum = sum of variances

mean TB test until cured = sum of means = 2.5 + 183 = 185.5

variance TB test until cured = (0.50)^2 + (14)^2 = 196.25

SD = 14.01

x = number of days from test until cured --- want P(x > 200)

use z = x - mean / SD, where mean = 185.5 and SD = 14.01

P(x > 200) = P(z > (200-185.5)/14.01) = P(z > 1.035) = 1 - P(z < 1.035) = 0.85