Degrees: Pharmacy, Biotechnology
Date: November 19, 2018
In a population that is exposed to two viruses strains $A$ and $B$ it is known that 2% of persons are immune only to virus $A$ and 4% are immune only to virus $B$. On the other hand it is known tha 91% of the population would be infected by some of the two viruses.
- What is the probability that a person is immune to the two viruses?
- What is the probability that a person immune to virus $A$ is infected by virus $B$?
- Are dependent the events of being immune to the two viruses?
1. $P(A\cap B)=0.09$
2. $P(\overline B|A)=0.1818$.
3. The events are dependent.
In a study about the blood pressure the systolic pressure of 2400 males older than 18 was measured. It was observed that 640 had a pressure greater than 14 mmHg and 1450 had between 10 and 14 mmHg. Assuming that the systolic pressure in males older than 18 is normally distributed,
- Compute the mean and the standard deviation.
- Compute how many males had a systolic pressure between 11 and 13 mmHg.
- Compute the value of the systolic pressure such that there was 300 males with a systolic pressure above it.
2. $P(10\leq X\leq 13)=0.3288$ and there are $789.0501$ persons with a systolic pressure between 11 and 13 mmHg.
3. 300 males have a systolic pressure above 15.2 mmHg.
The average number of people that enters the intensive care unit of a hospital in an 8-hours shift is $1.4$.
- Compute the probability that a day enter more than 3 persons in the ICU.
- Compute the probability that in a week there are more than one day with less than 3 persons entering the ICU.
2. Let $Y$ be the number of days in a week with less than 3 persons entering the ICU. $Y\sim B(7,0.2102)$ and $P(Y>1)=0.4513$.
Two hospitals use different tests $A$ and $B$ to detect a streptococcal infection. The tables below show the results of applying these tests in each hospital during the last year.
- Compute the probability of a correct diagnostic with test $A$.
- Compute the positive predicted value of test $A$.
- Compute the negative predicted value of test $B$.
- How can these tests be combined to reduce the risk of wrong diagnosis?
4. $NPV_A=0.9844$ and $PPV_B=0.8047$. Since $B$ has the higher negative predicted value and $A$ the higher positive predicted value, it is better to use test $B$ first to rule out the infection and then apply test $A$ only to individuals with a positive outome in test $B$, to confirm the infection.