Pharmacy exam 2018-11-19

Degrees: Pharmacy, Biotechnology
Date: November 19, 2018

Question 1

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?

Solution

Let $A$ and $B$ the events of being inmune to virus $A$ and $B$ respectively.

$P(A\cap B)=0.09$
$P(\overline B|A)=0.1818$.
The events are dependent.

Question 2

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.

Solution

Let $X$ be the systolic pressure, $X\sim N(12.5788, 2.2815)$.
$P(10\leq X\leq 13)=0.3288$ and there are $789.0501$ persons with a systolic pressure between 11 and 13 mmHg.
300 males have a systolic pressure above 15.2 mmHg.

Question 3

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.

Solution

Let $X$ be the number of persons that enter in the ICU in a day. $X\sim P(4.2)$ and $P(X>3)=0.6046$.
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 $P(Y>1)=0.4513$.

Question 4

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.

$$ \begin{array}{ccc} \mbox{First hospital} (A) & \quad & \mbox{Second hospital} (B) \newline \begin{array}{|l|r|r|} \hline & \mbox{Test} + & \mbox{Test} - \newline \hline \mbox{Infected} & 705 & 65 \newline \hline \mbox{Non infected} & 120 & 4110 \newline \hline \end{array} & & \begin{array}{|l|r|r|} \hline & \mbox{Test} + & \mbox{Test} - \newline \hline \mbox{Infected} & 1710 & 70 \newline \hline \mbox{Non infected} & 415 & 7805 \newline \hline \end{array} \end{array} $$

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?

Solution

$P(\mbox{Correct diagnotic})=0.963$.
$PPV_A=0.8545$.
$NPV_B=0.9911$.
$NPV_A=0.9844$ 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.

Last updated on Sep 15, 2020