Pharmacy exam 2020-11-23

Degrees: Pharmacy, Biotechnology
Date: November 23, 2020

Question 1

A test to detect the COVID19 was applied to 850 persons infected by COVID19 with a positive outcome in 800 of them, and it was also applied to 9150 non-infected persons with a positive outcome in 10% of them.

Compute the sensitivity and the specificity of the test.
Compute the positive and the negative predictive values and interpret them.
Compute the probability of a correct diagnostic.

Solution

Let $D$ the event corresponding to suffering COVID19 and $+$ and $-$the events corresponding to get a positive and a negative outcome respectively.

The sensitivity is $P(+|D) = 0.9412$ and specificity $P(-|\overline D) = 0.9$.
Positive predictive value $P(D|+) = 0.4665$ and negative predictive value $P(\overline D|-) = 0.994$. As the positive predictive value is less than 0.5 we can not use this test to confirm COVID19, but we can use it to rule it out with a strong confidence since the negative predictive value is pretty close to 1.
$P(D\cap +) + P(\overline D\cap -) = 0.9035$.

Question 2

A newborn baby affected by Moebius syndrome blinks, on average, twice a minute.

Compute the probability that a newborn blinks twice in half a minute.
In a hospital five children have been born with Moebius syndrome. Compute the probability that at least 3 of them blink in their first minute of life.
In which distribution is more representative the mean, in the number of times that a newborn blinks in a minute or in the number of times that a newborn blinks in half a minute?

Solution

Let $X$ be the number of times that a newborn blinks in half a minute, then $X\sim P(1)$ and $P(X=2)=0.1839$.
Let $Y$ be the number of newborns that blink in their first minute of life in a sample of 5 newborns, then $Y\sim B(5,0.8647)$ and $P(Y\geq 3)=0.98$.
Let $Z$ be the number of times that a newborn blinks in a minute, then $cv_z = 0.7071$ and $cv_x = 1$. Thus, the mean of $Z$ represents better since its coefficient of variation is smaller.

Question 3

The prolactin level in pregnant and non-pregnant females follows anormal distribution with different means but with the same variance.When the prolactin levels exceed 15 ng/ml, females secrete milk through their mammary glands. It is known that 95% of pregnant females secrete milk but only 1% of non-pregnant females secret milk.

If the median of the prolactin level in pregnant females is 16 ng/ml, what are the means and the standard deviation of the prolactin level in both populations?
Compute the percentage of pregnant females with a prolactin level between 15.5 and 17 ng/ml.
Compute the prolactin level such that 20% of pregnant females are above that level.

Solution

Let $X$ and $Y$ be the prolactin levels in pregnant and non-pregnant females respectively.

$\mu_x=16$ ng/ml, $\mu_y=13.5857$ ng/ml and $\sigma=0.608$ ng/ml.
$P(15.5<X<17) = 0.7446$, so 74.4583% of pregnant females.
$P_{80} = 16.5117$ ng/ml.

Question 4

An organism has the same chance of being infected by a virus and a bacteria. At the same time, the probability of being infected by a virus doubles when the organism has been previously infected by a bacteria. On the other hand, the probability of being infected by no pathogen (neither virus nor bacteria) is $0.52$.

What is the probability of being infected by a virus and a bacteria at the same time?
What is the probability of being infected by a bacteria if it has been infected by a virus?
What is the probability of being infected only by a virus?
Are the events of being infected by a virus an a bacteria independent?

Solution

Let $V$ and $B$ the events corresponding to be infected by a virus and a bacteria respectively.

$P(V\cap B) = 0.32$.
$P(B|V) = 0.8$.
$P(V\cap \overline B) = 0.08$.
The events are dependents since $P(V) = 0.4 \neq 0.8 = P(V|B)$.

Last updated on Jan 23, 2021