Pharmacy exam 2022-01-17

Degrees: Pharmacy, Biotechnology
Date: January 17, 2022

Question 1

A diagnostic test for a disease with a prevalence of 10% has a positive predictive value of 40% and negative predictive value of 95%.

  1. Compute the sensitivity and the specificity of the test.

  2. Compute the probability of a right diagnose.

  3. What must be the minimum sensitivity of the test to be able to diagnose the disease?

  1. Sensitivity $P(+|D)=0.571$ and specificity $P(-|\overline D)=0.9048$.

  2. $P(\mbox{Right diagnose}) = P(D \cap +) + P(\overline D \cap -) = 0.8714$.

  3. Minimum sensitivity to diagnose the disease $P(+|D)=0.857$.

Question 2

To study the effectiveness of two antigen tests for the COVID both tests have been applied to a sample of 100 persons. The table below shows the results:

$$ \begin{array}{ccr} \hline \mbox{Test $A$} & \mbox{Test $B$} & \mbox{Num persons}\newline \mbox{+} & \mbox{+} & 8\newline \mbox{+} & \mbox{-} & 2\newline \mbox{-} & \mbox{+} & 3\newline \mbox{-} & \mbox{-} & 87\newline \hline \end{array} $$

Define the following events and compute its probabilities:

  1. Get a $+$ in the test $A$.

  2. Get a $+$ in the test $A$ and a $-$ in the test $B$.

  3. Get a $+$ in some of the two tests.

  4. Get different results in the two tests.

  5. Get the same result in the two tests.

  6. Get a $+$ in the test $B$ if we got a $+$ in the test $A$.

Are the outcomes of the two tests independent?

Let $A$ and $B$ the events of getting positive outcomes in the tests $A$ and $B$ respectively.

  1. $P(A)=0.1$.

  2. $P(A\cap \overline B)=0.02$.

  3. $P(A\cup B) = 0.13$.

  4. $P(A\cap \overline B) + P(\overline A \cap B) = 0.05$.

  5. $P(A\cap B) + P(\overline A \cap \overline B)= 0.95$.

  6. $P(B|A) = 0.8$.

As $P(B|A)\neq P(B)$ the events are dependent.

Question 3

It is known that the life of a battery for a peacemaker follows a normal distribution. It has been observed that 20% of the batteries last more than 15 years, while 10% last less than 12 years.

  1. Compute the mean and the standard deviation of the battery life.

  2. Compute the fourth decile of the battery life.

  3. If we take a sample of 5 batteries, what is the probability that more than half of them last between 13 and 14 years?

  4. If we take a sample of 100 batteries, what is the probability that some of them last less than 11 years?

Let $X$ be the duration of a battery. Then $X\sim N(\mu,\sigma)$.

  1. $\mu = 13.8108$ years and $\sigma = 1.413$ years.

  2. $D_4 = 13.4528$ years.

  3. Let $Y$ be the number of batteries lasting between 13 and 14 years in a sample of 5 batteries. Then $Y\sim B(5,0.2702)$ and $P(Y>2.5)=0.0209$.

  4. Let $U$ be the number of batteries lasting less than 11 years in a sample of 100 batteries. Then $U\sim B(100, 0.0233)\approx P(2.3335)$ and $P(U\geq 1)=0.903$.

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Last updated on Jan 23, 2022

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