Physiotherapy exam 2021-05-05
Degrees: Physiotherapy
Date: May 5, 2021
Probability and random variables
Question 1
The average number of injuries in an international tennis tournament is 2.
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Compute the probability that in an international tennis tournament there are more than 2 injuries.
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If a tennis circuit has 6 international tournaments, what is the probability that there are no injuries in some of them?
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Let $X$ be the number of injuries in a tournament, then $X\sim P(2)$ and $P(X>2)=0.3233$.
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Let $Y$ be the number of tournaments in the tennis circuit with no injuries, then $Y\sim B(6,0.1353)$ and $P(Y>0)=0.5821$.
Question 2
The tables below corresponds to two tests $A$ and $B$ to detect an injury that have been applied to the same sample.
$$ \begin{array}{lcc} \hline \mbox{Test A} & \mbox{Injury} & \mbox{No injury} \newline \mbox{Outcome } + & 87 & 14 \newline \mbox{Outcome }- & 33 & 866 \newline \hline \end{array} \qquad \begin{array}{lcc} \hline \mbox{Test B}& \mbox{Injury} & \mbox{No injury} \newline \mbox{Outcome }+ & 104 & 115 \newline \mbox{Outcome }- & 16 & 765 \newline \hline \end{array} $$
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Which test is more sensitive? Which one is more specific?
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According to the predictive values, which test is better to diagnose the injury? Which one is better to rule out the injury?
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Assuming that both tests are independent, what is the probability of getting a right diagnose with both tests if we apply both tests to a healthy person?
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Assuming that both tests are independent, what is the probability of getting at least a positive outcome if we apply both tests to a random person?
Let $D$ the event of suffering the injury, and $+$ and $-$ the events of getting a positive and a negative outcome in the test, respectively.
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Test $A$: sen = $0.725$ and spe = $0.9841$.
Test $B$: sen = $0.8667$ and spe = $0.8693$.
Thus, test $A$ is more specific and test $B$ is more sensitive. -
Test $A$: PPV = $0.8614$ and NPV = $0.9633$.
Test $B$: PPV = $0.4749$ and NPV = $0.9795$.
Thus, test $A$ is better to diagnose the injury and test $B$ is better to rule out the injury. -
$P(-_A\cap -_B | \overline{D}) = 0.8555$.
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$P(+_A\cup +_B) = 0.2979$.
Question 3
A study tries to determine the effect of a low fat diet in the lifetime of rats. The rats where divided into two groups, one with a normal diet and another with a low fat diet. It is assumed that the lifetimes of both groups are normally distributed with the same variance but different mean. If 20% of rats with normal diet lived more than 12 months, 5% less than 8 months, and 85% of rats with low fat diet lived more than 11 months,
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Compute the means and the standard deviation of the lifetime of rats following a normal diet and a low fat diet?
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If 40% of the rats were under a normal diet, and 60% of rats under a low fat diet, what is the probability that a random rat die before 9 months?
Let $X$ be the life time of a random rat, and let $X_1$ and $X_2$ be the lifetime of rats with a normal diet and a low fat diet respectively,
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$\mu_1=10.6461$ months, $\mu_2=12.6673$ months and $s=1.6087$ months.
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$P(X<9)=0.068$.