Pharmacy exam 2019-12-16

Degrees: Pharmacy, Biotechnology
Date: December 16, 2019

Question 1

The table below summarizes the time (in minutes) required to remove anesthesia after a surgery in a sample of 50 patients.

$\begin{array}{cr} Time & Patients \\ 10 - 30 & 2 \\ 30 - 45 & 11 \\ 45 - 60 & 18 \\ 60 - 90 & 9 \\ 90 - 120 & 8 \\ 120 - 180 & 2 \end{array}$

Are there some outliers in the sample?
Compute the mean. Is it representative?
If according to a postoperative protocol the 15% of patients that require more time to remove the anesthesia must be monitored, above what time should a patient be monitored?
If we apply a drug that is anesthesia antagonist, it is known that the time required to remove the anesthesia decreases a 25%. How will the time decrease affect the representativeness of the mean?
If it is known that another type of anesthesia $B$ has mean 50 minutes and standard deviation 15 minutes, what time is relatively greater, 70 minutes with this type of anesthesia or 60 minutes with the type $B$ ?.

Use the following sums for the computations:
$\sum x_{i} n_{i} = 3212.5$ min, $\sum x_{i}^{2} n_{i} = 249706.25$ min $^{2}$ ,
$\sum (x_{i} - \bar{x})^{3} n_{i} = 1400531.25$ min $^{3}$ y
$\sum (x_{i} - \bar{x})^{4} n_{i} = 143958437.7$ min $^{4}$ .

Solution

$Q_{1} = 44.3182$ , $Q_{3} = 81.6667$ , $I Q R = 37.3485$ , $f_{1} = - 11.7045$ and $f_{2} = 137.6894$ . Since the last class contains values above the upper fence, there could be outliers.
$\bar{x} = 64.25$ min, $s^{2} = 866.0625$ min $^{2}$ , $s = 29.4289$ min and $c v = 0.458$ . Thus the representativity of the mean is moderate.
$P_{85} = 99.375$ min.
Applying the linear transformation $y = 0.75 x$ , $\bar{y} = 48.1875$ min, $s_{y} = 22.0717$ min and $c v = 0.458$ . Thus the representativity of the mean is the same.
Standard score in first anesthesia: $z (70) = 0.1954$ . Standard score in anesthesia $B$ : $z (60) = 0.6667$ . Thus, 60 min with anesthesia $B$ is relatively greater.

Question 2

The table below summarizes the scores of a group of 10 students in three practical exams of Maths.

$\begin{array}{rrr} Exam 1 (X) & Exam 2 (Y) & Exam 3 (Z) \\ 5.5 & 3.2 & 5.0 \\ 7.5 & 6.5 & 2.0 \\ 2.5 & 4.0 & 1.0 \\ 6.0 & 4.0 & 6.0 \\ 8.0 & 7.5 & 6.0 \\ 4.0 & 3.5 & 1.0 \\ 7.0 & 5.5 & 4.0 \\ 9.5 & 10.0 & 9.0 \\ 10.0 & 9.5 & 8.0 \\ 1.0 & 3.0 & 0.5 \end{array}$

Which two scores are more linearly correlated?
Using linear models, what are the expected scores of the second and third exams for a student with a score $6.5$ in the first exam?

Use the following sums for the computations:
$\sum x_{i} = 61$ , $\sum y_{i} = 56.7$ , $\sum z_{i} = 42.5$ ,
$\sum x_{i}^{2} = 449$ , $\sum y_{i}^{2} = 382.49$ , $\sum z_{i}^{2} = 264.25$ ,
$\sum x_{i} y_{j} = 405.85$ , $\sum x_{i} z_{j} = 327$ , $\sum y_{j} z_{j} = 295$ .

Solution

$\bar{x} = 6.1$ , $s_{x}^{2} = 7.69$ , $\bar{y} = 5.67$ , $s_{y}^{2} = 6.1001$ , $\bar{z} = 4.25$ , $s_{z}^{2} = 8.3625$ , $s_{x y} = 5.998$ , $s_{x z} = 6.775$ , $s_{y z} = 5.4025$ , $r_{x y}^{2} = 0.7669$ , $r_{x z}^{2} = 0.7138$ and $r_{y z}^{2} = 0.5722$ . Thus, the two variables more linearly related are $X$ and $Y$ , since their coefficient of determination is greater.
Regression line of $Y$ on $X$ : $y = 0.9122 + 0.78 x$ and $y (6.5) = 5.982$ . Regression line of $Z$ on $X$ : $z = - 1.1242 + 0.881 x$ and $z (6.5) = 4.6024$ .

Question 3

To study the association between the osteoporosis and the gender a random sample of people between 65 and 70 years old was taken. The following table summarize the results

$\begin{array}{lcc} Osteoporosis & Not osteoporosis \\ Women & 480 & 2320 \\ Men & 255 & 1505 \end{array}$

Compute the prevalence of the osteoporosis in the population.
Compute the relative risk of osteoporosis in females with respect to males and interpret it.
Compute the odds ratio of osteoporosis in females with respect to males and interpret it.
Which of the two measures is most suitable to study the association between the osteoporosis and the gender?

Solution

Let $D$ be the event of suffering osteoporosis.

Prevalence: $P (D) = 0.1612$ .
$R R (D) = 1.1832$ . Thus, the risk of suffering osteoporosis in women is higher than in men but not to much. There is no strong association between the osteoporosis and the gender.
$O R (D) = 1.2211$ . Thus, the odds of suffering osteoporosis in women is higher than in men but not to much.
Since we can compute the prevalence of $D$ , both statistics are suitable, but relative risk is easier to interpret.

Question 4

The risks of getting the flu in two cities $A$ and $B$ with the same population size are 14% and 8% respectively.

Compute the probability of having more than 2 persons getting the flu in a random sample of 10 persons of the city $A$ .
Compute the probability of having more than 2 and less than 5 persons getting the flu in a random sample of 50 persons of the city $B$ .
Compute the probability of having 2 persons getting the flu in a random sample of 8 persons of the two cities.
Compute the probability of having some person getting the flu in a random sample of 5 persons that have been living in both cities.

Solution

Let $X$ be the number of persons with flu in a sample of 10 persons from $A$ , then $X \sim B (10, 0.14)$ and $P (X > 2) = 0.1545$ .
Let $Y$ be the number of persons with flu in a sample of 50 persons from $B$ , then $Y \sim B (50, 0.08) \approx P (4)$ and $P (2 < Y < 5) = 0.3907$ .
Let $Z$ be the number of persons with flu in a sample of 8 persons from $A$ and $B$ , then $Z \sim B (8, 0.11)$ and $P (Z = 2) = 0.1684$ .
Let $U$ be the number of persons with flu in a sample of 5 persons living in both cities, then $U \sim B (5, 0.2088)$ and $P (U > 0) = 0.69$ .

Question 5

In a study about the cholesterol two samples of 10000 males and 10000 females was taken. It was observed that 3420 males and 1234 females had a cholesterol level above 230 mg/dl, and that 4936 males had a cholesterol level between 210 and 230 mg/dl. Assuming that the cholesterol level in males and females follows a normal distribution with the same standard deviation, compute:

The means and the standard deviation of the distributions of cholesterol level in males and females.
The percentage of males with cholesterol level between 200 and 240 mg/dl.
The interquartile range of the cholesterol level of females.

Solution

Let $X$ be cholesterol level in males and $Y$ the cholesterol level in females, then $X \sim N (224.1164, 14.4556)$ and $X \sim N (213.2581, 14.4556)$ .
$P (200 \leq X \leq 240) = 0.8164$ .
$I Q R = 19.5003$ mg/dl.

Last updated on Sep 15, 2020