Pharmacy exam 2019-10-14

Degrees: Pharmacy, Biotechnology
Date: October 14, 2019

Question 1

It has been measured the systolic blood pressure (in mmHg) in two groups of 100 persons of two populations $A$ and $B$ . The table below summarize the results.

$\begin{array}{lrr} Systolic blood pressure & Num persons A & Num persons B \\ (80, 90] & 4 & 6 \\ (90, 100] & 10 & 18 \\ (100, 110] & 28 & 30 \\ (110, 120] & 24 & 26 \\ (120, 130] & 16 & 10 \\ (130, 140] & 10 & 7 \\ (140, 150] & 6 & 2 \\ (150, 160] & 2 & 1 \end{array}$

Which of the two systolic blood pressure distributions is less asymmetric? Which one has a higher kurtosis? According to skewness and kurtosis can we assume that populations $A$ and $B$ are normal?
In which group is more representative the mean of the systolic blood pressure?
Compute the value of the systolic blood pressure such that 30% of persons of the group of population $A$ are above it?
Which systolic blood pressure is relatively higher, 132 mmHg in the group of population $A$ , or 130 mmHg in the group of population $B$ ?
If we measure the systolic blood pressure of the group of population $A$ with another tensiometer, and the new pressure obtained ( $Y$ ) is related with the first one ( $X$ ) according to the equation $y = 0.98 x - 1.4$ , in which distribution, $X$ or $Y$ , is more representative the mean?

Use the following sums for the computations:
Group $A$ : $\sum x_{i} n_{i} = 11520$ mmHg, $\sum x_{i}^{2} n_{i} = 1351700$ mmHg $^{2}$ , $\sum (x_{i} - \bar{x})^{3} n_{i} = 155241.6$ mmHg $^{3}$ and $\sum (x_{i} - \bar{x})^{4} n_{i} = 16729903.52$ mmHg $^{4}$ .
Group $B$ : $\sum x_{i} n_{i} = 11000$ mmHg, $\sum x_{i}^{2} n_{i} = 1230300$ mmHg $^{2}$ , $\sum (x_{i} - \bar{x})^{3} n_{i} = 165000$ mmHg $^{3}$ and $\sum (x_{i} - \bar{x})^{4} n_{i} = 13632500$ mmHg $^{4}$ .

Solution

Group $A$ : $\bar{x} = 115.2$ mmHg, $s^{2} = 245.96$ mmHg $^{2}$ , $s = 15.6831$ mmHg, $g_{1 A} = 0.4024$ and $g_{2 A} = - 0.2346$ . Group $B$ : $\bar{x} = 110$ mmHg, $s^{2} = 203$ mmHg $^{2}$ , $s = 14.2478$ mmHg, $g_{1 B} = 0.5705$ and $g_{2 B} = 0.3081$ . Thus the distribution of the population $A$ group is less asymmetric since $g_{1 A}$ is closer to 0 than $g_{1 B}$ and the populaton $B$ group has a higher kurtosis since $g_{2 B} > g_{2 A}$ . Both populations can be cosidered normal since $g_{1}$ and $g_{2}$ are between -2 and 2.
$c v_{A} = 0.1361$ and $c v_{B} = 0.1295$ , thus, the mean of group $B$ is a little bit more representative since its coef. of variation is smaller than the one of group $A$ .
$P_{70} \approx 125$ mmHg.
The standard scores are $z_{A} (132) = 1.0712$ and $z_{B} (130) = 1.4037$ . Thus, 130 mmHg in group $B$ is relatively higher than 132 mmHg in group $A$ .
$\bar{y} = 111.496$ , $s_{y} = 15.3694$ and $c v_{y} = 0.1378$ . Thus the mean of $X$ is more representative than the mean of $Y$ since $c v_{x} < c v_{y}$ .

Question 2

In a symmetric distribution the mean is 15, the first quartile 12 and the maximum value is 25.

Draw the box and whiskers plot.
Could an hypothetical value of 2 be considered an outlier in this distribution?

Solution

$Q_{1} = 12$ , $Q_{2} = 15$ , $Q_{3} = 18$ , $I Q R = 6$ , $f_{1} = 3$ , $f_{2} = 27$ , $M i n = 5$ and $M a x = 25$ .

Yes, because $2 < f_{1}$ .

Question 3

A pharmaceutical company is trying three different analgesics to determine if there is a relation among the time required for them to take effect. The three analgesics were administered to a sample of 20 patients and the time it took for them to take effect was recorded. The following sums summarize the results, where $X$ , $Y$ and $Z$ are the times for the three analgesics.

$\sum x_{i} = 668$ min, $\sum y_{i} = 855$ min, $\sum z_{i} = 1466$ min,
$\sum x_{i}^{2} = 25056$ min $^{2}$ , $\sum y_{i}^{2} = 42161$ min $^{2}$ , $\sum z_{i}^{2} = 123904$ min $^{2}$ ,
$\sum x_{i} y_{j} = 31522$ min $^{2}$ , $\sum y_{j} z_{j} = 54895$ min $^{2}$ .

Is there a linear relation between the times $X$ and $Y$ ? And between $Y$ and $Z$ ? How are these linear relationships?
According to the regression line, how much will the time $X$ increase for every minute that time $Y$ increases?
If we want to predict the time $Y$ using a linear regression model, ¿which of the two times $X$ or $Z$ is the most suitable? Why?
Using the chosen linear regression model in the previous question, predict the value of $Y$ for a value of $X$ or $Z$ of 40 minutes.
If the correlation coefficient between the times $X$ and $Z$ is $r = - 0.69$ , compute the regression line of $X$ on $Z$ .

Solution

$\bar{x} = 33.4$ min, $s_{x}^{2} = 137.24$ min $^{2}$ , $\bar{y} = 42.75$ min, $s_{y}^{2} = 280.4875$ min $^{2}$ , $\bar{z} = 73.3$ min, $s_{z}^{2} = 822.31$ min $^{2}$ , $s_{x y} = 148.25$ min $^{2}$ and $s_{y z} = - 388.825$ min $^{2}$ . Thus, there is a direct linear relation between $X$ and $Y$ and an inverse linear relation between $Y$ and $Z$ .
$b_{x y} = 0.5285$ min.
$r_{x y}^{2} = 0.5709$ and $r_{y z}^{2} = 0.6555$ , thus the regression line of $Y$ on $Z$ explains better $Y$ than the regression line of $Y$ on $X$ since $r_{y z}^{2} > r_{x y}^{2}$ .
Regression line of $Y$ on $Z$ : $y = 77.4095 + - 0.4728 z$ and $y (40) = 58.4957$ .
$s_{x z} = - 231.7967$ and the regression line of $X$ on $Z$ is $x = 54.0622 + - 0.2819 z$ .

Last updated on Sep 15, 2020