Pharmacy exam 2018-12-17

Degrees: Pharmacy, Biotechnology
Date: December 17, 2018

Question 1

The chart below represents the cumulative distribution of the number of daily defective drugs produced by a machine in a sample of 40 days.

plot of chunk question1
  1. Construct the frequency table of the number of defective drugs.
  2. Draw the box and whiskers plot of the number of defective drugs.
  3. Study the symmetry of the distribution of the number of defective drugs.
  4. If the number of defective drugs produced by a second machine follows the equation $y=3x+2$, where $x$ and $y$ are the number of defective drugs with the first and the second machines respectively, in which machine is more representative the mean of the number of defective drugs?
  5. Which number of defective drugs is relatively smaller, 3 drugs in the first machine or 9 in the second one?

  1. $$\begin{array}{|c|r|r|r|r|} \hline \mbox{Defective drugs} & n_i & f_i & N_i & F_i\newline \hline 0 & 1 & 0.025 & 1 & 0.025\newline 1 & 3 & 0.075 & 4 & 0.100\newline 2 & 6 & 0.150 & 10 & 0.250\newline 3 & 7 & 0.175 & 17 & 0.425\newline 4 & 8 & 0.200 & 25 & 0.625\newline 5 & 6 & 0.150 & 31 & 0.775\newline 6 & 5 & 0.125 & 36 & 0.900\newline 7 & 2 & 0.050 & 38 & 0.950\newline 8 & 1 & 0.025 & 39 & 0.975\newline 9 & 1 & 0.025 & 40 & 1.000\newline \hline \end{array} $$
plot of chunk boxplot
  1. $\bar x=3.975$ drugs, $s_x=1.9936$ drugs and $g_1=0.3184$. Thus the distribution is a little bit right-skewed.
  2. $cv_x=0.5015$, $\bar y=13.925$ drugs, $s_y=5.9808$ drugs and $cv_y=0.4295$. Thus, the mean of $y$ is more representative than the mean of $x$ since its coef. of variation is smaller.
  3. $z_x=-0.4891$ and $z_y=-0.8235$, therefore 9 defective drugs in the $y$ machine is relatively smaller.

Question 2

A pharmaceutical laboratory produces two models of blood pressure monitor, one for the arm and the other for the wrist. To compare the accuracy of both blood pressure monitors, a quality control has been conducted with a sample of 20 patients, getting the following results:

$\sum x_i=265.4$ mmHg, $\sum y_i=262.5$ mmHg , $\sum z_i=262.4$ mmHg,
$\sum x_i^2=3701.14$ mmHg$^2$, $\sum y_i^2=3629.41$ mmHg$^2$, $\sum z_i^2=3615.38$ mmHg$^2$,
$\sum x_iy_j=3658.28$ mmHg$^2$, $\sum x_iz_j=3655.95$ mmHg$^2$, $\sum y_jz_j=3613.97$ mmHg$^2$.

Where $X$ is the blood pressure with the arm monitor, $Y$ with the wrist monitor and $Z$ the real blood pressure.

  1. Which blood pressure monitor predicts better the real blood pressure with a linear regression model?
  2. If a patient has a real blood pressure of $13.5$ mmHg, what is the expected blood pressure given by the arm blood pressure monitor?

  1. Blood pressure with the arm monitor: $\bar x=13.27$ mmHg, $s^2_x=8.9641$ mmHg².
    Blood pressure with the wrist monitor: $\bar y=13.125$ mmHg, $s^2_y=9.2049$ mmHg².
    Real blood pressure: $\bar z=13.12$ mmHg, $s^2_z=8.6346$ mmHg². $s_{xz}=8.6951$ mmHg², $s_{yz}=8.4985$ mmHg², $r^2_{xz}=0.9768$ and $r^2_{yz}=0.9087$.
    Thus, the arm monitor predicts better the real pressure with a linear regression model since its linear coef. of determination is greater.
  2. Regression line of $X$ on $Z$: $x=0.0581+1.007z$.
    Prediction: $x(13.5)=13.6527$ mmHg.

Question 3

The regression line of $Y$ on $X$ is $y=1.2x-0.6$.

  1. Which of the following lines can not be the regression line of $X$ on $Y$. Justify the answer.
    1. $x=0.9y-0.6$
    2. $x=-0.7y+0.4$
    3. $x=0.8y-0.7$
    4. $x=-0.6y-0.5$
    5. $x=0.4y-0.6$
    6. $x=-0.5y+0.9$
  2. Considering only the ones that can be the regression line of $X$ on $Y$, which one will give better predictions? Justify the answer.

  1. (b), (d) and (f) are not possible because the slope is negative, and (a) is not possible because the coef. of determination is greater than 1.
  2. (c) gives better predictions because its coef. of determination is greater.

Question 4

In an epidemiological study a sample of 400 persons with breast cancer was drawn and another sample of 1200 persons without breast cancer. In the sample of persons with breast cancer there was 180 smokers, while in the sample of persons without breast cancer there was 1140 non-smokers.

  1. Compute the relative risk of developing cancer smoking and interpret it.
  2. Compute the odds ratio of developing cancer smoking and interpret it.

Let $C$ be the event of having cancer.

  1. $RR(C)=4.6364$. That means that the probability of having cancer smoking is $4.6364$ times higher than non-smoking.
  2. $OR(C)=15.5455$. As is posibive there is a direct association between smoking and having cancer. The odds of having cancer smoking is more than 15 times greater than non-smoking.

Question 5

We want to develop a diagnostic test to rule out a disease when the outcome of the test is negative (negative predictive value) with a probability 90% at least. It is known that the prevalence of the disease in the population is 15% and the sensitivity of the test is set to 80%.

  1. What must be the minimum specificity of the test?
  2. Using the previous specificity, compute the probability of a correct diagnostic.
  3. If we apply the same test two times to the same patient with negative outcomes, what is the probability of ruling out the disease?

Let $D$ be the event of having the disease and $+$ and $-$ the events of getting a positive and a negative outcome in the diagnostic test respectively.

  1. Minimum specificity $P(-|\overline{D})=0.3176$.
  2. $P(TP) + P(TN) = P(D\cap +) + P(\overline{D}\cap -) = 0.12+0.27 = 0.39$.
  3. $P(\overline{D}| -_1\cap -_2)=0.9346$.

Question 6

It is known that in a city one out of 20 persons, in average, has blood type $AB$.

  1. If we draw randomly 200 blood donors, what is the probability of having at least 5 with blood type $AB$?
  2. If we draw randomly 10 blood donors, what is the probability of having more than 8 with blood type different of $AB$?

  1. Let $X$ be the number of donors with blood type $AB$ in a sample of 200 blood donors. Then $X\sim B(200,1/20)\approx P(10)$, and $P(X\geq 5)=0.9707$.
  2. Let $Y$ be the number of donors with no blood type $AB$ in a sample of 10 blood donors. Then $Y\sim B(10,19/20)$, and $P(Y>8)=0.9139$.

Question 7

In a course there are 150 females and 80 males. It is known that the distribution of scores of females and males are normal with the same standard deviation. It is also known that there are 120 females and 56 males with a score greater than 5, and 36 males with a score between 5 and 7.

  1. Compute the means and standard deviations of the distributions of scores of females and males.
  2. How many females will have a score between 4.5 and 8?
  3. Above what score will be 10% of females?

Let $X$ be score of a random male in the course and $Y$ the score of a random female in the course. Then $X\sim N(\mu_x,\sigma)$ and $Y\sim N(\mu_y,\sigma)$.

  1. $\mu_x=5.87$, $\mu_y=6.41$ and $\sigma=1.68$.
  2. $P(4.5\leq Y\leq 8) = 0.7018$, that is, $105.27$ females.
  3. $P_{90}=8.8$.
Exam Statistics Biostatistics
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Last updated on Mar 8, 2021

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