3.76 | Let A be the event that the difference of the automated Diastolic blood pressure has increased by bigger or equal than 10mmHG and B the event that the difference of the manual Diastolic blood pressure is bigger or equal than 10mmHG. Let N be the number of measurements. From Table 3.9, we see that the probability of A intersected with B is 6/N and the probability of B is P[B]=13/N and the probability of A is P[A]=21/N. The sensitivity is P[A | B] = P[ A intersect B ]/P[B] = (6/N)/(13/N) = 6/13 = 0.462. |
3.77 | The specificity of the automated Diastolic blood pressure measurements is P[A^{c} | B^{c}] = (51/N)/(66/N) = 51/66 = 0.773. |
3.78 | With the notation from above the predictive values
positive (PV+) and predictive values negative (PV-) are PV+ = P[ B | A] = P[B intersect A]/P[A]=6/21 = 0.286 PV- = P[ B^{c} | A^{c}] = 51/58 = 0.879 |
3.83 | The probabilities in the prevalence columns are P[Y_{i}]. The conditional probabilities P[X_{i}|Y_{j}] are given in the matrix of Table 3.11. Look up the entry in the 4'th row and 5'th column: P[X_{5}|Y_{4}] = 0.10. |
3.87 | Let X_{7} be the screening criterion for Y_{2}. The sensitivity is P[X_{y} | Y_{2}] = 0.7. |
3.88 | P[X_{7}] = 0.43325, and P[X_{7}^{c}| Y_{2}^{c} ] = 0.605 |
3.89 | P[Y_{2} | X_{1} intersect X_{7}] = 0.187 |
4.10 |
While commen sense tells the right answer 3=0.2*15 immediatly,
it is useful to formulate the problem precicely using the language
of probability theory. Let D be the group of all grade school students and A be the population of grade school students who develop influenca. Assume D contains N students. The probability of A, P[A] is known to be 0.2. Let B be the members of the class under consideration and C is the subset of students in B which develop influenca. We have C = A intersect B. It is an assumption that B is independent of A. Therefore, we have P[C] = P[A intersect B] = P[A] P[B] = 0.2 * P[B] = 0.2 * 15/N. The expected number of students is N P[C ] = 0.2*15 = 3. |
4.14 | X is the Binominal distributed random variable giving the number of inner-city newbotrns with HIV positive test results. The probability p to be HIV positive is 30/3741. The probability 1-p to be HIV negative is 3711/3741. The Binominal formula (which assumes that each birth event is independent from the others) gives P[X=5] = 3741!/(5! 3736!) p^{5} (1-p)^{3711} = 0.1573. |
4.15 | P[X > 4] = 1- P[X < 5].
This probability can be determined easily with Mathematica
(see
a guide how to use Mathematica at Harvard from any computer. The
computer does not need to have Mathematica installed but have access to
the Web). << Statistics`DiscreteDistributions` N[1- CDF[BinomialDistribution[500, 30/3741], 4]]which gives 0.373. |
4.20 | As in 4.14. The Binominal formula gives P[X_{3}=5] = 0.0004 |
Last update, 11/29/2001 |