The PCR test tests whether a patient has contracted Covid-19. The PCR test is said to be positive if
it indicates that genetic material that only comes from the virus is present in a saliva sample, and the
PCR test is said to be negative if it does not indicate that genetic material that only comes from the
virus is present in a saliva sample. The PCR test does not always indicate the correct result. As part
of a training program, staff at a testing lab applied the PCR test to 500 saliva samples known to
contain the genetic material from the virus. The PCR test was positive for 489 of those saliva
samples and negative for the other 11 samples. As part of the same training program, the staff also
applied the PCR test to 500 other saliva samples known to not contain the genetic material from the
virus. The PCR test was positive for 37 of those saliva samples and negative for the other 463
samples.
a. When a new saliva sample arrives at the lab, it will be tested to determine whether the genetic
material from the virus is present. Using the data from the training program, estimate the
probability that the PCR test would be positive when it is applied to a saliva sample that does not
contain the genetic material from the virus.
b. Among the saliva samples examined in the training program that provided positive PCR results for
Coivd-19, what proportion actually contained genetic material from the virus?
c. When a saliva sample yields a positive PCR test result, two more PCR tests are performed on the
same saliva sample. If at least one of the two additional PCR tests is positive, the saliva sample is
subjected to a more expensive and more accurate test to make a definitive determination of whether
the genetic material is present in the sample. Repeated PCR tests on the same sample are generally
assumed to be independent. Under the assumption of independence, what is the probability that a
new saliva sample that comes into the lab will be subjected to the more expensive test if that sample
does not contain the genetic material from the virus?