Question

1) The reliability of a particular skin test for tuberculosis (TB) is as follows: the sensitivity of the test is 0.9 (the test comes back positive 90% of the time if the subject has TB) and the specificity of the test is 0.95 (if the subject does not have TB, the test comes back negative 95% of the time). In a large population, only 0.3% (0.003) of the people have TB. A person is selected at random and given the test, which comes back positive. What is the probability that that person actually has TB? (hint: use a hypothetical 100,000 table.)

2)The American Medical Association (AMA) wishes to determine the proportion of doctors who are considering leaving the profession because of the rapidly increasing number of lawsuits against doctors. How large a sample should be taken to find the answer within a margin of error of ±3% at the 96% confidence level? Use an estimate of ?=̂ 0.5 in your calculation.

Please show work.

Answer 1

Question 1

Sensitivity = 0.9

Specificity = 0.95

P(People Have TB) = 0.3% = 0.003

P(Positive Test) = P(Person has disease) * P(Found positive) + P(Person doesn't have disease) * P(Found Positive)

= 0.003 * 0.90 + (1 - 0.003) * (1 - 0.95) = 0.05255

P(Person actually have TB) = (0.003 * 0.90)/0.05255 = 0.05138

Question 2

Here required sample size = n

Margin of error = 3% = 0.03

p^ = 0.5

standard error = sqrt(0.5 * 0.5/n)

confidence level= 96%

critical test statistic = NORMSINV(0.98)= 2.0537

0.03 = 2.0537 * sqrt(0.5 * 0.5/n)

n = (2.0537/0.03)2  * 0.52

n = 1171.64 or 1172