Twelve percent of the job applicants for a large company have recently smoked marijuana. The company gives all applicants a drug test. The test is not perfect. The test has a sensitivity of 98%- meaning that 98% of those who have recently smoked marijuana will test positive, the other 2% will test negative. The test has a specificity of 96%- meaning that if someone has not recently smoked marijuana, the test will correctly yield a negative test result 96% of the time.
Let,
M = applicant has recently smoked marijuana
C = applicant has not recently smoked marijuana (clean)
X = positive test (test indicates that the person has smoked marijuana recently)
N = negative test (test indicates that the person has not recently smoked marijuana)
Using the following table:
M = smoked C = hasn't recently smoked totals
X = test is positive
N = test is negative
totals 1.0000
Show your answers rounded to four decimal places if rounding is necessary.
a) What proportion of the applicants get an incorrect test result?
b) What proportion of the applicants test positive?
c) What proportion of those who test positive have not recently smoked marijuana?
d) What proportion of those who pass the drug test really shouldn't have?
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a) To calculate the proportion of applicants that get an incorrect test result, we need to add the proportion of false positives (C who test positive) and false negatives (M who test negative).
False positives (C who test positive) can be calculated as (1 - specificity) * proportion of clean applicants:
False_positives = (1 - 0.96) * 0.88 = 0.0352
False negatives (M who test negative) can be calculated as (1 - sensitivity) * proportion of applicants who smoked:
False_negatives = (1 - 0.98) * 0.12 = 0.0024
Proportion of incorrect test results = False_positives + False_negatives
Proportion of incorrect test results = 0.0352 + 0.0024 = 0.0376
So, the proportion of applicants who get an incorrect test result is 0.0376.
b) To calculate the proportion of applicants who test positive, we need to add the proportion of true positives (M who test positive) and false positives (C who test positive).
True positives (M who test positive) can be calculated as sensitivity * proportion of applicants who smoked:
True_positives = 0.98 * 0.12 = 0.1176
Proportion of applicants who test positive = True_positives + False_positives
Proportion of applicants who test positive = 0.1176 + 0.0352 = 0.1528
So, the proportion of applicants who test positive is 0.1528.
c) To calculate the proportion of those who test positive and have not recently smoked marijuana, we need to divide false positives (C who test positive) by the total number of applicants who test positive:
Proportion of those who test positive and have not recently smoked marijuana = False_positives / Proportion of applicants who test positive
Proportion of those who test positive and have not recently smoked marijuana = 0.0352 / 0.1528 = 0.2301
So, the proportion of those who test positive and have not recently smoked marijuana is 0.2301.
d) To calculate the proportion of those who pass the drug test but shouldn't have (clean applicants who test negative), we need to subtract false negatives (M who test negative) from the proportion of clean applicants:
Proportion of those who pass the drug test but shouldn't have = proportion of clean applicants - False_negatives
Proportion of those who pass the drug test but shouldn't have = 0.88 - 0.0024 = 0.8776
So, the proportion of those who pass the drug test but shouldn't have is 0.8776.
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