Mammography and clinical breast examination are the two major techniques used to screen for breast cancer. However, as true for any screening test, they are not completely accurate. If it is determined, based on mammographic findings, that there is a possibility of breast cancer, this is usually confirmed or disconfirmed by a subsequent biopsy. A false positive test is a mammogram positive test that is disconfirmed by biopsy. The data in Table 1 were reported in a paper concerning breast cancer screening (Elmore, et al, New England Journal of Medicine 1998; 338(16): 1089-1096). Table 1 False positive breast cancer screening tests over a 10-year period # Screening tests # False positive tests 9762 631 1. What is the estimated probability of a false positive test? 2. Suppose 10 women are given mammograms. What is the probability that at least 1 woman will have a false positive test? 3. Another aspect of mammographic screening is the presence of false negatives. A false negative is a screen negative woman who actually has breast cancer. False negatives usually occur because the tumors are small and hard to detect. It is estimated that among women with breast cancer, 2% of all screening tests are false negatives. Suppose that there are 100 women who have breast cancer. What is the probability that at least 3 of them will not be detected by a mammogram?
Mammography and clinical breast examination are the two major techniques used to screen for breast cancer. However, as true for any screening test, they are not completely accurate. If it is determined, based on mammographic findings, that there is a possibility of breast cancer, this is usually confirmed or disconfirmed by a subsequent biopsy. A false positive test is a mammogram positive test that is disconfirmed by biopsy. The data in Table 1 were reported in a paper concerning breast cancer screening (Elmore, et al, New England Journal of Medicine 1998; 338(16): 1089-1096). Table 1 False positive breast cancer screening tests over a 10-year period # Screening tests # False positive tests 9762 631 1. What is the estimated probability of a false positive test? 2. Suppose 10 women are given mammograms. What is the probability that at least 1 woman will have a false positive test? 3. Another aspect of mammographic screening is the presence of false negatives. A false negative is a screen negative woman who actually has breast cancer. False negatives usually occur because the tumors are small and hard to detect. It is estimated that among women with breast cancer, 2% of all screening tests are false negatives. Suppose that there are 100 women who have breast cancer. What is the probability that at least 3 of them will not be detected by a mammogram?
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Please show detailed step-by-step explanation and solution for each problem (problem #2, problem #3, problem #4, problem #5, and problem #6).
Please use the results of problems #2 and problems #3 to solve and show your work for problems #5 and problems #6 (just like the hint stated).
Transcribed Image Text:Mammography and clinical breast examination are the two major techniques used to screen
for breast cancer. However, as true for any screening test, they are not completely accurate.
If it is determined, based on mammographic findings, that there is a possibility of breast
cancer, this is usually confirmed or disconfirmed by a subsequent biopsy. A false positive test
is a mammogram positive test that is disconfirmed by biopsy. The data in Table 1 were
reported in a paper concerning breast cancer screening (Elmore, et al, New England Journal
of Medicine 1998; 338(16): 1089-1096).
Table 1 False positive breast cancer screening tests over a 10-year period
# Screening tests
# False positive tests
9762
631
1. What is the estimated probability of a false positive test?
2. Suppose 10 women are given mammograms. What is the probability that at least 1 woman will
have a false positive test?
3. Another aspect of mammographic screening is the presence of false negatives. A false negative
is a screen negative woman who actually has breast cancer.
False negatives usually occur because the tumors are small and hard to detect. It is estimated
that among women with breast cancer, 2% of all screening tests are false negatives.
Suppose that there are 100 women who have breast cancer. What is the probability that at least
3 of them will not be detected by a mammogram?
4. Write down a computer command in either Excel, Stata, or R to determine the quantity in
problem #3. (Only one is necessary). Specify the name of the command and all associated
parameters to be able to perform the computation.
5. Suppose that 5% of all mammograms are obtained from women who truly have breast cancer.
What is the proportion of mammograms that will yield test positive results?
Hint: Subdivide the probability of a test positive mammogram into two mutually exclusive
components of {test positive mammograms with breast cancer}, and {test positive
mammograms without breast cancer}, and use the results of problem #2 and problem #3 to
evaluate each of these components.
6. What proportion of the test positives will be confirmed breast cancer cases?
What is another name for this quantity?
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