QUESTION 4. (Data file included: No): According to a 2018 article in Esquire magazine, approximately 70% of males over age 70 will develop cancerous cells in their bowel. Bowel cancer is second only to skin cancer as the most common form of cancer for males in New Zealand. One of the most common tests for the detection of bowel cancer is the bowel-specific antigen (BSA) test. However, this test is known to have a high false-positive rate (tests that come back positive for cancer when no cancer is present). Suppose there is a .04 probability that a male patient has bowel cancer before testing. The probability of a false-positive test is .85, and the probability of a false-negative (no indication of cancer when cancer is actually present) is .30. a. What is the probability that the male patient has prostate cancer if the BSA test comes back positive? b. What is the probability that the male patient has prostate cancer if the BSA test comes back negative? c. For older men, the prior probability of having cancer increases. Suppose that the prior probability of the male patient is .35 rather than .04. What is the probability that the male patient has bowel cancer if the BSA test comes back positive? What is the probability that the male patient has bowel cancer if the BSA test comes back negative?
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
QUESTION 4. (Data file included: No): According to a 2018 article in Esquire magazine, approximately
70% of males over age 70 will develop cancerous cells in their bowel. Bowel cancer is second only to
skin cancer as the most common form of cancer for males in New Zealand. One of the most common
tests for the detection of bowel cancer is the bowel-specific antigen (BSA) test. However, this test is
known to have a high false-positive rate (tests that come back positive for cancer when no cancer is
present). Suppose there is a .04 probability that a male patient has bowel cancer before testing. The
probability of a false-positive test is .85, and the probability of a false-negative (no indication of cancer
when cancer is actually present) is .30.
a. What is the probability that the male patient has prostate cancer if the BSA test comes back
positive?
b. What is the probability that the male patient has prostate cancer if the BSA test comes back
negative?
c. For older men, the prior probability of having cancer increases. Suppose that the prior
probability of the male patient is .35 rather than .04. What is the probability that the male
patient has bowel cancer if the BSA test comes back positive? What is the probability that the
male patient has bowel cancer if the BSA test comes back negative?
d. What can you infer about the BSA test from the results of parts (a), (b) and (c)?
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