Download Section 4-4 Statistical Paradoxes

4.4 Statistical Paradoxes
LEARNING GOAL
Investigate a few common paradoxes that arise in
statistics, such as how it is possible that most people who
fail a “90% accurate” polygraph test may actually be
telling the truth.
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Better in Each Case, But Worse Overall
It is possible for something to appear better in each of
two or more group comparisons but actually be worse
overall. This occurs because of the way in which the
overall results are divided into unequally sized
groups.
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Slide 4.4- 2
EXAMPLE 1 Who Played Better?
Table 4.7 gives the shooting
performance of two players in
each half of a basketball game.
Shaq had a higher shooting
percentage in both the first half (40% to 25%) and the second
half (75% to 70%). Can Shaq claim that he had the better game?
Solution: No, and we can see why by looking at the overall
game statistics. Shaq made a total of 7 baskets (4 in the first half
and 3 in the second half) on 14 shots (10 in the first half and 4
in the second half), for an overall shooting percentage of 7/14 =
50%. Vince made a total of 8 baskets on 14 shots, for an overall
shooting percentage of 8/14 = 57.1%.
Surprisingly, even though Shaq had a higher shooting
percentage in both halves, Vince had a better overall shooting
percentage for the game.
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Does a Positive Mammogram Mean Cancer?
We often associate tumors with cancers, but most tumors are not
cancers. Medically, any kind of abnormal swelling or tissue
growth is considered a tumor.
A tumor caused by cancer is said to be malignant (or cancerous);
all others are said to be benign.
About 1 in 100 breast tumors turns out to be malignant.
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Slide 4.4- 4
Suppose a patient’s mammogram comes back positive.
Mammograms are not perfect, so the positive result does not
necessarily mean that she has breast cancer.
Let’s assume that the mammogram screening is 85% accurate:
It will correctly identify 85% of malignant tumors as malignant
and 85% of benign tumors as benign.
Because the mammogram screening is 85% accurate, most
people guess that the positive result means that the patient
probably has cancer.
Consider a study in which mammograms are given to 10,000
women with breast tumors. Assuming that 1% of tumors are
malignant, 1% × 10,000 = 100 of the women actually have
cancer; the remaining 9,900 women have benign tumors.
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Slide 4.4- 5
Table 4.8 summarizes the mammogram results.
• The mammogram screening
correctly identifies 85% of
the 100 malignant tumors as
malignant. Thus, it gives
positive (malignant) results
for 85 of the malignant
tumors; these cases are called true positives.
In the other 15 malignant cases, the result is negative, even though the
women actually have cancer; these cases are false negatives.
• The mammogram screening correctly identifies 85% of the 9,900 benign
tumors as benign. Thus, it gives negative (benign) results for 85% × 9,900 =
8,415 of the benign tumors; these cases are true negatives.
The remaining 9,900 – 8,415 = 1,485 women get positive results in which
the mammogram incorrectly identifies their tumors as malignant; these cases
are false positives.
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Slide 4.4- 6
Overall, the mammogram
screening gives positive
results to 85 women
who actually have cancer
and to 1,485 women who
do not have cancer.
The total number of positive
results is 85 + 1,485 = 1,570. Because only 85 of these are true
positives (the rest are false positives), the chance that a positive
result really means cancer is only 85/1,570 = 0.054, or 5.4%.
Therefore, when a patient’s mammogram comes back positive,
there’s still only a small chance that she has cancer.
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Slide 4.4- 7
By the Way ...
The accuracy of breast cancer screening is
rapidly improving; newer technologies,
including digital mammograms and
ultrasounds, appear to achieve accuracies
near 98%. The most definitive test for
cancer is a biopsy, though even biopsies
can miss cancers if they are not taken with
sufficient care. If you have negative tests but
are still concerned about an abnormality,
ask for a second opinion. It may save your
life.
Copyright © 2009 Pearson Education, Inc.
Slide 4.4- 8
EXAMPLE 2 False Negatives
Suppose you are a doctor
seeing a patient with a
breast tumor. Her
mammogram comes back
negative. Based on the
numbers in Table 4.8,
what is the chance that she has cancer?
Solution: For the 10,000 cases summarized in Table 4.8, the
mammograms are negative for 15 women with cancer and for
8,415 women with benign tumors. The total number of negative
results is 15 + 8,415 = 8,430. Thus, the fraction of women with
cancer who have false negatives is 15/8,430 = 0.0018, or
slightly less than 2 in 1,000. In other words, the chance that a
woman with a negative mammogram has cancer is only about 2
in 1,000.
Copyright © 2009 Pearson Education, Inc.
Slide 4.4- 9
TIME OUT TO THINK
While the chance of cancer with a negative mammogram
is small, it is not zero. Therefore, it might seem like a
good idea to biopsy all tumors, just to be sure. However,
biopsies involve surgery, which means they can be
painful and expensive, among other things. Given these
facts, do you think that biopsies should be routine for all
tumors? Should they be routine for cases of positive
mammograms? Defend your opinion.
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Slide 4.4- 10
Polygraphs and Drug Tests
Suppose the government gives the polygraph test to 1,000
applicants for sensitive security jobs. Further suppose that 990
of these 1,000 people tell the truth on their polygraph test,
while only 10 people lie. For a test that is 90% accurate, we
find the following results:
• Of the 10 people who lie, the polygraph correctly identifies
90%, meaning that 9 fail the test (they are identified as liars)
and 1 passes.
• Of the 990 people who tell the truth, the polygraph correctly
identifies 90%, meaning that 90% × 990 = 891 truthful
people pass the test and the other 10% × 990 = 99 truthful
people fail the test.
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Slide 4.4- 11
Figure 4.16 A tree diagram summarizes results of a 90% accurate polygraph
test for 1,000 people, of whom only 10 are lying.
The total number of people who fail the test is 9 + 99 = 108.
Of these, only 9 were actually liars; the other 99 were
falsely accused of lying.
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Slide 4.4- 12
That is, 99 out of 108, or 99/108 = 91.7%, of the people
who fail the test were actually telling the truth.
Assuming the government rejects applicants who fail the
polygraph test, then almost 92% of the rejected
applicants were actually being truthful and may have
been highly qualified for the jobs.
Copyright © 2009 Pearson Education, Inc.
Slide 4.4- 13
TIME OUT TO THINK
Imagine that you are falsely accused of a crime. The
police suggest that, if you are truly innocent, you should
agree to take a polygraph test. Would you do it? Why or
why not?
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Slide 4.4- 14
EXAMPLE 3 High School Drug Testing
All athletes participating in a regional high school track and field
championship must provide a urine sample for a drug test. Those
who fail are eliminated from the meet and suspended from
competition for the following year. Studies show that, at the
laboratory selected, the drug tests are 95% accurate.
Assume that 4% of the athletes actually use drugs. What fraction
of the athletes who fail the test are falsely accused and therefore
suspended without cause?
Solution: The easiest way to answer this question is by using
some sample numbers. Suppose there are 1,000 athletes in the
meet. Then 4%, or 40 athletes, actually use drugs; the remaining
960 athletes do not use drugs.
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Slide 4.4- 15
EXAMPLE 3 High School Drug Testing
Solution: (cont.)
In that case, the 95% accurate drug test should return the
following results:
• 95% of the 40 athletes who use drugs, or 0.95 × 40 = 38
athletes, fail the test. The other 2 athletes who use drugs pass
the test.
• 95% of the 960 athletes who do not use drugs pass the test,
but 5% of these 960, or 0.05 × 960 = 48 athletes, fail.
The total number of athletes who fail the test is 38 + 48 = 86.
But 48 of these athletes who fail the test, or 48/86 = 56%, are
actually nonusers.
Despite the 95% accuracy of the drug test, more than half of the
suspended students are innocent of drug use.
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The End
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