TB or Not TB?
A Develop Understanding Task
Part I
If you ever had chicken pox, measles, rubella, or other viruses, then for the
rest of your life you will have antibodies for those diseases in your blood,
and a simple blood test will be able to detect the presence of those
antibodies. In a similar way, if someone has tuberculosis, then they will
have antibodies for tuberculosis in their blood.
Blood tests are designed to detect the presence of specific antibodies, but
the tests are not perfect. Sometimes, even if there are no antibodies present, other factors will
trigger the test to come back positive. So, a positive test result isn’t absolute evidence that the
person has the disease.
To test for tuberculosis (TB) patients are administered a skin test which involves injecting a bubble
of serum underneath the skin, then several days later, observing the skin to see if the person has
shown a reaction to the serum. A skin reaction indicates the presence of TB antibodies. Suppose
that a young adult with no prior evidence of tuberculosis has a skin reaction to the serum. The
person wonders “Do I really have tuberculosis?” Below are the test results from a large sample of
people from countries with insufficient health care.
Test is Positive
Has tuberculosis
antibodies (TB)
361
Doesn’t have
tuberculosis
antibodies (TB)
Total
Test is Negative
380
558
423
Total
Complete the missing parts of the table. Based on this large sample find the following values:
1) What percent of the population of people represented on this table have tuberculosis antibodies?
2) What percent of individuals represented on this table will have a positive test result?
How accurate is the tuberculosis (TB) test?
Part II
Identify the cells in the table that show counts for the number of individuals who got inaccurate test
results. Put a big frowny face on those cells.
To answer the person who asks “Do I really have tuberculosis?” we will explore some probabilities.
1) Out of all those who tested positive, what percent of individuals actually had tuberculosis
antibodies? (The test accurately detected the disease.)
2) Out of all those who tested negative, what percent of individuals didn’t have TB antibodies? (The
test accurately said they were TB free.)
3) What percent of those with tuberculosis tested negative? (The test said they were TB free, but
they really had the disease. This is called a false negative.)
4) What percent of those who tested positive, didn’t have tuberculosis (The test
said they had TB, but they really didn’t. This is called a false positive.)
5) Which is worse, a false negative or a false positive? Why?
6) Most health tests are designed to be very sensitive, so sensitive that they often come back with
false positive results. Why do you suppose tests are planned to be that way?
7) What will you answer the young adult who tested positive and asks, “Do I really have TB?”
TB or Not TB?: – Teacher Notes
A Develop Understanding Task
Purpose:
Surfaces ideas such as:
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a review of 2-way tables and marginal (row and column) totals along with joint frequencies
(interior cells of the table) which have already been encountered in Secondary Math 1.
proportions may be more useful than counts when comparing data, but counts give actual
values and the size of sample spaces.
using 2-way tables to calculate probabilities/rates/proportions from table values.
conditional probabilities reveal information about subgroups that may not be immediately
apparent from casual examination of the table.
using the results of the exploration to promote the need for probability notation.
using data and probabilities to make recommendations and decisions.
Core Standards Focus:
S.CP.6: Find the conditional probability of A given B as the fraction of B’s outcomes that also belong
to A, and interpret the answer in terms of the model.
Related Standards: S.CP.3, S.CP. 4, S.CP.5
Materials needed: “TB or Not TB?” task sheets, calculators, a method for displaying student
responses such as giant post-it posters or document cameras
Launch (Whole Class):
Discuss the prevalence of tuberculosis (TB) in countries with inadequate health care. Infection
rates in some countries are as high as 30%, and consequently affects people traveling from country
to country and is deadly to many individuals, especially those with compromised immune systems.
All food handlers are required to take the TB skin test, which involves injecting a bubble of serum
underneath the skin, then several days later, observing the skin to see if the person has shown a
reaction to the serum, which indicates the presence of TB antibodies.
 Review information about tuberculosis: http://en.wikipedia.org/wiki/Tuberculosis
Famous people with tuberculosis: http://en.wikipedia.org/wiki/List_of_tuberculosis_cases
Explore (Small Group):
Distribute page 1 of the task sheet. If the pages are stapled, separate the papers. Allow time for
participants to think about how they would approach the question “Do I really have tuberculosis?”
As participants explore the data table they will discover that high numbers of TB skin tests result in
a false positive result.
After a sufficient time for exploration, distribute page 2 of the task, which provides guided
questions.
Discuss (Whole Class):
1. How does exploring conditional probabilities allow for investigation of the accuracy of
medical tests?
Students may have difficulty with identifying the wording of conditional probabilities,
for instance: “out of” and “given”.
As results are shared to the class, drive the need for labeling their results with correct
“units”, which will demonstrate the efficiency of probability notation.
On a giant sticky write, using correct notation, the answers to questions #1-4 on
“Tuberculosis or Not, Part II”.
For example: “The probability of having TB given that an individual tests positive”
= P(TB | positive) = 361/423 = 0.853 = 85.3%
2. How do we define “accuracy” in context of these medical tests?
3. Discuss number sense, and what the chance of a false positive really means. Is the
probability of a false positive a high chance or a low chance? How low is low? Compare to
lottery probabilities.
4. In this setting a false negative is much worse than a false positive. In what settings might a
false positive be worse? How would you feel if you got a false positive?
5. The rates are based on large population values. What do you tell an individual who asks
“Do I have TB?”