Lesson Building Two-Way Tables to Calculate Probability
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1 STATWAY STUDENT HANDOUT Lesson Building Two-Way Tables to Calculate Probability STUDENT NAME DATE INTRODUCTION Oftentimes when interpreting probability, you need to think very carefully about how the outcome is defined. A simple change in wording can have a large impact on the estimated probability. What Is the Question? The National Basketball Association (NBA) consists of 30 professional basketball teams. Consider the following: The chance of a man over 6 feet tall playing in the NBA. The chance of an NBA player being over 6 feet tall. 1 Are these two chances about the same? If not, which do you think is larger? Tell why you think so. 2 We could use a fraction to estimate each of these chances. To determine the chance that a man over 6 feet tall would play in the NBA, what would go in the denominator? What would go in the numerator? Would this fraction be large or small? 3 To determine the chance that a man playing in the NBA is over 6 feet tall, what would go in the denominator? What would go in the numerator? Would this fraction be large or small?
2 STATWAY STUDENT HANDOUT 2 Now consider the following two statements: The chance of a positive drug test for a person who is a drug user. The chance a person being a drug user if he or she had a positive drug test. Just as with the two statements associated with NBA players, there is a big difference between these two statements. Evaluating Drug Screening Many companies now require job applicants to undergo drug-screening tests prior to employment. These tests are not always accurate. Sometimes the test is positive even when the person being tested is not using drugs (called a false positive), and sometimes the test is negative for a person who is using drugs (a false negative). When new screening methods are introduced, it is important to evaluate how likely these kinds of errors are. Three different methods of screening for illegal drug use were compared in a study. 1 Each method was tested using a large number of blood samples from individuals whose drug-use status was known so that false-positive and false-negative rates could be estimated. The false-positive and false-negative percentages for these three methods are shown in the following table. Method False-Positive Percentage False-Negative Percentage Method Method Method TRY THESE 1 None of the three methods would be considered acceptable for employment drug screening. Why do you think this is so? 1 Lynch, K. L., Breaud, A. R., Vandenberghe, H., Wu, A. H. B., & Clarke, W. (2010). Performance evaluation of three liquid chromatography mass spectrometry methods for broad-spectrum drug screening. Clinica Chimica Acta, 411,
3 STATWAY STUDENT HANDOUT 3 2 If you had to recommend one method for employment drug screening, which would you select? Why did you choose this method? For the rest of these questions, you will focus on Method 3, which has a false-positive percentage of 35.7% and a false-negative percentage of 4.8%. 3 If this test is used on people who actually use illegal drugs, what percentage of the tests will correctly indicate that they use drugs? 4 If this test is employed on people who do not use drugs, what percentage of the tests will correctly indicate that they do not use drugs? 5 Suppose 1% of the people sent for screening by a potential employer are actually using illegal drugs (but unknown to the employer). The employer screens these people with Method 3, so you know there will be some false positives as well as some real positives. Suppose 100 people test positive. Make your best guess of how many are actually illegal drug users? Now let s see how good your guess was. One way to evaluate the percentage of people testing positive who are actually drug users is to construct a two-way table. Imagine a hypothetical population of 10,000 people who will be screened using Method 3.
4 STATWAY STUDENT HANDOUT 4 6 Assuming that 1% of these people are actually drug users, how many drug users are there in the population? How many nonusers are there? Enter these numbers in the two-way table. Remember that 1% = 0.01 = 1/100. Positive test Negative test Total Drug user Not a drug user Total 10,000 7 Take a look at those people who are not drug users. If 35.7% of them test positive (false positive), how many people in this group test positive? Remember that if you are counting people, you must use a whole number. Enter this number in the appropriate location in the table. 8 Take a look at those who are drug users. If 4.8% of them falsely test negative, about how many of these people test negative? Enter this number in the appropriate location in the table. 9 Use addition and subtraction to fill in the remaining entries and row totals for the table. 10 Of those who test positive, what proportion are actually drug users? (Hint: Is this a joint, marginal, or conditional proportion?) Does this surprise you? Was your guess from Question 5 close to the actual percentage? 11 Of those who test negative, what proportion are actually not drug users?
5 STATWAY STUDENT HANDOUT 5 12 Next, let s take a look at what happens if the percentage of the population who are drug users is 10% rather than 1%. Again, considering a population of 10,000 people and assuming that 10% of these people are actual drug users, how many drug users are there in the population? How many nonusers are there? Enter these numbers as the row totals in the table below. Positive test Negative test Total Drug user Not a drug user Total 10, Take a look at those people who are not drug users. If 35.7% of them test positive, how many of them test positive? Enter this number in the appropriate cell in the table. 14 Take a look at those who are drug users. If 4.8% of them test negative, how many of them test negative? Enter this number in the appropriate cell in the table. 15 Fill in the remaining cells and marginal totals for the table. 16 Of those who test positive, what proportion are actually drug users? Does this surprise you? Was your guess from Question (c) close to the actual percentage? 17 Of those who test negative, what proportion are actually not drug users?
6 STATWAY STUDENT HANDOUT 6 Cancer Screening Interpreting the results of medical test can be a challenge, even for doctors. In a study to see whether doctors are correct in their interpretations, a researcher posed a question similar to the one that follows to 160 doctors. 2 Suppose mammograms are used to screen for breast cancer in a particular city, and the following information is known. 1% of the women in the city have breast cancer. If a woman has breast cancer, the probability that the mammogram is positive is 90%. (sensitivity) If a woman does not have breast cancer, the probability that the mammogram is positive is 9%. (false-positive rate) TRY THESE 18 A woman has a mammogram, and the result is positive. She wants to know the chance that she has breast cancer in light of the positive mammogram. Which of the following do you think is the best answer? A The chance that she has breast cancer is about 81%. B C Out of 100 women with a positive mammogram, about 90 have breast cancer. Out of 100 women with a positive mammogram, about 10 have breast cancer. D The chance that she has breast cancer is about 1%. Surprisingly, only 21% of the doctors in a study answered a question like this correctly. Working with a partner, use the table approach to determine the best answer to this question. Notice that even though there are a large number of false positives, your assessment of the chance that a woman has breast cancer is still greater if you know she has tested positive than if you do not know the result of the mammogram. 2 Gigerenzer, G., Gaissmaier, W., Kurz-Milcke, E., Schwartz, L. M., & Woloshin, S. (2008). Helping doctors and patients make sense of health statistics. Psychological Science in the Public Interest, 8,
7 STATWAY STUDENT HANDOUT 7 NEXT STEPS Our intuition about chance and risk is often not very good, especially when dealing with conditional proportions. The activities in this lesson were designed to help you think about chance and risk in a systematic way. The technique of translating probability information into a hypothetical 10,000 table provides a powerful tool for computing and interpreting probabilities.
8 STATWAY STUDENT HANDOUT 8 TAKE IT HOME 1 Consider the breast cancer example from the lesson. What if the prevalence of breast cancer was 10%? Use the table approach to determine the proportion of those testing positive that actually have breast cancer. 2 Blood donors are usually screened for HIV, both for the safety of the blood supply and for the benefit of the donor. The test, called an ELISA, tests positive 97.5% of the time if the donor actually has HIV. If the donor does not have HIV, the ELISA test correctly indicates that the person does not have the disease 92.6% of the time. About 0.2% of college students have HIV. After a blood drive at a college, the lab calls and tells a student that the test has indicated he has HIV. A Use what you have learned in this lesson to determine the probability that the student actually has HIV. Because the prevalence of HIV is so small, it may be better to use 100,000 as your total population. B Considering the answers to questions 1 and 2, should you be more concerned about a positive test for a rare disease or a common disease? Explain your answer.
9 STATWAY STUDENT HANDOUT 9 3 When a tennis player serves (hits the ball to the opponent to start play), if the first serve lands in, the point is played until one player loses the point. If the first serve is out, the server can then hit a second serve, and that serve is played until one player loses the point. Suppose a tennis player wins 76% of the points for which his first serve lands in (so the chance of winning the point if the first serve is in is 0.76) and 43% of the points in which his first serve does not land in. Also suppose his first serve lands in for 64% of the points he serves. Use this information to complete the hypothetical 10,000 table below and to determine the chance that this player wins a point when he serves. Wins point Does not win point Total First serve in First serve not in Total 10, This lesson is part of STATWAY, A Pathway Through College Statistics, which is a product of a Carnegie Networked Improvement Community that seeks to advance student success. Version 1.0, A Pathway Through Statistics, Statway was created by the Charles A. Dana Center at the University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching. This version 1.5 and all subsequent versions, result from the continuous improvement efforts of the Carnegie Networked Improvement Community. The network brings together community college faculty and staff, designers, researchers and developers. It is an open-resource research and development community that seeks to harvest the wisdom of its diverse participants in systematic and disciplined inquiries to improve developmental mathematics instruction. For more information on the Statway Networked Improvement Community, please visit carnegiefoundation.org. For the most recent version of instructional materials, visit Statway.org/kernel STATWAY and the Carnegie Foundation logo are trademarks of the Carnegie Foundation for the Advancement of Teaching. A Pathway Through College Statistics may be used as provided in the CC BY license, but neither the Statway trademark nor the Carnegie Foundation logo may be used without the prior written consent of the Carnegie Foundation.
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