1.10: Percentages and Probabilities

INTRODUCTION

In this collaboration, we will be making quantitative statements about data that can be displayed in two-way frequency tables. Two-way frequency tables are used to display data based on two categorical variables, where each cell shows how many observations fall in each category.

Consider the following example. Out of 170 students who took a Chemistry 201 test,

• 20 students passed it even though they didn’t complete the corresponding Test Review assignment provided by the instructor
• 80 students passed after completing the corresponding Test Review assignment provided by the instructor
• 60 students didn’t pass it, and they didn’t complete the corresponding Test Review assignment provided by the instructor
• 10 students didn’t pass it even though they completed the corresponding Test Review assignment provided by the instructor

The given dataset of 170 students can be displayed in the following two-way frequency table.

In your group, can you think of other examples of data that can be displayed in a two-way frequency table like the one above?

SPECIFIC OBJECTIVES

By the end of this collaboration, you should understand that

• a percent has different uses, including being used to express the likelihood (or probability) of a certain event.
• it is important to select the correct comparison value and reference value when calculating percentages.

By the end of this collaboration, you should be able to

• extract relevant information from a table.
• select the appropriate values to calculate probabilities.

PROBLEM SITUATION: USING PERCENTAGES TO DESCRIBE THE ACCURACY OF MEDICAL TESTS

Some athletes use performance-enhancing drugs (PEDs) to improve how they do in sports. Schools, sports leagues, and other sports organizations usually do not allow the use of PEDs. These groups can administer athletes a blood or urine test to determine if the athletes are using drugs. In this problem situation, you will calculate the probability that such a test gives correct results and the probability that such a test gives incorrect results. Probability means the chance that something happens. Probabilities are often reported in percentages (%).

Five hundred athletes have undergone a test to determine if they use PEDs. A positive (+) test result indicates that the athlete is using a PED. A negative (–) test result indicates the athlete is not using these drugs. However, this test is not 100% accurate. This means that some errors may be present in the test results. The table below shows how often the test correctly determines if athletes used PEDs.

Use the numbers in the table to answer the following questions. Be careful of what numbers you use for the numerator and denominator in your calculations.

(1) The table is missing one row total and one column total. Discuss with your group what the missing numbers should be. Fill in the missing totals (a) and (b).

(2) You might want to know: “If an athlete is using PEDs, what is the probability that this test gives a positive result?” Use the steps below to answer that question.

(a) How many athletes are using PEDs?

(b) How many of the athletes using PEDs received a positive test result?

(c) If an athlete is using PEDs, what is the probability that this test gives a positive result?

(3) The table shows that one athlete who was using PEDs received a negative test result. This means the test incorrectly identified this single athlete. What percentage of athletes who are using PEDs incorrectly test negative?

Hint: Think about the ratio of the number of incorrect negative results compared to the number of athletes who were using PEDs.

(4) You might want to know: “If an athlete is not using PEDs, what is the probability that this test gives a negative result?” Use the steps below to answer that question.

(a) How many athletes are not using PEDs?

(b) How many of the athletes not using PEDs received a negative test result?

(c) If an athlete is not using PEDs, what is the probability that this test gives a negative result?

(5) Think about a situation in which a school principal receives a positive result on a PED test for a student athlete. Answer these questions:

(a) What percentage of athletes not using PEDs incorrectly test positive?

Hint: Think about the ratio of the number of incorrect positive results compared to the number of those who are not using PEDS.

(b) How should the principal think about this percentage? What recommendation would you make to the principal about what she should do when she receives a positive result on a PED test for a student athlete?

(6) Are your results from Question 2(c) and Question 3 related? Are your results from Question 4(c) and Question 5(a) related? If yes, briefly explain.

(7) Refer to the table to answer questions (a) - (c).

(a) You might also want to know: “What percentage of all positive test results are from athletes who are actually using PEDs?” Round your answer to one decimal place.

Hint: Think about the ratio of the number of correct positive results compared to the number of all positive test results.

(b) Now answer “What percentage of all positive test results are from athletes who are NOT using PEDs?”

(c) Does your answer to part (b) surprise you? Does this make you rethink your answer for Question 5(b) above? Now what recommendation would you make to the principal about what she should do when she receives a positive result on a PED test for a student athlete?

(8) You might also want to know: “What percentage of all negative test results are from athletes who are not using PEDs?” Hint: Think about the ratio of the number of correct negative results compared to the number of all negative test results.

Use the table to answer that question. Round to one decimal place.

(9) You can use a percentage to show how accurate the PED test was. A test is accurate when it produces very few mistakes or errors. Pick one percentage that you think best describes how accurate the test was. Explain what this percentage says about the test and why you picked this percentage.

(10) Now, think about how to use a percentage to show how inaccurate the test was. A test is inaccurate if it produces many errors. Pick one percentage to show how inaccurate the test was. Discuss in your group what this percentage says about the test and why you picked this percentage.

FURTHER APPLICATIONS

You encountered the following dataset in this lesson:

You will encounter a similar dataset in Question 4 of Exercise 1.9. Here is the dataset for easy reference:

(11) In this question, you should refer to the dataset from the collaboration as “Population 1 (P1),” and the dataset from Exercise 1.9, Question 4, as “Population 2 (P2).”

(a) A prevalence rate is the percentage of people in a population who have a certain disease or behave in a certain way. Find the prevalence rate of athletes using PEDs for P1 and P2. Another way to say this is, “What percentage of the population of athletes used PEDs?” Put your answers in the first row of the following table.

(b) Record your answers for P1 and P2 in (a) - (j) in the table above.

(c) What does the information in the table indicate about how prevalence can affect test outcomes? Discuss in your group. Write your answer using the Writing Principle.

MAKING CONNECTIONS

Record the important mathematical ideas from the discussion.