4.8: Corequisite- Percentages, Division, Extracting Relevant Information
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- 148612
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)SPECIFIC OBJECTIVES
By the end of this lesson, you should understand that
- percentages involve a numerator (comparison value) and a denominator (reference value).
- 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 lesson, you should be able to
- correctly identify the quantities involved in a verbal statement about percentages.
- convert between ratios and percentages.
- convert between the decimal representation of a number and a percent.
- read and use information presented in a two-way table.
- extract relevant information from a table.
- select the appropriate values to calculate probabilities.
PROBLEM SITUATION 1: THE LANGUAGE OF PERCENTAGES
World Health Organization (WHO) is the part of the United Nations that oversees health issues in the world. The WHO leads numerous studies on tobacco use around the world. In its study called, “Gender and Tobacco,” the organization learned that tobacco use among females is increasing.24 For example, recent research shows that just as many young girls smoke as young boys. The report is filled with information about percentages of females who smoke, percentages of males who smoke, and the percentage of smokers who start smoking by age 10. The language used to describe this information can be difficult to understand; therefore, pay close attention to the language used to describe a percent at the beginning of this lesson. This will help you to understand the findings reported in the WHO study.
Consider the following two quantities:
- Quantity 1 (Q1): The percentage of females who smoke.
- Quantity 2 (Q2): The percentage of smokers who are female.
(1) (a) Are these two quantities equal (Q1 = Q2)?
(b) Could Q1 be greater than Q2 (Q1 > Q2)?
(c) Could Q1 be less than Q2 (Q1 < Q2)?
(d) Explain your reasoning.
(2) What information would you need to calculate these percentages?
Smoking on Two College Campuses
Questions 3 and 4 present two situations with data about smoking. You can use these situations to test your ideas from Questions 1 and 2.
(3) Suppose a study on smoking was conducted at Midland University. The following table indicates the results of the study.
| Males | Females | |
| Smokers | 4,572 | 5,362 |
| Nonsmokers | 10,284 | 12,736 |
(a) What percentage of females smoke at Midland University? Round your answer to the nearest percent. Hint: How many females are involved in the study? How many females are smokers? Think about adding rows and columns.
(b) What percentage of smokers at Midland University are female? Round your answer to the nearest percent.
(4) Suppose a study was conducted at Northwest College. The following table indicates the results of the study.
| Males | Females | |
| Smokers | 1,256 | 536 |
| Nonsmokers | 1,028 | 1,053 |
(a) What percentage of females smoke at Northwest College? Round your answer to the nearest percent.
(b) What percentage of smokers at Northwest College are female? Round your answer to the nearest percent.
(c) A newspaper stated that 40% of the male students at Northwest College are smokers. Is that claim reasonable? Explain why or why not.
(5) In 2021, the World Health Organization conducted a study about the prevalence of tobacco use worldwide.25 The organization reported that 2.1% of the adult females in China smoke tobacco products. In the United States, 15.2% of adult females smoke. Using this information answer the following questions:
(a) Out of 100 adult females in China, about how many are smokers?
(b) Out of 1,000 adult females in China, about how many are smokers?
(c) Out of 100 adult females in the United States, about how many are smokers?
(d) Out of 1,000 adult females in the United States, about how many are smokers?
(e) Suppose you read that 590 out of 1,000 males in China smoke. Based on this data, what percentage of males in China smoke? Round to the nearest whole percent.
(f) Are there more female smokers in China or the United States?
(6) A teacher has collected data on the grades his students received in his morning and afternoon classes. The following tables show two different ways to represent the same data as percentages.
| Grade A | Grade B | Grade C | Grade D | Grade F | |
| Morning Class | 12.5% | 25.0% | 37.5% | 6.3% | 18.7% |
| Afternoon Class | 14.3% | 20.0% | 37.1% | 8.6% | 20.0% |
| Grade A | Grade B | Grade C | Grade D | Grade F | |
| Morning Class | 44.4% | 53.3% | 48.0% | 40.0% | 46.2% |
| Afternoon Class | 55.6% | 46.7% | 52.0% | 60.0% | 53.8% |
(a) Which table could be used to answer the following question: What percentage of the students who received an “A” is in the morning class? Explain your answer.
(b) Which table could be used to answer the following question: What percentage of the students in the morning class received an “A”? Explain your answer.
PROBLEM SITUATION 2: 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.
| Athletes Using PEDs | Athletes Not Using PEDs | Total | |
| Positive Test Result | 9 | 5 | (a) |
| Negative Test Result | 1 | 485 | 486 |
| Total | 10 | (b) | 500 |
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.
(7) The table is missing one row total (a) and one column total (b). Fill in the missing totals.
(8) 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 this 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?
(9) 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.
(10) You might want to know: “If an athlete is not using PEDs, what is the probability that this test gives a negative result?” Work through (a) - (c) below to answer this 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? Record your answer as a percentage in the space below. Round to the nearest whole percent.
(11) Think about a situation in which a school principal receives a positive result on a PED test for a student athlete. Refer back to the table from earlier. 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?
(12) Are your results from Question 8(c) and Question 9 related? Are your results from Question 10(c) and Question 11(a) related? If yes, briefly explain.
(13) For (a) - (c) below, refer back to the table from earlier.
(a) You might also want to know: “What percentage of all positive test results are from athletes who are using PEDs?” Use the table to answer that question. 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?" Record your answer as a percentage in the space below. Round your answer to one decimal place.
(c) Does your answer to part (b) surprise you? Does this make you rethink your answer for Question 11(b)? 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?
(14) You might also want to know: “What percentage of all negative test results are from athletes who are not using PEDs?” Use the table to answer that question. Hint: Think about the ratio of the number of correct negative results compared to the number of all negative test results.
(15) 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.
(16) 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. What does this percentage say about the test and why you picked this percentage?
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