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Math 30011 Kent State University – Stark Campus Spring 2009 – March 18, 2009 – Dr. Kasturiarachi Basic Probability & Statistics Lab #2: Probability NAME: PARTNER: In this Lab we will explore the concepts of probability in several different ways. The first Activity #1 uses a two-way table to find probabilities of dependent events. You will have to recall the formulas for conditional probability. P(A∩B) = P(A) ∙ P(B) for independent events A, B. P(A∩B) = P(A) ∙ P(B|A) for dependent events A, B. The third Activity #2 involves two readings taken from medical journals. You have to briefly analyze the conclusions drawn from these studies. The second Activity #3 is an experiment that explores what happens to chance during independent and dependent trials. A small prize will be given to the person with the closest guess. At the end of the lab you will have to hand in the completed lab. Check all your calculations, sign below and hand in. Thanks. SIGNATURE: ________________________________________ Activity #1: A Study in Diagnostic Testing You will be given 10 minutes to complete this activity. Work on this activity with at least one other student. A female student is interested in determining whether or not she should PANIC about the positive result she received when performing a home pregnancy test. To answer her question, she finds the following data on the accuracy of the pregnancy test she used when performed on 1,000 college-aged women. Test Result True Status Pregnant (+) Not Pregnant (-) Total Pregnant (+) 48 2 50 Not Pregnant (-) 38 912 950 Total 86 914 1000 1. What percentage of the women in the sample of 1,000 women were pregnant? 2. What percentage of the women in the sample of 1,000 women tested positive? 3. Given a woman is pregnant, what is the probability that she gets a positive pregnancy test? 4. Given a women receives a positive pregnancy test, what is the probability that she is truly pregnant? 5. Based on your above answers, argue one way or the other as to whether the student should or should not panic? Can you recommend the next step she should take? Activity #2: Relative measures in Risk Management You will be given 15 minutes to complete this activity. Read the following two news articles, and then answer the questions that follow on the back. Article A --- Plaque Breaks and Heart Attacks ORLANDO, Fla., Nov. 10 (Reuters)-Researchers say they have found out why sudden strenuous exercise can bring on a heart attack in men with blocked arteries. The fatty plaque that blocks the arteries of some people with heart disease is delicate and liable to rupture, they said. Bits of plaque then block the arteries leading to the heart, causing heart attacks, the researchers said on Sunday at the meeting here of the American Heart Association. So men with high cholesterol and other evidence of heart disease who do not exercise regularly should be careful about strenuous activity. Dr. Renu Virmani of the Armed Forces Institute of Pathology in Washington and her colleagues studied 146 sedentary men, most of them in their 50's, with heart disease. The researchers found that those who died during exercise were more than three times as likely to have evidence of ruptured plaque than men who died during normal daily activities or while resting. The plaque burst in 68 percent of the men who died while engaged in strenuous activity, compared with 21 percent of the men who died during normal daily activities. Dr. Virmani said the men who were doing exercise, like mowing the lawn, doing heavy lifting, playing basketball or having sex, had more vulnerable plaque. Men who have coronary heart disease and high cholesterol should be careful about undertaking vigorous physical activity, and they should try to lower their cholesterol through diet or drugs before beginning any type of exercise program, Dr. Virmani said. (New York Times, 11/11/97) Article B --- Being Thin While Pregnant is Seen as Best By the Associated Press Women who are very thin when they become pregnant for the first time are far more likely to have healthy babies than obese women or even those of normal weight, a Swedish study has found. The study, being published today in the New England Journal of Medicine, contradicts the common belief that heavier women are more likely to have healthy, fullterm babies. Although the thinnest women are more likely to have underweight babies, they are no more likely to give birth prematurely or to have their babies die in the first week after birth, the study found. "Advising lean women to gain weight before becoming pregnant may not be justified," the researchers concluded in the study, led by Dr. Sven Cnattingius of the Karolinska Institute in Stockholm. The study looked at the medical records of 167,750 Swedish women who gave birth in 1992 and 1993. The women were divided into four groups based on their body-mass index: lean, normal, overweight, or obese. For a woman who is 5 feet 5 inches, for example, a weight of less than 120 pounds is lean; a weight of 120 to 149 pounds is normal; a weight of 150 to 179 pounds is overweight, and a weight greater than 180 pounds is obese. The researchers then looked for problems: stillbirths after 28 weeks of pregnancy, infant deaths in the first week after birth, premature deliveries and low birth weight. The mother's weight before pregnancy proved most important in women having their first babies. Among first-time mothers, the study found, normal women had twice the risk of stillbirths compared with lean women; overweight women had three times the risk, and obese women had four times the risk. Among women who already had children, only the most overweight women had a greater risk of stillbirths than the thinnest. The babies of normal, overweight and obese firsttime mothers were twice as likely to die in the first week of life as the babies of lean women. (New York Times, 1/15/98) 1. 2. In article A, what allows the researchers to claim that "those who died during exercise were more than three times as likely to have evidence of ruptured plaque than men who died during normal activities or while resting."? In article B, the journalist reported "the babies of normal, overweight and obese first-time mothers were twice as likely to die in the first week of life as babies of lean women." Related to this statement, what isn't reported in Article B that was reported in Article A? Does it matter whether "this" is not reported? Activity #3: Your chances in Dependent and Independent Trials Prologue There are many situations in which researchers and ordinary people want to know if a certain event will occur. For example, an environmentalist wants to know if the probability that the water in a well is polluted, a quality control manager wants to know the probability of producing a defective item during a continuous production process, or a pollster wants to know the probability that voters favor a certain ballot issue. Since we cannot test all wells, check all products, and poll all voters, we use experiments to estimate probabilities. Objective What is the probability that a tossed thumb tack lands with point down? To answer this question we will perform the experiment in two different ways and find the probabilities. Question 1: Guess the probability (two decimals) of landing point down if trials are independent? ____________ Question 2: Guess the probability (two decimals) of landing point down if trials are dependent? ____________ 1. Collect Data for independent trails: (a). Put a single thumb tack in the cup and toss it 30 times and record the number of times it points down. (b). Copy the values from the other 6 groups. You should now have 210 total tosses. (c). Fill the last row to determine the relative frequency probabilities. # of tosses (independent) 30 60 90 120 150 180 210 # landing point down Relative frequency probability 2. Collect Data for dependent trails: (a). Put a 10 thumb tacks in the cup and toss it 3 times and record the number of times it points down. (b). Copy the values from the other 6 groups. You should now have 210 total tosses. (c). Fill the last row to determine the relative frequency probabilities. # of tosses (dependent) 30 60 90 120 150 # landing point down Relative frequency probability 3. In your opinion, which of the above experiments gives you a more accurate probability? 180 210