220 CHAPTER 5 Discrete Probability Distributions a. This procedure does satisfy the requirements for a binomial distribution, as shown below. 1. The number of trials (10) is fixed. 2. The 10 trials are independent because the probability of any random adult smartphone owner being cashless is not affected by the results from the other randomly selected adults. 3. Each of the 10 trials has two categories of outcomes: The selected person is either cashless or is not. 4. For each randomly selected adult smartphone owner, there is a 0.05 probability that this person is cashless, and that probability remains the same for each of the ten selected people. b. Having concluded that the given procedure does result in a binomial distribution, we now proceed to identify the values of n, x, p, and q. 1. With ten randomly selected adults, we have n = 10. 2. We want the probability of exactly two who are cashless, so x = 2. 3. The probability of success (getting someone who is cashless) for one selection is 0.05, so p = 0.05. 4. The probability of failure (not getting someone who is cashless) is 0.95, so q = 0.95. Consistent Notation Again, it is very important to be sure that x and p both refer to the same concept of “success.” In this example, we use x to count the number of people who are cashless, so p must be the probability that the selected person is cashless. Therefore, x and p do use the same concept of success: being cashless. SOLUTION YOUR TURN. Do Exercise 5 “Pew Survey.” “How Statistics Can Help Save Failing Hearts” A New York Times article by David Leonhardt featured the headline of “How Statistics Can Help Save Failing Hearts.” Leonhardt writes that patients have the best chance of recovery if their clogged arteries are opened within two hours of a heart attack. In 2005, the U.S. Department of Health and Human Services began posting hospital data on its website www.hospitalcompare.hhs.gov, and it included the percentage of heart attack patients who received treatment for blocked arteries within two hours of arrival at the hospital. Not wanting to be embarrassed by poor data, doctors and hospitals are reducing the time it takes to unblock those clogged arteries. Leonhardt writes about the University of California, San Francisco Medical Center, which cut its time in half from almost three hours to about 90 minutes. Effective use of simple statistics can save lives. C H l S F i Treating Dependent Events as Independent When selecting a sample (as in a survey), we usually sample without replacement. Sampling without replacement results in dependent events, which violates a requirement of a binomial distribution. However, we can often treat the events as if they were independent by applying the following 5% guideline introduced in Section 4-2: 5% Guideline for Cumbersome Calculations When sampling without replacement and the sample size is no more than 5% of the size of the population, treat the selections as being independent (even though they are actually dependent). Methods for Finding Binomial Probabilities We now proceed with three methods for finding the probabilities corresponding to the random variable x in a binomial distribution. The first method involves calculations using the binomial probability formula and is the basis for the other two methods. The second method involves the use of software or a calculator, and the third method involves the use of the Appendix Table A-1. (With technology so widespread, such tables are becoming obsolete.) If using technology that automatically produces binomial probabilities, we recommend that you solve one or two exercises using Method 1 to better understand the basis for the calculations.
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