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Chapter 5: Problem 10
If \(x\) has a binomial distribution with \(p=.5\), will the shape of the probability distribution be symmetric, skewed to the left, or skewed to the right?
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Insulin-dependent diabetes (IDD) is a common chronic disorder of children. This disease occurs most frequently in persons of northern European descent, but the incidence ranges from a low of \(1-2\) cases per 100,000 per year to a high of more than 40 per 100,000 in parts of Finland. \({ }^{11}\) Let us assume that an area in Europe has an incidence of 5 cases per 100,000 per year. a. Can the distribution of the number of cases of IDD in this area be approximated by a Poisson distribution? If so, what is the mean? b. What is the probability that the number of cases of IDD in this area is less than or equal to 3 per \(100,000 ?\) c. What is the probability that the number of cases is greater than or equal to 3 but less than or equal to 7 per \(100,000 ?\) d. Would you expect to observe 10 or more cases of IDD per 100,000 in this area in a given year? Why or why not?
Evidence shows that the probability that a driver will be involved in a serious automobile accident during a given year is .01. A particular corporation employs 100 full-time traveling sales reps. Based on this evidence, use the Poisson approximation to the binomial distribution to find the probability that exactly two of the sales reps will be involved in a serious automobile accident during the coming year.
Consider the medical payment problem in Exercise 5.27 in a more realistic setting. Of all patients admitted to a medical clinic, \(30 \%\) fail to pay their bills and the debts are eventually forgiven. If the clinic treats 2000 different patients over a period of 1 year, what is the mean (expected) number of debts that have to be forgiven? If \(x\) is the number of forgiven debts in the group of 2000 patients, find the variance and standard deviation of \(x\). What can you say about the probability that \(x\) will exceed \(700 ?\) (HINT: Use the values of \(\mu\) and \(\sigma,\) along with Tchebysheff's Theorem, to answer this question.)
Under what conditions can the Poisson random variable be used to approximate the probabilities associated with the binomial random variable? What application does the Poisson distribution have other than to estimate certain binomial probabilities?
A subject is taught to do a task in two different ways. Studies have shown that when subjected to mental strain and asked to perform the task, the subject most often reverts to the method first learned, regardless of whether it was easier or more difficult. If the probability that a subject returns to the first method learned is .8 and six subjects are tested, what is the probability that at least five of the subjects revert to their first learned method when asked to perform their task under stress?
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