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Chapter 5: Q4 (page 220)
Using the same SAT questions described in Exercise 2, is 20 a significantly high number of correct answers for someone making random guesses?
20 is not a significantly high number of correct answers.
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Composite Sampling. Exercises 33 and 34 involve the method of composite sampling, whereby a medical testing laboratory saves time and money by combining blood samples for tests so that only one test is conducted for several people. A combined sample tests positive if at least one person has the disease. If a combined sample tests positive, then individual blood tests are used to identify the individual with the disease or disorder.
HIV It is estimated that worldwide, 1% of those aged 15鈥�49 are infected with the human immunodeficiency virus (HIV) (based on data from the National Institutes of Health). In tests for HIV, blood samples from 36 people are combined. What is the probability that the combined sample tests positive for HIV? Is it unlikely for such a combined sample to test positive?
Expected Value for Life Insurance There is a 0.9986 probability that a randomly selected 30-year-old male lives through the year (based on data from the U.S. Department of Health and Human Services). A Fidelity life insurance company charges \(161 for insuring that the male will live through the year. If the male does not survive the year, the policy pays out \)100,000 as a death benefit.
a. From the perspective of the 30-year-old male, what are the monetary values corresponding to the two events of surviving the year and not surviving?
b. If a 30-year-old male purchases the policy, what is his expected value?
c. Can the insurance company expect to make a profit from many such policies? Why?
In Exercises 21鈥�25, refer to the accompanying table, which describes the numbers of adults in groups of five who reported sleepwalking (based on data from 鈥淧revalence and Comorbidity of Nocturnal Wandering In the U.S. Adult General Population,鈥� by Ohayon et al., Neurology, Vol. 78, No. 20).
Using Probabilities for Identifying Significant Events
a.Find the probability of getting exactly 1 sleepwalker among 5 adults.
b. Find the probability of getting 1 or fewer sleepwalkers among 5 adults.
c. Which probability is relevant for determining whether 1 is a significantly lownumber of sleepwalkers among 5 adults: the result from part (a) or part (b)?
d. Is 1 a significantly low number of sleepwalkers among 5 adults? Why or why not?
x | P(x) |
0 | 0.172 |
1 | 0.363 |
2 | 0.306 |
3 | 0.129 |
4 | 0.027 |
5 | 0.002 |
In Exercises 25鈥�28, find the probabilities and answer the questions.
Vision Correction A survey sponsored by the Vision Council showed that 79% of adults need correction (eyeglasses, contacts, surgery, etc.) for their eyesight. If 20 adults are randomly selected, find the probability that at least 19 of them need correction for their eyesight. Is 19 a significantly high number of adults requiring eyesight correction?
Hypergeometric Distribution If we sample from a small finite population without replacement, the binomial distribution should not be used because the events are not independent. If sampling is done without replacement and the outcomes belong to one of two types, we can use the hypergeometric distribution. If a population has A objects of one type (such as lottery numbers you selected), while the remaining B objects are of the other type (such as lottery numbers you didn鈥檛 select), and if n objects are sampled without replacement (such as six drawn lottery numbers), then the probability of getting x objects of type A and n - x objects of type B is
\(P\left( x \right) = \frac{{A!}}{{\left( {A - x} \right)!x!}} \times \frac{{B!}}{{\left( {B - n + x} \right)!\left( {n - x} \right)!}} \div \frac{{\left( {A + B} \right)!}}{{\left( {A + B - n} \right)!n!}}\)
In New Jersey鈥檚 Pick 6 lottery game, a bettor selects six numbers from 1 to 49 (without repetition), and a winning six-number combination is later randomly selected. Find the probabilities of getting exactly two winning numbers with one ticket. (Hint: Use A = 6, B = 43, n = 6, and x = 2.)
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