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Statistical Literacy Which of the following are continuous variables, and which are discrete? (a) Speed of an airplane (b) Age of a college professor chosen at random (c) Number of books in the college bookstore (d) Weight of a football player chosen at random (e) Number of lightning strikes in Rocky Mountain National Park on a given day

Statistical Literacy Consider the probability distribution of a random variable \(x .\) Is the expected value of the distribution necessarily one of the possible values of \(x\) ? Explain or give an example.

Basic Computation: Binomial Distribution Consider a binomial experiment with \(n=7\) trials where the probability of success on a single trial is \(p=0.30\). (a) Find \(P(r=0)\). (b) Find \(P(r \geq 1)\) by using the complement rule.

Agriculture: Apples Approximately \(3.6 \%\) of all (untreated) Jonathan apples had bitter pit in a study conducted by the botanists Ratkowsky and Martin (Source: Australian Journal of Agricultural Research, Vol. 25, pp. \(783-790)\). (Bitter pit is a disease of apples resulting in a soggy core, which can be caused either by overwatering the apple tree or by a calcium deficiency in the soil.) Let \(n\) be a random variable that represents the first Jonathan apple chosen at random that has bitter pit. (a) Write out a formula for the probability distribution of the random variable \(n\). (b) Find the probabilities that \(n=3, n=5\), and \(n=12\). (c) Find the probability that \(n \geq 5\). (d) What is the expected number of apples that must be examined to find the first one with bitter pit? Hint: Use \(\mu\) for the geometric distribution and round.

Fundraiser: Hiking Club The college hiking club is having a fundraiser to buy new equipment for fall and winter outings. The club is selling Chinese fortune cookies at a price of \(\$ 1\) per cookie. Each cookie contains a piece of paper with a different number written on it. A random drawing will determine which number is the winner of a dinner for two at a local Chinese restaurant. The dinner is valued at \(\$ 35\). Since the fortune cookies were donated to the club, we can ignore the cost of the cookies. The club sold 719 cookies before the drawing. (a) Lisa bought 15 cookies. What is the probability she will win the dinner for two? What is the probability she will not win? (b) Interpretation Lisa's expected earnings can be found by multiplying the value of the dinner by the probability that she will win. What are Lisa's expected earnings? How much did she effectively contribute to the hiking club?

Binomial Probabilities: Multiple-Cboice Quiz Richard has just been given a 10 -question multiple-choice quiz in his history class. Each question has five answers, of which only one is correct. Since Richard has not attended class recently, he doesn't know any of the answers. Assuming that Richard guesses on all 10 questions, find the indicated probabilities. (a) What is the probability that he will answer all questions correctly? (b) What is the probability that he will answer all questions incorrectly? (c) What is the probability that he will answer at least one of the questions correctly? Compute this probability two ways. First, use the rule for mutually exclusive events and the probabilities shown in Table 3 of Appendix II. Then use the fact that \(P(r \geq 1)=1-P(r=0)\). Compare the two results. Should they be equal? Are they equal? If not, how do you account for the difference? (d) What is the probability that Richard will answer at least half the questions correctly?

Criminal Justice: Jury Duty Have you ever tried to get out of jury duty? About \(25 \%\) of those called will find an excuse (work, poor health, travel out of town, etc.) to avoid jury duty (Source: Bernice Kanner, Are You Normal?, St. Martin's Press, New York). If 12 people are called for jury duty, (a) what is the probability that all 12 will be available to serve on the jury? (b) what is the probability that 6 or more will not be available to serve on the jury? (c) Find the expected number of those available to serve on the jury. What is the standard deviation? (d) Quota Problem How many people \(n\) must the jury commissioner contact to be \(95.9 \%\) sure of finding at least 12 people who are available to serve?

Criminal Justice: Convictions Innocent until proven guilty? In Japanese criminal trials, about \(95 \%\) of the defendants are found guilty. In the United States, about \(60 \%\) of the defendants are found guilty in criminal trials (Source: The Book of Risks, by Larry Laudan, John Wiley and Sons). Suppose you are a news reporter following seven criminal trials. (a) If the trials were in Japan, what is the probability that all the defendants would be found guilty? What is this probability if the trials were in the United States? (b) Of the seven trials, what is the expected number of guilty verdicts in Japan? What is the expected number in the United States? What is the standard deviation in each case? (c) Quota Problem As a U.S. news reporter, how many trials \(n\) would you need to cover to be at least \(99 \%\) sure of two or more convictions? How many trials \(n\) would you need if you covered trials in Japan?