91Ó°ÊÓ

Which of the following are continuous variables, and which are discrete? (a) Number of traffic fatalities per year in the state of Florida (b) Distance a golf ball travels after being hit with a driver (c) Time required to drive from home to college on any given day (d) Number of ships in Pearl Harbor on any given day (e) Your weight before breakfast each morning

Consider a binomial experiment with \(n=20\) trials and \(p=0.40\) (a) Find the expected value and the standard deviation of the distribution. (b) Would it be unusual to obtain fewer than 3 successes? Explain. Confirm your answer by looking at the binomial probability distribution table.

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.

Susan is taking Western Civilization this semester on a pass/fail basis. The department teaching the course has a history of passing \(77 \%\) of the students in Western Civilization each term. Let \(n=1\) \(2,3, \ldots\) represent the number of times a student takes western civilization until the first passing grade is received. (Assume the trials are independent.) (a) Write out a formula for the probability distribution of the random variable \(n\) (b) What is the probability that Susan passes on the first try \((n=1) ?\) (c) What is the probability that Susan first passes on the second try \((n=2) ?\) (d) What is the probability that Susan needs three or more tries to pass western civilization? (e) What is the expected number of attempts at western civilization Susan must make to have her (first) pass? Hint: Use \(\mu\) for the geometric distribution and round.

Consider a binomial distribution with \(n=10\) trials and the probability of success on a single trial \(p=0.85\) (a) Is the distribution skewed left, skewed right, or symmetric? (b) Compute the expected number of successes in 10 trials. (c) Given the high probability of success \(p\) on a single trial, would you expect \(P(r \leq 3)\) to be very high or very low? Explain. (d) Given the high probability of success \(p\) on a single trial, would you expect \(P(r \geq 8)\) to be very high or very low? Explain.

Jim is a 60 -year-old Anglo male in reasonably good health. He wants to take out a 50,000 dollar term (i.e., straight death benefit) life insurance policy until he is \(65 .\) The policy will expire on his 65 th birthday. The probability of death in a given year is provided by the Vital Statistics Section of the Statistical Abstract of the United States (116th edition). $$\begin{array}{|l|ccccc|} \hline x=\text { age } & 60 & 61 & 62 & 63 & 64 \\\ \hline P( \text { death at this age) } & 0.01191 & 0.01292 & 0.01396 & 0.01503 & 0.01613 \\ \hline \end{array}$$ Jim is applying to Big Rock Insurance Company for his term insurance policy. (a) What is the probability that Jim will die in his 60 th year? Using this probability and the 50,000 dollar death benefit, what is the expected cost to Big Rock Insurance? (b) Repeat part (a) for years \(61,62,63,\) and \(64 .\) What would be the total expected cost to Big Rock Insurance over the years 60 through \(64 ?\) (c) If Big Rock Insurance wants to make a profit of 700 dollar above the expected total cost paid out for Jim's death, how much should it charge for the policy? (d) If Big Rock Insurance Company charges 5000 dollar for the policy, how much profit does the company expect to make?

Aldrich Ames is a convicted traitor who leaked American secrets to a foreign power. Yet Ames took routine lie detector tests and each time passed them. How can this be done? Recognizing control questions, employing unusual breathing patterns, biting one's tongue at the right time, pressing one's toes hard to the floor, and counting backward by 7 are countermeasures that are difficult to detect but can change the results of a polygraph examination (Source: Lies' Liesi't Lies\%' The Psychology of Deceit, by C. V. Ford, professor of psychiatry, University of Alabama). In fact, it is reported in Professor Ford's book that after only 20 minutes of instruction by "Buzz" Fay (a prison inmate), \(85 \%\) of those trained were able to pass the polygraph examination even when guilty of a crime. Suppose that a random sample of nine students (in a psychology laboratory) are told a "secret" and then given instructions on how to pass the polygraph examination without revealing their knowledge of the secret. What is the probability that (a) all the students are able to pass the polygraph examination? (b) more than half the students are able to pass the polygraph examination? (c) no more than four of the students are able to pass the polygraph examination? (d) all the students fail the polygraph examination?

Are your finances, buying habits, medical records, and phone calls really private? A real concern for many adults is that computers and the Internet are reducing privacy. A survey conducted by Peter D. Hart Research Associates for the Shell Poll was reported in USA Today. According to the survey, \(37 \%\) of adults are concerned that employers are monitoring phone calls. Use the binomial distribution formula to calculate the probability that (a) out of five adults, none is concerned that employers are monitoring phone calls. (b) out of five adults, all are concerned that employers are monitoring phone calls. (c) out of five adults, exactly three are concerned that employers are monitoring phone calls.