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According to an article on Yahoo.com on February 19,2012, the average salary of actuaries in the U.S. is \(\$ 98,620\) a year (http://education.yahoo.net/articles/careers_for_shy_people_2.htm?kid=1KWO3). Suppose that currently the distribution of annual salaries of all actuaries in the U.S. is approximately normal with a mean of \(\$ 98,620\) and a standard deviation of \(\$ 18,000\). How much would an actuary have to be paid in order to be in the highest-paid \(10 \%\) of all actuaries?

At Jen and Perry Ice Cream Company, a machine fills 1-pound cartons of Top Flavor ice cream. The machine can be set to dispense, on average, any amount of ice cream into these cartons. However, the machine does not put exactly the same amount of ice cream into each carton; it varies from carton to carton. It is known that the amount of ice cream put into each such carton has a normal distribution with a standard deviation of 18 ounce. The quality control inspector wants to set the machine such that at least \(90 \%\) of the cartons have more than 16 ounces of ice cream. What should be the mean amount of ice cream put into these cartons by this machine?

A soft-drink vending machine is supposed to pour 8 ounces of the drink into a paper cup. However, the actual amount poured into a cup varies. The amount poured into a cup follows a normal distribution with a mean that can be set to any desired amount by adjusting the machine. The standard deviation of the amount poured is always \(.07\) ounce regardless of the mean amount. If the owner of the machine wants to be \(99 \%\) sure that the amount in each cup is 8 ounces or more, to what level should she set the mean?

According to the College Board (http://professionals.collegeboard.com/gateway), the mean SAT mathematics score for all college-bound seniors was 511 in 2011 . Suppose that this is true for the current population of college-bound seniors. Furthermore, assume that \(17 \%\) of college-bound seniors scored below 410 in this test. Assume that the distribution of SAT mathematics scores for college-bound seniors is approximately normal. a. Find the standard deviation of the mathematics SAT scores for college-bound seniors. b. Find the percentage of college-bound seniors whose mathematics SAT scores were above 660 .

A charter bus company is advertising a singles outing on a bus that holds 60 passengers. The company has found that, on average, \(10 \%\) of ticket holders do not show up for such trips; hence, the company routinely overbooks such trips. Assume that passengers act independently of one another. a. If the company sells 65 tickets, what is the probability that the bus can hold all the ticket holders who actually show up? In other words, find the probability that 60 or fewer passengers show up. b. What is the largest number of tickets the company can sell and still be at least \(95 \%\) sure that the bus can hold all the ticket holders who actually show up?