STATISTICS 401 PRACTICE PROBLEMS FOR FINAL EXAM Spring 2000 1 The probability that an American engineering firm will establish a branch office in Toronto is 7 the probability that it will establish a branch office in Mexico City is 4 and the probability it will establish a branch office in at least one of the cities is 8 Define events A and B as follows A American engineering firm will establish a branch office in Toronto B American engineering firm will establish a branch office in Mexico City It may be useful to draw Venn diagrams For parts a c what is the probability that a branch office will be established in a in both cities b neither of the cities c exactly one of the cities d Are events A and B independent Why or why not 2 From a load of 50 Panasonic Tape recorders 35 are destined for New York and 15 for Boston If two are shipped to Trenton by mistake and the selection is random what are the probabilities that a both tape recorders should have gone to New York b one should have gone to New York and one to Boston 3 From 5 biologists 4 chemists and 3 physicists a how many committees of size 4 can be formed b how many committees containing 2 biologists 1 chemist and 1 physicist can be formed 4 Seventy percent of the light aircraft that disappear while in flight in a certain country are subsequently discovered Of the aircraft that are discovered 60 have an emergency locator whereas 90 of the aircraft not discovered do not have such a locator Define events A and B as follows A light aircraft that disappears is discovered B light aircraft that disappears has an emergency locator Suppose a light aircraft has disappeared a What is the probability that it has an emergency locator and it will not be discovered b What is the probability that it has an emergency locator c If it has an emergency locator what is the probability that it will not be discovered 5 Grafting the uniting of the stem of one plant with the stem or root of another is widely used commercially to grow the stem of one variety that produces fine fruit on the root system of another variety with a hardy root system For example most sweet oranges grow on trees grafted to the root of a sour orange variety Suppose that each graft fails independently with probability 3 a Write the formula for the p m f for X the number of grafts that fail in a series of five trials b Plot the cumulative distribution function F c Suppose that the cost of each failed graft is 9 00 Find i the probability that the cost from failed grafts will exceed 20 00 in the five trials ii the expected cost from failed grafts in the five trials iii the variance of the cost from the failed grafts in the five trials 6 Let X denote the amount of time for which the Stat 401 book on two hour reserve at McAllister is checked out by a randomly selected student and suppose that X has density function f x 5x 0 x 2 0 otherwise What is the probability that the book is checked out between 5 and 1 5 hours 7 The finished inside diameter of a piston ring is normally distributed with a mean of 10 cm and a standard deviation of 0 03 cm a Above what value of inside diameter will 85 08 of the piston rings fall b What is the probability that the diameter of a randomly selected piston will be less than 10 06 8 Suppose your waiting time for the bus in the morning has mean 3 minutes and variance 28 minutes while the waiting time in the evening has mean 6 minutes and variance 4 In 25 a typical week you take the bus 5 times in the morning and 3 times in the evening a Calculate the expected value of the total waiting time in a typical week b Calculate the variance of the total waiting time in a typical week State your assumptions Hint Set Xi ith morning waiting time where i 1 5 and Yj jth evening waiting time where j 1 2 3 9 A question relating to a study of the echo location system for bats is how far apart are the bat and an insect when the bat first senses the insect The technical problems for measuring this are complex and so only 11 data points Y1 Y11 were obtained It is given that y 48 36 and s2 327 05 a Construct a 95 confidence interval for the mean distance b What assumption did you use 10 An appliance manufacturer is considering the purchase of a new machine for cutting sheet metal parts If 0 is the average number of metal parts cut per hour by her old machine and is the corresponding average for the new machine the manufacturer wants to test the null hypothesis H0 0 against a suitable alternative What should the alternative be if a she does not want to buy the new machine unless there is evidence it is more productive than the old one b she wants to buy the new machine which has some other nice features unless there is evidence it is less productive than the old one c For each of the two cases above state whether she should buy the new machine if the null hypothesis gets rejected d Suppose that 0 9 5 cuts per hour and she decides to test H0 9 5 against Ha 9 5 at the 05 level She gets the machine manufacturer to lend her a new machine and she measures the number of cuts made by the machine in fifty 50 randomly selected one hour time periods The summary statistics for this random sample are x 9 8 and s 1 095 Would you reject H0 11 A study of the electromechanical protection devices used in electrical power systems showed that of 193 devices that failed when tested 75 where due to mechanical parts failures a Find a 95 confidence interval for p the proportion of failures due to mechanical causes b How large a sample is required to estimate p within 0 03 i e to have a confidence interval of length 0 06 with 95 confidence i when you have no idea of what p might be ii when a preliminary study investigations leads you to believe that p 39 12 A manufacturer of automatic washers provides a particular model in one of three colors white almond or blue Of the first 1000 washers sold it is noted that 400 of the washers were white The manufacturer wants to know if this is evidence that more than a third of consumers prefer white washers a Set up the appropriate null and alternative hypotheses for the manufacturer b Carry out a test of your null hypothesis versus your alternative hypothesis at the 05 level Is there evidence that more than one third of consumers prefer …