Show all work in a neat and organized fashion. Clearly indicate your answers.
10 points possible.
Use the standard normal distribution table to obtain any required z-values or areas, showing all work.
Always use the most appropriate probability distribution that we have studied.
1. (10 pts.) (a) Find the probability that a person flipping a fair coin gets the third head on the seventh flip.
(b) Find the probability that a person flipping a fair coin gets the first head on the fourth flip.
2. (10 pts.) A certain drug treatment cures 90% of hookworm in children (ficticious but realistic value; see Hutchison, et al., Clinical Trial of Mebendazole, a broad-spectrum anthelminthic, British Medical Journal 2, (1975) 309-310). Suppose that 20 children suffering from hookworm are to be treated, and that the children can be regarded as a random sample from the population.
(a) Find the probability that all 20 will be cured.
(b) Find the probability that exactly 90% will be cured.
3. (10 pts.) To avoid detection at customs, a traveler has placed 6 narcotic tablets in a bottle containing 9 vitamin pills that are similar in appearance. If the customs official selects 3 different tablets at random for analysis, what is the probability that the traveler will be arrested for illegal possession of narcotics?
4. (10 pts.) An important part of the customer-service responsibilities of a telephone company relate to the speed with which troubles in residential service can be repaired. Suppose that past data indicate that the likelihood is 0.70 that troubles in a residential service can be repaired on the same day. Of 50 troubles reported in a given day, what is the approximate probability that at least 25 will be repaired by the end of the day?
5. (10 pts.) A rescue squad spokesperson claims that the average time between calls is 20 minutes. What is the probability that the time between two consecutive calls is greater than 1.5 hours?
6. (10 pts.) A completely randomized design will be perfomed on 6 subjects. Let A, B, C, D, E, and F denote the subjects. Two subjects will be assigned at random to treatment 1, and the other four will get treatment 2.
(a) List all possible assignments of subjects to treatments.
(b) Assume A, B, C, and D give successes, while E and F give failures. Find all possible 2 by 2 contingency tables (using ``treatment number'' for the rows), and find the probability that each table occurs.
7. (10 pts.) Suppose the sediment density (g/cm) of a randomly selected specimen from a certain region has a mean of 2.65 g/cm and a standard deviation of 0.85 g/cm (suggested in ``Modeling Sediment and Water Column Interactions for Hydrophobic Pollutants,'' Water Research, 1984: 1169-1174).
(a) If a random sample of 35 specimens is selected, what is the probability that the average sediment density is at most 3.00 g/cm?
(b) There is an 85 percent chance that the average sediment density of 35 specimens will be above what value?
8. (10 pts.) Let X = the time it takes a read/write head to locate a desired record on a computer disk memory device once the head has been positioned over the correct track. If the disks rotate once every 25 millisec, a reasonable assumption is that X is uniformly distributed on the interval from 0 to 25 millisec. Compute the probability that X is between 5 and 15 millisec.
9. (10 pts.) The article ``Characterization of Room Temperature Daming in Aluminum-Indium Alloys'' (Metallurgical Trans., 1993: 1611-1619) suggests that Al matrix grain size (mm) for an alloy consisting of 2% indium could be modeled with a normal distribution with a mean value of 96 mm and standard deviation 14 mm.
(a) What interval, symmetrically distributed around the mean, includes 87% of all grain sizes?
(b) If four grain sizes are independently selected, what is the probability that at least one is greater than 100 mm?
10. (10 pts.) In studying sleep patterns in humans, five stages of sleep (drowsiness, light, intermediate, deep, REM) are recognized by using the electroencephalogram. Intermediate sleep is characterized by the presence of high-amplitude waves averaging about 2 waves per second. What is the probability that during intermediate sleep, exactly 7 such waves will appear in a 5-second period?