Math 346 Exam 2 Show all work in a neat and organized fashion ...

Math 346
Exam 2
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 broadspectrum
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?
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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 (µm) for an alloy consisting of 2% indium could be modeled
with a normal distribution with a mean value of 96 µm and standard deviation
14 µm.
(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 µm?
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?
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