MATH 384. Guía 10.
1. For a car traveling 30 miles per hour (mph), the distance required to brake to a stop is
normally distributed with a mean of 50 feet and a standard deviation of 8 feet. Suppose
you are traveling 30 mph in a residential area and a car moves abruptly into your path at
a distance of 60 feet.
a. If you apply your brakes, what is the probability that you will brake to a stop within 40
feet or less? Within 50 feet or less?
b. If the only way to avoid a collision is to brake to a stop, what is the probability that
you will avoid the collision?
2. Suppose you must establish regulations concerning the maximum number of people
who can occupy an elevator. A study of elevator occupancies indicates that, if eight
people occupy the elevator, the probability distribution of the total weight of the eight
people has a mean equal to 1200 pounds and a variance equal to 9800 lbs? What is the
probability that the total weight of eight people exceeds 1300 pounds? 1500 pounds?
(Assume that the probability distribution is approximately normal.)
3. The discharge of suspended solids from a phosphate mine is normally distributed, with
a mean daily discharge of 27 milligrams per liter (mgl/l) and a standard deviation of 14
mgl/l. What proportion of days will the daily discharge exceed 50 mgl/l?
4. An experimenter wondered if the stem diameters of the dicot sunflower would change
depending on whether the plant was left to sway freely in the wind or was artificially
supported. Suppose that the unsupported stem diameters at the base of a particular
species of sunflower plant have a normal distribution with an average diameter of 35
millimeters (mm) and a standard deviation of 3 mm.
a. What is the probability that a sunflower plant will have a basal diameter of more than
40 mm?
b. If two sunflower plants are randomly selected, what is the probability that both plants
will have a basal diameter of more than 40 mm?
c. Within what limits would you expect the basal diameters to lie, with probability 0.95?
d. What diameter represents the 90th percentile of the distribution of diameters?
5. The number of times x an adult human breathes per minute when at rest depends on
the age of the human and varies greatly from person to person. Suppose the probability
distribution for x is approximately normal, with the mean equal to 16 and the standard
deviation equal to 4. If a person is selected at random and the number x of breaths per
minute while at rest is recorded, what is the probability that x will exceed 22?
6. One method of arriving at economic forecasts is to use a consensus approach. A
forecast is obtained from each of a large number of analysts, and the average of these
individual forecasts is the consensus forecast. Suppose the individual 2003 January prime
interest rate forecasts of all economic analysts are approximately normally distributed,
with the mean equal to 7.5% and the standard deviation equal to 1.3%. If a single analyst
is randomly selected from among this group, what is the probability that the analyst's
forecast of the prime interest rate will take on these values?
a. Exceed 10%
b. Be less than 7%
7. How does the IRS decide on the percentage of income tax returns to audit for each
state? Suppose they do it by randomly selecting 50 values from a normal distribution
with a mean equal to 1.55% and a standard deviation equal to 0.45%. (Computer
programs are available for this type of sampling.)
a. What is the probability that a particular state will have more than 2.5% of its income
tax returns audited?
b. What is the probability that a state will have less than 1 % of its income tax returns
audited?
8. Suppose the numbers of a particular type of bacteria in samples of 1 milliliter (ml) of
drinking water tend to be approximately normally distributed, with a mean of 85 and a
standard deviation of 9. What is the probability that a given 1-ml sample will contain
more than 100 bacteria?
9. A normal random variable x has mean  = 10 and standard deviation  = 2. Find the
probabilities of these x-values:
a. x > 13.5
b. x < 8.2
c. 9.4 < x < 10.6
10. A normal random variable x has mean  = 1.20 and standard deviation  = .15. Find
the probabilities of these x-values:
a. 1.00 < x < 1.10
b. x > 1.38
c. 1.35 < x < 1.50
11. A normal random variable x has an unknown mean  and standard deviation  = 2. If
the probability that x exceeds 7.5 is .8023, find  .
12. A normal random variable x has mean 35 and standard deviation 10. Find a value of
x that has area 0.01 to its right. This is the 99th percentile of this normal distribution.
13. A normal random variable x has mean 50 and standard deviation 15. Would it be
unusual to observe the value x = 0? Explain your answer.
14. A normal random variable x has an unknown mean and standard deviation. The
probability that x exceeds 4 is 0.9772, and the probability that x exceeds 5 is 0.9332.
Find  and  .
15. The meat department at a local supennarket specifically prepares its" 1-pound"
packages of ground beef so that there will be a variety of weights, some slightly more
and some slightly less than I pound. Suppose that the weights of these "1-pound"
packages are normally distributed with a mean of 1.00 pound and a standard deviation of
.15 pound.
a. What proportion of the packages will weigh more than I pound?
b. What proportion of the packages will weigh between .95 and 1.05 pounds?
c. What is the probability that a randomly selected package of ground beef will weigh
less than .80 pound?
d. Would it be unusual to find a package of ground beef that weighs 1.45 pounds? How
would you explain such a large package?
16. Human heights are one of many biological random variables that can be modeled by
the normal distribution. Assume the heights of men have a mean of 69 inches with a
standard deviation of 3.5 inches.
a. What proportion of all men will be taller than 6'0"? (HINT: Convert the measurements
to inches.)
b. What is the probability that a randomly selected man will be between 5'8" and 6'1"
tall?
c. President Bush is 6'0" tall. Is this an unusual height?
d. Of the 36 elected presidents from 1856 to the present, 17 were 6' 0" or taller. Would
you consider this to be unusual, given the proportion found in part a?
17. The diameters of Douglas firs grown at a Christmas tree farm are normally
distributed with a mean of 4 inches and a standard deviation of 1.5 inches.
a. What proportion of the trees will have diameters between 3 and 5 inches?
b. What proportion of the trees will have diameters less than 3 inches?
c. Your Christmas tree stand will expand to a diameter of 6 inches. What proportion of
the trees will not fit in your Christmas tree stand?
18. Cerebral blood flow (CBF) in the brains of healthy people is normally distributed
with a mean of 74 and a standard deviation of 16.
a. What proportion of healthy people will have CBF readings between 60 and 80?
b. What proportion of healthy people will have CBF readings above 100?
c. If a person has a CBF reading below 40, he is classified as at risk for a stroke. What
proportion of healthy people will mistakenly be diagnosed as "at risk"?