**Self Test 4 for Probability
and Statistics**

This test was constructed by Dr.
Carol Dahl and Arturo Vasquez based on the Schaum's Outline *Theory and
Problems of Probability and Statistics* by Murray R. Spiegel. If you need
more review refer also to this outline.

** 1. True False.** If 10% of the wells drilled by an oil
company in a field contain natural gas, the probability that out of 4 wells
drilled at random

(a) 1 will be defective is 29.16%

(b) less than 2 nails will be defective is 94.77%

** 2. True
False. **The probability that a sizable amount of raw diamonds (0.5% of the weight
of a rock load) will be extracted from a given ore load from a certain diamond
mine in

a.
9 will
have a sizable content of raw diamonds is 4.031%

b.
all ten
will have a sizable content of raw diamonds is 0.6047%

c.
greater
than 8 loads will have a sizable diamond content is 4.6357%

**The Normal Distribution**

** 3. True
False. **The number of ore diggers who are necessary for burrowing a mineshaft
follows a normal distribution with mean 20 and variance 25, N(20,25). The probability
that the number of diggers is between the mean and 24 is 0.523.

** 4. True
False.** A fuel for a certain airplane is
going to content a percentage (called X) of a particular additive. The
specifications require that X is between 30 and 35 percent. The refinery, which
produces that fuel, will obtain a net profit according to the following
function of X:

_{}

If X follows a normal distribution with mean 33 and standard deviation of 9, the expected profit E[T] 6.74 cents per gallon (Contributed by Arturo Vasquez).

** 5. True
False. **In the north of

** 6. True
False.** The mean weight of 600 male
employees at a certain oil company is 151 lb and the standard deviation is 15
lb. Assuming that the weights are normally distributed, the number of employees
that weigh (a) between 120 and 155 lb is 360, (b) the number of employees that weight
more than 185 lb is 6 (Contributed by Claudio Valencia).

** 7. True
False. **A study indicated that the copper mines in

**The Poisson
Distribution**

** 8. True
False.** Ten percent of the bulldozers used in a mine site tend to fail. Find the
probability that in a sample of 10 bulldozers chosen at random exactly two
fail. The probability is 0.15 (Contributed by Arturo Vasquez).

** 9. True
False.** The average number of cars that randomly arrives to a gas station is 24
per hour. The probability that 5 cars arrive in a period of 12 minutes is 20%
(Contributed by Arturo Vasquez).

**Central Limit Theorem**

** 10. True
False.** A multinational mining company expects to receive an average of 50
environmental claims made by the regulatory agencies of the countries where it
operates. The actual number of claims is a random variable and it is well
described by a Poisson distribution. The value of each environmental claim is a
random variable, independent of all other claims, with a mean of $500,000 and a
standard deviation of $100,000. a) The probability of having 60 claims is 4.21%
b) If the company actually has 55 claims, the probability that the total value
of those claims will be more than $29,000,000 is 2.17% (Contributed by Arturo
Vasquez).

** 11. True
False.** A group of excavator machines in a copper mine can extract ore loads
that have an average weight of 300 tons and a standard deviation of 50 tons.
The probability that 25 ore loads taken at random can exceed the capacity of an
ore processing facility which can process 8,200 tons is 2% (Contributed by
Arturo Vasquez).

**Multinomial and
Hypergeometric Distributions**

** 12. True
False.** The Peruvian Tax Administration is auditing the income tax returns of
several mining companies because it suspects that some of these companies
didn't fill their tax forms correctly (they probably didn't consider the
windfalls earned after a mineral price boom which occurred last year). The
probability that a tax return was properly filled out is 60%. The probability
that a tax return has an error which favors the taxpayer is 20%, the
probability of having a tax return with errors which favors the government is
10%, and the probability of having a tax return with both errors is 10%. The
General Tax Controller chooses ten tax returns at random. The probability that
5 tax returns were properly filled out, 3 have errors which favor the
taxpayers, 1 has errors which favor the government, and 1 has both kinds of
errors is 5% (Contributed by Arturo Vasquez).

** 14. True False.
**The trucks that carry ores in a certain mine site have to pass by a certain
control point each 8 minutes. A supervisor can unexpectedly arrive to the
control point to verify the transportation of the valuable ore. If X is the
random variable which represents the time that the supervisor has to wait until
a truck passes by the control point, a) the probability that the supervisor
waits for a truck less than 5 minutes is 1/8 b) Thye probability that the
supervisor waits for a truck more than two minutes is 1/4. (Contributed by
Arturo Vasquez).

** 15. True
False**. The mean (a) and the variance (b)
of the uniform density, which is defined by

f(x) = (b-a)^{-1}, a ≤
x ≤ b;

= 0, otherwise.

Are:

(a) m =1/2(a+b)

(b) (b) s^{2}=
(b-a)^{2}/12.

(Contributed by Joe Mazumdar)

** 16. True False.** The graph of the chi-square
distribution with 5 degrees of freedom is shown in the figure below. The values
χ

(a) the
area to the right of χ^{2}_{2} = 0.05,

(b) the
total area to the left of χ^{2}_{1} and to the right of
χ^{2}_{2} = 0.05,

(c) the
area to the left of χ^{2}_{1} = 0.10,

(d) the
area to the right of χ^{2}_{2} = 0.01.

Are:

(a) 11.1

(b) 0.831 and 12.8

(c) 1.61

(d) 15.1

(Contributed by Joe Mazumdar)

** 17. True
False. **A petroleum engineer is modeling the number of wells with oil in a
certain field by using a statistical model. He realized that the number of
wells with oil 'X' in the field follows a normal distribution with mean 7 and
variance 4. Besides, the number of wells with water is defined by the following
random variable: (X - 7)

__18. ____True __** False.**
The graph of Student's t distribution with 9 degrees of freedom is shown in the
figure above. The value of t

(a) the
area to the right of t_{1 }= 0.05.

(b) the
area to the left of -t_{1} and to the right of t_{1 }= 0.05

(c) the
area between -t_{1 }and t_{1 }is 0.99

(d) the
area to the left of -t_{1} is 0.01

(e) the
area to the left of t_{1} is 0.90.

Are:

(a) 1.83

(b) 2.26

(c) 3.25

(d) -2.82

(e) 1.38

(Contributed by Joe Mazumdar)

** 19. True False.
**Two mechanical engineers are evaluating the relation between the diameter
and thickness in a sample of pipelines in order to analyze the probability of
having a leakage. They realize that the diameter 'D' follows a normal
distribution with a mean of 2 meters and a standard deviation of 0.5 meters,
and the thickness 'K' follows a Chi-square distribution with a mean of 20
inches. It is known that the probability of having a leakage depends on a
variable defined by the ratio 'L' of the standardized diameter and the square
root of the thickness divided by its mean. What is distribution function to
model the probability of having a leakage? The distribution function is unknown
(Contributed by Arturo Vasquez)

** 20. True False. **Using the table for the F
distribution, the values of (a) F

Are:

(a) 4.96

(b) 0.257

(c) 2.54

(d) 0.325

(Contributed by Joe Mazumdar)