Applied Statistics
Midterm 2
Name:
ID:
Carefully Read The Instructions!
Instructions: This exam will last 50 minutes and consists of 6 problems and one
extra credit problem. Provide solutions to the problems in the space provided. All
solutions must be sufficiently justified to receive credit, for example, the correct
solution without justification may not receive any credit. You may use a calculator
on this exam. You may use one page of handwritten notes. Good Luck!
Advice: If you get stuck on a problem don’t panic! Move on and come back to it
later.
Page Number Points
1
2
3
4
5
6
Extra Credit
1. (10 points) The weight of adult males in the United States is known to follow
a normal distribution with mean 194lbs and standard deviation 35lbs.
(a) What is the probability that a randomly selected adult male from the
United States will weigh more than 220lbs?
(b) Let X denote the weight of a randomly selected adult male from the
United States. Compute ξ so that P (X > ξ) = .90.
(c) Compute c so that P (−c ≤ X − 194 ≤ c) = .99 where X is defined in
part (b).
2. (10 points) Suppose X has cumulative distribution function

:x<1
 0
ln(x) : 1 ≤ x < e
F (x) =

1
: e ≤ x,
where ln(x) denotes the natural logarithm and e denotes the base of the
natural logarithm.
(a) Compute the 80th percentile of the distribution of X
(b) Compute E(X)
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(c) Compute V ar(X)
(d) Compute P (X > 1.5)
3. (10 points) Brittney owns an ice cream company that makes a cherry ice
cream containing chocolate chips and cherries. Suppose Y is the number of
chocolate chips in a random 3 ounce scoop of this cherry ice cream, and X is
the number of cherries in a random 3 ounce scoop. X and Y have joint
probability mass function:
Y
p(x,y) 2
3
4
X
1
.2 .15 .1
2
.15 .15 .25
(a) Compute P (X + Y ≥ 5), the probability that their are at least 5 pieces of
chocolate/cherries in a random 3 ounce scoop.
(b) Compute V ar(Y ).
4. (12 points) Kyle works at a battery factory.
(a) Suppose that Kyle independently tests 13 randomly selected batteries in
order to find defects. The probability that any one battery will be
defective is .02.
i. What is the probability that Kyle will find 3 defective batteries?
ii. What is the expected number of defective batteries he will find?
(b) Kyle is also asked to test batteries as they come off the assembly line
until he finds 3 defective batteries. The probability that any one battery
is defective is again .02. What is the probability that it takes 7 tests in
order for Kyle to find 3 defective batteries?
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(c) Kyle accidently put 4 defective batteries into a shipment of 50 batteries.
Whoops! Fortunately, it is known that these are the only defective
batteries among the 50. In order to find these defective batteries, Kyle
tests 20 batteries in the shipment at random.
i. What is the probability that Kyle will find 2 or less defective
batteries?
ii. What is the standard deviation of the number of defective batteries
he will find?
Normal Q−Q Plot
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Sample Quantiles
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Theoretical Quantiles
5. (6 points) The figure above shows the normal probability plot (normal QQ
plot) of a random variable generated from 60 independent simulations.
Discuss the plot and what conclusions can be drawn about the distribution of
the random variable used to generate the plot.
6. An ant colony is known to contain 100,000 ants, of which 10,000 are soldier
ants. Suppose a sample of 500 ants is drawn from the population in such a
way that any subset of 500 ants is equally likely.
(a) (2 points) What is the expected value of the number of solider ants in the
sample?
(b) (8 points) Approximate the probability that the sample contains more
than 40 solider ants.
7. (5 points) (Extra Credit) Suppose X ∼ N (0, 1). Derive an expression for EX k
for all k ≥ 1