Download Chapter 7: Random Variables

Chapter 7: Random Variables – Review Questions
Use the following to answer question 1:
A psychologist studied the number of puzzles subjects were able to solve in a five-minute period
while listening to soothing music. Let X be the number of puzzles completed successfully by a
subject. The psychologist found that X had the following probability distribution:
Value of X
Probability
1
0.2
2
0.4
3
0.3
4
0.1
1. Referring to the information above, the probability that a randomly chosen subject
completes at least three puzzles in the five-minute period while listening to soothing
music is
A) 0.3. B) 0.4. C) 0.6. D) 0.7. E) 0.9.
Use the following to answer questions 2 through 4:
Let the random variable X be a randomly generated number with the uniform probability density
curve given below.
2. Referring to the information above, P(X = 0.25) is
A) 0. B) 0.025. C) 0.25. D) 0.75. E) 1.
3. Referring to the information above, P(X  0) has value
A) 0. B) 0.1. C) 0.5. D) 1.
E) The value cannot be determined since X must be greater than 0.
4. Referring to the information above, P(0.7 < X < 1.1) has value
A) 0.30. B) 0.40. C) 0.60. D) 0.70. E) 1.1.
5. A random variable is
A) a hypothetical list of the possible outcomes of a random phenomenon.
B) any phenomenon in which outcomes are equally likely.
C) any number that changes in a predictable way in the long run.
D) a variable used to represent the outcome of a random phenomenon.
E) a variable whose value is a numerical outcome associated with a random
phenomenon.
Chapter 7: Random Variables
6. Suppose X is a continuous random variable taking values between 0 and 2 and having
the probability density curve below.
P(1 ≤ X ≤ 2) has value
A) 0.75. B) 0.50. C) 0.33.
D) 0.25.
E) 0.
Use the following to answer questions 7 and 8:
Let the random variable X represent the profit made on a randomly selected day by a certain
store. Assume X is normal with a mean of $360 and a standard deviation of $50.
7. Referring to the information above, the value of P(X > $400) is
A) 0.1587. B) 0.2119. C) 0.2881. C) 0.7881. E) 0.8450.
8. Referring to the information above, the probability is approximately 0.6 that on a
randomly selected day the store will make less than
A) $0.30. B) $347.40.
C) $361.30. D) $372.60. E) $390.00.
Use the following to answer questions 9 and 10:
In a particular game, a fair die is tossed. If the number of spots showing is either 4 or 5, you win
$1; if number of spots showing is 6, you win $4; and if the number of spots showing is 1, 2, or 3,
you win nothing. Let X be the amount that you win on a single play of the game.
9. Referring to the information above, the expected value of X is
A) $0. B) $1. C) $1.33. D) $2.50. E) $4.
10. Referring to the information above, the variance of X is
A) 1. B) 1.414. C) 3/2. D) 2. E) 13/6.
Use the following to answer questions 11 and 12:
Suppose there are three balls in a box. On one of the balls is the number 1, on another is the
number 2, and on the third is the number 3. You select two balls at random and without
replacement from the box and note the two numbers observed. The sample space S consists of
the three equally likely outcomes {(1, 2), (1, 3), (2, 3)}. Let X be the sum of the numbers on the
two balls selected.
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Chapter 7: Random Variables
11. Referring to the information above, the mean of X is
A) 8/6. B) 2. C) 14/6. D) 4. E) 26/6.
12. Referring to the information above, the variance of X is
A) 1/3. B) 2/3. C) 0.816. D) 1. E) 4.
13. In a particular game, a ball is randomly chosen from a box that contains three red balls,
one green ball, and six blue balls. If a red ball is selected, you win $2; if a green ball is
selected, you win $4; and if a blue ball is selected, you win nothing. Let X be the
amount that you win. The expected value of X is
A) $0.10. B) $1. C) $2. D) $3. E) $4.
Use the following to answer questions 14 through 17:
The weight of medium-sized tomatoes selected at random from a bin at the local supermarket is a
random variable with mean  = 10 ounces and standard deviation  = 1 ounce.
14. Suppose we pick four tomatoes from the bin at random and put them in a bag. The
weight of the bag is a random variable with a mean of
A) 2.5 ounces. B) 4 ounces. C) 10 ounces. D) 40 ounces. E) 41 ounces.
15. Suppose we pick four tomatoes from the bin at random and put them in a bag. The
weight of the bag is a random variable with a standard deviation (in ounces) of
A) 0.25. B) 0.50. C) 0.71. D) 2. E) 4.
16. The weight of a tomato in pounds (1 pound = 16 ounces) is a random variable with
standard deviation
A) 1/256 pounds. B) 1/16 pounds. C) 1 pound. D) 16 pounds.
E) 256 pounds.
17. Suppose we pick two tomatoes at random from the bin. The difference in the weights of
the two tomatoes selected (the weight of the first tomato minus the weight of the second
tomato) is a random variable with a standard deviation (in ounces) of
A) 0. B) 1. C) 1.41. D) 2. E) 4.
18. Suppose X is a random variable with mean X and standard deviation X. Suppose Y is a
random variable with mean Y and standard deviation Y. The mean of X + Y is
A) X + Y.
B) (X/X) + (Y/Y).
C) X + Y, but only if X and Y are independent.
D) (X + Y)/(X+Y).
E) (X/X) + (Y/Y), but only if X and Y are independent.
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Chapter 7: Random Variables
19. Suppose X is a random variable with mean X and standard deviation X. Suppose Y is a
random variable with mean Y and standard deviation Y. The variance of X + Y is
A) X + Y.
B) (X)2 + (Y)2.
C) X + Y, but only if X and Y are independent.
D) (X)2 + (Y)2, but only if X and Y are independent.
E) X + Y)2.
20. A random variable X has mean X and standard deviation X. Suppose n independent
observations of X are taken and the average X of these n observations is computed. If n
is very large, the law of large numbers implies
A) that X will be close to X.
B) that X will be approximately normally distributed.
C) that the standard deviation of X will be close to X.
D) that X will be uniformly distributed.
E) all of the above.
21. I toss a fair coin a large number of times. Assuming tosses are independent, which of the
following is true?
A) Once the number of flips is large enough (usually about 10,000), the number of
heads will always be exactly half of the total number of tosses. For example, after
10,000 tosses I should have 5000 heads.
B) The proportion of heads will be about ½, and this proportion will tend to get closer
and closer to ½ as the number of tosses increases.
C) As the number of tosses increases, any run of heads will be balanced by a
corresponding run of tails so that the overall proportion of heads is ½.
D) If the number of heads is greater than the number of tails for the first 5000 tosses,
then the number of tails will be greater than the number of heads for the next 5000
tosses.
E) All of the above.
Use the following to answer questions 22 through 24:
A small store keeps track of the number X of customers that make a purchase during the first
hour that the store is open each day. Based on the records, X has the following probability
distribution.
X
P(X)
0
0.1
1
0.1
2
0.1
3
0.1
4
0.6
22. Referring to the information above, the mean number of customers that make a purchase
during the first hour that the store is open is
A) 1.
B) 2. C) 2.5. D) 3. E) 4.
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Chapter 7: Random Variables
23. Referring to the information above, the standard deviation of the number of customers
that make a purchase during the first hour that the store is open is
A) 1.2.
B) 1.4. C) 2. D) 3. E) 4.
24. Referring to the information above, suppose the store is open seven days a week from
8:00 AM to 5:30 PM. The mean number of customers that make a purchase during the
first hour that the store is open during a one-week period is
A) 3. B) 7. C) 9. D) 21. E) 28.
Use the following to answer questions 25 and 26:
The weight of medium-sized tomatoes selected at random from a bin at the local supermarket is a
normal random variable with mean  = 10 ounces and standard deviation  = 1 ounce. Suppose
we pick two tomatoes at random from the bin, so the weights of the tomatoes are independent.
25. Referring to the information above, the difference in the weights of the two tomatoes
selected (the weight of first tomato minus the weight of the second tomato) is a random
variable with which of the following distributions?
A) N(0, 0.5). B) N(0, 1.41). C) N(0, 2). D) N(0, 4). E) Uniform with mean
0.
26. Referring to the information above, the probability that the difference in the weights of
the two tomatoes exceeds 2 ounces is closest to
A) 0.017. B) 0.068. C) 0.079. D) 0.159. E) 0.921.
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Chapter 7: Random Variables
Answer Key
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