Name: ______________________ Class: _________________ Date: _________
ID: A
AP Statistics - Random Variables (Multiple Choice)
Multiple Choice ( 1.5 minutes each, 4 points each)
Identify the choice that best completes the statement or answers the question.
____
1. A marketing survey compiled data on the number of personal computers in households. If X = the
number of computers in a randomly-selected household, and we omit the rare cases of more than 5
computers, then X has the following distribution:
X
P(X)
0
0.24
1
0.37
2
0.20
3
0.11
4
0.05
5
0.03
What is the probability that a randomly chosen household has at least two personal computers?
A. 0.19
B. 0.20
C. 0.29
D. 0.39
E. 0.61
____
2. A random variable X has a probability distribution as follows:
X
P(X)
0
2k
1
3k
2
13k
3
2k
Where k is a positive constant. The probability P(X < 2.0) is equal to
A. 0.90.
B. 0.25.
C. 0.65.
D. 0.15.
E. 1.00.
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Name: ______________________
____
ID: A
3. Roll a fair 8-sided die 10 times. The probability of getting exactly 3 sevens in those 10 rolls is given by
A.
B.
C.
D.
E.
____
4. X and Y are independent random variables, and a and b are constants. Which one of the following
statements is true?
A.
B.
C.
D.
E.
____
5. Let the random variable X represent the profit made on a randomly selected day by a certain store.
Assume that X is approximately normal with mean $360 and standard deviation $50. What is P(X >
$400)?
A. 0.2119
B. 0.2881
C. 0.5319
D. 0.7881
E. 0.8450
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Name: ______________________
ID: A
____
6. A dealer in the Sands Casino in Las Vegas selects 40 cards from a standard deck of 52 cards. Let Y be the
number of red cards (hearts or diamonds) in the 40 cards selected. Which of the following best describes
this setting?
A. Y has a binomial distribution with n = 40 observations and probability of success p =
0.5.
B. Y has a binomial distribution with n = 40 observations and probability of success p =
0.5, provided the deck is shuffled well.
C. Y has a binomial distribution with n = 40 observations and probability of success p =
0.5, provided that after selecting a card it is replaced in the deck and the deck is
shuffled well before the next card is selected.
D. Y has a geometric distribution with n = 40 observations and probability of success p =
0.5.
E. Y has a geometric distribution with n = 52 observations and probability of success p =
0.5.
____
7. In a large population of college students, 20% of the students have experienced feelings of math anxiety.
If you take a random sample of 10 students from this population, the mean and standard deviation of the
number of students in the sample who have experienced math anxiety is:
A. µ = 1.6; σ = 1.414
B. µ = 1.6; σ = 1.265
C. µ = 2; σ = 1.6
D. µ = 2; σ = 1.414
E. µ = 2; σ = 1.265
____
8. Which of the following is a true statement?
A. The binomial setting requires that there are only two possible outcomes for each trial,
while the geometric setting permits more than two outcomes.
B. A geometric random variable takes on integer values from 0 to n.
C. If X is a geometric random variable and the probability of success is 0.85, then the
probability distribution of X will be skewed left, since 0.85 is closer to 1 than to 0.
D. An important difference between binomial and geometric random variables is that
there is a fixed number of trials in a binomial setting, and the number of trials varies
in a geometric setting.
E. The distribution of every binomial random variable is skewed right.
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Name: ______________________
____
ID: A
9. In the town of Tower Hill, the number of cell phones in a household is a random variable W with the
following distribution:
W
P(W)
The
A.
B.
C.
D.
E.
0
0.1
1
0.1
2
0.25
3
0.3
4
0.2
5
0.05
probability that a randomly-selected household has at least two cell phones is
0.20.
0.25.
0.55.
0.70.
0.80.
____ 10. A rock concert producer has scheduled an outdoor concert. If it is warm that day, she expects to make a
$20,000 profit. If it is cool that day, she expects to make a $5000 profit. If it is very cold that day, she
expects to suffer a $12,000 loss. Based upon historical records, the weather office has estimated the
chances of a warm day to be 0.60; the chances of a cool day to be 0.25. What is the producer’s expected
profit?
A. $5,000
B. $11,450
C. $13,000
D. $13,250
E. $15,050
____ 11. A set of 10 cards consists of 5 red cards and 5 black cards. The cards are shuffled thoroughly and you
turn cards over, one at a time, beginning with the top card. Let Y be the number of cards you turn over
until you observe the first red card. The random variable Y has which of the following probability
distributions?
A. the Normal distribution with mean 5
B. the binomial distribution with p = 0.5
C. the geometric distribution with probability of success 0.5
D. the uniform distribution that takes value 1 on the interval from 0 to 1
E. none of the above
____ 12. In order for the random variable X to have a geometric distribution, which of the following conditions
must X satisfy?
I.
II. The number of trials is fixed.
III. Trials are independent.
IV. The probability of success has to be the same for each trial.
V. All outcomes in the sample space are equally likely.
A. III and IV
B. II, III, IV, and V
C. I and III
D. I, III, and V
E. II and III
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Name: ______________________
ID: A
____ 13. A randomly chosen subject arrives for a study of exercise and fitness. Consider these statements.
I. After 10 minutes on an exercise bicycle, you ask the subject to rate his or her effort on the Rate of
Perceived Exertion (RPE) scale. RPE ranges in whole-number steps from 6 (no exertion at all) to 20
(maximum exertion).
II. You measure VO2, the maximum volume of oxygen consumed per minute during exercise. VO2 is
generally between 2.5 liters per minute and 6 liters per minute.
III. You measure the maximum heart rate (beats per minute).
The statement(s) that describe a discrete random variable are
A. I.
B. II.
C. I, III.
D. I, II, III.
E. None of the statements describe a discrete random variable.
____ 14. Let the random variable X represent the amount of money Dan makes doing lawn care in a randomly
selected week in the summer. Assume that X is Normal with mean $240 and standard deviation $60.
The probability is approximately 0.6 that, in a randomly selected week, Dan will make less than
A. $144
B. $216
C. $255
D. $30
E. $360
____ 15. A vending machine operator has determined that the number of candy bars sold per week by a certain
machine is a random variable with mean 125 and standard deviation 7. His profit on each bar sold is
$0.25, and it costs him $5.00 per week to maintain the machine and rent the space for it. What are the
mean and standard deviation for Y = the profit he earns from this machine in a randomly-selected week?
A. Mean = 31.25, Standard deviation –$3.25
B. Mean = 31.25, Standard deviation $1.25
C. Mean = 31.25, Standard deviation $1.75
D. Mean = 26.25, Standard deviation $1.25
E. Mean = 26.25, Standard deviation $1.75
____ 16. Which of the following random variables is geometric?
A. The number of phone calls received in a one-hour period
B. The number of times I have to roll a six-sided die to get two 5s.
C. The number of digits I will read beginning at a randomly selected starting point in a
table of random digits until I find a 7.
D. The number of 7s in a row of 40 random digits.
E. All four of the above are geometric random variables.
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ID: A
AP Statistics - Random Variables (Multiple Choice)
Answer Section
MULTIPLE CHOICE
1. ANS: D
/D/Correct!
PTS: 4
2. ANS: B
/B/Correct!
REF: Test 6A
PTS: 4
REF: Test 6A
3. ANS: B
/B/Correct! Binomial probability formula: P(k successes in n trials when p = success in one trial) is
.
PTS: 4
REF: Test 6A
4. ANS: B
/B/Correct! When adding or subtracting random variables, variances add but standard deviations do not.
PTS: 4
5. ANS: A
REF: Test 6A
/A/Correct!
PTS: 4
REF: Test 6A
6. ANS: C
/C/Correct! The dealer must sample with replacement in order to meet the independence requirement for
the binomial distribution.
PTS: 4
7. ANS: E
REF: Test 6A
/E/Correct!
;
PTS: 4
REF: Test 6A
8. ANS: D
/D/Correct! (d) is the essential difference between the binomial setting and the geometric setting. None
of the other statements are true.
PTS: 4
REF: Test 6A
1
ID: A
9. ANS: E
/E/Correct!
PTS: 4
10. ANS: B
/B/Correct!
REF: Test 6B
PTS: 4
REF: Test 6B
11. ANS: E
/E/Correct! Since we are sampling without replacement, draws from the deck are not independent,
violating one property of both binomial and geometric distributions.
PTS: 4
REF: Test 6B
12. ANS: A
/A/Correct! Statements II and III are conditions for the geometric setting. (Statement II is only true for
the binomial setting).
PTS: 4
REF: Test 6B
13. ANS: C
/C/Correct! Variables in I and III take on discrete integer values, the variable in II is continuous (though
actual values are limited by the measurement process, there are no explicit gaps between values).
PTS: 4
REF: Test 6C
14. ANS: C
/C/Correct! z-value for 0.60 is 0.25.
PTS: 4
15. ANS: E
/E/Correct!
REF: Test 6C
PTS: 4
REF: Test 6C
16. ANS: C
/C/Correct! Fits all conditions for geometric setting.
PTS: 4
REF: Test 6C
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