Quiz 6.1A
AP Statistics
Name:
1. The probability distribution below is for the random variable X = number of mice caught in
traps during a single night in small apartment building.
X
P(X)
0
1
2
3
4
5
0.12 0.20 0.31 0.14 0.16 0.07
(a) Make a histogram of this probability distribution in the grid:
(b) Describe P X 2 in words and find its value.
(c) Express the event “trapping at least one mouse” in terms of X and find its probability.
2. The total sales on a randomly-selected day at Joy’s Toy Shop can be represented by the
continuous random variable S, which has a Normal distribution with a mean of $3600 and a
standard deviation of $500. Find and interpret P S $4000 .
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3. Joe the barber charges $32 for a shave and haircut and $20 for just a haircut. Based on
experience, he determines that the probability that a randomly selected customer comes in
for a shave and haircut is 0.85, the rest of his customers come in for just a haircut. Let J =
what Joe charges a randomly-selected customer.
(a) Give the probability distribution for J.
(b) Find and interpret the mean of J,
J
.
(c) Find and interpret the standard deviation of J,
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Quiz 6.1B
AP Statistics
Name:
1. The probability distribution below is for the random variable X = number of medical tests
performed on a randomly selected outpatient at a certain hospital.
X
P(X)
0
1
2
3
4
5
0.33 0.20 0.18 0.14 0.12 0.03
(a) Make a histogram of this probability distribution in the grid:
(b) Describe P X 3 in words and find its value.
(c) Express the event “performing at least two tests” in terms of X and find its probability.
2. The mean height of players in the National Basketball Association is about 79 inches and
the standard deviation is 3.5 inches. Assume the distribution of heights is approximately
Normal. Let H = the height of a randomly-selected NBA player. Find and interpret
P H 74 .
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3. Man Hong is running the balloon darts game at the school fair. He has blown up hundreds of
balloons with notes about prize tickets inside them. Twelve percent of the notes say “You
win 5 tickets,” twenty percent say “You win 3 tickets,” and the rest say “Sorry, try again!”
After each play, he replaces the popped balloon with another one bearing the same note. Let
T = the number of tickets won by a randomly selected player of this game.
(a) Give the probability distribution for T.
(b) Find and interpret the mean of T,
T
.
(c) Find and interpret the standard deviation of T,
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Quiz 6.1C
AP Statistics
Name:
1. A four sided die, shaped like an asymmetrical tetrahedron, has the following roll
probabilities.
Number on Die
Probability
1
0.4
2
0.3
3
0.2
4
0.1
Let X = the result of a single roll.
(a) Find P 1 X 4
(b) Find P X 3
(c) Describe P x 3| x 2 in words and find its value.
(d) Find the smallest value A for which P X A 0.6
(e) If T = the sum of two rolls, find P T 4
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1. (Continued) Below is a copy of the table from the first page, showing the probability
distribution of X = the number rolled on an asymmetrical four-sided die.
X
P(X)
1
0.4
2
0.3
3
0.2
4
0.1
(f) Find and interpret the mean and standard deviation of X.
2. As of December 2008, various polls indicate that 35% of people who use the internet have
profiles on at least one social networking site. We will discover later in the text that if you
take a sample of 50 internet users, the proportion of the sample who have profiles on social
networking sites can be considered a random variable. Moreover, assuming the 35% is
accurate, this random variable will be approximately Normally distributed with a mean of
0.35 and a standard deviation of 0.067. What is the probability that the proportion of a
sample of size 50 who have profiles on social networking sites is greater than 0.5?
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Quiz 6.2A
AP Statistics
Name:
1. The manager of a children’s puppet theatre has determined that the number of adult tickets he
sells for a Saturday afternoon show is a random variable with a mean of 28.3 tickets and a
standard deviation of 5.3 tickets. The mean number of children’s tickets he sells is 42.5, with
a standard deviation of 8.1.
(a) The adult tickets sell for $10. Let A = the money he collects from adult tickets on a
random Saturday. What are the mean and standard deviation of A?
(b) The children’s tickets sell for $6. Let T = the money he collects from all ticket sales
(adults and children) on a random Saturday. Assume (unrealistically, perhaps) that the
number of tickets sold to adults is independent of the number sold to children. What are
the mean and standard deviation of T?
(c) It costs $300 for the manager to put on each puppet show. Let P = the profit from a
random Saturday’s show. What are the mean and standard deviation of P?
2. Mr. Voss and Mr. Cull bowl every Tuesday night. Over the past few years, Mr. Voss’s scores
have been approximately Normally distributed with a mean of 212 and a standard deviation
of 31. During the same period, Mr. Cull’s scores have also been approximately Normally
distributed with a mean of 230 and a standard deviation of 40. Assuming their scores are
independent, what is the probability that Mr. Voss scores higher than Mr. Cull on a
randomly-selected Tuesday night?
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Quiz 6.2B
AP Statistics
Name:
1. A casino operator has invented a new game of “skill” and chance called Grab All You Can.
Here’s how it works: a contestant reaches his right hand into a jar of dimes and grabs as
many as he can in one handful. Then he does the same thing with his left hand in a jar of
quarters. Research with many volunteers has determined that the mean number of dimes
drawn is 68 with a standard deviation of 9.5, and the mean number of quarters is 42, with a
standard deviation of 5.8.
(a) If D = the amount of money, in dollars, that a randomly-selected contestant grabs from
the “dime grab,” find the mean and standard deviation of D.
(b) If T = the total amount of money, in dollars, that a contestant grabs from both jars, find
the mean and standard deviation of T.
(c) The casino operator plans to charge $20 for one round of play (that is, one grab from each
jar). If G = how much the contestant gains from one round of play, find the mean and
standard deviation of G.
2. Suppose that the mean height of policemen is 70 inches with a standard deviation of 3
inches. And suppose that the mean height for policewomen is 65 inches with a standard
deviation of 2.5 inches.
If heights of policemen and policewomen are Normally distributed, find the probability that a
randomly selected policewoman is taller than a randomly selected policeman.
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Quiz 6.2C
AP Statistics
Name:
1. The rules for means and variances allow you to find the mean and variance of a sum of
random variables without first finding the distribution of the sum, which is usually much
harder to do.
(a) A single toss of a balanced coin results in either 0 or 1 head, each with probability 1/2.
What are the mean and standard deviation of the number of heads?
(b) Toss a coin four times. Use the rules for means and variances to find the mean and
standard deviation of the total number of heads.
2. You have two instruments with which to measure the height of a tower. If the true height is
100 meters, measurements with the first instrument vary with mean 100 meters and standard
deviation 1.2 meters. Measurements with the second instrument vary with mean 100 meters
and standard deviation 0.65 meters. You make one measurement with each instrument.
Your results are X1 for the first and X2 for the second and the measurements are independent.
(a) To combine the two measurements, you might average them, Y
1
2 X 1 X 2 . What are the
mean and standard deviation of Y?
(b) It makes sense to give more weight to the less variable measurement because it is more
likely to be closer to the truth. Statistical theory says that to make the standard deviation
as small as possible you should weight the two measurements inversely proportional to
their variances. The variance of X2 is very close to half the variance of X1, so X2 should
get twice the weight of X1. That is, use
W
1
3X1
2
3X2
What are the mean and standard deviation of W?
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3. Lamar and Lawrence run a two-person lawn-care service. They have been caring for Mr.
Johnson’s very large lawn for several years, and they have found that the time it takes Lamar
to mow the lawn itself is approximately Normally distributed with a mean of 105 minutes
and a standard deviation of 10 minutes. Meanwhile, the time it takes for Lawrence to use the
edger and string trimmer to attend to details is also Normally distributed with a mean of 98
minutes and a standard deviation of 15 minutes. They prefer to finish their jobs within 5
minutes of each other. What is the probability that this happens, assuming their finish times
are independent?
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Quiz 6.3A
AP Statistics
Name:
1. Determine whether each random variable described below satisfies the conditions for a
binomial setting, a geometric setting, or neither. Support your conclusion in each case.
(a) Suppose that one of every 100 people in a large community is infected with HIV. You
want to identify an HIV-positive person to include in a study of an experimental new
drug. How many individuals would you expect to have to interview in order to find the
first person who is HIV-positive?
(b) Deal seven cards from a standard deck of 52 cards. Let H = the number of hearts dealt.
2. Research suggests that about 24% of 12-year-olds in the United States can pick out the state
of Colorado on a map.
(a) What is the probability that you must sample exactly 5 twelve-year-olds to find the first
one who can pick out Colorado on a map?
(b) What is the probability that you must sample 5 or more twelve-year-olds to find the first
one who can pick out Colorado on a map?
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3. An online poll reported that 20% of respondents subscribe to the “five-second rule.” That is,
they would eat a piece of food that fell onto the kitchen floor if it was picked up within five
seconds. Let’s assume this figure is accurate for the entire U.S. population, and we select 15
people at random from this population.
(a) Determine the probability that exactly 3 of the 15 people subscribe to the “five-second
rule.”
(b) Find the probability that less than 4 people out of 15 subscribe to the “five-second rule.”
(c) Let F = the number of people in our sample of 15 who subscribe to the “five-second
rule.” Find the mean and standard deviation for F.
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Quiz 6.3B
AP Statistics
Name:
1. Determine whether each random variable described below satisfies the conditions for a
binomial setting, a geometric setting, or neither. Support your conclusion in each case.
(a) A high school principal goes to 10 different classrooms and randomly selects one student
from each class. X = the number of female students in his group of 10 students.
(b) You are on Interstate 80 in Pennsylvania, counting the occupants in every fifth car
you pass. Let Z = the number of cars you pass before you see one with more than two
occupants.
2. A manufacturer produces a large number of toasters. From past experience, the manufacturer
knows that approximately 2% are defective. In a quality control procedure, we randomly
select 20 toasters for testing.
(a) Determine the probability that exactly one of the toasters is defective.
(b) Find the probability that at most two of the toasters are defective.
(c) Let X = the number of defective toasters in the sample of 20. Find the mean and standard
deviation of X.
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3. Suppose that 20% of a herd of cows is infected with a particular disease.
(a) What is the probability that the first diseased cow is the 3rd cow tested?
(b) What is the probability that 4 or more cows would need to be tested until a diseased cow
was found?
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Quiz 6.3C
AP Statistics
Name:
1. Determine whether each random variable described below satisfies the conditions for a
binomial setting, a geometric setting, or neither. Support your conclusion in each case.
(a) Draw a card from a standard deck of 52 playing cards, observe the card, return the card to
the deck, and shuffle. Count the number of times you draw a card in this manner until
you observe a jack.
(b) Joey buys a Virginia lottery ticket every week. X is the number of times in a year that he
wins a prize.
2. When a computerized generator is used to generate random digits, the probability that any
particular digit in the set {0, 1, 2, . . . , 9} is generated on any individual trial is 1/10 = 0.1.
Suppose that we are generating digits one at a time and are interested in tracking occurrences
of the digit 0.
(a) Determine the probability that the first 0 occurs as the fifth random digit generated.
(b) How many random digits would you expect to have to generate in order to observe the
first 0?
(c) Let X = number of digits selected until first zero
is encountered. Construct a probability
distribution histogram for X = 1 through X = 5.
Use the grid provided.
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3. Suppose there are 1100 students in your high school, and 28% of them take Spanish. You
select a sample of 50 student in the school, and you want to calculate the probability that 15
or more of the students in your sample take Spanish. Which condition for the binomial
setting has been violated here, and why does the binomial distribution do a good job of
estimating this probability anyway?
4. A fair coin is flipped 20 times.
(a) Determine the probability that the coin comes up tails exactly 15 times.
(b) Let X = the number of tails in the 20 flips. Find the mean and standard deviation of X.
(c) Find the probability that X takes a value within 1 standard deviation of its mean.
Chapter 6 Solutions
Quiz 6.1A
1. (a) See histogram at right. (b) The probability that two or
more mice are caught during a single night; 0.68. (c)
P x 1 0.88 .
2.
0.3
0.2
4000 3600
P z 0.8 0.2119.
500
This is the probability that the total sales on a
randomly-selected day exceeds $4000.
3. (a)
J
20
32
P S $4000 P z
P(J)
0.15
0.1
0
1 2 3 4 5
0.85
(b) J = the mean amount of money Joe can expect to make per
customer in the long run = 20 0.15 32 0.85 $30.20.
(c) J = the average distance from the mean ($30.20) for each
individual customer =
.15 20 30.20 2 .85 32 30.20 2 $4.28.
Quiz 6.1B
1. (a) See histogram at right. (b) The probability that an
outpatient
undergoes no more than 3 medical tests; 0.85. (c)
P x 2 0.47 .
74 79
P z 1.43 0.9236.
2. P H 74 P z
3.5
This is the probability that a randomly-selected NBA player’s
height is greater than 74 inches.
3.
T
0
3
5
P(T) 0.68 0.20 0.12
(b) T = the mean number of tickets won per contestant in the
long
run = 0 0.68 3 0.2 5 0.12 1.2 tickets. (c) T = the
average
distance from the mean (1.2) for each individual contestant
= 0.68 0 1.2 2 .2 3 1.2 2 .12 5 1.2 2 1.833 tickets.
0.3
0.2
0.1
0
1 2 3 4 5
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Quiz 6.1C
1. (a) P 1 X 4 P X 2 or X 3 0.3 0.2 0.5 . (b)
P X 3 1 P X 3 1 0.2 0.8 (c) The probability of rolling a 3, given that the roll is 2 or greater;
0 .2 1
0. 6 3.
(d) P X 3 0.4 0.3, so A = 3. (e) The event T = 4 can happen three ways:
{1, 3}, {3, 1}, and {2, 2}. Probabilities for these events are, respectively,
0.4 0.2 0.08; 0.2 0.4 0.08; and 0.3 0.3 0.09. Hence the total probability is 0.25.
(f)
.4 1 2 2 .3 2 2 2 .2 3 2 2 .1 4 2 2
X
X
1 0.4 2 0.3 3 0.2 4 0.1 2 ;
X
1
is the expected mean roll if the die is rolled many times, or the expected long-run value of a
single roll. X is the variability in the number rolled in the long run.
0.5 0.35
2. P X 0.5
P
z
P z 2.24 0.0125.
0.067
Quiz 6.2A
1. (a)
10 28.3 $283;
A
A
10 5.3 $53. (b)
T
10 28.3 6 42.5 $538;
10 5.3 2 6 8.1 2 $71.91. (c) P $538 $300 $238 ; T $71.91 (not changed
by subtracting a constant). 2. Let D = difference in scores between Mr. Cull and Mr. Voss.
T
Then
18 and
D
312
D
402
50.61.
0 18
PD 0 P z
P z .36 0.3594.
50.6
Quiz 6.2B
1. (a)
D
T
0.10 68 $6.80;
0.10 9.5
contestant);
G
2
D
0.10 9.5 $0.95. (b)
0.25 5.8 2 $1.73. (c)
G
T
0.10 68 0.25 42 $17.30;
$17.30 $20 $2.70 (a net loss for the
$1.73 (not changed by subtracting a constant). 2.
between random male and random female. Then
0 5
PD 0 P z
D
5 and
D
Let D = difference in height
32
2.52
3.91.
P z 1.28 0.1003.
3.91
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Quiz 6.2C
1. (a)
.5;
.5 0 .5 2 .5 1 .5 2
H
.52 .52 .52 .52
1
2 1
1.2
2
2
X
.
H
Y
1
Y
3
1.2
2
1. 2. (a)
2
0.65
2
1
Y
0.5
(b)
H
1
2 100
2 100
0.682
(b)
1
Y
100 ;
2
100
3
0.65
2
.5 .5 .5 .5 2;
100 100 ;
3
0.590 . 3. We are interested in the difference between their job
3
completion times, so D 105 98 7min utes; D
102 152
minutes means ±5 minutes. Therefore
5 7 P .67 z . 11 0.2048.
P 5 D 5
P 57z
18.03
18.03
18.03 . To be within 5
Quiz 6.3A
1. (a) This is a geometric setting: Binary outcomes (HIV or not), Independent trials (one
person’s condition does not influence condition of next randomly-selected individual), we are
counting the number of Trials to the first HIV case, and the probability of Success—finding a
person who is HIV-positive—is always 0.01. (b) This is neither binomial nor geometric,
because each trial is not independent of previous trials since it is done without replacement: the
suit of the first card influences the probability of a heart on the second card, and so on.
2. (a) Geometric setting with p= 0.24: 0.76 4 0.24 0.0801. (b) Equivalent to
P No successes in first 4 trials
0.76 4
0.3336 ; or
P x 5 1 P x 4 1 geomcdf 0.24,4 0.3336.
3. (a) F is binomial with n = 15 and p = 0.20. P F 3
15
3
0.2 3 0.8 12 0.2501
(b) P F 4 P F 3 binomcdf 15,.2,3 0.6482
(c)
F
np 15 0.20 3 ,
F
np 1 p
15 0.2 0.8 1.55 .
Quiz 6.3B
1. (a) This is neither binomial or geometric, because the probability of success (selecting a
female student) is probably different in each trial, unless every classroom has exactly the same
proportion of females, which is unlikely. (b) This is a geometric setting: Binary outcomes
(more than two occupants or not), Independent trials (the number of occupants of one car does
not influence the number of occupants in the next randomly-selected car, we are counting the
number of Trials to the first car with more than two occupants, and the probability of
Success— finding a car with more than two occupants—is always the same.
2. (a) X is binomial with n = 20 and p = 0.02. P X 1 20 0.02 1 0.98 19 0.2725
1
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(b) P X 2 binomcdf 20,.02,2 0.9929
(c) X np 20 0.02 0.4; X np 1 p 20 0.02 0.98 0.63 . 3. (a) 0.8 2 0.2 0.128 . (b)
Equivalent to
P No diseased cows in first 3 trials 0.8 3 0.512 ; or P X 4
1 P X 3 1 geomcdf 0.2,3 0.512.
Quiz 6.3C
1. (a) This is a geometric setting: Binary outcomes (card is jack or not), Independent trials
(since we are sampling with replacement, the result for one card does not influence the result for
any other), we are counting the number of Trials to the first jack, and the probability of
Success—finding a jack—is always 1/13. (b) This is a binomial setting: Binary outcomes (Joey
wins or loses), Independent trials (whether he wins this week does not influence whether he
wins next week), Number of trials is fixed at 52 (weeks), and the probability of Success—
however minuscule—does not change.
1 1
2. (a) Geometric probability: 0.9 4 0.1 0.0656 . (b) Geometric mean; X
p . 1 10 .
(c) Histogram below. 3. The condition of independence has been violated, since the probability
of selecting a student who takes Spanish changes if one or more previously-sampled students
take Spanish. For example, if all 50 students in the sample are taking Spanish, then the
308
probability of picking someone who is enrolled in Spanish changes from
0.28 for the
1100
258
first person to
0.2457 for the last one. Nevertheless, the binomial distribution is
1050
reasonably accurate because the sample is less than 10% of the population.
4. (a) Binomial distribution with n = 20 and p = 0.5. P X 15 20 0.5 15
(b)
X
np 20 0.5 10 ;
X
np 1 p
0.5
5
0.0148
15
20 0.5 0.5 2.236
(c)
P 7.76 X 12.24 P 8 X 12 binomcdf 20,0.5,12 binomcdf 20,0.5,7 0.7368 .
0.10
0.05
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