Ch. 8 Review
IB Statistics
1. 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
probability that exactly 2 students have experienced math anxiety is
A) 0.302
B) 0.2634
C) 0.2013
D) 0.5
E) 1
2. Refer to the previous problem. The standard deviation of the number of students in the
sample who have experienced math anxiety is
A) 0.016
B) 1.265
C) 0.253
D) 1
E) 0.207
3. In a certain large population, 40% of households have a total annual income of at least
$70,000. A simple random sample of 4 of these households is selected. What is the
probability that 2 or more of the households in the survey have an annual income of at
least $70,000?
A) 0.3456
B) 0.4
C) 0.5
D) 0.5248
E) The answer cannot be computed from the information given.
4. A factory makes silicon chips for use in computers. It is known that about 90% of the
chips meet specifications. Every hour a sample of 18 chips is selected at random for
testing. Assume a binomial distribution is valid. Suppose we collect a large number of
these samples of 18 chips and determine the number meeting specifications in each
sample. What is the approximate mean of the number of chips meeting specifications?
A) 16.2 B) 1.62 C) 4.02 D) 16 E) not enough information.
5. A college basketball player makes 80% of his free throws. Over the course of the season
he will attempt 100 free throws. Assuming free throw attempts are independent, what is
the approximate probability that he makes at least 86 of these attempts?
A) 0.9196
B) 0.8
C) 0.5546
D) 0.2
E) 0.0804
6. Which of the following random variables is geometric?
A) The number of phone calls received in a one-hour period.
B) The number of cards I need to deal from a deck of 52 cards that has been
thoroughly shuffled so that at least one of the cards is a heart.
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 length of time a traffic light must be green so that 5 cars can make a left turn.
E) All of the above.
7. Which of the following are true statements?
I.
II.
III.
A)
B)
C)
D)
E)
The expected value of a geometric random variable is determined by the formula
(1 – p)n-1p.
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.
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.
I only
II only
III only
I, II, and III
None of the above are true.
8. In a group of 10 college students, 4 are business majors. You choose 3 of the 10
students at random and ask their major. The distribution of the number of business
majors you choose is
A) binomial with n = 10 and p = 0.4
B) binomial with n = 3 and p = 0.4
C) not binomial and not geometric.
D) geometric with p = 0.4
E) geometric with p = 0.4 and n = 10
9. Government statistics tell us that 2 out of every 3 American adults are overweight. Let
X = number of American adults that are overweight. How large would an SRS of
American adults need to be in order for it to be safe to assume that the sampling
distribution of X is approximately Normal?
A) 3 B) 9 C) 15 D) 18 E) 30
10. Dillon is a lifetime 70% free throw shooter. He shoots 7 free throws in a game. If the
shots are independent of each other, the probability that he makes the first 5 and misses
the last 2 is about
A) 0.635 B) 0.318 C) 0.015 D) 0.49 E) 0.35
11. For which of the following counts would a binomial probability model be reasonable?
A) The number of traffic tickets written by each police officer in a large city during
one month.
B) The number of hearts in a hand of five cards dealt from a standard 52-card deck that
has been thoroughly shuffled.
C) The number of 7’s in a randomly selected set of five random digits from a table of
random digits.
D) The number of phone calls received in a one-hour period.
E) All of the above.
12. A set of 10 cards consists of five red cards and five black cards. The cards are shuffled
thoroughly and I choose one at random, observe its color, and replace it in the set. The
cards are thoroughly reshuffled and I again choose a card at random, observe its color,
and replace it in the set. This is done a total of 4 times. Let X be the number of red cards
observed in these 4 trials. The mean of X is
A) 4 B) 2 C) 1 D) 0.5 E) 0.1
13. An airplane has a front door and a rear door that are both opened to allow passengers to
exit when the plane lands. The plane has 100 passengers seated. The number of
passengers exiting through the front door should have
A) a binomial distribution with mean 50
B) a binomial distribution with 100 trials but success probability not equal to 0.5
C) a binomial distribution with 50 trials but success probability not equal to 0.5
D) a normal distribution with a standard deviation of 5
E) none of the above distributions
14. A small class has 10 students. Five of the students are male and five are female. I write
the name of each student on a 3-by-5 card. The cards are shuffled thoroughly and I
choose one at random, observe the name of the student, and replace it in the set. The
cards are thoroughly reshuffled and I again choose a card at random, observe the name,
and replace it in the set. This is done a total of four times. Let X be the number of cards
observed in these four trials with a name corresponding to a male student. The random
variable X has which of the following probability distributions?
A) The normal distribution with mean 2 and variance 1
B) The binomial distribution with parameters n = 4 and p = 0.5
C) The binomial distribution with parameters n = 4 and p = 0.1
D) The uniform distribution on 0, 1, 2, 3, and 4
E) None of the above
15. If X is binomial with parameters n = 9 and p = 1/3, the mean X of X is
A) 6 B) 3 C) 2 D) 1.414 E) 1.732
16. If X is binomial with parameters n = 9 and p = 1/3, the standard deviation X of X is
A) 6 B) 3 C) 2 D) 1.414 E) 1.732
17. In a certain game of chance, your chances of winning are 0.2. If you play the game five
times and outcomes are independent, the probability that you win at most once is
A) 0.0819 B) 0.2 C) 0.3277 D) 0.4096 E) 0.7373
18. In a certain game of chance, your chances of winning are 0.2. If you play the game five
times and outcomes are independent, the probability that you win all five times is
A) 1 B) 0.6723 C) 0.3277 D) 0.04 E) 0.00032
19. In a certain game of chance, your chances of winning are 0.2. You play the game five
times and outcomes are independent. Suppose it costs $1 to play the game each time.
Each time you win, you receive $4 (for a net gain of $3). Each time you lose, you
receive nothing (for a net loss of $1). Your expected winnings for five plays are
A) $3 B) $1 C) $0 D) –$1 E) –$2
Use the following to answer questions 20-22:
A survey asks a random sample of 2000 adults in Alabama if they support an increase in the state
sales tax from 5% to 6%, with the additional revenue going to education. Let X denote the
number in the sample that say they support the increase. Suppose that only 30% of all adults in
Alabama support the increase.
20. The mean of X is
A) 5% B) 0.4 C) 0.75
D) 30
21. The standard deviation of X is
A) 30 B) 1.278 C) 18.97
E) 600
D) 20.49
E) 600
22. The probability that X is more than 610 is about
A) 0.3034 B) 0.489 C) 0.7356 D) 0.8112
E) 1
23. 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 card (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 and p = 0.5
B) Y has a binomial distribution with n = 40 and p = 0.5, provided that the deck is
shuffled well.
C) Y has a binomial distribution with n = 40 and p = 0.5, provided 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 normal distribution with mean p = 0.5
E) Y has a geometric distribution with n = 40 and p = 0.5
24. A cell phone manufacturer claims that 92% of the cell phones of a certain model are free
of defects. Assuming that this claim is accurate, how many cell phones would you
expect to have to test until you find a defective phone?
A) 2, because it has to be a whole number.
B) 8
C) 12.5
D) 92
E) 93
25. The probability that a three-year-old battery still works is 0.8. A cassette recorder
requires four working batteries to operate. The state of batteries can be regarded as
independent, and four three-year-old batteries are selected for the cassette recorder.
What is the probability that the cassette recorder operates?
A) 0.9984 B) 0.8 C) 0.5904 D) 0.4096 E) not enough information.
26. Twenty percent of all trucks undergoing a certain inspection will fail the inspection.
Assume that trucks are independently undergoing this inspection, one at a time. The
expected number of trucks inspected before a truck fails inspection is
A) 2 B) 4 C) 5 D) 20 E) not enough information.
27. Two percent of circuit boards manufactured by a particular company are defective. If
circuit bards are randomly selected for testing, the probability that the number of circuit
boards inspected until a defective one is found is greater than 10 is
A) 1.024 × 107 B) 5.12 × 107 C) 0.1829 D) 0.8171 E) not enough information.
28. A random sample of 15 people is taken from a population in which 40% favor a
particular political stand. What is the probability that exactly 6 individuals in the sample
favor this political stand?
A) 0.6098 B) 0.5 C) 0.4 D) 0.2066 E) 0.0041
29. Experience has shown that a certain lie detector will show a positive reading (indicates a
lie) 10% of the time when a person is telling the truth and 95% of the time when a
person is lying. Suppose that a random sample of 5 suspects is subjected to a lie detector
test regarding a recent one-person crime. The probability of observing no positive
reading if all suspects plead innocent and are telling the truth is about
A) 0.409
B) 0.735
C) 0.00001
D) 0.591
E) 0.99999
30. Which of the following is NOT an assumption of the binomial distribution?
A) All trials must be identical.
B) All trials must be independent.
C) Each trial must be classified as a success or failure.
D) The number of successes in the trials is counted.
E) The probability of success is equal to 0.5 in all trials.
31. Brenna is a lifetime 60% free throw shooter. Suppose this probability is the same for
each free throw she attempts, and free throw attempts are independent. The probability
that it takes more than 3 free throws before she makes her first free throw is
A) 0.936
B) 0.6
C) 0.064
D) less than 0.0001
E) none of the above
32. It has been estimated that about 30% of frozen chickens contain enough salmonella
bacteria to cause illness if improperl7 cooked. A consumer purchases 12 frozen
chickens. What is the approximate probability that the consumer will have more than 6
contaminated chickens?
A) 0.961 B) 0.118 C) 0.882 D) 0.039 E) 0.079
33. Refer to the previous question. Suppose a supermarket buys 1000 chickens from a
supplier. The number of frozen chickens that may be contaminated that are within two
standard deviations of the mean is between A and B. The numbers A and B are
A) (90, 510)
B) (290.8, 309.2)
C) (0, 730)
D) (271, 329)
E) (255, 345)
34. Megan makes 90% of her free throws. Suppose this probability is the same for each free
throw she attempts, and free throw attempts are independent. The probability that she
makes all of her first four free throws and then misses her fifth attempt this season is
A) 0.93439 B) 0.08192 C) 0.06561 D) 0.00128 E) 0.00032
35. Suppose X has a geometric distribution with probability 0.3 of success and 0.7 of failure
on each observation. The probability that X = 4 is
A) 0.0081 B) 0.0189 C) 0.1029 D) 0.2401 E) none of these
36. If X has a binomial distribution with n = 400 and p = 0.4, the Normal approximation for
the binomial probability of the event {155 < X < 175} is
A) 0.6552 B) 0.6429 C) 0.6078 D) 0.6201 E) 0.632
37. In a test of ESP (extrasensory perception), the experimenter looks at cards that are
hidden from the subject. Each card contains a star, a circle, a wavy line, or a square. The
subject tries to read the experimenter’s mind and name the shape on each card. If the
subject is just guessing, what is the expected number of guesses before the subject gets
his first correct guess?
A) 5 B) 4 C) 3 D) 2.5 E) 0.25
38. Suppose X has a geometric distribution with probability 0.3 of success and 0.7 of failure
on each observation. The mean and variance of X are
A) mean = 0.3, variance = 0.21
D) mean = 3.33, variance = 7.78
B) mean = 1.43, variance = 0.78
E) mean = 1.43, variance = 0.61
C) mean = 3.33, variance = 2.79
39. Caitlin makes 75% of her free throws. Suppose this probability is the same for each free
throw she attempts, and free throw attempts are independent. The probability that she
makes all of her first five free throws and then misses her sixth attempt is about
A) 0.75
B) 0.0593
C) 0.5835
D) 0.00073
E) 1
40. In order for the random variable X to have a geometric distribution, which of the
following conditions must X satisfy?
A)
B)
C)
D)
E)
I.
np ≥ 10 and n(1 – p) ≥ 10
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.
The n observations have to be a random sample.
III and IV
II, III, IV, and V
I and III
I, III, and V
II, and III
41. 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 X be
the number of cards you turn over until you observe the first red card. The probability
that X is greater than 2 is
A) 0.125 B) 0.5 C) 0.25 D) 0.945
E) cannot be done, trials are not independent.
42. Suppose we select an SRS of size n = 100 from a large population having proportion p
of successes. Let X be the number of successes in the sample. For which value of p
would it be safe to assume the sampling distribution of X is approximately Normal?
A) 0.01 B) 1/9 C) 0.975 D) 0.9999 E) all of these
43. Suppose we roll a fair die 10 times. The probability that an even number occurs exactly
the same number of times as an odd number on the 10 rolls is
A) 0.1667 B) 0.2461 C) 0.3125 D) 0.5 E) none of these
44. In a test of ESP (extrasensory perception), the experimenter looks at cards that are
hidden from the subject. Each card contains a star, a circle, a wavy line, or a square. The
subject tries to read the experimenter’s mind and name the shape on each card. What is
the probability that the subject gets the first four correct before giving a wrong answer if
he is just guessing?
A) 0.00293 B) 0.00391 C) 0.06328 D) 0.0791 E) 0.31641
45. Seventeen people have been exposed to a particular disease. Each one independently has
a 40% chance of contracting the disease. A hospital has the capacity to handle 10 cases
of the disease. What is the probability that the hospital’s capacity will be exceeded?
A) 0.965 B) 0.035 C) 0.989 D) 0.011 E) 0.736
46. Refer to the previous problem. Planners need to have enough beds available to handle a
proportion of all outbreaks. Suppose a typical outbreak has 100 people exposed, each
with a 40% chance of coming down with the disease. Which is not correct?
A) This scenario satisfies the assumption of a binomial distribution.
B) About 95% of the time, between 30 and 50 people will contract the disease.
C) Almost all of the time, between 25 and 55 people will contract the disease.
D) On average, about 40 people will contract the disease.
E) Almost all the time, less than 40 people will be infected.
47. There are 10 patients on the neonatal ward of a local hospital who are monitored by 2
staff members. If the probability of a patient requiring emergency attention by a staff
member is 0.3, what is the probability that there will be sufficient staff to attend all
emergencies? Assume that emergencies occur independently.
A) 0.3828 B) 0.3 C) 0.09 D) 0.91 E) 0.6172
48. In 1989 Newsweek reported that 60% of young children have blood lead levels that
could impair their neurological development. Assuming that a class in a school is a
random sample from the population of all children at risk, the probability that more than
3 children have to be tested until one is found to have a blood lead level that may impair
development is
A) 0.064 B) 0.096 C) 0.64 D) 0.16 E) 0.88
Extra Hints
*Be able to explain the difference between “binompdf” and “binomcdf”.
*Be able to explain the difference between “geometpdf” and “geometcdf”.
*Memorize all formulas from the notes.
*Know the properties of the binomial and geometric distributions.
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