Test 2 – Version A
STAT 3090
Fall 2016
Multiple Choice: (Questions 1 - 20) Answer the following questions on the scantron provided using a #2 pencil.
Bubble the response that best answers the question. Each multiple choice correct response is worth 3 points. For
your record, also circle your choice on your exam since the scantron will not be returned to you. Only the
responses recorded on your scantron will be graded.
1.
Suppose that 75% of the students at Clemson university wear orange color T-shirts on Friday. If
1000 students are selected at random from this university, what are the mean and standard
deviation of the random variable X = number of selected students who wear the orange color
t-shirts on Friday? X is a binomial random variable
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A) Mean = 750; standard deviation = 13.69
B) Mean = 75; standard deviation = 187.5
C) Mean = 75; standard deviation = 13.69
D) Mean = 750; standard deviation = 187.5
2.
Which of the following will provide the smallest standard deviation of the sampling distribution
of the sample mean, thus resulting in the sample mean differing the least from sample to sample?
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A) Random sample of size 100 from a population with 𝜇𝜇 = 25 and 𝜎𝜎 = 8
B) Random sample of size 20 from a population with 𝜇𝜇 = 15 and 𝜎𝜎 = 8
C) Random sample of size 1000 from a population with 𝜇𝜇 = 25 and 𝜎𝜎 = 15
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D) Random sample of size 100 from a population with 𝜇𝜇 = 15 and 𝜎𝜎 = 12
3.
A gasoline tank for a certain car is designed to hold 15 gallons of gas. Suppose that the random
variable X = actual capacity of a randomly selected tank has a distribution that is well approximated
by a normal curve with mean 15.0 gallons and standard deviation 0.1 gallon. Which of the
following probability statements represents the probability that a randomly selected tank will hold
at most 14.8 gallons?
A) 𝑃𝑃(𝑋𝑋 ≤ 14.8)
B) 𝑃𝑃(𝑍𝑍 < −2.0)
C) 𝑃𝑃(𝑋𝑋 < 14.8)
D) All the above
1
Test 2 – Version A
4.
STAT 3090
Fall 2016
Suppose the standard deviation of the sampling distribution of 𝑋𝑋� is A, which of the following would
be the standard deviation if the sample size were only a quarter of the size?
A) 2A
B) 0.5A
C) 4A
5.
Ke
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D) 0.25A
The quality manager of a fortune cookie company believes that a larger than acceptable proportion
of paper fortunes being used are blank. Suppose a sample of 290 fortune cookies was taken. If the
true proportion of fortunes that were blank is 0.02, what is the probability that the sample
proportion is at most 0.03?
A) 0.1119
B) 5.8
D) 0.02
Let X denote the number of bars of service on your cell phone whenever you are at an intersection
with the following probabilities:
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6.
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C) 0.8888
x
P(X=x)
0
0.1
1
0.15
2
0.25
3
0.25
4
0.15
5
0.1
What is the expected value of X?
A) 2.5
B) 1
C) 0.53
D) 0.0.5
2
Test 2 – Version A
7.
STAT 3090
Fall 2016
On average, the number of pumpkin spiced lattes ordered at a local coffee shop during the month of
October is 17.2 per hour. Consider the random variable X that represents the number of pumpkin
spiced lattes ordered per day in the month of October. The random variable X has which of the
following probability distributions?
A) A poisson distribution
B) A binomial distribution
D) None of the above
8.
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C) A normal distribution
The thickness, x, of a protective coating applied to glass shower doors designed to ensure the
longevity and clarity of the glass follows a uniform distribution over the interval from 10 to 40
microns. Find the probability that the coating is less than 20 microns thick.
A) 0.3333
B) 0.6667
D) 0.3000
Which of the following situations describes a binomial random variable?
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9.
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C) 0.5000
A) You play two games against the same opponent. The probability that you win the first game
is 0.4. If you win the first game, the probability you also win the second game is 0.2. You
are interested in the probability that you win both games?
B) A new restaurant opening in Greenwich Village has a 30% chance of survival during their
first year. Assume restaurant openings are independent. You count the number of
restaurants that survive out of the 16 new restaurants that open this year.
C) You are training your dog, Sophie to catch a ball. You count the number of tosses before
Sophie catches her first ball. Assume that Sophie’s attempts to catch the ball are
independent.
D) You wish to obtain three out of the five prizes that are offered in Cocoa Krackers cereal
boxes. You purchase Cocoa Krackers until all three prizes are obtained.
3
Test 2 – Version A
10.
STAT 3090
Fall 2016
A roulette wheel has 38 slots, 18 are red, 18 are black, and 2 are green. Each slot is equally likely to
be selected. You play five games and always bet on red.
Approximately, what is the probability that you will win at least one game?
5
B) � � 0.47371 (1 − 0.4737)1
1
5
C) 1 - � � 0.47370 (1 − 0.4737)5
0
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5
A) � � 0.47370 (1 − 0.4737)5
0
5
5
D) � � 0.47370 (1 − 0.4737)5 + � � 0.47371 (1 − 0.4737)1
0
1
Determine n and p for the following situation:
A student is taking a 15 question multiple-choice Anthropology exam in which each question has
four choices. Assuming that she has no knowledge of the correct answer to any of the questions, she
has decided on a strategy in which she will place four balls (marked A, B, C, and D) into a
box. She randomly selects one ball for each question and replaces the ball in the box. The
marking on the ball will determine her answer to the question. This is a binomial situation with:
A) n = 4, p = 0.5
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B) n = 4, p=0.25
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11.
C) n = 15, p=0.5
D) n=15, p=0.25
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Test 2 – Version A
12.
STAT 3090
Fall 2016
In a batch of batteries 5% are defective. A random sample of 80 batteries is to be taken from a large
production of batteries. Let X be the number of defective batteries out of 80. X is a binomial
random variable. Which of the following is the best interpretation of the standard deviation of
X?
A) It is expected that there will be 4 defective batteries in a sample of 80 batteries.
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B) It is expected that the number of defective batteries in a sample of 80 batteries will be
between 5 and 6.
C) The number of batteries that are defective in repeated samples of 80 batteries in the long run
will typically vary from the mean by approximately 1.95 batteries.
D) The number of batteries that are defective in one single sample of 80 batteries once selected
will typically vary from the mean by approximately 1.95 batteries.
A)
23 𝑒𝑒 −2
B)
32 𝑒𝑒 −3
C)
32 𝑒𝑒 −3
D)
32 𝑒𝑒 −3
3!
+
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On average there are 3 typing errors per page of text. What is the probability that on any given
page of text there are 2 typing errors?
32 𝑒𝑒 −3
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13.
2!
2!
2!
+
1!
32 𝑒𝑒 −3
1!
+
32 𝑒𝑒 −3
0!
5
Test 2 – Version A
14.
STAT 3090
Fall 2016
Suppose a sample of 400 people is used to perform a taste test. If the true proportion of the
population that prefer Pepsi is 0.5. What is the standard deviation of the sampling distribution of
proportion of people that prefer Pepsi in a sample of 400?
A) 100
B) 10
C) 0.025
15.
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D) 0.000625
The average GPA at a particular school is 2.89 with a standard deviation 0.63. A random sample of
55 students is collected. Let 𝑋𝑋� denote average GPA from a sample of 55 students. What can be said
�?
about 𝑿𝑿
A) The sampling distribution of 𝑋𝑋� may not be normally distributed because the sample size is too
small.
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B) The sampling distribution of 𝑋𝑋� is normally distributed because it comes from a normally
distributed population.
C) The sampling distribution of 𝑋𝑋� is approximately normally distributed because the sample size
is small enough.
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D) The sampling distribution of 𝑋𝑋� is approximately normally distributed because the sample size
is large enough.
16.
Which of the following statement(s) is/are true for standard normal distribution?
I.
II.
III.
The total area under a probability distribution is equal to 1
The probability associated with one particular value P(X=x) = 0
The distribution has a median of 0.
A) Only I
B) Only I and II
C) Only I and III
D) Only II and III
E) All above are true
6
Test 2 – Version A
17.
STAT 3090
Fall 2016
Let Z be a standard normal random variable. What is the probability that Z will greater than 1.23?
A) 0.8907
B) 0.1093
C) 0.3907
D) 0.6093
Suppose Y = the number of broken eggs in a randomly selected carton of one dozen eggs. The
probability distribution of Y is as follows:
Y
P(y)
0
0.65
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18.
1
0.20
2
0.10
3
0.04
4
?
What is the probability that the number of broken eggs is at least 2?
A) 0.85
C) 0.15
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D) 0.05
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B) 0.95
19.
Let Z be a standard normal random variable. Find the value of 𝑍𝑍 such that 0.67 of the area under the
curve lies to the right of 𝑍𝑍.
A) 𝑧𝑧 = −0.44
B) 𝑧𝑧 = 0.44
C) 𝑧𝑧 = 0.7486
D) 𝑧𝑧 = 0.6293
7
Test 2 – Version A
20.
STAT 3090
Fall 2016
Coliform bacteria are randomly distributed in a certain Arizona river at an average concentration of
0.5 per 10 cc of water. A test tube contains 10 cc of liquid. Let X be the number of coliform
bacteria per test tube of water. X is a Poisson random variable. What is the standard deviation of
the number of coliform bacteria per test tube of water?
A) 0.5
B) 0.71
C) 3.16
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Ke
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D) 10
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Test 2 – Version A
STAT 3090
Fall 2016
Free Response: The Free Response questions will count 40% of your total grade. Read each question carefully. In
order to receive full credit you must show legible and logical (relevant) justification which supports your final
answer. You MUST show your work. Answers with no justification will receive no credit.
1. (5 pts) Explain which of the conditions for a binomial experiment is NOT met for the following random
variable.
A football team plays 12 games in its regular season. X = number of games the team wins.
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The probability of success does not remain constant from trial to trial – for each game the team does not
have the same chance of winning it is dependent on the opponent
OR
The trials are not independent – if the team wins one game that may influence the motivation of the team
and change the chance of winning the next game
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[2 pts – one of correct conditions stated; 3 pts – explanation in context of problem]
2. (4 pts) Suppose that the time students wait for a bus can be described by a uniform random variable X,
where X is between 0 and 20 minutes. On the axes below draw the probability density function for the
random variable X. Make sure to use proper labels.
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𝑓𝑓(𝑥𝑥)
1/20 =
0.05
0
20
𝑋𝑋
[2 pts for 0 to 20 labeled; 2 pts for 0.05 labeled]
9
Test 2 – Version A
STAT 3090
Fall 2016
3. Suppose the pulse rates of women is normally distributed with a mean of 75 and a standard deviation of
8.
A) (5 pts) What is the probability that a randomly selected woman has a pulse rate greater than 92?
Provide the probability statement (ie, P(…)), show work, and provide value to 4 decimal places.
Work shown may be in calculator syntax as long as appropriate parameters are properly labeled.
Ke
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𝑋𝑋 = 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑜𝑜𝑜𝑜 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤
𝑋𝑋 − 𝜇𝜇 92 − 75
� = 𝑃𝑃(𝑍𝑍 > 2.125)
𝑃𝑃(𝑋𝑋 > 92) = 𝑃𝑃 �
>
𝜎𝜎
8
= 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 (𝐿𝐿𝐿𝐿 = 2.125, 𝑈𝑈𝑈𝑈 = 1𝐸𝐸99, 𝜇𝜇 = 0, 𝜎𝜎 = 1) = 0.0168
[2 pts for probability statement; 3 points for work show and value] (-0.5 for minor arithmetic errors)
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B) (5 pts) What is the probability that 15 randomly selected woman has an average pulse rate less than
80? Provide the probability statement (ie, P(…)), show work, and provide value to 4 decimal places.
Work shown may be in calculator syntax as long as appropriate parameters are properly labeled.
𝑋𝑋� − 𝜇𝜇𝑋𝑋 80 − 75
<
� = 𝑃𝑃(𝑍𝑍 < 2.42)
𝑃𝑃 (𝑋𝑋� < 80) = 𝑃𝑃 � 𝜎𝜎
8�
� 𝑛𝑛
√
√15
= 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 (𝐿𝐿𝐿𝐿 = −1𝐸𝐸99, 𝑈𝑈𝑈𝑈 = 2.42, 𝜇𝜇 = 0, 𝜎𝜎 = 1) = 0.9922
[2 pts for probability statement; 3 points for work show and value] (-0.5 for minor arithmetic errors)
10
Test 2 – Version A
STAT 3090
Fall 2016
4. (5 pts) The Graduate Record Examination (GRE) is a standardized test that students usually take before
entering graduate school. According to a publication by the Educational Testing Service, the scores on
the verbal portion of the GRE are approximately normally distributed with mean 462 points and
standard deviation 119 points. What score would a student need to achieve in order to be at the 90th
percentile? Work shown may be in calculator syntax as long as appropriate parameters are properly
labeled.
X = GRE score
Ke
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invNorm(area = 0.90,μ=0, σ=1) = 1.28
1.28 =
𝑋𝑋 − 462
119
𝑋𝑋 = 614.32
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[3 points for z-score; 2 points for X value] (Don’t have to label μ and σ if designating as a z-score –
mean 0 and standard deviation of 1 implied)
5. (5 pts) Airline passengers arrive randomly and independently at the passenger-screening facility at a
major international airport. The mean arrival rate is 10 passengers per minute. Let X be the number of
passengers that arrive within one-minute. X then follows a Poisson distribution. What is the
probability that exactly 40 passengers arrive within 4 minutes. Provide the probability statement (ie,
P(…)), show work, and provide value to 4 decimal places. Work shown may be in calculator syntax as
long as appropriate parameters are properly labeled.
Y = number of passengers within 4 minutes
10
𝜆𝜆𝑌𝑌
=
1 𝑚𝑚𝑚𝑚𝑚𝑚 4 𝑚𝑚𝑚𝑚𝑚𝑚
4040 𝑒𝑒 −40
= 0.0629
40!
= 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑑𝑑𝑑𝑑(𝜆𝜆 = 40, 𝑥𝑥 = 40)
𝑃𝑃 (𝑌𝑌 = 40) =
[2 pts probability statement; 2 pts for changing lambda; 1 point for work shown]
11
Test 2 – Version A
STAT 3090
Fall 2016
Ke
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6. (5 pts) A study conducted by the Pew Research Center showed that 75% of 18 to 34 years olds living
with their parents say they contributed to household expenses. Suppose that a random sample of fifteen
18 to 34 year olds living with their parents is selected and asked if they contribute to household
expenses. Let X be the number that say they contribute to household expenses out of the 15. X is a
binomial random variable. What is the probability that between 12 and 14 (inclusive) of those selected
say they contribute to household expenses? Provide the probability statement (ie, P(…)), show work,
and provide value to 4 decimal places. Work shown may be in calculator syntax as long as appropriate
parameters are properly labeled.
er
𝑃𝑃(12 ≤ 𝑋𝑋 ≤ 14) = 𝑃𝑃(𝑋𝑋 = 12) + 𝑃𝑃(𝑋𝑋 = 13) + 𝑃𝑃(𝑋𝑋 = 14)
15
15
15
= � � 0.7512 0.253 + � � 0.7513 0.252 + � � 0.7514 0.251
13
14
12
= 0.225199 + 0.155907 + 0.066817
= 0.4479
= 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 (𝑛𝑛 = 15, 𝑝𝑝 = 0.75, 𝑥𝑥 = 14) − 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏(𝑛𝑛 = 15, 𝑝𝑝
= 0.75, 𝑥𝑥 = 11)
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[2 pts probability statement; 3 points work]
12
Test 2 – Version A
STAT 3090
Fall 2016
7. (5 pts) Suppose that fund-raisers at a university call recent graduates to request donations for campus
outreach programs. They report the following information for last year’s graduates:
Size of donation
Proportion of calls
$0
0.45
$10
0.30
$25
0.20
$50
0.05
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Consider the random variable X = amount donation for a person selected at random from the population
of last year’s graduates of this university. What is the standard deviation of X? Show work, provide
answer to 2 decimal places and provide correct units.
𝜇𝜇 = (0)(0.45) + (10)(0.30) + (25)(0.20) + (50)(0.05) = $10.50
𝜎𝜎 2 = [(0)2 (0.45) + (10)2 (0.30) + (25)2 (0.20) + (50)2 (0.05)] − 10.52
= 169.75
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𝜎𝜎 = √169.75 = $13.03
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[-3 if forget to subtract mean squared; -2 for forgetting to take square root; -1 for forgetting unit]
Correct SCANTRON: If your scantron is correctly bubbled with a #2 pencil, with your correct XID, your correct test
version, AND the front of your test is completed with your signature on the academic integrity statement, you earn 1
point.
END OF TEST
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