Download CHAPTER 9 SECTION 2: SAMPLING DISTRIBUTIONS

CHAPTER 9 SECTION 2: SAMPLING DISTRIBUTIONS
MULTIPLE CHOICE
90. Given that X is a binomial random variable with very large n, the binomial probability P(X  5) is
approximated by the area under a normal curve to the right of
a. 4.5
b. 5.5
c. 4
d. 6
ANS: A
PTS: 1
REF: SECTION 9.2
91. Given that X is a binomial random variable with very large n, the binomial probability P(X = 5) is
approximated by the area under a normal curve between
a. 5 and 5
b. 4 and 6
c. 4.5 and 5.5
d. None of these choices.
ANS: C
PTS: 1
REF: SECTION 9.2
92. Given that X is a binomial random variable with very large n, the binomial probability P(X  5) is
approximated by the area under a normal curve to the left of
a. 5
b. 5
c. 5.5
d. 4.5
ANS: C
PTS: 1
REF: SECTION 9.2
93. As a general rule, the normal distribution is used to approximate the sampling distribution of the
sample proportion only if:
a. the sample size n is greater than 30.
b. the population proportion p is close to 0.50.
c. the underlying population is normal.
d. np and n(1  p) are both greater than or equal to 5.
ANS: D
PTS: 1
REF: SECTION 9.2
94. Given a binomial distribution with n trials and probability p of a success on any trial, a conventional
rule of thumb is that the normal distribution will provide an adequate approximation of the binomial
distribution if
a. np  5 and n(1  p)  5
b. np  5 and np(1  p)  5
c. np  5 and n(1  p)  5
d. None of these choices.
ANS: A
PTS: 1
REF: SECTION 9.2
95. A sample of size 200 is taken at random from an infinite population. Given that the population
proportion is 0.60, the probability that the sample proportion is greater than 0.58 is:
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a.
b.
c.
d.
0.281
0.719
0.580
0.762
ANS: A
PTS: 1
REF: SECTION 9.2
96. A sample of size 200 is taken at random from an infinite population. Given that the population
proportion is 0.60, the probability that the sample proportion is less than 0.58 is
a. 0.281
b. 0.719
c. 0.580
d. 0.762
ANS: B
PTS: 1
REF: SECTION 9.2
97. A sample of size 200 is taken at random from an infinite population. Given that the population
proportion is 0.60, the probability that the sample proportion is between 0.58 and 0.62 is:
a. 0.4314
b. 0.0320
c. 0.0200
d. None of these choices.
ANS: A
PTS: 1
REF: SECTION 9.2
98. Suppose that the probability p of success on any trail of a binomial distribution equals 0.90. Then for
which of the following number of trials, n, would the normal distribution provide a good
approximation to the binomial distribution?
a. 35
b. 45
c. 55
d. All of these choices are true.
ANS: C
PTS: 1
REF: SECTION 9.2
99. A sample of 250 observations is selected at random from an infinite population. Given that the
population proportion is .25, the standard error of the sampling distribution of the sample proportion
is:
a. 0.0274
b. 0.5000
c. 0.0316
d. 0.0548
ANS: A
PTS: 1
REF: SECTION 9.2
100. The standard error of the sample proportion gets larger as:
a. p approaches 0
b. p approaches 0.50
c. p approaches 1.00
d. None of these choices.
ANS: B
PTS: 1
REF: SECTION 9.2
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101. The standard deviation of is also called the:
a. standard error of the sample proportion.
b. standard deviation of the population.
c. standard deviation of the binomial.
d. None of these choices.
ANS: A
PTS: 1
REF: SECTION 9.2
TRUE/FALSE
102. The mean of the sampling distribution of the sample proportion , when the sample size n = 100 and
the population proportion p = 0.92, is 92.0.
ANS: F
PTS: 1
REF: SECTION 9.2
103. The standard error of the sampling distribution of the sample proportion , when the sample size n =
100 and the population proportion p = 0.30, is 0.0021.
ANS: F
PTS: 1
REF: SECTION 9.2
104. Recall the rule of thumb used to indicate when the normal distribution is a good approximation of the
sampling distribution for the sample proportion . For the combination n = 50, p = 0.05, the rule is
satisfied.
ANS: F
PTS: 1
REF: SECTION 9.2
105. In an effort to identify the true proportion of college freshman who are under 18 years of age, a
random sample of 500 freshmen was taken. Fifty of them were under the age of 18. The value 0.10 is a
point estimate of the true proportion of freshman under age 18.
ANS: T
PTS: 1
REF: SECTION 9.2
106. As a general rule, the normal distribution is used to approximate the sampling distribution of the
sample proportion only if the sample size n is greater than or equal to 30.
ANS: F
PTS: 1
REF: SECTION 9.2
107. If a simple random sample of 300 observations is taken from a population whose proportion p = 0.6,
then the expected value of the sample proportion is 0.60.
ANS: T
PTS: 1
REF: SECTION 9.2
108. In general, the binomial probability P(X = x) is approximated by the area under a normal curve
between x  .5 and x + .5.
ANS: T
PTS: 1
REF: SECTION 9.2
109. In general, the binomial probability P(X  x) is approximated by the area under the normal curve to the
left of x + .5.
ANS: T
PTS: 1
REF: SECTION 9.2
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copied, or distributed without the prior consent of the publisher.
110. In general, the binomial probability P(X  x) is approximated by the area under the normal curve to the
left of x  .5.
ANS: F
PTS: 1
REF: SECTION 9.2
COMPLETION
111. The estimator of the probability of success in a binomial distribution is: ____________________.
ANS:
the sample proportion
PTS: 1
REF: SECTION 9.2
112. Under certain conditions where n is large enough, you can approximate the ____________________
distribution using the ____________________ distribution.
ANS: binomial; normal
PTS: 1
REF: SECTION 9.2
113. The continuity correction factor adds and subtracts the number ____________________ to/from x to
find P(X = x).
ANS: 0.5
PTS: 1
REF: SECTION 9.2
114. To find the binomial probability P(X  4) we calculate the area under the normal curve to the
____________________ (left/right) of the number ____________________.
ANS: left; 4.5
PTS: 1
REF: SECTION 9.2
115. The expected value of
is ____________________.
ANS: p
PTS: 1
REF: SECTION 9.2
116. The variance of
is ____________________.
ANS:
PTS: 1
REF: SECTION 9.2
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copied, or distributed without the prior consent of the publisher.
117. The standard deviation of
is also called the ____________________ of the sample proportion.
ANS: standard error
PTS: 1
118.
REF: SECTION 9.2
is approximately normally distributed provided that ____________________ and
____________________ are both greater than or equal to 5.
ANS:
np; n(1  p)
n(1  p); np
PTS: 1
119.
REF: SECTION 9.2
has an approximate normal distribution provided that np and n(1  p) are both greater than or equal
to ____________________.
ANS:
5
five
PTS: 1
REF: SECTION 9.2
120. The normal approximation to the binomial distribution gets better and better as
____________________ increases.
ANS:
n
the sample size
the number of trials
PTS: 1
REF: SECTION 9.2
SHORT ANSWER
121. The probability of success on any trial of a binomial experiment is 20%. Find the probability that the
proportion of success in a sample of 400 is:
a. less than 18%.
b. more than 18%.
c. between 18% and 22%.
ANS:
a. 0.1587
b. 0.8413
c. 0.6826
PTS: 1
REF: SECTION 9.2
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copied, or distributed without the prior consent of the publisher.
122. Suppose it is known that 60% of students at a particular college are smokers. A sample of 500 students
from the college is selected at random. Approximate the probability that at least 280 of these students
are smokers.
ANS:
0.97
PTS: 1
REF: SECTION 9.2
123. What is the probability that you get more than 200 heads if you flip a coin 400 times?
ANS:
0.500. No correction factor is needed
PTS: 1
REF: SECTION 9.2
John Kerry
Al Gore, the former Vice President of the United States, believes that the proportion of voters who will
vote for John Kerry in the year 2004 presidential elections is 0.65. A sample of 500 voters is selected
at random.
124. {John Kerry Narrative} Assume that Gore is correct and p = 0.65. What is the sampling distribution of
the sample proportion ? Explain.
ANS:
Approximately normal, since np = 325 and n(1  p) = 175 are both greater than or equal to 5.
PTS: 1
REF: SECTION 9.2
125. {John Kerry Narrative} Find the expected value and the standard deviation of the sample proportion .
ANS:
E( ) = 0.65, and
PTS: 1
= 0.0213
REF: SECTION 9.2
126. {John Kerry Narrative} What is the probability that the number of voters in the sample who will vote
for John Kerry in the year 2004 is between 340 and 350?
ANS:
0.0699
PTS: 1
REF: SECTION 9.2
127. Let X be a binomial random variable with n = 100 and p = 0.7. Approximate the following
probabilities, using the normal distribution.
a. P(X = 75)
b. P(X  70)
c. P(X > 70)
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ANS:
a. 0.05
b. 0.54
c. 0.46
PTS: 1
REF: SECTION 9.2
Rental Store
A videocassette rental store wants to know what proportion of its customers are under age 21. A
simple random sample of 500 customers was taken, and 375 of them were under age 21. Presume that
the true population proportion of customers under age 21 is 0.68.
128. {Rental Store Narrative} Describe the shape of the sampling distribution of proportion of customers
who are under age 21.
ANS:
Since np  5, and n(1  p)  5, the sampling distribution of
PTS: 1
is approximately normal.
REF: SECTION 9.2
129. {Rental Store Narrative} Find the mean and standard deviation of .
ANS:
= p = 0.68, and
PTS: 1
REF: SECTION 9.2
130. {Rental Store Narrative} What is the probability that the sample proportion
proportion of customers who are under age 21?
is within 0.03 of the true
ANS:
0.8502
PTS: 1
REF: SECTION 9.2
131. Given a binomial random variable with n = 15 and p = .40, find the exact probabilities of the following
events and their normal approximations.
a. X = 6
b. X  9
c. X  10
ANS:
a. Exact and approximated probabilities are 0.207 and 0.2052, respectively.
b. Exact and approximated probabilities are 0.095 and 0.0934, respectively.
c. Exact and approximated probabilities are 0.991 and 0.9911, respectively.
PTS: 1
REF: SECTION 9.2
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copied, or distributed without the prior consent of the publisher.
Graduate Assistantships
The chairman of the statistics department in a certain college believes that 70% of the department's
graduate assistantships are given to international students. A random sample of 50 graduate assistants
is taken.
132. {Graduate Assistantships Narrative} Assume that the chairman is correct and p = 0.70. What is the
sampling distribution of the sample proportion ? Explain.
ANS:
Approximately normal, since np = 35 and n(1  p) = 15 are both greater than or equal to 5.
PTS: 1
REF: SECTION 9.2
133. {Graduate Assistantships Narrative} Find the expected value and the standard error of the sampling
distribution of .
ANS:
E( ) = 0.70, and
PTS: 1
= 0.0648
REF: SECTION 9.2
134. {Graduate Assistantships Narrative} What is the probability that the sample proportion
0.65 and 0.73?
is between
ANS:
0.4566
PTS: 1
REF: SECTION 9.2
135. {Graduate Assistantships Narrative} What is the probability that the sample proportion
.05 of the population proportion p?
is within
ANS:
0.5588
PTS: 1
REF: SECTION 9.2
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copied, or distributed without the prior consent of the publisher.