1. The standard error of sample mean is large when the observations in the population are spread out (large ), but
that the standard error can be reduced by taking a smaller sample.
a. True
b. False
2. A sample in
...
1. The standard error of sample mean is large when the observations in the population are spread out (large ), but
that the standard error can be reduced by taking a smaller sample.
a. True
b. False
2. A sample in which the sampling units are chosen from the population by means of a random mechanism is a
a. probability sample b. judgmental sample
c. stratified sample d. systematic sample
3. A judgmental sample is a sample in which the
a. sampling units are chosen using a random number table
b. quality of sampling units judged
c. sampling units are chosen according to the sampler’s judgment
d. sampling units are all biased and vocal about it
4. Potential sample members, called sampling units, are:
a. people b. companies
c. households d. All of these options
5. In sampling, a population is:
a. the set of all humans
b. the set of all members about which a study intends to make inferences
c. any group of test subjects
d. a random group of individuals, households, cities or countries
6. A list of all members of the population is called a:
a. sampling unit b. probability sample
c. frame d. relevant population
7. A list of all members of the population from which we can choose a sample is called a frame, and the potential sample
members are called sampling units.
a. True
b. False
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8. A probability sample is a sample in which the sampling units are chosen from the population by means of a random
mechanism such as a random number table.
a. True
b. False
9. A sample chosen in such a way that every possible subset of same size has an equal chance of being selected is
called a(n)
a. interval estimation b. point estimation
c. simple random sample. d. statistic
10. The sampling method in which a population is divided into blocks and then selected by choosing a random mechanism
is called a
a. random sampling b. systematic sampling
c. stratified sampling d. cluster sampling
11. Which of the following is not a consideration when determining appropriate sample size?
a. The cost of sampling b. The timely collection of the data
c. Interviewer fatigue d. The likelihood of nonsampling error
12. Identifiable subpopulations within a population are called:
a. clusters
b. samples
c. blocks
d. strata
e. None of these options
13. The defining property of a simple random sample is that:
a. every sample has the same chance of being chosen
b. the easiest method to access samples are chosen
c. the fewest samples are chosen
d. every fourth subject is chosen as a sample
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14. Selecting a random sample from each identifiable subgroup within a population is called:
a. random sampling
b. systematic sampling
c. stratified sampling
d. cluster sampling
e. None of these options
15. Which of the following are reasons for why simple random sampling is used infrequently in real applications?
a. Samples can be spread over a large geographic region
b. Simple random sampling requires that all sampling units be identified prior to sampling
c. Simple random sampling can result in underrepresentation or overrepresentation of certain segments of the
population
d. All of these options
16. With proportional sample sizes:
a. The proportion of a stratum in the sample is independent of the proportion of that stratum in the population
b. The proportion of a stratum in the sample is the same as the proportion of that stratum in the population
c. The proportion of a stratum in the sample is greater than the proportion of that stratum in the population
d. The proportion of a stratum in the sample is less than the proportion of that stratum in the population
17. Simple random samples are typically used in real applications.
a. True
b. False
18. A simple random sample is one where each member of the population has a known chance (this may differ from one
member to another) or probability of being chosen.
a. True
b. False
19. In systematic sampling, one of the first k members is selected randomly, and then every kth member after this one is
selected. The value k is called the sampling interval and equals the ratio N / n, where N is the population size and n is
the desired sample size.
a. True
b. False
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20. In stratified sampling, the population is divided into relatively homogeneous subsets called strata, and then random
samples are taken from each stratum.
a. True
b. False
21. In stratified sampling with proportional sample sizes, the proportion of each stratum selected differs from stratum to
stratum.
a. True
b. False
22. In cluster sampling, the population is divided into subsets called clusters (such as cities or city blocks), and then a
random sample of the clusters is selected. Once the clusters are selected, we typically sample all of the members in
each selected cluster.
a. True
b. False
23. Which of the following statements correctly describe estimation?
a. It is the process of inferring the values of known population parameters from those of unknown sample statistics.
b. It is the process of inferring the values of unknown sample statistics from those of known population parameters.
c. It is the process of inferring the values of known sample statistics from those of unknown population parameters.
d. It is the process of inferring the values of unknown population parameters from those of known sample statistics.
24. A sampling error is the result of:
a. measurement error b. nonresponse bias
c. nontruthful responses d. bad luck
25. The standard deviation of is usually called the
a. standard error of the mean b. standard error of the sample
c. standard error of the population d. randomized standard error
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26. When a portion of the sample does not respond to the survey, results.
a. measurement error
b. nonresponse bias
c. sampling error
d. systematic failure
e. nonlinear error
27. The accuracy of the point estimate is measured by its:
a. standard deviation b. standard error
c. sampling error d. nonsampling error
28. The sampling mean is the estimate for the population mean .
a. random b. point
c. simple d. interval
29. Non-truthful response is a particular problem when:
a. sensitive questions are asked. b. surveys are anonymous.
c. interviewers are not trained. d. the sample is from an unusual population.
30. Measurement error occurs when:
a. a portion of the sample does not respond to the survey
b. the sample responses are not clear
c. the responses to question do not reflect what the investigator had in mind
d. the investigator does not correctly tally all responses
31. The two basic sources for error when using random sampling are:
a. sampling and selection
b. identification and selection
c. sampling and nonsampling
d. bias and randomness
e. linear and nonlinear
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32. Sampling error is evident when:
a. a question is poorly worded
b. the sample is too small
c. the sample is not random
d. the sample mean differs from the population mean
33. The opportunity for sampling error is decreased by:
a. larger sample sizes b. smaller sample sizes
c. affluent samples d. educated samples
34. The theorem that states that the sampling distribution of the sample mean is approximately normal when the
sample size n is reasonably large is known as the:
a. central limit theorem b. central tendency theorem
c. simple random sample theorem d. point estimate theorem
35. There is approximately % chance that any particular will be within two standard deviations of the population mean (
).
a. 90 b. 95
c. 99 d. 99.7
36. Which of the following statements are correct?
a. A point estimate is an estimate of the range of a population parameter
b. A point estimate is a single value estimate of the value of a population parameter
c. A point estimate is an unbiased estimator if its standard deviation is the same as the actual value of the
population standard deviation
d. All of these options
37. An unbiased estimator is a:
a. sample statistic used to approximate a population parameter
b. sample statistic, which has an expected value equal to the value of the population parameter
c. sample statistic whose value is usually less than the population parameter
d. standard error of the mean
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38. Which of the following statements are correct?
a. A confidence interval describes a range of values that is likely not to include the actual population parameter
b. A confidence interval is an estimate of the range for a sample statistic.
c. A confidence interval is an estimate of the range of possible values for a population parameter.
d. None of these options
39. The approximate standard error of the sample mean is calculated as:
a. b.
c. d.
40. The approximate 95% confidence interval for a population mean is:
a. b.
c. d.
41. The finite population correction factor, , should generally be used when:
a. N is any finite size
b. n is less than 5% of the population size N
c. n is greater than 5% of the population size N
d. n is any finite size
42. The Central Limit Theorem (CLT) is generally valid for:
a. n > 5
b. n > 10
c. n > 20
d. n > 30
e. any size n
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43. The averaging effect means that as you average more and more observations from a given distribution, the variance of
the average
a. increases
b. decreases
c. is unaffected
d. could either increase, decrease or stay the same
44. When we sample less than 5% of the population, the finite population correction factor; fpc = , is used
to modify the formula for the standard error of the sample mean.
a. True
b. False
45. A point estimate is a single numeric value, a “best guess” of a population parameter, calculated from the sample data.
a. True
b. False
46. The difference between the point estimate and the true value of the population parameter being estimated is called the
estimation error.
a. True
b. False
47. A confidence interval is an interval calculated from the population data, where we strongly believe the true value of the
population parameter lies.
a. True
b. False
48. The sampling distribution of any point estimate (such as the sample mean or proportion) is the distribution of the point
estimates we would obtain from all possible samples of a given size drawn from the population.
a. True
b. False
49. An unbiased estimate is a point estimate such that the mean of its sampling distribution is equal to the true value of
the population parameter being estimated.
a. True
b. False
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50. The standard error of an estimate is the standard deviation of the sampling distribution of the estimate. It measures how
much estimates from different samples vary.
a. True
b. False
51. Ideally, we prefer estimates that have large standard errors.
a. True
b. False
52. It is customary to approximate the standard error of the sample mean by substituting the sample standard
deviation s for in the formula: SE( ) = .
a. True
b. False
53. An estimator is said to be unbiased if the mean of its sampling distribution equals the value of the population parameter
being estimated.
a. True
b. False
54. Estimation is the process of inferring the value of an unknown population parameter using data from a random sample
a. True
b. False
55. The Central Limit Theorem (CLT) states that the sampling distribution of the mean is approximately normal, no matter
what the distribution of the population, so long as the sample size is large enough.
a. True
b. False
56. The averaging effect says that as you average more and more observations from a given distribution, the variance of the
average increases.
a. True
b. False
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57. We can measure the accuracy of judgmental samples by applying some simple rules of probability. This way,
judgmental samples are not likely to contain our built-in biases.
a. True
b. False
58. The probability of being chosen in a simple random sample of size n from a population of size N is:
a. 1/N b. N – 1/n
c. N/n d. n/N
59. The key to using stratified sampling is:
a. identifying the strata b. selecting the appropriate strata
c. defining the strata d. randomizing the strata
60. If systematic sampling is chosen as the sampling technique, it is probably because:
a. Systematic sampling has better statistical properties than simple random sampling
b. Systematic sampling is more convenient
c. Systematic sampling always results in more representative sampling than simple random sampling
d. None of these options
61. The primary advantage of cluster sampling is sampling convenience (and possibly less cost). The downside, however, is
that the inferences drawn from a cluster sample can be less accurate, for a given sample size, than for other sampling
plans.
a. True
b. False
62. If a simple random sample of size n is chosen from a population of size N, then each member of the population has
probability N / n of being chosen in the sample.
a. True
b. False
63. Simple random sampling can result in under-representation or over-representation of certain segments of the population.
This is one of several reasons that simple random samples are almost never used in real applications.
a. True
b. False
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64. Stratified samples are typically not used in real applications because they provide less accurate estimates of
population parameters for a given sampling cost.
a. True
b. False
65. Cluster sampling is often less convenient and more costly than other random sampling methods.
a. True
b. False
66. One obvious advantage of stratified sampling is that we obtain separate estimates within each stratum – which we
would not obtain if we took a simple random sample from the entire population. A more important advantage is that we
can increase the accuracy of the resulting population estimates by using appropriately defined strata.
a. True
b. False
67. Systematic sampling is generally similar to simple random sampling in its statistical properties.
a. True
b. False
68. The mean of the sampling distribution of always equals
a. the population mean b. / n
c. the population standard deviation d. / n
69. The opportunity for nonsampling error is increased by:
a. larger sample sizes b. smaller sample sizes
c. affluent samples d. educated samples
70. The reason the Central Limit Theorem (CLT) is such an important result in statistics is because:
a. The CLT allows us to assume that the population distribution is approximately normal, provided n is reasonably
large
b. The CLT allows us to estimate the population mean without knowing the exact form of the population distribution,
provided n is reasonably large
c. The CLT allows us to construct confidence intervals for the population mean without knowing the exact form of
the population distribution, provided n is reasonably large
d. All of these options
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71. A sample of size 20 is selected at random from a population of size N. If the finite population correction factor is 0.9418,
then N must be 169.
a. True
b. False
72. The sampling distribution of the mean will have the same mean as the original population from which the samples were
drawn.
a. True
b. False
73. The sampling distribution of the mean will have the same standard deviation as the original population from which the
samples were drawn.
a. True
b. False
74. The randomized response technique is a way of getting at sensitive information to avoid estimation errors due to
nontruthful responses.
a. True
b. False
75. Voluntary response bias occurs when the responses to questions do not reflect what the investigator had in mind.
a. True
b. False
76. If the sample size is greater than 30, the Central Limit Theorem (CLT) will always apply.
a. True
b. False
77. The Central Limit Theorem (CLT) says that as long as the sample size is reasonably large, there is about a 95%
chance that the magnitude of the sampling error for the mean will be no more than two standard errors.
a. True
b. False
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78. The size of a sample can be selected by first determining the desired standard error and then using the formula
to calculate n.
a. True
b. False
79. Consider the frame of 50 full-time employees of Computer Technologies, Inc (CTI). CTI’s human resources manager has
collected annual salary figures for all employees and she has calculated a mean of $47,723, a median of $41,082 and a
standard deviation of $24,167. A simple random sample of 10 employees is presented below (salary is in $1,000’s).
Compute the mean, median, and standard deviation for the sample and compare these statistics with the measures for
the entire company.
Employee 1 2 3 4 5 6 7 8 9 10
Salary 38.8 46.7 61.1 49.6 58.5 78.8 36.7 46.5 47.6 56.7
Current annual salaries:
Number of years of post-secondary education:
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80. (A) Compute the mean, median, and standard deviation of the annual salaries for the 52 employees in the given frame.
(B) Use Excel to choose a systematic sample of size 13 from the frame of annual salaries.
(C) Compute the mean, median, and standard deviation of the annual salaries for the 13 employees included in your
systematic sample in (B)
(D) Compare your statistics in (C) with your computed descriptive measures for the frame in (A). Is your systematic
sample representative of the frame with respect to the annual salary variable?
(E) Assume that we wish to stratify these employees by the number of years of post-secondary education, select such
a stratified sample of size 15 with approximately proportional sample sizes.
(F) Compute the mean, median, and standard deviation of the annual salaries for the 15 employees included in your
stratified sample in (E).
(G) Compare these statistics in (F) with your computed descriptive measures for the frame obtained in (A). Is your
stratified sample representative of the frame with respect to the annual salary variable?
(C)
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(D) After generating the summary measures for both the frame and the sample, we can conclude that the
sample does not represent the frame well. The mean, median, and standard deviation of the frame are all
much smaller than the mean, median, and standard deviation of the sample.
(E) This portion of the solution involves several steps. First, we noted the total sample size needed.
Second, we developed the strata we will use to separate the given frame: in this case we placed every two
years in a new stratum as shown below. Next, we generated a column labeled "Category", to place a
number between 1 and 5 next to the salary that corresponds with the stratum of that number. For
example, if the annual salary was of a person who only had 2 years of education beyond secondary
education, then a number 2 for Stratum 2 was placed next to the salary. The "Category" column was
generated using an IF statement. We then unstacked the categories in order to count the number of
salaries in each stratum. This was done by using StatTools's Data Utilities/Unstack function. Once this
was completed, we used Excel's COUNT function to count the number of values in each stratum and then
generated proportional numbers for each stratum with respect to the size of the given population. Once
the proportions are generated, we used an Excel's random number function to assign a random number to
each salary. Then, by using Excel to sort the salaries in each stratum by their random number (in this
case by ascending number) we selected the salaries in each stratum that will be included in the sample.
These salaries are shown below.
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(F)
(G) When looking at the mean, median, and standard deviation of both the sample and population, we can
conclude that the stratified sample represents the population fairly well, although the summary measures
are all slightly lower than those of the population.
81. A sales manager for a company that makes commercial ovens for restaurants is interested in estimating the average
number of restaurants in all metropolitan areas across the entire country. He does not have access to the data for each
metropolitan location, so he had decided to select a sample that will be representative of all such areas, and will use a
sample size of 30. Do you believe that simple random sampling is the best approach to obtaining a representative
subset of the metropolitan areas in the given frame? Explain. If not, recommend how the sales manager might proceed
to select a better sample of size 30 from this data?
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Suppose that the average weekly earnings for employees in general automotive repair shops is $450, and that the
standard deviation for the weekly earnings for such employees is $50. A sample of 100 such employees is selected at
random.
82. (A) Find the mean and standard deviation of the sampling distribution of the average weekly earnings in the sample.
(B) Find probability that the mean of the sample is less than $445.
(C) Find the probability that the mean of the sample is between $445 and $455.
(D) Find the probability that the mean of the sample is greater than $460.
(E) Explain why the assumption of normality about the distribution of the average weekly earnings for employees was
not involved in the answers to (A) through (D).
ANSWER: (A) E( ) = = 450, and SE( ) = = 5
(B) P( < 445) = P(Z < -1) = 0.5000 – 0.3413 = 0.1587
(C) P(445< <455) = P(-1.0 < Z < 1.0) = 2(0.3413) = 0.6826
(D) P( > 460) = P(Z > 2.0) = 0.5000 – 0.4772 = 0.0228
(E) The sample size is large; n = 100 is greater than 30, so the distribution of the average weekly
earnings for employees is at least approximately normal.
Suppose that you are an entrepreneur interested in establishing a new Internet-based auction service. Furthermore,
suppose that you have gathered basic demographic information on a large number of Internet users. You currently have
information on 1000 individuals related to their gender, age, education, marital status, annual household income, and
number of people in household. Assume that these individuals were carefully selected through stratified sampling.
83. (A) To assess potential interest in your proposed enterprise, you would like to conduct telephone interviews with a
representative subset of the 1000 Internet users. How would you proceed to stratify the given frame of 1000 individuals
to choose 50 for telephone interviews? Explain your approach.
(B) Explain how you could apply cluster sampling to obtain a sample size of 50 from this frame. What are the
advantages and disadvantages of employing cluster sampling in this case?
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An editor of a local newspaper is concerned with the number of errors that are found in the daily paper. In order to
understand the extent of this problem, the editor would like to estimate the average number of errors in the daily paper.
The frame in this case is the number of errors found in the daily paper for the past six months (180 issues).
84. (A) What sample size would be required for the production personnel to be approximately 95% sure that their estimate
of the average number of errors per issue is within 4 errors of the true mean? Assume that the editor’s best estimate of
the population standard deviation ( ) is 10 errors per issue.
(B) How does your answer to (A) change if the editor wants the estimate to be within 3 errors of the actual
population mean? Explain the difference in your answers to (A) and (B).
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85. (A) Using these 50 students as the frame, use Excel to generate a simple random sample of size 10 from this frame.
(B) Compute the mean scores in the frame and the simple random sample you generated in (A).
(C) Compare the mean scores you computed in (B). Is your simple random sample a good representative of the frame?
Why or why not?
(D) Using these 50 students as the frame, use Excel to generate a systematic sample of size 10 from this frame.
(E) Compare the mean scores in the frame with that in the systematic sample in (D). What do you conclude?
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A university bookstore manager is mildly concerned about the number of textbooks that were under-ordered and thus
unavailable two days after the beginning of classes. The manager instructs an employee to pick a random number, go
to the place where that number book is shelved, examine the next 50 titles, and record how many titles are unavailable.
86. (A) Technically, this process does not yield a random sample of the books in the store. Why not?
(B) How could a truly random sample be obtained?
87. (A) Do you think she has obtained a true random sample?
(B) What average price could she report, based on the above sample?
(C) What average price range could she report, based on the above sample?
(D) Do you see any issues with reporting the range calculated for (C)?
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· The selling price of each house (in thousands of dollars)
· The size of each house (in hundreds of square feet)
· The number of bedrooms in each house
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88. (A) Suppose that Sally wishes to examine a representative subset of these 60 houses that has been stratified by the
number of bedrooms. Use Excel to assist her by finding such a stratified sample of size 10 with proportional sample
sizes.
(B) Explain how Sally could apply cluster sampling in selecting a sample of size 15 from this frame.
(C) What are the advantages and disadvantages of employing cluster sampling in this case?
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$75.30 $614.11 $696.34 $572.08
$748.23 $21.20 $99.79 $1,233.38
$530.40 $378.37 $596.14 $239.65
$2,995.38 $1,069.06 $929.80 $259.98
$123.65 $68.92 $192.35 $754.45
$309.00 $163.31 $71.75 $904.92
$40.70 $161.12 $459.38 $171.48
$402.81 $157.44 $41.81 $87.08
$489.97 $468.12 $400.57 $319.40
$533.82 $1,801.35 $1,666.50 $37.16
$85.92 $91.43 $193.14 $106.95
$214.62 $10.62 $582.18 $39.65
$123.66 $76.33 $291.73 $398.48
$659.18 $101.24 $1,740.47 $322.26
$1,509.34 $1,599.04 $358.62 $492.05
$1,052.68 $596.33 $100.54 $1,288.70
$421.46 $1,799.51 $581.21 $571.63
$180.58 $98.82 $358.68 $38.93
$874.78 $2,761.93 $750.44 $376.60
$269.48 $456.79 $216.81 $305.49
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89. (A) What sample size would be required for the auditors to be approximately 95% sure that their estimate of the
average savings account balance at this bank is within $150 of the true mean? Assume that their best estimate of the
population standard deviation is $300.
(B) Choose a simple random sample of the size found in (A).
(C) Compute the observed sampling error based on the sample you have drawn from the population. How
does the actual sampling error compare to the maximum possible probable absolute error established in
(A)? Explain
ANSWER:
(A)
(B) The simple random sample of size 16 was generated using StatTool’s Random Sample tool in the
Data Utilities section. Next, the VLOOKUP function was used to place the appropriate balances next to
the customers that were selected to be included in the sample. The following sample was obtained.
Customer Balance
40 456.79
51 193.14
63 239.65
37 1799.51
8 402.81
20 269.48
42 99.79
39 2761.93
78 38.93
3 530.40
35 1599.04
64 259.98
14 659.18
32 10.62
11 85.92
68 87.08
(C)
Based on the above sample (results will differ):
The sample mean = $593.39
The frame mean = $537.31
The sampling error is the difference between the sample mean and the frame mean. In this
case, the sampling error is $56.08, which is much less than the maximum probable absolute
error of $150. This is the case because the maximum probable absolute error is, by definition,
the largest possible amount that will still give 95% certainty. As illustrated here, the observed
sampling error is smaller than the largest possible error.
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90. A cannery claims that its sardine cans have a net weight of 8 oz., with a standard deviation of 0.1 oz. You take a simple
random sample of 30 cans and encounter a sample mean of 7.85 oz. Are you inclined to believe the claim?
ANSWER:
The sampling distribution of is normal (since n 30) with mean and standard deviation given by E( )
= = 8, and SE( ) = = 0.0183, respectively.
Therefore, P( < 7.8) = P(Z < -8.2) = 0. If the claim were true, such a sample would not be
encountered. The cannery management is not telling the truth.
The manager of a local fast-food restaurant is interested in improving service provided to customers who use the
restaurant’s drive-up window. As a first step in the process, the manager asks his assistant to record the time (in
minutes) it takes to serve a large number of customers at the final window in the facility’s drive-up system. The given
frame in this case is 200 customer service times observed during the busiest hour of the day for this fast-food
restaurant. The frame of 200 service times yielded a mean of 0.881. A simple random sample of 10 from this frame is
presented below.
Customer 1 2 3 4 5 6 7 8 9 10
Service time 1.02 1.18 0.95 0.90 0.85 1.10 0.75 0.60 1.25 1.00
91. (A) Compute the point estimate of the population mean from the sample above. What is the sampling error in this case?
Assume that the population consists of the given 200 customer service times.
(B) Compute the point estimate of the population standard deviation from the sample above.
(C) Should you use the finite population correction (fpc) factor to estimate the standard error of ? Explain. If your
answer is yes, what is the value of the fpc?
(D) Determine a good approximation to the standard error of the mean in this case.
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Chapter 07
A battery manufacturer wants to estimate the average number of defective (or dead) batteries contained in a box
shipped by the company. Production personnel at this company have recorded the number of defective batteries found
in each of the 2000 boxes shipped in the past week.
92. (A) What sample size would be required for the production personnel to be approximately 95% sure that their estimate
of the average number of defective batteries per box is within 0.3 unit of the true mean? Assume that the best estimate
of the population standard deviation ( ) is 0.9 defective batteries per box.
(B) How does your answer to (A) change if the production personnel want their estimate to be within 0.5
unit of the actual population mean? Evaluate the tradeoff between required accuracy and sample size
requirement for this case and the case in (A).
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