Download Chapter 7 - Practice Problems 3

Chapter 7 - Practice Problems 3
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Determine the margin of error in estimating the population mean, μ.
1) Based on a sample of 36 randomly selected years, a 90% confidence interval for the mean annual
precipitation in one city is from 46.3 inches to 49.7 inches. Find the margin of error.
1)
A) There is not enough information to find the margin of error.
B) 3.4 inches
C) 1.7 inches
D) 0.47
Find the necessary sample size.
2) Scores on a certain test are normally distributed with a variance of 14. A researcher wishes to
estimate the mean score achieved by all adults on the test. Find the sample size needed to assure
with 98 percent confidence that the sample mean will not differ from the population mean by more
than 2 units.
A) 11
B) 39
C) 20
D) 267
3) You wish to estimate the mean weight of machine components of a certain type and you require a
92% degree of confidence that the sample mean will be in error by no more than 0.008 g. Find the
sample size required. A pilot study showed that the population standard deviation is estimated to
be 0.06 g.
A) 11
B) 173
C) 14
D) 112
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
4) Based on a sample of size 25, a researcher obtains an estimate of 64.4 inches for the mean
height of all women aged 30-40. At the 95% confidence level, the margin of error is
1.2 inches. Do you agree with the interpretation below? If not, explain why not and give a
correct interpretation.
4)
Researcherʹs interpretation: ʺWe can be 95% confident that the height of a randomly
selected woman will differ from 64.4 inches by at most 1.2 inches.ʺ
5) A sample mean is used to estimate a population mean. To obtain a margin of error of 1.5 at
a confidence level of 95%, a sample size of 120 is needed. Would the required sample size
be larger or smaller if the researcher wished to
5)
(a) increase the confidence level while keeping the same margin of error?
(b) decrease the margin of error while keeping the same confidence level?
Explain your answers.
6) Compare the basic properties of t-curves with the basic properties of the standard normal
curve. In what ways are t-curves similar to the standard normal curve? In what ways are
they different?
1
2)
6)
3)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
7) Which of the following statements regarding t-curves is/are true?
7)
A. The total area under a t-curve with 10 degrees of freedom is greater than the area under the
standard normal curve.
B. The t-curve with 10 degrees of freedom is flatter and wider than the standard normal curve.
C. The t-curve with 10 degrees of freedom more closely resembles the standard normal curve than
the t-curve with 20 degrees of freedom.
A) A
B) C
C) B
D) B, C
Solve the problem.
8) A 90% confidence interval for the mean percentage of airline reservations being canceled on the
day of the flight is (1.3%, 5.1%). What is the point estimate of the mean percentage of reservations
that are canceled on the day of the flight?
A) 2.55%
B) 1.90%
C) 3.8%
D) 3.20%
9) Forty-five CEOs from the electronics industry were randomly sampled and a 95% confidence
interval for the average salary of all electronics CEOs was constructed. The interval was
($129,997, $149,612). At what confidence level are the inferences derived from this information
valid?
A) 5%
B) 47.5%
C) 0.95%
8)
9)
D) 95%
10) Forty-five CEOs from the electronics industry were randomly sampled and a 90% confidence
interval for the average salary of all electronics CEOs was constructed. The interval was
($95,136, $110,372). Give a practical interpretation of the interval above.
10)
A) 90% of the electronics industry CEOs have salaries that fall between $95,136 to $110,372.
B) We are 90% confident that the mean salary of all the electronics industry CEOs falls in the
interval $95,136 to $110,372.
C) 90% of the sampled CEOs salaries fell in the interval $95,136 to $110,372.
D) We are 90% confident that the mean salary of the sampled CEOs falls in the interval $95,136
to $110,372.
11) Forty-five CEOs from the electronics industry were randomly sampled and a 99% confidence
interval for the average salary of all electronics CEOs was constructed. The interval was
($142,281, $153,625). To make more useful inferences from the data, it is desired to reduce the width
of the confidence interval. What will result in a reduced interval width?
A) Decrease the sample size and increase the confidence level.
B) Decrease the sample size and decrease the confidence level.
C) Increase the sample size and decrease the confidence level.
D) Increase the sample size and increase the confidence level.
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11)
12) At Great State University there are many sections of Introductory Statistics. All students are given
a list of selected homework problems for each section. Some students choose to do the homework
and some do not. The scores of the first test from a random sample of 100 students who do their
homework were analyzed. The scores had a mean of 82.4 with a standard deviation of 5.6. The true
mean score of the students doing their homework is 72.1. Is there evidence at the 95% confidence
level that the true mean test score for the students doing their homework differs from 72.1?
12)
A) Yes. 72.1 is more than one standard deviation from 82.4.
B) No. 72.1 is within the 95% confidence interval.
C) Yes. 72.1 falls outside of the 95% confidence interval.
D) No. 72.1 is outside of the 95% confidence interval, but we need to also find the 95%
confidence level of the scores of the students neglecting their homework.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
13) The numbers of advertisements seen or heard in one week for 30 randomly selected
people in the United States are listed below. Construct a 95% confidence interval for the
true mean number of advertisements.
598
481
734
494
298
588
441
135
590
595
846
540
728
764
673
690
317
727
684
649
545
486
732
486
735
582
702
13)
808
677
703
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
14) Find the value t0 such that the following statement is true: P(-t0 ≤ t ≤ t0 ) = .99 where df = 9.
A) 2.2821
B) 3.250
C) 1.833
14)
D) 2.262
15) Suppose a 98% confidence interval for μ turns out to be (1,000, 2,100). If this interval was based on
a sample of size n = 23, explain what assumptions are necessary for this interval to be valid.
A) The population must have an approximately normal distribution.
B) The sampling distribution of the sample mean must have a normal distribution.
C) The sampling distribution must be biased with 22 degrees of freedom.
D) The population must have an approximate t distribution.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
16) A marketing research company needs to estimate the average total compensation of CEOs
in the service industry. Data were randomly collected from 18 CEOs and the 98%
confidence interval was calculated to be ($2,181,260, $5,836,180). Based on the interval
above, do you believe the average total compensation of CEOs in the service industry is
$1,500,000?
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16)
15)
Answer Key
Testname: CH 7 SET 3
1)
2)
3)
4)
C
C
B
The researcherʹs interpretation is not correct since the margin of error refers to the difference between the estimate
and the true mean, not to the difference between the estimate and the height of an individual woman. The correct
interpretation is: We can be 95% confident that the estimate of 64.4 inches differs from the true mean by at most
1.2 inches.
5) In both cases, a larger sample would be needed.
6) For both t-curves and the standard normal curve:
- The total area under the curve is 1.
- The curve is bell-shaped.
- The curve is symmetrical about 0.
- The curve extends indefinitely in both directions, approaching, but never touching, the horizontal axis as it does so.
The t-curves are flatter and wider than the standard normal curve, though as the number of degrees of freedom
increases, the t-curves looks increasingly like the standard normal curve.
7) C
8) D
9) D
10) B
11) C
12) C
13) (543.8, 658. 0)
14) B
15) A
16) No, the average total compensation is not $1,500,000 and we are 98% sure of this statement. This is because the value
$1,500,000 is not contained in the 98% confidence interval for μ.
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