Chapter 11—Estimation: Describing a single population
MULTIPLE CHOICE
1. The term 1 – refers to:
A.
the probability that a confidence interval does not contain the population parameter.
B.
the level of confidence that a confidence interval does not contain the population
parameter.
C.
the probability that a confidence interval contains the population parameter.
D.
the level of confidence that a confidence interval contains the population parameter.
2. The letter in the formula for constructing a confidence interval estimate of the population mean is:
A.
the level of confidence.
B.
the probability that a confidence interval will contain the population mean.
C.
the probability that a confidence interval will not contain the population mean.
D.
the area in the lower tail of the sampling distribution of the sample mean.
3. A 90% confidence interval estimate of the population mean can be interpreted to mean that:
A.
if we repeatedly draw samples of the same size from the same population, 90% of the
values of the sample means
x
will result in a confidence interval that includes the
population mean .
B.
there is a 90% probability that the population mean will lie between the lower confidence
limit (LCL) and the upper confidence limit (UCL).
C.
we are 90% confident that we have selected a sample whose range of values does not
contain the population mean .
D.
We are 90% confident that 10% the values of the sample means
x
will result in a
confidence interval that includes the population mean .
4. Which of the following is not a characteristic of a good estimator?
A.
Biasedness
B.
Consistency
C.
Relative efficiency
D.
Unbiasedness
5. The width of a confidence interval estimate of the population mean widens when the:
A.
level of confidence increases.
B.
sample size increases.
C.
population standard deviation decreases.
D.
Sample mean gets further from the population mean.
6. Which of the following statements is false?
A.
The width of a confidence interval estimate of the population mean narrows when the
sample size increases.
B.
The width of a confidence interval estimate of the population mean narrows when the
value of the sample mean increases.
C.
The width of a confidence interval estimate of the population mean widens when the
confidence level increases.
D.
All of the above statements are true.
7. Which of the following statements is false?
A.
The sample size needed to estimate a population mean is directly proportional to the
population variance.
B.
The sample size needed to estimate a population mean is directly proportional to the
square of the standard normal cutoff value z.
C.
The sample size needed to estimate a population mean is directly proportional to the
square of the maximum allowable error W.
D.
All of the above statements are true.
8. A 90% confidence interval estimate for a population mean is determined to be 43.78 to 52.19. If the
confidence level is increased to 95%, the confidence interval :
A.
becomes wider.
B.
remains the same.
C.
becomes narrower.
D.
None of the above answers is correct.
9. The z value for a 96.6% confidence interval estimate is:
A.
2.12.
B.
1.82.
C.
2.00.
D.
1.96.
10. In developing an interval estimate for a population mean, the population standard deviation was
assumed to be 10. The interval estimate was 50.92 ± 2.14. Had equaled 20, the interval estimate
would have been:
A.
60.92 ± 2.14.
B.
50.92 ± 12.14.
C.
101.84 ± 4.28.
D.
50.92 ± 4.28.
11. In developing an interval estimate for a population mean, a sample of 50 observations was used. The
interval estimate was 19.76 ± 1.32. Had the sample size been 200 instead of 50, the interval estimate
would have been:
A.
19.76 ± 0.33.
B.
19.76 ± 0.66.
C.
9.88 ± 1.32.
D.
4.94 ± 1.32.
12. After constructing a confidence interval estimate for a population mean, you believe that the interval is
useless because it is too wide. In order to correct this problem without changing the level of
confidence, you need to:
A.
increase the population standard deviation.
B.
reduce the population mean.
C.
reduce the sample size.
D.
increase the sample size.
13. In developing an interval estimate at 87.4% for a population mean, the value of z to use is:
A.
1.15.
B.
0.32.
C.
1.53.
D.
0.16.
14. The sample size needed to estimate a population mean to within 2 units with a 95% confidence when
the population standard deviation equals 8 is:
A.
9.
B.
61.
C.
62.
D.
8.
15. A random sample of 64 observations has a mean of 30. The population variance is assumed to be 9.
The 85.3% confidence interval estimate for the population mean (to the third decimal place) is:
A.
28.369 ± 31.631.
B.
29.456 ± 30.544.
C.
28.560 ± 31.440.
D.
29.383 ± 30.617.
16. In developing an interval estimate for a population mean, the interval estimate was 62.84 to 69.46. The
population standard deviation was assumed to be 6.50, and a sample of 100 observations was used.
The mean of the sample was:
A.
56.34.
B.
62.96.
C.
13.24.
D.
66.15.
17. A point estimate is defined as:
A.
the average of the sample values.
B.
the average of the population values.
C.
a single value of an estimator.
D.
an interval within which the population parameter is believed to lie .
18. An unbiased estimator of a population parameter is defined as:
A.
an estimator which is always equal to the parameter.
B.
an estimator whose variance is equal to one.
C.
an estimator whose expected value is equal to the parameter.
D.
an estimator whose variance goes to zero as the sample size goes to infinity.
19. An estimator is said to be consistent if:
A.
it is an unbiased estimator.
B.
the variance of the estimator is close to one.
C.
the expected value of the estimator is known and positive.
D.
it is an unbiased estimator and the difference between the estimator and the population
parameter grows smaller as the sample size grows larger.
20. If there are two unbiased estimators of a population parameter, the one whose variance is smaller is
said to be:
A.
a biased estimator.
B.
relatively efficient.
C.
consistent.
D.
relatively unbiased.
21. Which of the following statements is (are) correct?
A.
The sample mean is an unbiased estimator of the population mean.
B.
The sample proportion is an unbiased estimator of the population proportion.
C.
The sample mean is a consistent estimator.
D.
All of the above statements are correct.
22. Which of the following statements is (are) true?
A.
The sample mean is relatively more efficient than the sample median.
B.
The sample median is relatively more efficient than the sample mean.
C.
The sample variance is relatively more efficient than the sample variance.
D.
All of the above statements are true.
23. The problem with relying on a point estimate of a population parameter is that:
A.
it has no variance.
B.
it might be unbiased.
C.
it might not be relatively efficient.
D.
it does not tell us how close or far the point estimate might be from the parameter.
24. A confidence interval is defined as:
A.
a point estimate plus or minus a specific level of confidence.
B.
a lower and upper confidence limit associated with a specific level of confidence.
C.
an interval that has a 95% probability of containing the population parameter.
D.
a lower and upper confidence limit that has a 95% probability of containing the population
parameter.
25. As its name suggests, the objective of estimation is to determine the approximate value of:
A.
a population parameter on the basis of a sample statistic.
B.
a sample statistic on the basis of a population parameter.
C.
the sample mean.
D.
the sample variance.
26. The sample variance
2
s
is an unbiased estimator of the population variance
2
when the denominator
of
2
s
is:
A.
n.
B.
n – 1.
C.
n.
D.
n + 1.
27. Which of the following assumptions must be true in order to use the formula
nzx /
2/
to find a
confidence interval estimate of the population mean?
A.
The population variance is known.
B.
The population mean is known.
C.
The population is normally distributed.
D.
The confidence level is greater than 90%.
28. In the formula
nzx /
2/
, the
2/
refers to:
A.
the probability that the confidence interval will contain the population mean.
B.
the probability that the confidence interval will not contain the population mean.
C.
the area in the lower tail or upper tail of the sampling distribution of the sample mean.
D.
the level of confidence.
29. Which of the following is not a part of the formula for constructing a confidence interval estimate of
the population mean?
A.
A point estimate of the population mean.
B.
The standard error of the sampling distribution of the sample mean.
C.
The confidence level.
D.
The value of the population mean.
30. The smaller the level of confidence used in constructing a confidence interval estimate of the
population mean, the:
A.
more likely that the confidence interval will contain the population mean.
B.
wider the confidence interval.
C.
narrower the confidence interval.
D.
larger the sample required to estimate the population mean to within a certain error bound.
31. A robust estimator is one that:
A.
is unbiased and symmetrical about zero.
B.
is consistent and is also mound-shaped.
C.
is efficient and less spread out.
D.
is not sensitive to moderate departure from the assumption of normality in the population.
32. Which of the following statements is false?
A.
The t-distribution is symmetric about zero.
B.
The t-distribution is more spread out than the standard normal distribution.
C.
As the number of degrees of freedom gets smaller, the t–distribution’s dispersion gets
smaller.
D.
The t-distribution is mound-shaped.
33. The student t-distribution approaches the normal distribution as the:
A.
number of degrees of freedom increases.
B.
number of degrees of freedom decreases.
C.
sample size decreases.
D.
population size increases.
34. As the number of degrees of freedom for a t-distribution increases:
A.
the dispersion of the distribution decreases.
B.
the shape of the distribution becomes narrower and taller.
C.
the t-distribution becomes more and more similar to the standard normal distribution.
D.
All of the above statements are correct.
35. For statistical inference about the mean of a single population when the population standard deviation
is unknown, the number of degrees for freedom for the t-distribution is equal to n – 1 because we lose
one degree of freedom by using the:
A.
sample mean as an estimate of the population mean.
B.
sample standard deviation as an estimate of the population standard deviation.
C.
sample proportion as an estimate of the population proportion.
D.
sample size as an estimate of the population size.
36. In constructing a confidence interval for the population mean when the population variance is
unknown, which of the following assumptions is required when using the following formula?
A.
The sample size is greater than 30.
B.
The population variance is known.
C.
The population is normal.
D.
The sample is drawn from a positively skewed distribution.
37. For a sample of size 30 taken from a normally distributed population with standard deviation equal to
5, a 95% confidence interval for the population mean would require the use of:
A.
z = 1.96.
B.
z = 1.645.
C.
t = 2.045.
D.
t = 1.699.
38. Which of the following is true about the t-distribution?
A.
It approaches the normal distribution as the number of degrees of freedom increases.
B.
It assumes that the population is normally distributed.
C.
It is more spread out than the standard normal distribution.
D.
All of the above statements are true.
39. A random sample of size 15 taken from a normally distributed population revealed a sample mean of
75 and a sample variance of 25. The upper limit of a 95% confidence interval for the population mean
would equal:
A.
77.769.
B.
72.231.
C.
72.727.
D.
77.273.
40. A random sample of size 20 taken from a normally distributed population resulted in a sample
variance of 32. The lower limit of a 90% confidence interval for the population variance would be:
A.
52.185.
B.
20.375.
C.
20.170.
D.
54.931.
41. In selecting the sample size to estimate the population proportion p, if we have no knowledge of even
the approximate values of the sample proportion
, we:
A.
take another sample and estimate p.
B.
take two more samples and find the average of their
values.
C.
let
= 0.50.
D.
let
= 0.95.
42. Under which of the following circumstances is it impossible to construct a confidence interval for the
population mean?
A.
A non-normal population with a large sample and an unknown population variance.
B.
A normal population with a large sample and a known population variance.
C.
A non-normal population with a small sample and an unknown population variance.
D.
A normal population with a small sample and an unknown population variance.
43. The use of the standard normal distribution for constructing a confidence interval estimate for the
population proportion p requires that:
A.
n
and n(1 –
) are both greater than 5, where
denotes the sample proportion.
B.
np and n(1 – p) are both greater than 5.
C.
n
and n(p +
) are both greater than 5.
D.
the sample size is greater than 5.
44. The upper limit of a confidence interval at the 99% level of confidence for the population proportion if
a sample of size 100 had 40 successes is:
A.
0.3040.
B.
0.4047.
C.
0.4960.
D.
0.4806.
45. A random sample of size n has been selected from a normally distributed population whose standard
deviation is s. In estimating an interval for the population mean, the t-distribution should be used
instead of the z–test if:
A.
n 30 and is known.
B.
is unknown and estimated by s, and the population is normal.
C.
Population is normal and is known.
D.
Both A and C are correct.
46. From a sample of 300 items, 15 are defective. The point estimate of the population proportion
defective will be:
A.
0.05.
B.
15.
C.
0.20.
D.
0.05–0.10.
47. After you calculate the sample size needed to estimate a population proportion to within 0.05, your
statistics lecturer tells you the maximum allowable error must be reduced to just 0.025. If the original
calculation led to a sample size of 400, the sample size will now have to be:
A.
800.
B.
200.
C.
4000.
D.
1600.
48. If the standard error of the sampling distribution of the sample proportion is 0.0229 for samples of size
400, then the population proportion must be either:
A.
0.4 or 0.6.
B.
0.5 or 0.5.
C.
0.2 or 0.8.
D.
0.3 or 0.7.
49. A sample of size 200 is to be taken at random from an infinite population. Given that the population
proportion is 0.60, the probability that the sample proportion will be greater than 0.58 is:
A.
0.281.
B.
0.719.
C.
0.580.
D.
0.762.
50. A sample of size 300 is to be taken at random from an infinite population. Given that the population
proportion is 0.70, the probability that the sample proportion will be smaller than 0.75 is:
A.
0.9706.
B.
0.4772.
C.
0.4706.
D.
0.9772.
51. If the standard error of the sampling distribution of the sample proportion is 0.0337 for samples of size
200, then the population proportion must be:
A.
0.25.
B.
0.75.
C.
either 0.20 or 0.80.
D.
either 0.35 or 0.65.
E.
either 0.30 or 0.70.
52. A sample of 250 observations is to be selected at random from an infinite population. Given that the
population proportion is 0.25, the standard error of the sampling distribution of the sample proportion
is:
A.
0.0274.
B.
0.50.
C.
0.0316.
D.
0.0548.
TRUE/FALSE
1. An interval estimate is a range of values within which the actual value of a population parameter falls.
2. A confidence interval is an interval estimate for which there is a specified degree of certainty that the
actual value of the population parameter will fall within the interval.
3. An interval estimate is an estimate of the range for a sample statistic.
4. An unbiased estimator of a population parameter is an estimator whose expected value is equal to the
population parameter to be estimated.
5. The sample proportion is an unbiased estimator of the population proportion.
6. The sample standard deviation is an unbiased estimator of the population standard deviation.
7. Knowing that an estimator is unbiased only assures us that its expected value equals the parameter, but
it does not tell us how close the estimator is to the parameter.
8. An unbiased estimator is said to be consistent if the difference between the estimator and the
parameter grows smaller as the sample size grows larger.
9. The probability that a confidence interval includes the parameter of interest is either 1 or 0.
10. The sample mean is a consistent estimator of the population mean
.
11. The sample proportion
is a consistent estimator of the population proportion p because it is unbiased
and the variance of
is p(1 – p)/n, which grows smaller as n grows larger.
12. The upper limit of the 90% confidence interval for
, given that n = 64,
x
= 70 and
= 20, is 65.89.
13. The lower and upper limits of the 68.26% confidence interval for the population mean
, given that n
= 64,
x
= 110 and
= 8, are 109 and 111, respectively.
14. An unbiased estimator is said to be consistent if the difference between the estimator and the
parameter grows larger as the sample size grows larger.
15. An unbiased estimator is relatively efficient compared to another unbiased estimator of the same
parameter if it has smaller variance.
16. The range of a confidence interval is a measure of the expected sampling error.
17. The difference between the sample statistic and actual value of the population parameter is the
confidence level of the estimate.
18. The sample variance
2
s
is an unbiased estimator of the population variance
2
when the denominator
of
2
s
is n – 1.