Chapter 7
Confidence Intervals
True/False
1. The t distribution always has n degrees of freedom.
Answer: False Difficulty: Easy
2. Assuming the same level of significance α, as the sample size increases, the value of tα/2 approaches the value of zα/2.
Answer: True Difficulty: Medium
3. When constructing a confidence interval for a sample proportion, the t distribution is appropriate if the sample size is small.
Answer: False Difficulty: Medium
4. When the population is normally distributed and the population standard deviation s is unknown, then for any sample size n, the sampling distribution of is based on the z distribution.
Answer: False Difficulty: Medium (REF)
5. When the sample size and sample standard deviation remain the same, a 99% confidence interval for a population mean, m will be narrower than the 95% confidence interval for m.
Answer: False Difficulty: Medium (REF)
6. When the level of confidence and sample standard deviation remain the same, a confidence interval for a population mean based on a sample of n = 100 will be narrower than a confidence interval for a population mean based on a sample of n = 50.
Answer: True Difficulty: Medium
7. When the level of confidence and the sample size remain the same, a confidence interval for a population mean m will be wider, when the sample standard deviation s is small than when s is large.
Answer: False Difficulty: Medium
8. When the level of confidence and sample proportion remain the same, a confidence interval for a population proportion p based on a sample of n = 100 will be wider than a confidence interval for p based on a sample of n = 400.
Answer: True Difficulty: Medium
9. When the level of confidence and sample size remain the same, a confidence interval for a population proportion p will be narrower when is larger than when is smaller.
Answer: False Difficulty: Medium (REF)
10. When solving for the sample size needed to compute a 95% confidence interval for a population proportion “p”, having a given error bound “B”, we choose a value of that makes as small as reasonably possible.
Answer: False Difficulty: Medium (REF)
11. When determining the sample size n, if the value found for n is 79.2, we would choose to sample 79 observations.
Answer: False Difficulty: Medium (REF)
Multiple Choice
12. The t distribution approaches the _______________ as the sample size ___________.
A) Binomial, increases
B) Binomial, decreases
C) Z, decreases
D) Z, increases
Answer: D Difficulty: Medium (REF)
13. The width of a confidence interval will be:
A) Narrower for 99% confidence than 95% confidence.
B) Wider for a sample size of 100 than for a sample size of 50.
C) Narrower for 90% confidence than 95% confidence.
D) Wider when the sample standard deviation (s) is small than when s is large.
Answer: C Difficulty: Medium
14. As standard deviation increases, samples size _____________ to achieve a specified level of confidence.
A) Increases
B) Decreases
C) Remains the same
Answer: A Difficulty: Medium
15. When determining the sample size, if the value found is not an integer initially, the next highest integer value will ____________ be chosen.
A) Always
B) Sometimes
C) Never
Answer: A Difficulty: Medium
16. When constructing a confidence interval for a population mean, if a population is normally distributed and a small sample is taken, then the distribution of is based on _____ distribution.
A) z
B) t
C) Neither
D) Both A and B
Answer: B Difficulty: Medium
17. A confidence interval increases in width as
A) The level of confidence increases
B) n decreases
C) s increases
D) All of the above
Answer: D Difficulty: Medium (REF)
18. The width of a confidence interval will be:
A) Narrower for 98% confidence than for 90% confidence.
B) Wider for a sample size of 64 than for a sample size of 36.
C) Wider for a 99% confidence than for 95% confidence
D) Narrower for sample size of 25 than for a sample size of 36.
E) None of the above
Answer: C Difficulty: Medium
19. A Research and Development Laboratory researcher for a paint company is measuring the level a certain chemical contained in a certain type of paint. If the paint contains too much of this chemical, the quality of the paint will be compromised. On the average, each can of paint contains 10% of the chemical, How many cans of paint should the sample contain if the researcher wants to be 98% certain of being within 1% of the true proportion of this chemical?
A) 4887
B) 1107
C) 26
D) 645
Answer: A Difficulty: Medium (AS)
20. Which of the following is an advantage of confidence interval estimate over a point estimate for a population parameter?
A) Interval estimates are more precise than point estimates.
B) Interval estimates are less accurate than point estimates.
C) Interval estimates are both more accurate and more precise than point estimates.
D) Interval estimates take into account the fact that the statistic being used to estimate the population parameter is a random variable.
Answer: D Difficulty: Medium
21. When the sample size and sample standard deviation remain the same, a 99% confidence interval for a population mean, m will be _________________ the 95% confidence interval for m.
A) Wider than
B) Narrower than
C) Equal to
Answer: A Difficulty: Medium (REF)
22. When the level of confidence and sample standard deviation remain the same, a confidence interval for a population mean based on a sample of n = 100 will be ______________ a confidence interval for a population mean based on a sample of n = 50.
A) Wider than
B) Narrower than
C) Equal to
Answer: B Difficulty: Medium
23. When the level of confidence and the sample size remain the same, a confidence interval for a population mean m will be ________________, when the sample standard deviation s is small than when s is large.
A) Wider
B) Narrower
C) Neither A nor B, they will be the same
Answer: B Difficulty: Medium
24. When the sample size and the sample proportion remain the same, a 90% confidence interval for a population proportion p will be ______________ the 99% confidence interval for p.
A) Wider than
B) Narrower than
C) Equal to
Answer: B Difficulty: Medium
25. When the level of confidence and sample proportion remain the same, a confidence interval for a population proportion p based on a sample of n = 100 will be ______________ a confidence interval for p based on a sample of n = 400.
A) Wider than
B) Narrower than
C) Equal to
Answer: A Difficulty: Medium
26. When the level of confidence and sample size remain the same, a confidence interval for a population proportion p will be ______________ when is larger than when is smaller.
A) Wider
B) Narrower
C) Neither A nor B, they will be the same
Answer: A Difficulty: Medium
27. When the population is normally distributed, population standard deviation s is unknown, and the sample size is n = 15; the confidence interval for the population mean m is based on the:
A) z (normal) distribution
B) t distribution
C) Binomial distribution
D) Poisson Distribution
E) None of the above
Answer: B Difficulty: Medium
28. When solving for the sample size needed to compute a 95% confidence interval for a population proportion “p”, having a given error bound “B”, we choose a value of that makes:
A) as small as reasonably possible
B) as large as reasonably possible
C) as close to .5 as reasonably possible
D) as close to .25 as reasonably possible
E) Both B and D are correct
Answer: E Difficulty: Medium (REF)
29. When a confidence interval for a population proportion is constructed for a sample size n = 30 and the value of = .4, the interval is based on the:
A) z distribution
B) t distribution
C) exponential distribution
D) Poisson distribution
E) None of the above
Answer: A Difficulty: Medium
30. There is little difference between the values of tα/2 and zα/2 when the sample:
A) size is small
B) size is large
C) mean is small
D) mean is large
E) standard deviation is small
Answer: B Difficulty: Medium
31. Assuming the same level of significance a, as the sample size increases, the value of ta/2 _____ approaches the value of .
A) Always
B) Sometimes
C) Never
Answer: A Difficulty: Medium
32. In a manufacturing process a random sample of 9 bolts manufactured has a mean length of 3 inches with a variance of .09. What is the 90% confidence interval for the true mean length of the bolt?
A) 2.8355 to 3.1645
B) 2.5065 to 3.4935
C) 2.4420 to 3.5580
D) 2.8140 to 3.8160
E) 2.9442 to 3.0558
Answer: D Difficulty: Hard
33. In a manufacturing process a random sample of 9 bolts manufactured has a mean length of 3 inches with a standard deviation of .3 inches. What is the 95% confidence interval for the true mean length of the bolt?
A) 2.804 to 3.196
B) 2.308 to 3.692
C) 2.770 to 3.231
D) 2.412 to 3.588
E) 2.814 to 3.186
Answer: C Difficulty: Hard
34. In a manufacturing process a random sample of 36 bolts manufactured has a mean length of 3 inches with a standard deviation of .3 inches. What is the 99% confidence interval for the true mean length of the bolt?
A) 2.902 to 3.098
B) 2.884 to 3.117
C) 2.871 to 3.129
D) 2.228 to 3.772
E) 2.902 to 3.098
Answer: C Difficulty: Medium
35. The internal auditing staff of a local manufacturing company performs a sample audit each quarter to estimate the proportion of accounts that are delinquent (more than 90 days overdue). For this quarter, the auditing staff randomly selected 400 customer accounts and found that 80 of these accounts were delinquent. What is the 95% confidence interval for the proportion of all delinquent customer accounts at this manufacturing company?
A) .1608 to .2392
B) .1992 to .2008
C) .1671 to .2329
D) .1485 to .2515
E) .1714 to .2286
Answer: A Difficulty: Hard
36. The internal auditing staff of a local manufacturing company performs a sample audit each quarter to estimate the proportion of accounts that are current (between 0 and 60 days after billing). The historical records show that over the past 8 years 70 percent of the accounts have been current. Determine the sample size needed in order to be 99% confident that the sample proportion of the current customer accounts is within .03 of the true proportion of all current accounts for this company.
A) 1842
B) 1548
C) 897
D) 632
E) 1267
Answer: B Difficulty: Hard
37. In a manufacturing process, we are interested in measuring the average length of a certain type of bolt. Past data indicates that the standard deviation is .25 inches. How many bolts should be sampled in order to make us 95% confident that the sample mean bolt length is within .02 inches of the true mean bolt length?
A) 25
B) 49
C) 423
D) 601
E) 1225
Answer: D Difficulty: Hard
38. In a manufacturing process, we are interested in measuring the average length of a certain type of bolt. Based on a preliminary sample of 9 bolts, the sample standard deviation is .3 inches. How many bolts should be sampled in order to make us 95% confident that the sample mean bolt length is within .02 inches of the true mean bolt length?
A) 864.36
B) 80
C) 1470
D) 3989
E) 1197
Answer: E Difficulty: Hard
Fill-in-the-Blank
39. In the construction of a confidence interval, as the confidence level required in estimating the mean increases, the width of the confidence interval ______________.
Answer: increases Difficulty: Medium
40. As the sample size n increases, the width of the confidence interval _______________.
Answer: decreases Difficulty: Medium
41. When establishing the confidence interval for the average weight of a cereal box, assume that the population standard deviation is known to be 2 ounces, and based on a sample the average weight of a sample of 20 boxes is 16 ounces. The appropriate test statistics to use is ________.
Answer: Z Difficulty: Medium
42. As the significance level, α increases, the width of the confidence interval _______________.
Answer: decreases Difficulty: Medium
43. As the standard deviation, (s) decreases, the width of the confidence interval _______________.
Answer: decreases Difficulty: Medium
44. As the stated confidence level decreases, the width of the confidence interval _______________.
Answer: decreases Difficulty: Medium
45. As the margin of error decreases, the width of the confidence interval _______________.
Answer: decreases Difficulty: Medium
46. If everything else is held constant, decreasing the margin of error, __________ the required sample size.
Answer: increases Difficulty: Medium
47. A confidence interval for the population mean is an interval constructed around the _____.
Answer: Sample mean Difficulty: Medium
Essay
48. A random sample of size 30 from a normal population yields = 32.8 and s = 4.51. Construct a 95 percent confidence interval for .
Answer: (31.19, 34.41)
Difficulty: Medium
49. A sample set of weights in pounds are 1.01, .95, 1.03, 1.04, .97, .97, .99, 1.01, and 1.03. Assume the population of weights are normally distributed. Find a 99 percent confidence interval for the mean population weight.
Answer: (.965, 1.035)
Difficulty: Hard
50. A sample of 8 items has an average fat content of 18.6 grams and a standard deviation of 2.4 grams. Assuming a normal distribution, construct a 99 percent confidence interval for .
Answer: (15.63, 21.57)
Difficulty: Medium
51. A sample of 12 items yields = 48.5 grams and s = 1.5 grams. Assuming a normal distribution, construct a 90 percent confidence interval for the population mean weight.
Answer: (47.722, 49.278)
Difficulty: Medium
52. A sample of 100 items has a standard deviation of 5.1 and a mean of 21.6. Construct a 95 percent confidence interval for.
Answer: (20.6, 22.6)
Difficulty: Medium
53. In a survey of 1,000 people, 420 are opposed to the tax increase. Construct a 95 percent confidence interval for the proportion of those people opposed to the tax increase.