Homework answers / question archive / University of Texas, Arlington - BSTAT 3321 CHAPTER 8: CONFIDENCE INTERVAL ESTIMATION 1)The width of a confidence interval estimate for a proportion will be narrower for 99% confidence than for 95% confidence

University of Texas, Arlington - BSTAT 3321 CHAPTER 8: CONFIDENCE INTERVAL ESTIMATION 1)The width of a confidence interval estimate for a proportion will be narrower for 99% confidence than for 95% confidence

Statistics

Share With

---

University of Texas, Arlington - BSTAT 3321

CHAPTER 8: CONFIDENCE INTERVAL ESTIMATION

1)The width of a confidence interval estimate for a proportion will be

1. 1. narrower for 99% confidence than for 95% confidence.
   2. wider for a sample size of 100 than for a sample size of 50.
   3. narrower for 90% confidence than for 95% confidence.
   4. narrower when the sample proportion is 0.50 than when the sample proportion is 0.20.

 

 

2. When determining the sample size for a proportion for a given level of confidence and sampling error, the closer to 0.50 that p is estimated to be, the __________ the sample size required.
   1. smaller
   2. larger
   3. The sample size is not affected.
   4. The effect cannot be determined from the information given.

 

 

3. A 99% confidence interval estimate can be interpreted to mean that
   1. if all possible samples are taken and confidence interval estimates are developed, 99% of them would include the true population mean somewhere within their interval.
   2. we have 99% confidence that we have selected a sample whose interval does include the population mean.
   3. both of the above
   4. none of the above

 

 

4. If you were constructing a 99% confidence interval of the population mean based on a sample of n = 25, where the standard deviation of the sample s = 0.05, the critical value of t will be
   1. 2.7970.
   2. 2.7874.
   3. 2.4922.
   4. 2.4851.

 

 

5. Which of the following is NOT true about the Student’s t distribution?
   1. It has more area in the tails and less in the center than does the normal distribution.
   2. It is used to construct confidence intervals for the population mean when the population standard deviation is known.
   3. It is bell shaped and symmetrical.
   4. As the number of degrees of freedom increases, the t distribution approaches the normal distribution.

 

 

6. True or False:  The t distribution is used to construct confidence intervals for the population mean when the population standard deviation is unknown.

 

 

7. The t distribution
   1. assumes the population is normally distributed.
   2. approaches the normal distribution as the sample size increases.
   3. has more area in the tails than does the normal distribution.
   4. all of the above

 

 

8. It is desired to estimate the average total compensation of CEOs in the Service industry. Data were randomly collected from 18 CEOs and the 97% confidence interval was calculated to be ($2,181,260, $5,836,180). Which of the following interpretations is correct?
   1. 97% of the sampled total compensation values fell between $2,181,260 and $5,836,180.
   2. We are 97% confident that the mean of the sampled CEOs falls in the interval $2,181,260 to $5,836,180.
   3. In the population of Service industry CEOs, 97% of them will have total compensations that fall in the interval $2,181,260 to $5,836,180.
   4. We are 97% confident that the average total compensation of all CEOs in the Service industry falls in the interval $2,181,260 to $5,836,180.

 

 

9. It is desired to estimate the average total compensation of CEOs in the Service industry. Data were randomly collected from 18 CEOs and the 97% 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 more than $3,000,000?
   1. Yes, and I am 97% confident of it.
   2. Yes, and I am 78% confident of it.
   3. I am 97% confident that the average compensation is $3,000,000.
   4. I cannot conclude that the average exceeds $3,000,000 at the 97% confidence level.

 

 

10. A confidence interval was used to estimate the proportion of statistics students that are females. A random sample of 72 statistics students generated the following 90% confidence interval: (0.438, 0.642). Based on the interval above, is the population proportion of females equal to 0.60?
    1. No, and we are 90% sure of it.
    2. No. The proportion is 54.17%.
    3. Maybe. 0.60 is a believable value of the population proportion based on the information above.
    4. Yes, and we are 90% sure of it.

 

 

11. A confidence interval was used to estimate the proportion of statistics students that are female. A random sample of 72 statistics students generated the following 90% confidence interval: (0.438, 0.642). Using the information above, what size sample would be necessary if we wanted to estimate the true proportion to within ±0.08 using 95% confidence?
    1. 105
    2. 150
    3. 420
    4. 597

 

 

 

12. When determining the sample size necessary for estimating the true population mean, which factor is NOT considered when sampling with replacement?
    1. the population size
    2. the population standard deviation
    3. the level of confidence desired in the estimate
    4. the allowable or tolerable sampling error

 

 

 

13. Suppose a 95% confidence interval for  turns out to be (1,000, 2,100). Give a definition of what it means to be “95% confident” in an inference.
    1. In repeated sampling, the population parameter would fall in the given interval 95% of the time.
    2. In repeated sampling, 95% of the intervals constructed would contain the population mean.
    3. 95% of the observations in the entire population fall in the given interval.
    4. 95% of the observations in the sample fall in the given interval.

 

 

14. Suppose a 95% confidence interval for  turns out to be (1,000, 2,100). To make more useful inferences from the data, it is desired to reduce the width of the confidence interval. Which of the following will result in a reduced interval width?
    1. Increase the sample size.
    2. Decrease the confidence level.
    3. Increase the sample size and decrease the confidence level.
    4. Increase the confidence level and decrease the sample size.

 

 

 

15. Suppose a 95% confidence interval for  has been constructed. If it is decided to take a larger sample and to decrease the confidence level of the interval, then the resulting interval width would                                     . (Assume that the sample statistics gathered would not change very much for the new sample.)
    1. be larger than the current interval width
    2. be narrower than the current interval width
    3. be the same as the current interval width
    4. be unknown until actual sample sizes and reliability levels were determined

 

 

16. In the construction of confidence intervals, if all other quantities are unchanged, an increase in the sample size will lead to a                                interval.
    1. narrower
    2. wider
    3. less significant
    4. biased

 

 

17. A major department store chain is interested in estimating the average amount its credit card customers spent on their first visit to the chain’s new store in the mall. Fifteen credit card accounts were randomly sampled and analyzed with the following results:  and . Assuming the distribution of the amount spent on their first visit is approximately normal, what is the shape of the sampling distribution of the sample mean that will be used to create the desired confidence interval for ?
    1. approximately normal with a mean of $50.50
    2. a standard normal distribution
    3. a t distribution with 15 degrees of freedom
    4. a t distribution with 14 degrees of freedom

 

 

 

18. A major department store chain is interested in estimating the average amount its credit card customers spent on their first visit to the chain’s new store in the mall. Fifteen credit card accounts were randomly sampled and analyzed with the following results:  and . Construct a 95% confidence interval for the average amount its credit card customers spent on their first visit to the chain’s new store in the mall, assuming that the amount spent follows a normal distribution.
    1. $50.50  $9.09
    2. $50.50  $10.12
    3. $50.50  $11.00
    4. $50.50  $11.08

 

 

19. Private colleges and universities rely on money contributed by individuals and corporations for their operating expenses. Much of this money is put into a fund called an endowment, and the college spends only the interest earned by the fund. A recent survey of 8 private colleges in the United States revealed the following endowments (in millions of dollars): 60.2, 47.0, 235.1, 490.0, 122.6, 177.5, 95.4, and 220.0. What value will be used as the point estimate for the mean endowment of all private colleges in the United States?
    1. $1,447.8
    2. $180.975
    3. $143.042
    4. $8

 

 

20. Private colleges and universities rely on money contributed by individuals and corporations for their operating expenses. Much of this money is put into a fund called an endowment, and the college spends only the interest earned by the fund. A recent survey of 8 private colleges in the United States revealed the following endowments (in millions of dollars): 60.2, 47.0, 235.1, 490.0, 122.6, 177.5, 95.4, and 220.0. Summary statistics yield  and . Calculate a 95% confidence interval for the mean endowment of all the private colleges in the United States, assuming a normal distribution for the endowments.
    1. $180.975  $94.066
    2. $180.975  $99.123
    3. $180.975  $116.621
    4. $180.975  $119.605

 

 

21. A university system enrolling hundreds of thousands of students is considering a change in the way students pay for their education. Presently the students pay $55 per credit hour. The university system administrators are contemplating charging each student a set fee of $750 per quarter, regardless of how many credit hours each takes. To see if this proposal would be economically feasible, the administrators would like to know how many credit hours, on the average, each student takes per quarter. A random sample of 250 students yields a mean of 14.1 credit hours per quarter and a standard deviation of 2.3 credit hours per quarter. Suppose the administration wanted to estimate the mean to within 0.1 hours at 95% reliability and assumed that the sample standard deviation provided a good estimate for the population standard deviation. How large a sample would they need to take?

 

 

22. As an aid to the establishment of personnel requirements, the director of a hospital wishes to estimate the mean number of people who are admitted to the emergency room during a 24-hour period. The director randomly selects 64 different 24-hour periods and determines the number of admissions for each. For this sample,  and s² = 25. Which of the following assumptions is necessary in order for a confidence interval to be valid?
    1. The population sampled from has an approximate normal distribution.
    2. The population sampled from has an approximate t distribution.
    3. The mean of the sample equals the mean of the population.
    4. None of these assumptions are necessary.

 

 

23. As an aid to the establishment of personnel requirements, the director of a hospital wishes to estimate the mean number of people who are admitted to the emergency room during a 24-hour period. The director randomly selects 64 different 24-hour periods and determines the number of admissions for each. For this sample,  and s² = 25. Estimate the mean number of admissions per 24-hour period with a 95% confidence interval.

 

 

 

24. As an aid to the establishment of personnel requirements, the director of a hospital wishes to estimate the mean number of people who are admitted to the emergency room during a 24-hour period. The director randomly selects 64 different 24-hour periods and determines the number of admissions for each. For this sample,  and s² = 25. If the director wishes to estimate the mean number of admissions per 24-hour period to within 1 admission with 99% reliability, what size sample should she choose?

 

 

25. A university dean is interested in determining the proportion of students who receive some sort of financial aid. Rather than examine the records for all students, the dean randomly selects 200 students and finds that 118 of them are receiving financial aid. Use a 90% confidence interval to estimate the true proportion of students who receive financial aid.

 

 

26. A university dean is interested in determining the proportion of students who receive some sort of financial aid. Rather than examine the records for all students, the dean randomly selects 200 students and finds that 118 of them are receiving financial aid. The 95% confidence interval for p is 0.59  0.07. Interpret this interval.
    1. We are 95% confident that the true proportion of all students receiving financial aid is between 0.52 and 0.66.
    2. 95% of the students get between 52% and 66% of their tuition paid for by financial aid.
    3. We are 95% confident that between 52% and 66% of the sampled students receive some sort of financial aid.
    4. We are 95% confident that 59% of the students are on some sort of financial aid.

 

 

 

27. A university dean is interested in determining the proportion of students who receive some sort of financial aid. Rather than examine the records for all students, the dean randomly selects 200 students and finds that 118 of them are receiving financial aid. If the dean wanted to estimate the proportion of all students receiving financial aid to within 3% with 99% reliability, how many students would need to be sampled?
    1. n = 1,844
    2. n = 1,784
    3. n = 1,503
    4. n = 1,435

 

 

28. An economist is interested in studying the incomes of consumers in a particular region. The population standard deviation is known to be $1,000. A random sample of 50 individuals resulted in an average income of $15,000. What is the upper end point in a 99% confidence interval for the average income?
    1. $15,052
    2. $15,141
    3. $15,330
    4. $15,364

 

 

29. An economist is interested in studying the incomes of consumers in a particular region. The population standard deviation is known to be $1,000. A random sample of 50 individuals resulted in an average income of $15,000. What is the width of the 90% confidence interval?
    1. $232.60
    2. $364.30
    3. $465.23
    4. $728.60

 

 

30. An economist is interested in studying the incomes of consumers in a particular region. The population standard deviation is known to be $1,000. A random sample of 50 individuals resulted in an average income of $15,000. What sample size would the economist need to use for a 95% confidence interval if the width of the interval should not be more than $100?
    1. n = 1537
    2. n = 385
    3. n = 40
    4. n = 20

 

 

31. The head librarian at the Library of Congress has asked her assistant for an interval estimate of the mean number of books checked out each day. The assistant provides the following interval estimate: from 740 to 920 books per day. What is an efficient, unbiased point estimate of the number of books checked out each day at the Library of Congress?
    1. 740
    2. 830
    3. 920
    4. 1,660

 

32. The head librarian at the Library of Congress has asked her assistant for an interval estimate of the mean number of books checked out each day. The assistant provides the following interval estimate: from 740 to 920 books per day. If the head librarian knows that the population standard deviation is 150 books checked out per day, approximately how large a sample did her assistant use to determine the interval estimate?
    1. 2
    2. 3
    3. 12
    4. It cannot be determined from the information given.

 

 

33. The head librarian at the Library of Congress has asked her assistant for an interval estimate of the mean number of books checked out each day. The assistant provides the following interval estimate: from 740 to 920 books per day. If the head librarian knows that the population standard deviation is 150 books checked out per day, and she asked her assistant for a 95% confidence interval, approximately how large a sample did her assistant use to determine the interval estimate?
    1. 125
    2. 13
    3. 11
    4. 4

 

 

34. The head librarian at the Library of Congress has asked her assistant for an interval estimate of the mean number of books checked out each day. The assistant provides the following interval estimate: from 740 to 920 books per day. If the head librarian knows that the population standard deviation is 150 books checked out per day, and she asked her assistant to use 25 days of data to construct the interval estimate, what confidence level can she attach to the interval estimate?
    1. 99.7%
    2. 99.0%
    3. 98.0%
    4. 95.4%

 

 

 

35. True or False: A race car driver tested his car for time from 0 to 60 mph, and in 20 tests obtained an average of 4.85 seconds with a standard deviation of 1.47 seconds. A 95% confidence interval for the 0 to 60 time is 4.52 seconds to 5.18 seconds.

 

 

 

36. True or False: Given a sample mean of 2.1 and a sample standard deviation of 0.7, a 90% confidence interval will have a width of 2.36.

 

37. True or False: Given a sample mean of 2.1 and a population standard deviation of 0.7, a 90% confidence interval will have a width of 2.36.

 

 

 

38. True or False: A sample size of 5 provides a sample mean of 9.6. If the population variance is known to be 5 and the population distribution is assumed to be normal, the lower limit for a 92% confidence interval is 7.85.

 

 

 

39. True or False: A random sample of 50 provides a sample mean of 31 with a standard deviation of s=14. The upper bound of a 90% confidence interval estimate of the population mean is 34.32.

 

 

 

40. True or False: In forming a 90% confidence interval for a population mean from a sample size of 22, the number of degrees of freedom from the t distribution equals 22.

 

 

 

41. True or False:  Other things being equal, as the confidence level for a confidence interval increases, the width of the interval increases.

 

 

 

42. True or False:  The t distribution allows the calculation of confidence intervals for means when the actual standard error is not known.

 

 

 

43. True or False:  The t distribution allows the calculation of confidence intervals for means for small samples when the population variance is not known, regardless of the shape of the distribution in the population.

 

 

 

44. True or False:  For a t distribution with 12 degrees of freedom, the area between – 2.6810 and 2.1788 is 0.980.

 

 

 

45. True or False:  A sample of 100 fuses from a very large shipment is found to have 10 that are defective. The 0.95 confidence interval would indicate that, for this shipment, the proportion of defective fuses is between 0 and 0.28.

 

 

 

46. True or False: The sample mean is a point estimate of the population mean.

 

 

 

47. True or False: The confidence interval estimate of the population mean is constructed around the sample mean.

 

 

 

48. True or False: The confidence interval estimate of the population proportion is constructed around the sample proportion.

 

 

 

 

49. True or False: A point estimate consists of a single sample statistic that is used to estimate the true population parameter.

 

 

 

50. True or False: The confidence interval obtained will always correctly estimate the population parameter.

 

 

 

 

51. True or False: Other things being equal, the confidence interval for the mean will be wider for 95% confidence than for 90% confidence.

 

 

 

52. True or False: The t distribution is used to develop a confidence interval estimate of the population mean when the population standard deviation is unknown.

 

 

53. True or False: The t distribution is used to develop a confidence interval estimate of the population proportion when the population standard deviation is unknown.

 

 

 

54. True or False: The standardized normal distribution is used to develop a confidence interval estimate of the population proportion regardless of whether the population standard deviation is known.

 

 

 

 

55. True or False: The standardized normal distribution is used to develop a confidence interval estimate of the population proportion when the sample size is sufficiently large.

 

 

 

56. True or False: The t distribution approaches the standardized normal distribution when the number of degrees of freedom increases.

 

 

 

57. True or False: In estimating the population mean with the population standard deviation unknown, if the sample size is 12, there will be 6 degrees of freedom.

 

 

 

58. True or False: The difference between the sample mean and the population mean is called the sampling error.

 

 

 

59. True or False: The difference between the sample proportion and the population proportion is called the sampling error.

 

 

60. True or False: The difference between the sample size and the population size is called the sampling error.

 

 

 

 

61. True or False: The difference between the upper limit of a confidence interval and the point estimate used in constructing the confidence interval is called the sampling error.

 

 

 

62. True or False: The difference between the lower limit of a confidence interval and the point estimate used in constructing the confidence interval is called the sampling error.

 

 

 

63. True or False: Sampling error equals to half the width of a confidence interval.

 

 

64. True or False: The width of a confidence interval equals twice the sampling error.

 

 

 

65. True or False: The sampling error can either be positive or negative.

 

 

 

66. True or False: A population parameter is used to estimate a confidence interval.

 

 

 

 

 

67. True or False: For a given data set and confidence level, the confidence interval will be wider for 95% confidence than for 90% confidence.

 

 

 

 

TABLE 8-1

 

A random sample of 100 stores from a large chain of 1,000 garden supply stores was selected to determine the average number of lawnmowers sold at an end-of-season clearance sale. The sample results indicated an average of 6 and a standard deviation of 2 lawnmowers sold. A 95% confidence interval (5.623 to 6.377) was established based on these results.

 

68. True or False: Referring to Table 8-1, if the population had consisted of 10,000 stores, the confidence interval estimate of the mean would have been wider in range.

 

 

 

69. True or False: Referring to Table 8-1, of all possible samples of 100 stores drawn from the population of 1,000 stores, 95% of the sample means will fall between 5.623 and 6.377 lawnmowers.

 

 

 

70. True or False: Referring to Table 8-1, of all possible samples of 100 stores taken from the population of 1,000 stores, 95% of the confidence intervals developed will contain the true population mean within the interval.

 

 

 

71. True or False: Referring to Table 8-1, there are 10 possible samples of 100 stores that can be selected out of the population of 1,000 stores.

 

 

 

72. True or False: Referring to Table 8-1, 95% of the stores have sold between 5.623 and 6.377 lawnmowers.

 

 

 

73. True or False: Referring to Table 8-1, we do not know for sure whether the true population mean is between 5.623 and 6.377 lawnmowers sold.

 

 

 

TABLE 8-2

 

The managers of a company are worried about the morale of their employees. In order to determine if a problem in this area exists, they decide to evaluate the attitudes of their employees with a standardized test. They select the Fortunato test of job satisfaction, which has a known standard deviation of 24 points. 

 

74. Referring to Table 8-2, they should sample ________ employees if they want to estimate the mean score of the employees within 5 points with 90% confidence.

 

 

 

75. Referring to Table 8-2, due to financial limitations, the managers decide to take a sample of 45 employees.  This yields a mean score of 88.0 points.  A 90% confidence interval would go from ________ to ________.

 

 

 

76. True or False: Referring to Table 8-2, this confidence interval is only valid if the scores on the Fortunato test are normally distributed.

 

 

 

 

TABLE 8-3

 

A quality control engineer is interested in the mean length of sheet insulation being cut automatically by machine. The desired length of the insulation is 12 feet. It is known that the standard deviation in the cutting length is 0.15 feet.  A sample of 70 cut sheets yields a mean length of 12.14 feet. This sample will be used to obtain a 99% confidence interval for the mean length cut by machine.

 

77. Referring to Table 8-3, the critical value to use in obtaining the confidence interval is ________.

 

 

 

78. Referring to Table 8-3, the confidence interval goes from ________ to ________.

 

 

 

79. True or False: Referring to Table 8-3, the confidence interval indicates that the machine is not working properly.

 

 

 

80. True or False: Referring to Table 8-3, the confidence interval is valid only if the lengths cut are normally distributed.

 

 

 

81. Referring to Table 8-3, suppose the engineer had decided to estimate the mean length to within 0.03 with 99% confidence. Then the sample size would be ________.

 

 

 

 

TABLE 8-4

 

To become an actuary, it is necessary to pass a series of 10 exams, including the most important one, an exam in probability and statistics. An insurance company wants to estimate the mean score on this exam for actuarial students who have enrolled in a special study program. They take a sample of 8 actuarial students in this program and determine that their scores are: 2, 5, 8, 8, 7, 6, 5, and 7. This sample will be used to calculate a 90% confidence interval for the mean score for actuarial students in the special study program.

 

82. Referring to Table 8-4, the mean of the sample is __________, while the standard deviation of the sample is __________.

 

 

 

83. Referring to Table 8-4, the confidence interval will be based on __________ degrees of freedom.

 

 

 

 

84. Referring to Table 8-4, the critical value used in constructing a 90% confidence interval is __________.

 

 

 

85. Referring to Table 8-4, a 90% confidence interval for the mean score of actuarial students in the special program is from __________ to __________.

 

 

 

86. True or False: Referring to Table 8-4, for the confidence interval to be valid, it is necessary that test scores of students in the special study program on the actuarial exam be normally distributed.

 

 

 

 

 

87. True or False: Referring to Table 8-4, it is possible that the confidence interval obtained will not contain the mean score for all actuarial students in the special class.

 

 

 

88. True or False: Referring to Table 8-4, if we use the same sample information to obtain a 95% confidence interval, the resulting interval would be narrower than the one obtained here with 90% confidence.

 

 

 

89. Suppose a department store wants to estimate the average age of the customers of its contemporary apparel department, correct to within 2 years, with level of confidence equal to 0.95.  Management believes that the standard deviation is 8 years. The sample size they should take is ________.

 

 

 

 

TABLE 8-5

 

The actual voltages of power packs labeled as 12 volts are as follows: 11.77, 11.90, 11.64, 11.84, 12.13, 11.99, and 11.77.

 

90. Referring to Table 8-5, a confidence interval for this sample would be based on the t distribution with __________ degrees of freedom.

 

 

 

91. Referring to Table 8-5, the critical value for a 99% confidence interval for this sample is __________.

 

 

 

 

92. Referring to Table 8-5, a 99% confidence interval for the mean voltage of the power packs is from __________ to __________.

 

 

93. True or False:  Referring to Table 8-5, a 95% confidence interval for the mean voltage of the power pack is wider than a 99% confidence interval.

 

 

 

 

94. True or False: Referring to Table 8-5, a 99% confidence interval will contain 99% of the voltages for all such power packs.

 

 

 

95. True or False: Referring to Table 8-5, a confidence interval estimate of the population mean would only be valid if the distribution of voltages is normal.

 

 

 

96. True or False: Referring to Table 8-5, a 90% confidence interval calculated from the same data would be narrower than a 99% confidence interval.

 

 

 

 

97. True or False: Referring to Table 8-5, it is possible that the 99% confidence interval calculated from the data will not contain the mean voltage for the sample.

 

 

 

 

 

98. True or False: Referring to Table 8-5, it is possible that the 99% confidence interval calculated from the data will not contain the mean voltage for the entire population.

 

 

 

 

TABLE 8-6

 

A sample of salary offers (in thousands of dollars) given to management majors is: 28, 31, 26, 32, 27, 28, 27, 30, 31, and 29. Using this data to obtain a 95% confidence interval resulted in an interval from 27.5 to 30.3.

 

99. True or False: Referring to Table 8-6, 95% of the time, the sample mean salary offer to management majors will fall between 27.5 and 30.3.

 

 

 

100. True or False: Referring to Table 8-6, 95% of the salary offers are between 27.5 and 30.3.

 

 

 

101. True or False: Referring to Table 8-6, it is possible that the mean of the population is between 27.5 and 30.3.

 

 

 

102. True or False: Referring to Table 8-6, it is possible that the mean of the population is not between 27.5 and 30.3.

 

 

 

103. True or False: Referring to Table 8-6, 95% of the sample means will fall between 27.5 and 30.3.

 

 

 

104. True or False: Referring to Table 8-6, 95% of all confidence intervals constructed similarly to this one will contain the mean of the population.

 

 

 

 

105. True or False: Referring to Table 8-6, a 99% confidence interval for the mean of the population from the same sample would be wider than 27.5 to 30.3.

 

 

 

 

106. True or False: Referring to Table 8-6, the confidence interval obtained is valid only if the distribution of the population of salary offers is normal.

 

 

 

 

107. A prison official wants to estimate the proportion of cases of recidivism. Examining the records of 250 convicts, the official determines that there are 65 cases of recidivism. A confidence interval will be obtained for the proportion of cases of recidivism. Part of this calculation includes the estimated standard error of the sample proportion. For this sample, the estimated standard error is __________.

 

 

 

108. A prison official wants to estimate the proportion of cases of recidivism. Examining the records of 250 convicts, the official determines that there are 65 cases of recidivism. A 99% confidence interval for the proportion of cases of recidivism would go from __________ to __________.

 

 

TABLE 8-7

 

After an extensive advertising campaign, the manager of a company wants to estimate the proportion of potential customers that recognize a new product. She samples 120 potential consumers and finds that 54 recognize this product. She uses this sample information to obtain a 95% confidence interval that goes from 0.36 to 0.54.

 

109. True or False: Referring to Table 8-7, the parameter of interest to the manager is the proportion of potential customers in this sample that recognize the new product.

 

 

 

 

110. True or False: Referring to Table 8-7, the parameter of interest is 54/120 = 0.45.

 

 

 

 

111. True or False: Referring to Table 8-7, this interval requires the use of the t distribution to obtain the confidence coefficient.

 

 

 

 

112. True or False: Referring to Table 8-7, this interval requires the assumption that the distribution of the number of people recognizing the product has a normal distribution.

 

 

 

 

113. True or False: Referring to Table 8-7, 95% of the time, the proportion of people that recognize the product will fall between 0.36 and 0.54.

 

 

 

114. True or False: Referring to Table 8-7, 95% of the time, the sample proportion of people that recognize the product will fall between 0.36 and 0.54.

 

 

 

115. True or False: Referring to Table 8-7, 95% of the people will recognize the product between 36% and 54% of the time.

 

 

 

116. True or False: Referring to Table 8-7, it is possible that the true proportion of people that recognize the product is between 0.36 and 0.54.

 

 

 

117. True or False: Referring to Table 8-7, it is possible that the true proportion of people that recognize the product is not between 0.36 and 0.54.

 

 

 

 

118. The head of a computer science department is interested in estimating the proportion of students entering the department who will choose the new computer engineering option. Suppose there is no information about the proportion of students who might choose the option. What size sample should the department head take if she wants to be 95% confident that the estimate is within 0.10 of the true proportion?

 

 

 

119. The head of a computer science department is interested in estimating the proportion of students entering the department who will choose the new computer engineering option. A preliminary sample indicates that the proportion will be around 0.25. Therefore, what size sample should the department head take if she wants to be 95% confident that the estimate is within 0.10 of the true proportion?

 

 

 

 

TABLE 8-8

 

A hotel chain wants to estimate the average number of rooms rented daily in each month.  The population of rooms rented daily is assumed to be normally distributed for each month with a standard deviation of 24 rooms.

 

120. Referring to Table 8-8, during January, a sample of 16 days has a sample mean of 48 rooms.  This information is used to calculate an interval estimate for the population mean to be from 40 to 56 rooms.  What is the level of confidence of this interval?

 

 

 

121. Referring to Table 8-8, during February, a sample of 25 days has a sample mean of 37 rooms.  Use this information to calculate a 92% confidence interval for the population mean.

 

 

 

 

122. The county clerk wants to estimate the proportion of retired voters who will need special election facilities.  The clerk wants to find a 95% confidence interval for the population proportion which extends at most 0.07 to either side of the sample proportion.  How large a sample must be taken to assure these conditions are met?

 

 

 

123. The county clerk wants to estimate the proportion of retired voters who will need special election facilities.  Suppose a sample of 400 retired voters was taken.  If 150 need special election facilities, calculate an 80% confidence interval for the population proportion.

 

 

 

TABLE 8-9

 

A wealthy real estate investor wants to decide whether it is a good investment to build a high-end shopping complex in a suburban county in Chicago.  His main concern is the total market value of the 3,605 houses in the suburban county.  From past experience, the standard deviation of market housing prices is estimated to be $40,500.  He commissioned a statistical consulting group to take a sample of 200 houses and obtained a sample average market price of $225,000 and a sample standard deviation of $38,700.  The consulting group also found that the average differences between market prices and appraised prices was $125,000, with a standard deviation of $3,400.  Also, the proportion of houses in the sample that were appraised for higher than the market prices was 0.24.

 

124. Referring to Table 8-9, if he wants a 95% confidence on estimating the true population average market price of the houses in the suburban county to be within $10,000, how large a sample will he need?

 

 

 

125. Referring to Table 8-9, what will be the 90% confidence interval for the average market price of the houses in the suburban county constructed by the consulting group?

 

 

 

126. Referring to Table 8-9, what will be the 90% confidence interval for the total market price of the houses in the suburban county constructed by the consulting group?

 

 

 

127. Referring to Table 8-9, what will be the 90% confidence interval for the total difference between the market prices and appraised prices of the houses in the suburban county constructed by the consulting group?

 

 

 

128. Referring to Table 8-9, what will be the 90% confidence interval for the population proportion of houses that will be appraised for higher than the market prices?

 

 

 

129. A quality control engineer is interested in estimating the proportion of defective items coming off a production line. In a sample of 300 items, 27 are defective. A 90% confidence interval for the proportion of defectives from this production line would go from __________ to __________.

 

 

 

TABLE 8-10

 

The president of a university would like to estimate the proportion of the student population who owns a personal computer. In a sample of 500 students, 417 own a personal computer.

 

130. True or False:  Referring to Table 8-10, the parameter of interest is the average number of students in the population who own a personal computer.

 

 

 

 

131. True or False:  Referring to Table 8-10, the parameter of interest is the proportion of the student population who own a personal computer.

 

 

 

 

132. Referring to Table 8-10, the critical value for a 99% confidence interval for this sample is __________.

 

 

 

133. Referring to Table 8-10, a 99% confidence interval for the proportion of the student population who own a personal computer is from __________ to __________.

 

 

 

134. Referring to Table 8-10, the sampling error of a 99% confidence interval for the proportion of the student population who own a personal computer is __________.

 

 

 

135. True or False:  Referring to Table 8-10, a 95% confidence interval for the proportion of the student population who own a personal computer is narrower than a 99% confidence interval.

 

 

 

 

136. True or False: Referring to Table 8-10, a 99% confidence interval will contain 99% of the student population who own a personal computer.

 

 

 

137. True or False: Referring to Table 8-10, a confidence interval estimate of the population proportion would only be valid if the distribution of the number of students who own a personal computer is normal.

 

 

 

138. True or False: Referring to Table 8-10, a 90% confidence interval calculated from the same data would be narrower than a 99% confidence interval.

 

 

 

 

139. True or False: Referring to Table 8-10, it is possible that the 99% confidence interval calculated from the data will not contain the sample proportion of students who own a personal computer.

 

 

 

 

140. True or False: Referring to Table 8-10, it is possible that the 99% confidence interval calculated from the data will not contain the proportion of the student population who own a personal computer.

 

 

 

 

141. True or False: Referring to Table 8-10, we are 99% confident that the average numbers of the student population who own a personal computer is between 0.7911 and 0.8769.

 

 

 

 

142. True or False: Referring to Table 8-10, we are 99% confident that between 79.11% and 87.69% of the student population own a personal computer.

 

 

 

 

TABLE 8-11

 

The superintendent of a unified school district of a small town wants to make sure that no more than 5% of the students skip more than 10 days of school in a year.  A random sample of 145 students showed that 12 students skipped more than 10 days of school last year.

 

143. Referring to Table 8-11, what is the critical value for the 95% one-sided confidence interval for the proportion of students who skipped more than 10 days of school last year?

 

 

 

144. Referring to Table 8-11, what is the upper bound of the 95% one-sided confidence interval for the proportion of students who skipped more than 10 days of school last year?

 

 

 

145. True or False: Referring to Table 8-11, the superintendent can conclude with a 95% level of confidence that no more than 5% of the students in the unified school district skipped more than 10 days of school last year.

 

 

 

TABLE 8-12

 

The president of a university is concerned that illicit drug use on campus is higher than the 1% acceptable level.  A random sample of 250 students revealed that 7 of them had used illicit drugs during the last 12 months.

 

146. Referring to Table 8-12, what is the critical value for the 90% one-sided confidence interval for the proportion of students who had used illicit drugs during the last 12 months?

 

 

 

 

147. Referring to Table 8-12, what is the upper bound of the 90% one-sided confidence interval for the proportion of students who had used illicit drugs during the last 12 months?

 

 

 

148. True or False: Referring to Table 8-12, the superintendent can be 90% confident that no more than 5% of the students at the university had used illicit drugs during the last 12 months.

 

 

 

 

149. True or False: Referring to Table 8-12, using the 90% one-sided confidence interval, the superintendent can be 95% confident that no more than 5% of the students at the university had used illicit drugs during the last 12 months.

 

 

 

150. True or False: Referring to Table 8-12, using the 90% one-sided confidence interval, the superintendent can be 85% confident that no more than 5% of the students at the university had used illicit drugs during the last 12 months.