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Chapter 9 Estimation and Confidence Intervals True/False 1. A point estimate is a single value used to estimate a population parameter. Answer: True Difficulty: Easy Goal: 1 AACSB: AS 2. A point estimate is a range of values used to estimate a population parameter. Answer: False Difficulty: Easy Goal: 1 3. An interval estimate is a single value used to estimate a population parameter. Answer: False Difficulty: Easy Goal: 2 AACSB: AS 4. An interval estimate is a range of values used to estimate a population parameter. Answer: True Difficulty: Easy Goal: 2 5. A sample of 2000 union members was selected and their opinions regarding a proposed management union contract were recorded. 1,600 members were in favor of it. A 95% confidence interval for the population proportion was computed to be 0.78 and 0.82. This indicates that about 80 out of 100 similarly constructed intervals would include the population proportion. Answer: False Difficulty: Medium Goal: 5 AACSB: AS 6. If the size of a sample equals the size of the population, we would not expect any error in estimating the population parameter. Answer: True Difficulty: Easy Goal: 1 7. We can expect some difference between sample statistics and the corresponding population parameters. This difference is called the sampling error. Answer: True Difficulty: Easy Goal: 1 8. The Central Limit Theorem proves that the sampling distribution of sample means tends to approximate a normal distribution when the sample size becomes larger.. Answer: True Difficulty: Easy Goal: 2 9. The 95 percent confidence interval states that 95 percent of the sample means of a specified sample size selected from a population will lie within plus and minus 1.96 standard deviations of the hypothesized population mean. Answer: True Difficulty: Easy Goal: 2 Statistical Techniques in Business & Economics, Lind/Marchal/Wathen, 13/e 137 10. One factor in determining the size of a sample is the degree of confidence selected. This is usually 0.95 or 0.99, but it may be any degree of confidence you specify. Answer: True Difficulty: Easy Goal: 6 11. One factor in determining the size of a sample is the maximum allowable error that you must decide on. It is the maximum error you will tolerate at a specified level of confidence. Answer: True Difficulty: Easy Goal: 6 12. The variation in the population as measured by the standard deviation has little or no effect in determining the size of a sample selected from the population. Answer: False Difficulty: Easy Goal: 6 13. The higher the degree of confidence, the larger the sample required to give a certain precision. Answer: True Difficulty: Medium Goal: 6 14. To determine the value of the standard error of the mean, the total error is divided by the sample size. Answer: False Difficulty: Medium Goal: 3 15. A distribution of sample means is normally distributed with a mean equal to the population mean and a standard deviation equal to the standard error of the mean. Answer: True Difficulty: Easy Goal: 2 16. To determine the size of a sample, the standard deviation of the population must be estimated by either taking a pilot survey or by approximating it based on knowledge of the population. Answer: True Difficulty: Medium Goal: 6 17. The t distribution is based on the assumption that the population of interest is normal or nearly normal. Answer: True Difficulty: Easy Goal: 4 18. The t distribution is a continuous distribution. Answer: True Difficulty: Easy Goal: 4 AACSB: AS 19. There is not one t distribution, but rather a "family" of t distributions. Answer: True Difficulty: Easy Goal: 4 20. The t distribution is positively skewed. Answer: False Difficulty: Easy Goal: 4 138 Test Bank, Chapter 9 21. All t distributions have a mean of zero and a standard deviation of 1. Answer: False Difficulty: Easy Goal: 4 22. The t distribution is more spread out and flatter at the center than the standard normal distribution. However, as the sample size increases, the t distribution curve approaches the standard normal distribution. Answer: True Difficulty: Easy Goal: 4 23. The Student t distribution has a greater spread than does the z distribution. As a result, the critical values of t for a given level of significance are larger in magnitude than the corresponding z critical values. Answer: True Difficulty: Easy Goal: 4 24. The test statistic t has n – 1 degrees of freedom. Answer: True Difficulty: Easy Goal: 4 25. William S. Gosset, a brewmaster, developed the t test for the Guiness Brewery in Ireland, who published it in 1908 using the pen name "Student." Answer: True Difficulty: Easy Goal: 4 26. The test statistic for a problem involving any sample size and an unknown population standard deviation is the Student's t distribution. Answer: True Difficulty: Easy Goal: 4 Multiple Choice 27. The mean number of travel days per year for salespeople employed by hardware distributors needs to be estimated with a 0.90 degree of confidence. For a small pilot study the mean was 150 days and the standard deviation was 14 days. If the population mean is estimated within two days, how many salespeople should be sampled? A) 134 B) 452 C) 511 D) 2100 Answer: A Difficulty: Hard Goal: 6 Statistical Techniques in Business & Economics, Lind/Marchal/Wathen, 13/e 139 28. The proportion of junior executives leaving large manufacturing companies within three years is to be estimated within 3 percent. The 0.95 degree of confidence is to be used. A study conducted several years ago revealed that the percent of junior executives leaving within three years was 21. To update this study, the files of how many junior executives should be studied? A) 594 B) 612 C) 709 D) 897 Answer: C Difficulty: Hard Goal: 6 29. There are 2,000 eligible voters in a precinct. 500 of the voters are selected at random and asked to indicate whether they planned to vote for the Democratic incumbent or the Republican challenger. Of the 500 surveyed, 350 said they were going to vote for the Democratic incumbent. Using the 0.99 confidence coefficient, what are the confidence limits for the proportion that plan to vote for the Democratic incumbent? A) 0.647 and 0.753 B) 0.612 and 0.712 C) 0.397 and 0.797 D) 0.826 and 0.926 Answer: A Difficulty: Hard Goal: 5 30. A random sample of 85 group leaders, supervisors, and similar personnel revealed that a person spent an average 6.5 years on the job before being promoted. The population standard deviation was 1.7 years. Using the 0.95 degree of confidence, what is the confidence interval for the population mean? A) 6.99 and 7.99 B) 4.15 and 7.15 C) 6.14 and 6.86 D) 6.49 and 7.49 Answer: C Difficulty: Hard Goal: 3 31. The mean weight of trucks traveling on a particular section of I-475 is not known. A state highway inspector needs an estimate of the mean. He selects a random sample of 49 trucks passing the weighing station and finds the mean is 15.8 tons. The population standard deviation is 3.8 tons. What is the 95 percent interval for the population mean? A) 14.7 and 16.9 B) 13.2 and 17.6 C) 10.0 and 20.0 D) 16.1 and 18.1 Answer: A Difficulty: Hard Goal: 3 140 Test Bank, Chapter 9 32. A bank wishes to estimate the mean balances owed by customers holding MasterCard. The population standard deviation is estimated to be $300. If a 98 percent confidence interval is used and an interval of $75 is desired, how many cardholders should be sampled? A) 44 B) 212 C) 629 D) 87 Answer: D Difficulty: Hard Goal: 6 33. Which of the following would be used as a point estimate for the population mean ( A) B) x n C) x D) s Answer: C Difficulty: Medium Goal: 1 34. Mileage tests were conducted on a randomly selected sample of 100 newly developed automobile tires. The average tread life was found to be 50,000 miles with a standard deviation of 3,500 miles. What is the best estimate of the average tread life in miles for the entire population of these tires? A) 50,000 B) 3,500 C) (50,000/100) D) (3,500/100) Answer: A Difficulty: Medium Goal: 1 AACSB: AS 35. For a given confidence interval, what is the interpretation of a 96% confidence level? A) 96% chance that the given interval includes the true value of the population parameter B) Approximately 96 out of 100 such intervals would include the true value of the population parameter C) 4% chance that the given interval does not include the true value of the population parameter D) Both "a" and "c" are true Answer: B Difficulty: Medium Goal: 2 36. Which statement(s) is/are correct about the t distribution? A) Mean = 0 B) Symmetric C) Based on degrees of freedom D) All of the above are correct E) None of the above is correct Answer: D Difficulty: Medium Goal: 4 Statistical Techniques in Business & Economics, Lind/Marchal/Wathen, 13/e 141 37. What kind of distribution is the t distribution? A) Continuous B) Discrete C) Subjective D) None of the above Answer: A Difficulty: Medium Goal: 4 38. How is the t distribution similar to the standard z distribution? A) Both are continuous distribution B) Both are bell-shaped C) Both are families of distributions D) A and B E) A, B and C Answer: D Difficulty: Easy Goal: 4 39. A sample of 20 is selected from the population. To determine the appropriate critical t-value what number of degrees of freedom should be used? A) 20 B) 19 C) 21 D) 25 Answer: B Difficulty: Medium Goal: 4 AACSB: AS 40. The t distribution is similar to the z distribution in all BUT one of the following characteristics. Which one is it? A) Continuous B) Symmetrical C) Bell-shaped D) t distribution's mean = 0 and standard deviation = 1 Answer: D Difficulty: Medium Goal: 4 41. Suppose 1,600 of 2,000 registered voters sampled said they planned to vote for the Republican candidate for president. Using the 0.95 degree of confidence, what is the interval estimate for the population proportion (to the nearest tenth of a percent)? A) 78.2% to 81.8% B) 69.2% to 86.4% C) 76.5% to 83.5% D) 77.7% to 82.3% Answer: A Difficulty: Hard Goal: 5 142 Test Bank, Chapter 9 42. Which of the following is NOT necessary to determine how large a sample to select from a population? A) Level of confidence in estimating the population parameter B) Size of the population C) Maximum allowable error in estimating the population parameter D) Estimate of the population variation E) All of the above are necessary Answer: B Difficulty: Medium Goal: 6 43. A sample mean is the best point estimate of the A) population standard deviation B) population median C) population mean D) the sample standard deviation Answer: C Difficulty: Easy Goal: 1 44. A sample standard deviation is the best point estimate of the A) population range B) population skewness C) population mode D) population standard deviation Answer: D Difficulty: Easy Goal: 1 45. A confidence interval for a population mean A) estimates the population range B) estimates a likely interval for a population mean C) estimates a likelihood or probability D) estimates the population standard deviation Answer: B Difficulty: Easy Goal: 2 46. A 95% confidence interval infers that the population mean A) is between 0 and 100% B) is within 1.96 standard deviations of the sample mean C) is within 1.96 standard errors of the sample mean D) is too large Answer: C Difficulty: Easy Goal: 2 47. When a confidence interval for a population mean is constructed from sample data, A) we can conclude that the population mean is in the interval B) we can conclude that the population mean is not in the interval C) we can conclude, with a stated level of confidence, that the population mean is in the interval D) we cannot make any inferences. Answer: C Difficulty: Easy Goal: 2 Statistical Techniques in Business & Economics, Lind/Marchal/Wathen, 13/e 143 48. The distribution of Student's t has A) a mean of zero and a standard deviation of one B) a mean of one and a standard deviation of one C) a mean of zero and a standard deviation that depends on the sample size D) a mean that depends on the sample size and a standard deviation of one Answer: C Difficulty: Easy Goal: 4 49. The distribution of Student's t is A) symmetrical B) negatively skewed C) positively skewed D) a discrete probability distribution. Answer: A Difficulty: Easy Goal: 4 50. When using Student's t to compute an interval estimate, A) we assume that the samples are collected from normally distributed populations B) we estimate the population standard deviation based on the sample standard deviation C) use the z distribution D) A and B only Answer: D Difficulty: Easy Goal: 4 51. A random sample of 42 college graduates who worked during their program revealed that a student spent an average 5.5 years on the job before being promoted. The sample standard deviation was 1.1 years. Using the 0.99 degree of confidence, what is the confidence interval for the population mean? A) 5.04 and 5.96 B) 5.06 and 5.94 C) 2.67 and 8.33 D) 4.40 and 6.60 Answer: A Difficulty: Hard Goal: 4 52. The mean weight of trucks traveling on a particular section of I-475 is not known. A state highway inspector needs an estimate of the mean. He selects a random sample of 49 trucks passing the weighing station and finds the mean is 15.8 tons. The sample standard deviation is 5.0 tons. What is the 95 percent interval for the population mean? A) 10.80 and 20.80 B) 14.40 and 17.20 C) 14.36 and 17.24 D) 14.00 and 17.00 Answer: C Difficulty: Hard Goal: 4 Fill-in-the-Blank 53. What estimate states the range within which a population parameter probably lies? _____________ Answer: interval Difficulty: Easy Goal: 2 144 Test Bank, Chapter 9 54. What is the interval within which a population parameter is expected to lie? __________________ Answer: confidence interval Difficulty: Medium Goal: 2 55. In order to construct the 99 percent confidence interval, how many standard errors of the mean from the hypothesized population mean must you go? __________ Answer: 3, or plus or minus 3 standard errors Difficulty: Medium Goal: 5 56. In order to construct the 68 percent confidence interval, how many standard errors of the mean from the hypothesized population mean must you go? __________ Answer: 1, or plus or minus 1 standard error Difficulty: Medium Goal: 5 57. In terms of the standard error of the mean, how wide is a 99 percent confidence interval?______ Answer: 6 standard errors Difficulty: Hard Goal: 5 58. In terms of the standard error of the mean, how wide is a 90 percent confidence interval?______ Answer: 2.56 standard errors Difficulty: Hard Goal: 5 59. What is the measure of confidence that one has in the interval estimate called? _____________________ Answer: degree or level of confidence Difficulty: Medium Goal: 2 60. What is reasoning from a sample or small group to the entire group or population called? ______________________ Answer: statistical inference Difficulty: Medium Goal: 2 61. A 95 percent confidence interval implies that about 95 out of 100 similarly constructed intervals will include the _____________ being estimated. Answer: parameter Difficulty: Medium Goal: 2 62. For a sampling distribution of the means, what percent of the means would be between ± 1.96 standard deviations? ______________ Answer: 95% Difficulty: Medium Goal: 2 63. Other things being equal, a smaller sample size would result in a wider confidence interval because the standard error of the mean would _______________. Answer: larger Difficulty: Medium Goal: 3 64. What does the t distribution approach as the sample size increases? ______________________ Answer: standard normal Z distribution Difficulty: Medium Goal: 4 Statistical Techniques in Business & Economics, Lind/Marchal/Wathen, 13/e 145 65. What happens to the computed value of t as the sample size increases? ______________ Answer: decreases Difficulty: Hard Goal: 4 Multiple Choice 66. When we use a confidence interval to reach a conclusion about the population mean, we are applying a type of reasoning or logic called A) descriptive statistics B) the normal distribution C) statistical inference D) graphics Answer: C Difficulty: Medium Goal: 2 67. A local company wants to evaluate their quality of service by surveying their customers. Their budget limits the number of surveys to 100. What is their maximum error of the estimated mean quality for a 95% level of confidence and an estimated standard deviation of 5? A) 0.9604 B) 0.9920 C) 1.96 D) 5% Answer: B Difficulty: Hard Goal: 6 68. A local retail company wants to estimate the mean amount spent. Their budget limits the number of surveys to 225. What is their maximum error of the estimated mean amount spent for a 99% level of confidence and an estimated standard deviation of $10.00? A) $10 B) $1.00 C) 1% D) $1.72 Answer: D Difficulty: Hard Goal: 6 Use the following to answer questions 69-71: Recently, a university surveyed recent graduates of the English Department for their starting salaries. Four hundred graduates returned the survey. The average salary was $25,000 with a standard deviation of $2,500. 69. What is the best point estimate of the population mean? A) $25,000 B) $2,500 C) 400 D) $62.5 Answer: A Difficulty: Easy Goal: 1 146 Test Bank, Chapter 9 70. What is the 95% confidence interval for the mean salary of all graduates from the English Department? A) [$22,500, $27,500] B) [$24,755, $25,245] C) [$24,988, $25,012] D) [$24,600, $25,600] Answer: B Difficulty: Medium Goal: 3 Refer To: 09_01 71. What do the confidence interval results mean? A) The population mean is in the interval B) The population mean is not in the interval C) The likelihood that any confidence interval based on a sample of 400 graduates will contain the population mean is 0.95 D) There is a 5% chance that the computed interval does not contain the population mean. Answer: C Difficulty: Medium Goal: 3 Use the following to answer questions 72-74: A survey of an urban university (population of 25,450) showed that 870 of 1100 students sampled supported a fee increase to fund improvements to the student recreation center. 72. Using the 95% level of confidence, what is the confidence interval? A) [0.767, 0.814] B) [0.759, 0.822] C) [0.771, 0.811] D) [0.714, 0.866] Answer: A Difficulty: Hard Goal: 5 73. Using the 99% level of confidence, what is the confidence interval? A) [0.751, 0.829] B) [0.758, 0.822] C) [0.767, 0.814] D) [0.771, 0.811] Answer: B Difficulty: Hard Goal: 5 74. If university officials say that at least 70% of the voting student population supporting the fee increase, what conclusion can be drawn based on a 95% level of confidence? A) 70% is not in the interval, need to take another sample. B) 70% is not in the interval, so assume it will not be supported. C) 70% is below the interval, so assume it will be supported. D) Since this was not based on population, cannot make conclusion. Answer: C Difficulty: Hard Goal: 5 Statistical Techniques in Business & Economics, Lind/Marchal/Wathen, 13/e 147 Use the following to answer questions 75-76: A group of statistics students decided to conduct a survey at their university to find the average (mean) amount of time students spent studying per week. 75. Assuming a standard deviation of 6 hours, what is the required sample size if the error should be less than ½ hour with a 95% level of confidence? A) 554 B) 130 C) 35 D) 393 Answer: A Difficulty: Hard Goal: 6 AACSB: AS 76. Assuming a standard deviation of 3 hours, what is the required sample size if the error should be less than ½ hour with a 99% level of confidence A) 196 B) 240 C) 15 D) 16 Answer: B Difficulty: Hard Goal: 6 Use the following to answer questions 77-80: A survey of 144 retail stores revealed that the average price of a microwave was $375 with a standard error of $20. 77. What is the 95% confidence interval to estimate the true cost of the microwave? A) $323.40 to $426.60 B) $328.40 to $421.60 C) $335.80 to $414.20 D) $335.46 to $414.54 Answer: D Difficulty: Medium Goal: 3 78. What is the 99% confidence interval to estimate the true cost of the microwave? A) $322.78 to $427.22 B) $328.40 to $421.60 C) $323.40 to $426.60 D) $335.82 to $414.28 Answer: A Difficulty: Medium Goal: 3 148 Test Bank, Chapter 9 79. If 90% and 95% confidence intervals were developed to estimate the true cost of the microwave, what similarities would they have? A) Point estimates B) Z-variates C) Standard errors D) Both “a” and “c” Answer: D Difficulty: Medium Goal: 3 80. If 95% and 98% confidence intervals were developed to estimate the true cost of the microwave, what differences would they have? A) Standard errors B) Interval widths C) Z-variates D) Both "b" and "c" Answer: D Difficulty: Medium Goal: 3 Use the following to answer questions 81-83: A student wanted to quickly construct a 95% confidence interval for the average age of students in her statistics class. She randomly selected 9 students. Their average age was 19.1 years with a standard deviation of 1.5 years. 81. What is the best point estimate for the population mean? A) 2.1 years B) 1.5 years C) 19.1 years D) 9 years Answer: C Difficulty: Medium Goal: 4 82. What is the 95% confidence interval for the population mean? A) [0.97, 3.27] B) [15.64, 22.56] C) [17.97, 20.23] D) [17.95, 20.25] Answer: D Difficulty: Medium Goal: 4 83. What is the 99% confidence interval for the population mean? A) [17.42, 20.78] B) [17.48, 20.72] C) [14.23, 23.98] D) [0.44, 3.80] Answer: A Difficulty: Medium Goal: 4 Statistical Techniques in Business & Economics, Lind/Marchal/Wathen, 13/e 149 Use the following to answer questions 84-86: A survey of 25 grocery stores revealed that the average price of a gallon of milk was $2.98 with a standard error of $0.10. 84. What is the 95% confidence interval to estimate the true cost of a gallon of milk? A) $2.81 to $3.15 B) $2.94 to $3.02 C) $2.77 to $3.19 D) $2.95 to $3.01 Answer: C Difficulty: Medium Goal: 4 85. What is the 98% confidence interval to estimate the true cost of a gallon of milk? A) $2.73 to $3.23 B) $2.85 to $3.11 C) $2.94 to $3.02 D) $2.95 to $ 3.01 Answer: A Difficulty: Medium Goal: 4 86. If 90% and 95% confidence intervals are developed to estimate the true cost of a gallon of milk, what similarities would they have? A) Point estimates B) t-statistics C) Standard errors D) Both "a" and "c" Answer: D Difficulty: Easy Goal: 4 87. A sample of 25 is selected from a known population of 100 elements. What is the finite population correction factor? A) 8.66 B) 75 C) 0.87 D) Cannot be determined Answer: C Difficulty: Medium Goal: 3 88. A sample of 50 is selected from a known population of 250 elements. The population standard deviation is 15. What is the standard error of the sample means using the finite population correction factor? A) 2.89 B) 1.90 C) 2.12 D) Cannot be determined Answer: B Difficulty: Medium Goal: 3 150 Test Bank, Chapter 9 89. A sample of 100 students is selected from a known population of 1000 students to construct a 95% confidence interval for the average SAT score. What correction factor should be used to compute the standard error? A) 0.949 B) 0.901 C) 1.96 D) Cannot be determined Answer: A Difficulty: Easy Goal: 4 90. A pharmaceutical company wanted to estimate the population mean of monthly sales for their 250 sales people. Forty sales people were randomly selected. Their mean monthly sales were $10,000 with a standard deviation of $1000. Construct a 95% confidence interval for the population mean. A) [9,690.1, 10,309.9] B) [9,715.5, 10,284.5] C) [8,040, 11,960] D) [8,000, 12,000] Answer: B Difficulty: Medium Goal: 4 Use the following to answer questions 91-92: A survey of an urban university (population of 25,450) showed that 750 of 1100 students sampled attended a home football game during the season. What inferences can be made about student attendance at football games? 91. Using the 99% level of confidence, what is the confidence interval? A) [0.767, 0.814] B) [0.0.6550, 0.7050] C) [0.6659, 0.6941] D) [0.0.6795, 0.6805] Answer: C Difficulty: Hard Goal: 5 92. Using the 90% level of confidence, what is the confidence interval? A) [0.7510, 0.8290] B) [0.6620, 0.6980] C) [0.6659, 0.6941] D) [0.0.6795, 0.6805] Answer: B Difficulty: Hard Goal: 5 Essay 93. A company wants to estimate next years total revenue. Why is an interval estimate preferred to a point estimate? Answer: An interval estimate provides a range of possible values. The range could be relatively small or large. This range provides an idea of the relative precision of the estimate. A point estimate does not provide this additional information. Difficulty: Hard Goal: 3 Statistical Techniques in Business & Economics, Lind/Marchal/Wathen, 13/e 151 94. Explain the concept of a confidence interval when estimating a population mean. Answer: A confidence interval is derived from the sampling distribution of the mean. When a sample is measured and a sample mean and standard deviation computed, a confidence interval can be computed. Whether the interval contains the population parameter is not known. However, the chosen level of confidence (for example, 95%) provides a probability statement that a certain percentage of samples (95%) will provide confidence intervals that include the population mean. Difficulty: Hard Goal 3 95. Deciding on the number of observations to include in a sample is an important decision when planning a research study. What information is required to make this decision? Answer: Three pieces of information are required. The researcher must decide on the relative precision required in the estimate, i.e., plus or minus 5 units. The researcher must decide on the confidence level for the interval estimate. The researcher must know or find an estimate of the variability in the population, i.e., the researcher must know or estimate the variance or standard deviation in the population. Difficulty: Hard Goal: 6 96. When deciding on the number of observations to include in a sample, what would favor larger sample sizes? Answer: Increased precision of the estimate (smaller range of the interval), greater variability in the population, and higher degree of confidence. Difficulty: Hard Goal: 6 152 Test Bank, Chapter 9