Sampling and Sampling Distributions: True/False Questions Explained

Chapter 5 Sampling and Sampling Distributions True/False Questions 1. A numerical measure of a sample is called a sample parameter. Answer: False Type: Concept Difficulty: Easy 2. A single numerical value used to estimate one particular value of the population is a point estimate. Answer: True Type: Concept Difficulty: Easy 3. The sample standard deviation, s, is a point estimate for  . Answer: True Type: Concept Difficulty: Easy 4. The standard error of the sample mean is equal to  . Answer: False Type: Concept Difficulty: Easy 5. A sample size of 30 or more values is large enough for the Central Limit Theorem to be effective. Answer: True Type: Concept Difficulty: Easy 6. As the sample size, n, increases, the sampling distribution of the sample proportion becomes less and less a normal distribution. Answer: False Type: Concept Difficulty: Easy 7. The sample mean is an unbiased estimate of the population mean. Answer: True Type: Concept Difficulty: Easy 8. A point estimate is a biased estimator if its expected value is equal to the population parameter it is estimating. Answer: False Type: Concept Difficulty: Easy 9. Estimator A is more efficient than estimator B, if Estimator A's variance is smaller than B's. Answer: True Type: Concept Difficulty: Medium Aczel, Complete Business Statistics, Sixth Edition 1

Chapter 5 Sampling and Sampling Distributions 10. The sample mean is a consistent estimator of  . Answer: True Type: Concept Difficulty: Medium 11. A discrete random variable can take on any value within its range and assigns probability zero to any one given value. Answer: False Type: Concept Difficulty: Easy 12. The Central Limit Theorem states that as the sample size increases, the distribution of the sample mean approaches a normal distribution with mean equal to the population mean and standard deviation equal to the population standard deviation. Answer: False Type: Concept Difficulty: Medium 13. Performing a random sample means to haphazardly select the sample. Answer: False Type: Concept Difficulty: Easy 14. Sampling error is the difference between the sample statistic and the population parameter. Answer: True Type: Concept Difficulty: Easy Multiple Choice Questions 15. Following is a random sample of the heights of students in a class: 5'3", 5'9”, 6'1”, 5'8”, 5'11”, 6'5", 6'2”, 5'7”, 5'1”, 6', 5'6”, 5'9”. Compute a point estimate of the mean and the standard deviation of the heights of the students. A) 5'9.8", 2.9" B) 5'10.3", 3.44" C) 5'9.8", 2.8" D) 5'10.3", 3.31" E) none of the above Answer: E Type: Computation Difficulty: Medium 2 Aczel, Complete Business Statistics, Sixth Edition

Chapter 5 Sampling and Sampling Distributions 16. A new shampoo is introduced in the market. The manufacturer interviews 25 different individuals for their opinion. Eighteen individuals like the shampoo. Compute a point estimate for the proportion of the population liking the shampoo. A) 0.36 B) 0.72 C) 0.28 D) 0.56 E) 0.64 Answer: B Type: Computation Difficulty: Easy 17. In a certain town, a random sample of executives have the following personal incomes (in thousands); 35, 43, 29, 55, 63, 72, 28, 33, 36, 41, 42, 57, 38, 30. Compute point estimates for the mean and the standard deviation of the incomes of the executives in the town. A) 44, 13.10 B) 44, 13.64 C) 43, 13.63 D) 45, 10.77 E) 43, 13.13 Answer: C Type: Computation Difficulty: Medium 18. In which of the following situations is the Central Limit Theorem not applicable? A) When the sample is small and the population is normal. B) When the sample is large and the population is normal. C) When the sample is large, above 30, and the population is not normal. D) When the sample is small, below 30, and the population is not normal. E) none of the above Answer: D Type: Concept Difficulty: Medium 19. A certain brand of electric bulbs has an average life of 300 hours with a standard deviation of 25. A random sample of 100 bulbs is tested. What is the probability that the sample mean will be less than 295? A) 0.0548 B) 0.1096 C) 0.4452 D) 0.0793 E) 0.0228 Answer: E Type: Computation Difficulty: Medium Aczel, Complete Business Statistics, Sixth Edition 3

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Chapter 5 Sampling and Sampling Distributions 20. When sampling from a population with a standard deviation of 14, using a sample size of 144, what is the probability that the sample mean will be at least 3 units away from the population mean? A) 0.0051 B) 0.0257 C) 0.2157 D) 0.0102 E) 0.0514 Answer: D Type: Computation Difficulty: Medium 21. A particular camera is sold at an average price of $190, with a standard deviation of $12. In a sample of 50 randomly selected stores, what is the probability that the sample mean falls within $4 of the population mean? A) 0.9913 B) 0.9825 C) 0.9816 D) 0.4909 E) 0.9818 Answer: C Type: Computation Difficulty: Medium 22. A bank has an average balance of $495 and a standard deviation of $102, in its checking accounts. What is the probability that 36 randomly selected accounts have an average balance greater than $500? A) 0.7718 B) 0.1141 C) 0.3859 D) 0.2282 E) 0.6141 Answer: C Type: Computation Difficulty: Medium 23. It is found that on average, 9% of the population recycles its garbage, with a standard deviation of 3% over the country. If a random sample of 49 households is taken, what is the probability that more than 4 households recycle their garbage? A) 0.4744 B) 0.9488 C) 0.0256 D) 0.9744 E) 0.0512 Answer: D Type: Computation Difficulty: Medium 4 Aczel, Complete Business Statistics, Sixth Edition

Chapter 5 Sampling and Sampling Distributions Aczel, Complete Business Statistics, Sixth Edition 5

Chapter 5 Sampling and Sampling Distributions 24. The standard deviation of a random variable is: A) always positive B) always negative C) sometimes positive and sometimes negative D) depends on the mean of the random variable E) none of the above Answer: A Type: Concept Difficulty: Easy 25. The standard deviation of the average number of cassettes a college student owns is 14. What minimum sample size of students will give a sample standard deviation of at most 4? A) 12 B) 4 C) 13 D) 3 E) 18 Answer: C Type: Computation Difficulty: Medium 26. The average advertising expenditure in the computer industry is found to be $15,000,000 annually, with a standard deviation of $3,000,000. In a sample of 35 companies, what is the probability that the sample mean will exceed 16.5 million? A) 0.4756 B) 0.0244 C) 0.0015 D) 0.0456 E) 0.9756 Answer: C Type: Computation Difficulty: Medium 27. Suppose that X is a normal random variable with mean 10 and standard deviation 4. Then the probability that X will be greater than 12 is: A) 0.1587 B) 0.3413 C) 0.3085 D) 0.4772 E) 0.1915 Answer: C Type: Computation Difficulty: Medium 6 Aczel, Complete Business Statistics, Sixth Edition

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Chapter 5 Sampling and Sampling Distributions 28. The distribution of X, the number of cars sold per day, where X can be 0, 1, 2, 3, 4, or 5, is: A) sometimes a normal distribution B) is normal if n, the number of days sampled, is at least 30. C) always a normal distribution D) never a normal distribution E) is a binomial distribution Answer: D Type: Concept Difficulty: Medium 29. A normal random variable is: A) always symmetric B) never symmetric C) sometimes symmetric D) symmetric if the mean is positive E) symmetric if the variance is large Answer: A Type: Concept Difficulty: Easy 30. Suppose that X is a normal random variable with mean 17 and standard deviation 10. The probability that the value of X will be between -2.6 and 36.6 is: A) 0 B) 0.99 C) 1 D) 0.95 E) 0.90 Answer: D Type: Computation Difficulty: Medium 31. When sampling from a normal population with mean 100 and standard deviation 12, using a sample size of 36, the probability that the sample mean will exceed 110 is: A) 0.2033 B) 0.7967 C) a very small number D) a very large number E) a number very close to 1 Answer: C Type: Computation Difficulty: Medium Aczel, Complete Business Statistics, Sixth Edition 7

Chapter 5 Sampling and Sampling Distributions 32. On the average, 18% of the population recycles their garbage, with a standard deviation of 4%. Out of a randomly selected 36 households, how many will recycle their garbage on average? A) 18 B) 6.48 C) 9 D) 5.04 E) none of the above Answer: B Type: Computation Difficulty: Easy 33. The average telephone bill in a locality is $70, with a standard deviation of $40. In a sample of 50 randomly selected phone connections, what is the probability that the sample average will exceed $75? A) 0.3106 B) 0.8694 C) 0.1894 D) 0.4483 E) none of the above Answer: C Type: Computation Difficulty: Medium 34. The average advertising expenditure in the computer industry is found to be $15,000,000 annually, with a standard deviation of $3,000,000. What is the probability that in a sample of 100 companies, the sample mean will be less than 14.5 million? A) 0.0475 B) 0.0004 C) 0.9996 D) 0.4325 E) none of the above Answer: A Type: Computation Difficulty: Hard 35. An estimator that contains all the information about the parameter it estimates is said to be: A) unbiased B) sufficient C) efficient D) consistent E) biased Answer: B Type: Concept Difficulty: Medium 8 Aczel, Complete Business Statistics, Sixth Edition

Chapter 5 Sampling and Sampling Distributions 36. Suppose that X is a normal random variable with mean 20 and standard deviation 3. What is the probability that a sample of 36 taken has a mean of at least 19? A) a number very close to 0. B) about 0.5 C) a negative number. D) a number very close to 1. E) 0.0228 Answer: E Type: Computation Difficulty: Medium 37. When the probability that an estimator is close to the parameter it estimates increases as the sample size, n, increases, that estimator is said to be: A) unbiased B) biased C) efficient D) consistent E) sufficient Answer: D Type: Concept Difficulty: Medium 38. A poll of 1,500 registered voters is taken. Of these, 600 say they will vote for candidate Smith. Compute a point estimate for the proportion of the population who will vote for this candidate. A) 0.60 B) 0.04 C) 0.40 D) 0.36 E) none of the above Answer: C Type: Computation Difficulty: Easy 39. Which of the following statements does not refer to the Central Limit Theorem? A) The expected value of the mean of the distribution of sample means is  . B) The distribution of sample means is approximately normally distributed. C) The Central Limit Theorem is true for any distribution, when the sample size is at least 30. D) The standard deviation of the distribution of sample means is equal to  . E) When a non-normal population is sampled, the distribution of sample means is still normally distributed, as long as the sample size is large. Answer: D Type: Concept Difficulty: Medium Aczel, Complete Business Statistics, Sixth Edition 9

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Chapter 5 Sampling and Sampling Distributions 40. A certain brand of tires has an average life of 45,000 miles, with standard deviation of 2,000 miles. A random sample of 36 tires is tested. What is the probability that this sample will have a mean of at least 44,000 miles? A) a number very close to zero. B) a number very close to one. C) about 0.5 D) 0.3085 E) none of the above Answer: B Type: Computation Difficulty: Medium 41. A population has a mean of 90 and a standard deviation of 15. A random sample of 100 is selected. What is the standard error? A) 15 B) 3.873 C) 1.5 D) 2.25 E) none of the above Answer: C Type: Computation Difficulty: Easy 42. The “no-show” rate at a popular resort hotel is 8%. If a random sample of 100 reservations is taken, what is the probability that the number of “no shows” will exceed 10? A) 0.0049 B) 0.1788 C) 0.2296 D) 0.7704 E) 0.8665 Answer: B Type: Computation Difficulty: Hard Use the following to answer questions 43-45: The reliability of a widely used microprocessor is typically expressed as time between failures (TBF). Suppose TBF is an exponentially distributed random variable that averages 3,000 cycles with a standard deviation of 54.776 cycles. 10 Aczel, Complete Business Statistics, Sixth Edition

Chapter 5 Sampling and Sampling Distributions 43. If a random sample of 64 of these microprocessors is taken, what is the probability that the sample average will be between 2,995 cycles and 3,005 cycles? A) 0.0718 B) 0.2052 C) 0.4176 D) 0.4654 E) 0.5346 Answer: E Type: Computation Difficulty: Medium 44. Suppose a shipment of these microprocessors will be rejected if, after the drawing of a random sample of size 36, the sample average is less than 2,998 cycles. What proportion of shipments will be rejected? A) 0.1230 B) 0.2959 C) 0.4129 D) 0.4404 E) 0.4641 Answer: C Type: Computation Difficulty: Medium 45. Suppose the purchaser of these microprocessors will draw a random sample of size 36, calculate a sample mean and standard deviation and then reject the shipment if the sample average falls below some agree-upon value. What value would ensure that approximately 95% of all shipments are accepted? A) 2,982.11 B) 2,985.07 C) 2,988.75 D) 3,014.96 E) 3,017.88 Answer: B Type: Computation Difficulty: Hard Aczel, Complete Business Statistics, Sixth Edition 11

Chapter 5 Sampling and Sampling Distributions 46. The Tightward Mutual Funds Company bills itself as “The Leader in Low-Fee Funds” based on a recent random sample of 16 of its funds that exhibited a sample average of 48 basis points (1 basis point is equivalent to a fee of 0.01%). Assume that the fees of Tightwad's competitors are normally distributed and average 50 basis points with a standard deviation of 12.5 basis points. What is the probability that a random sample of 16 funds from Tightwad's competitors would have a lower sample average than 48? A) 0.1003 B) 0.2611 C) 0.3745 D) 0.4364 E) None of the above Answer: B Type: Computation Difficulty: Medium Use the following to answer questions 47-48: A manufacturer and one of its suppliers are engaged in negotiations regarding the inspection plan for a component the supplier provides. Components will be shipped in very large batches, and random samples of 50 will be drawn for inspection. The manufacturer needs components to have an average length of 68 cm, with a standard deviation of 4 cm. Both manufacturer and supplier have agreed that there should be no more than a 5% chance of rejecting a shipment that conforms to this specification. 47. A policy in which a sample mean of __________ leads to a rejected shipment will ensure only there is approximately a 5% risk of a conforming shipment's being rejected. A) Less than 67.07 cm B) Less than 61.45 cm C) Greater than 68.93 cm D) Greater than 74.70 cm E) None of the above Answer: A Type: Computation Difficulty: Hard 12 Aczel, Complete Business Statistics, Sixth Edition

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Chapter 5 Sampling and Sampling Distributions 48. Suppose that a particular shipment actually has an average length of 69 cm, with a standard deviation of 16 cm. If the decision rule is to reject the shipment if its sample average is in excess of 71 cm, how likely is it that this shipment will be rejected? A) 0.6700 B) 0.4483 C) 0.2266 D) 0.1894 E) None of the above Answer: D Type: Computation Difficulty: Medium 49. If the Central Limit Theorem is applicable, this means that the sampling distribution of a ___________ population can be treated as normal since the ___________ is ___________. A) Non-normal; sample size; small B) Non-symmetrical; variance; large C) Platy-kurtotic; sample size; large D) Binomial; variance; large E) Non-normal; standard error; large Answer: C Type: Concept Difficulty: Medium 50. Transportation authorities consider a particular 16-passenger airplane to be overloaded if the total carried exceeds 2,640 pounds. Suppose that the load placed on a plane by a typical passenger is normally distributed with a mean of 155 pounds and a standard deviation of 30 pounds. If passengers are typically assigned randomly to flights, without regard for their particular load, what proportion of flights will be overloaded? A) 0.0918 B) 0.3707 C) 0.6293 D) 0.9082 E) Cannot be determined from information given Answer: A Type: Computation Difficulty: Hard Aczel, Complete Business Statistics, Sixth Edition 13

Chapter 5 Sampling and Sampling Distributions Short Answer Questions 51. Following is a random sample of the heights of students in a class: 5'3", 5"9", 6'1”, 5'8”, 5'11”, 6'5”, 6'2”, 5'7”, 5'11”, 6', 5'6", 5'9". Estimate the proportion of students with heights more than 6 feet. Answer: 1/4 - count number of heights over 6' = 3/12 = 1/4 Type: Computation Difficulty: Easy 52. In a certain town, a random sample of executives has the following personal incomes (in thousands); 35, 43, 29, 55, 63, 72, 28, 33, 36, 41, 42, 57, 38, 30. Find an estimate for the proportion of executives with income greater than $45,000. Answer: 2/7 - 4/14 = 2/7 Type: Computation Difficulty: Easy 53. In a casino, a person loses $260 at an average, with a standard deviation of $65. A group of 40 people go into the casino. What is the probability that the mean loss of this group is at least $250 per person? Answer: 0.8340 - z = (250 - 260)/(65/sq rt 40) = -0.97 from z table: 0.3340 is area between a = -0.97 and the mean, so P(z > -0.97) = 0.8340 Type: Computation Difficulty: Medium 54. In a casino, a person loses $260 at an average, with a standard deviation of $65. A group of 40 people go into the casino. What is the probability that this group loses less than $220 at an average? Answer: a number close to zero - z = (220 - 260)/65 sq rt 40 = 3.89 answer is a number very close to zero Type: Computation Difficulty: Medium 55. If the population mean is 108, the population variance is 529, and the sample size is 64, what is the probability that the sample mean is greater than 105? Answer: 0.8508 - z = (105 - 108)/(sq rt 529/8) = 1.04 from the normal table: 0.3508, so P(P z > -1.04) = 0.8508 Type: Computation Difficulty: Medium 56. A population has a mean of 45 and a standard deviation of 8. A random sample of 81 is selected. What is the expected value of the sample mean and its standard error? Answer: 45, 0.8889 - expected value is 45, same as population mean standard error is standard deviation/sq rt of n = 8/9 = 0.8889 Type: Computation Difficulty: Medium 14 Aczel, Complete Business Statistics, Sixth Edition

Chapter 5 Sampling and Sampling Distributions 57. A random sample of size 27 is collected and the mean is computed. What is the total number of degrees of freedom for deviations from the mean? Answer: 26 - n -1 = 27, 1 = 26 Type: Concept Difficulty: Easy 58. The average profit stores make on the sale of a particular computer is $227, with a standard deviation of $24. In a random sample of 50 retail outlets, what is the probability that the sample mean is between $220 and $225? Answer: 0.2579 - z = (220 - 227)/(24 sq rt 50) = -2.06, z = (225 - 227)/(24 sq rt 50) = -0.59 from table: 0.4803 - 0.2224 = 0.2579 Type: Computation Difficulty: Medium 59. The average profit stores make on the sale of a particular computer is $227, with a standard deviation of $24. In a random sample of 50 retail outlets, what is the probability that the average sample profit is more than $230? Answer: 0.1894 - z = (230 - 227)/)(sq rt 50) = 0.88 from table: 0.5 - 0.3106 = 0.1894 Type: Computation Difficulty: Medium 60. A professor is distributing 40 assignments to 23 students in a class. Each student will be given only one assignment. How many degrees of freedom does the professor have? Answer: 23 Type: Concept Difficulty: Medium 61. A bank has an average balance of $495 and a standard deviation of $102 in its checking accounts. What is the probability of a random sample of 100 accounts having a minimum mean balance of $485? Answer: 0.8365 - z = (485 - 495)/(102/10) = -0.98 from table: 0.3365 + 0.5 = 0.8365 Type: Computation Difficulty: Medium Use the following to answer questions 62-63: The following are the observed amounts in a random sample of 16 bottles taken from the production line: 9.92 10.12 10.08 10.01 9.87 9.92 9.93 9.96 ' 10.10 10.02 9.93 9.91 9.92 9.97 9.96 9.94 Aczel, Complete Business Statistics, Sixth Edition 15

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Chapter 5 Sampling and Sampling Distributions 62. Find the mean, variance, and standard deviation of these data. Answer: Mean: 9.9725; Variance: .00539; Standard Deviation: .0734 Type: Concept Difficulty: Easy 63. The bottling machine is designed to dispense an average of 10 ounces per bottle, with a standard deviation of .5 ounce. If the machine is operating properly, what is the probability of finding a sample of 16 bottles with a mean as small or smaller than the mean of this sample? Answer: .4129 Type: Concept Difficulty: Medium 64. Mature poppy plants are expected to have an average height of 16 inches, with a standard deviation of 3 inches. What's the probability that a sample of 64 poppies will have an average height of between 15 and 17 inches? Answer: 0.9924 Type: Concept Difficulty: Medium 65. Plasma display units undergoing a coating operation have an average required curing time of 83.5 seconds. Experience has been shown the required curing time to be normally distributed with a standard deviation of 12 seconds. If 10 units are drawn at random, what is the probability that the sample average for required drying time will be no less than 80 seconds but no more than 85 seconds? Answer: 0.4766 Type: Computation Difficulty: Medium 66. LaggardAir, which is burdened with one of the worst on-time records in the airline industry, still sees many of its flights depart on time. Suppose representatives from an air travelers' advocacy association are coming to examine LaggardAir's records. A random sample of 50 flight records will be drawn, and if 80% or more show an on- time departure, LaggardAir will be given the association's Punctuality Award. What is the probability that LaggardAir will be given the Punctuality Award even if its actual on-time rate is only 60%? Answer: 0.0019 Type: Computation Difficulty: Medium 67. Johnny Appleseed Grocery monitors its UPC data-linking process closely. Each item scanned at checkout is priced either correctly or incorrectly, depending on how well the UPC data-linking process is performing. Suppose a recent sample of 1,500 item scans indicated 16 that linked incorrectly. Assuming the data-linking process is 99.3% accurate, what was the probability of observing a mislink rate that high or higher? Answer: 0.0446 Type: Computation Difficulty: Medium 16 Aczel, Complete Business Statistics, Sixth Edition

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