ECO 045 - Practice Problems for Final Exam - Part1_Spring 2023

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1 Lehigh University ECO 045 Practice Problems for Final Exam – Part1 Yuval Erez Spring 2023 1. The records show that 8% of the items produced by a machine do not meet the specifications. What is the probability that a sample of 100 units contains? (Note: You can use the normal approximation to the binomial distribution to answer the following questions.) a. Five or more defective units? b. Ten or fewer defective units? c. Eleven or less defective units? ANSWER: a. 0.9015 b. 0.8212 c. 0.9015 2. An airline has determined that 20% of its international flights are not on time. What is the probability that of the next 80 international flights (Note: Here too you can use the normal approximation to the binomial distribution to answer the following questions.) a. fifteen or less will not be on time? b. eighteen or more will not be on time? c. exactly 17 will not be on time? ANSWER: a. 0.4443 b. 0.3372 c. 0.1071 3. Parameters are a. numerical characteristics of a sample. b. numerical characteristics of a population. c. the averages taken from a sample. d. numerical characteristics of either a sample or a population. ANSWER: b

2 4. A simple random sample of 100 observations was taken from a large population. The sample mean and the standard deviation were determined to be 80 and 12 respectively. The standard error of the mean is a. 1.20 b. 0.12 c. 8.00 d. 0.80 ANSWER: a 5. The closer the sample mean is to the population mean, a. the larger the sampling error. b. the smaller the sampling error. c. the sampling error equals 1. d. none of these alternatives is correct. ANSWER: b 6. As the sample size increases, the a. standard deviation of the population decreases. b. population mean increases. c. standard error of the mean decreases. d. standard error of the mean increases. ANSWER: c 7. The sample mean is the point estimator of a. μ b. σ c. d. ANSWER: a

3 8. Whenever the population has a normal probability distribution, the sampling distribution of is a normal probability distribution for a. large sample sizes. b. small sample sizes. c. any sample size. d. samples of size thirty or greater. ANSWER: c 9. The sampling error is the a. same as the standard error of the mean. b. difference between the value of the sample mean and the value of the population mean. c. error caused by selecting a bad sample. d. standard deviation multiplied by the sample size. ANSWER: b 10. A probability distribution of all possible values of a sample statistic is known as a. a sample statistic. b. a parameter. c. simple random sampling. d. a sampling distribution. ANSWER: d 11. The purpose of statistical inference is to provide information about the a. sample based upon information contained in the population. b. population based upon information contained in the sample. c. population based upon information contained in the population. d. mean of the sample based upon the mean of the population. ANSWER: b

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4 12. For a population with any distribution, the form of the sampling distribution of the sample mean is a. sometimes normal for all sample sizes. b. sometimes normal for large sample sizes. c. always normal for all sample sizes. d. always normal for large sample sizes. ANSWER: d 13. A sample of 24 observations is taken from a population that has 150 elements. The sampling distribution of is a. approximately normal because is always approximately normally distributed. b. approximately normal because the sample size is large in comparison to the population size. c. approximately normal because of the central limit theorem. d. normal if the population is normally distributed. ANSWER: d 14. Doubling the size of the sample will a. reduce the standard error of the mean to one-half its current value. b. reduce the standard error of the mean to approximately 70% of its current value. c. have no effect on the standard error of the mean. d. double the standard error of the mean. ANSWER: b 15. The following data was collected from a simple random sample of a population. 13 15 14 16 12 The point estimate of the population standard deviation is a. 2.500 b. 1.581 c. 2.000 d. 1.414 ANSWER: b

5 16. The following data was collected from a simple random sample of a population. 13 15 14 16 12 The mean of the population a. is 14 b. is 15 c. is 15.1581 d. could be any value ANSWER: d 17. The expected value of ࠵?̅ equals the mean of the population from which the sample is drawn a. only if the sample size is 30 or greater. b. only if the sample size is 50 or greater. c. only if the sample size is 100 or greater. d. for any sample size. ANSWER: d 18. All of the following are true about the standard error of the mean except a. it is larger than the standard deviation of the population. b. it decreases as the sample size increases. c. its value is influenced by the standard deviation of the population. d. it measures the variability in sample means. ANSWER: a 19. The basis for using a normal probability distribution to approximate the sampling distribution of ࠵?̅ is a. Chebyshev’s theorem. b. The empirical rule. c. The central limit theorem. d. Bayes' theorem. ANSWER: c

6 20. Below you are given the values obtained from a random sample of 4 observations taken from an infinite population. 32 34 35 39 a. Find a point estimate for μ . Is this an unbiased estimate of μ ? Explain. b. Find a point estimate for σ 2 . c. Find a point estimate for σ . d. What can be said about the sampling distribution of ? Be sure to discuss the expected value, the standard deviation, and the shape of the sampling distribution of . ANSWER: a. 35; Yes; E( ) = μ b. 8.667; c. 2.944 d. E( ) = μ , the standard deviation = sqrt( σ 2 /n) = σ /sqrt(n) , and the sampling distribution of is normally distributed IF the population is normally distributed. 21. A simple random sample of 100 observations was taken from a large population. The sample mean and the standard deviation were determined to be 80 and 12 respectively. The standard error of the mean is a. 1.20 b. 0.12 c. 8.00 d. 0.80 ANSWER: a 22. A population has a standard deviation of 16. If a sample of size 64 is selected from this population, what is the probability that the sample mean will be within ±2 of the population mean? a. 0.6826 b. 0.3413 c. -0.6826 d. Since the mean is not given, there is no answer to this question. ANSWER: a

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7 23. From a population of 200 elements, a sample of 49 elements is selected. It is determined that the sample mean is 56 and the sample standard deviation is 14. The standard error of the mean is a. 3 b. 2 c. greater than 2 d. less than 2 ANSWER: d (notice that n > 5% * N, so you need to use the finite population correction factor) 24. A population has a mean of 300 and a standard deviation of 18. A sample of 144 observations will be taken. The probability that the sample mean will be between 297 to 303 is a. 0.4332. b. 0.9544. c. 0.9332. d. 0.0668. ANSWER: b 25. A sample of 400 observations will be taken from an infinite population. The population proportion equals 0.8. The probability that the sample proportion will be greater than 0.83 is a. 0.4332 b. 0.9332 c. 0.0668 d. 0.5668 ANSWER: c 26. A population has a mean of 53 and a standard deviation of 21. A sample of 49 observations will be taken. The probability that the sample mean will be greater than 57.95 is a. 0 b. .0495 c. .4505 d. .9505 ANSWER: b

8 27. As a rule of thumb, the sampling distribution of the sample proportions can be approximated by a normal probability distribution whenever a. np ≥5, n ≥30. b. np ≥5 and n(1-p ) ≥5. c. n 30 and (1 - p) = 0.5. d. none of these alternatives is correct. ANSWER: b 28. In a local university, 40% of the students live in the dormitories. A random sample of 80 students is selected for a particular study. The probability that the sample proportion (the proportion living in the dormitories) is at least 0.30 is a. 0.4664 b. 0.9328 c. 0.9664 d. 0.0336 ANSWER: c 29. The expected value of ࠵?̅ equals the mean of the population from which the sample is drawn a. only if the sample size is 30 or greater. b. only if the sample size is 50 or greater. c. only if the sample size is 100 or greater. d. for any sample size. ANSWER: d 30. All of the following are true about the standard error of the mean except a. it is larger than the standard deviation of the population. b. it decreases as the sample size increases. c. its value is influenced by the standard deviation of the population. d. it measures the variability in sample means. ANSWER: a

9 31. A population of 1,000 students spends an average of $10.50 a day on dinner. The standard deviation of the expenditure is $3. A simple random sample of 64 students is taken. a. What are the expected value, standard deviation, and shape of the sampling distribution of the sample mean? b. What is the probability that these 64 students will spend a combined total of more than $715.21? (Hint: Convert the total expenditure to average expenditure). c. What is the probability that these 64 students will spend a combined total between $703.59 and $728.45? ANSWER: a. 10.5 0.363 normal b. 0.035897 c. 0.0857 32. The basis for using a normal probability distribution to approximate the sampling distribution of ࠵?̅ is a. Chebyshev’s theorem. b. The empirical rule. c. The central limit theorem. d. Bayes' theorem. ANSWER: c 33. In a restaurant, the proportion of people who order coffee with their dinner is .9. A simple random sample of 144 patrons of the restaurant is taken. a. What are the expected value, standard deviation, and shape of the sampling distribution of ? b. What is the probability that the proportion of people who will order coffee with their meal is between 0.85 and 0.875? c. What is the probability that the proportion of people who will order coffee with their meal is at least 0.945? ANSWER: a. 0.9; 0.025; normal b. 0.1359 c. 0.0359

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10 34. In interval estimation, as the sample size becomes larger, the interval estimate a. becomes narrower. b. becomes wider. c. remains the same, because the mean is not changing. d. gets closer to 1.96. ANSWER: a 35. When s is used to estimate σ , the margin of error is computed by using the a. normal distribution. b. t distribution. c. mean of the sample. d. mean of the population. ANSWER: b 36. From a (large) population with a variance of 900, a sample of 225 items is selected. At 95% confidence, the margin of error is a. 15. b. 2.0. c. 3.92. d. 4. ANSWER: c 37. In interval estimation, the t distribution is applicable only when a. the population has a mean of less than 30. b. the sample standard deviation is used to estimate the population standard deviation. c. the variance of the population is known. d. the mean of the population is unknown. ANSWER: b

11 38. From a population that is not normally distributed and whose standard deviation is not known, a sample of 6 items is selected to develop an interval estimate for the mean of the population ( μ ). a. The normal distribution can be used. b. The t distribution with 5 degrees of freedom must be used. c. The t distribution with 6 degrees of freedom must be used. d. The sample size must be increased. ANSWER: d 39. From a population that is normally distributed, a sample of 25 elements is selected and the standard deviation of the sample is computed. For the interval estimation of μ , the proper distribution to use is the a. normal distribution. b. t distribution with 25 degrees of freedom. c. t distribution with 26 degrees of freedom. d. t distribution with 24 degrees of freedom. ANSWER: d 40. As the sample size increases, the margin of error a. increases. b. decreases. c. stays the same. d. fluctuates depending on the mean. ANSWER: b 41. A 95% confidence interval for a population mean is determined to be 100 to 120. For the same data, if the confidence coefficient is reduced to .90, the confidence interval for μ a. becomes narrower. b. becomes wider. c. does not change. d. becomes 100.1 to 120.1. ANSWER: a

12 42. In general, higher confidence levels provide a. wider confidence intervals. b. narrower confidence intervals. c. a smaller standard error. d. unbiased estimates. ANSWER: a 43. A sample of 225 elements from a population with a standard deviation of 75 is selected. The sample mean is 180. The 95% confidence interval for μ is a. 105 to 225. b. 175 to 185. c. 171.78 to 188.22. d. 170.2 to 189.8. ANSWER: d 44. A random sample of 64 students at a university showed an average age of 25 years and a sample standard deviation of 2 years. The 98% confidence interval for the true average age of all students in the university is (Hint: You need to use the t-table here, since the population standard deviation is unknown) a. 20.5 to 29.5. b. 24.4 to 25.6. c. 23.0 to 27.0. d. 20.0 to 30.0. ANSWER: b 45. The sample size needed to provide a margin of error of 2 or less with a .95 probability when the population standard deviation equals 11 is a. 10. b. 11. c. 116. d. 117. ANSWER: d

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13 46. It is known that the population variance equals 484. With a .95 probability, the sample size that needs to be taken if the desired margin of error is 5 or less is a. 190. b. 74. c. 189. d. 75. ANSWER: d 47. The following random sample from a population whose values were normally distributed was collected. 10 8 11 11 The 95% confidence interval for μ is a. 8.52 to 11.48. b. 7.75 to 12.25. c. 9.25 to 10.75. d. 8.00 to 10.00. ANSWER: b 48. The following random sample from a population whose values were normally distributed was collected. 10 12 18 16 The 80% confidence interval for μ is a. 12.054 to 15.946. b. 10.108 to 17.892. c. 10.321 to 17.679. d. 11.009 to 16.991. ANSWER: d 49. In a random sample of 100 observations, ࠵? $ = .2. The 95.44% confidence interval for p is a. .122 to .278. b. .164 to .236. c. .134 to .266. d. .120 to .280. ANSWER: d

14 50. A random sample of 1000 people was taken. Four hundred fifty of the people in the sample favored Candidate A. The 95% confidence interval for the true proportion of people who favor Candidate A is a. .419 to .481. b. .40 to .50. c. .45 to .55. d. .424 to .476. ANSWER: a 51. A random sample of 64 SAT scores of students applying for merit scholarships showed an average of 1400 with a (sample!) standard deviation of 240. The 95% confidence interval for the population mean SAT score is a. 1340.06 to 1459.94. b. 1341.20 to 1458.80. c. 1349.93 to 1450.07. d. 1320.32 to 1479.68. ANSWER: a 52. A sample of 75 information systems managers had an average hourly income of $40.75 with a standard deviation of $7.00. The value of the margin of error at 95% confidence is a. 80.83. b. 7.00. c. .81. d. 1.61. ANSWER: d 53. We can use the normal distribution to make confidence interval estimates for the population proportion, p , when a. np > 5. b. n (1 - p ) > 5. c. p has a normal distribution. d. both np > 5 and n (1 - p ) > 5. ANSWER: d

15 54. The degrees of freedom associated with a t distribution are a function of the a. area in the upper tail. b. sample standard deviation. c. confidence coefficient. d. sample size. ANSWER: d 55. The margin of error in an interval estimate of the population mean is a function of all of the following except a. α . b. sample mean. c. sample size. d. variability of the population. ANSWER: b 56. A local university administers a comprehensive examination to the candidates for B.S. degrees in Business Administration. Five examinations are selected at random and scored. The scores are shown below. Scores 80 90 91 62 77 a. Compute the mean and the standard deviation of the sample. b. Compute the margin of error at 95% confidence. Assume the population is normally distributed. c. Develop a 95% confidence interval estimate for the mean of the population. ANSWER: a. Mean = 80, s = 11.77 (rounded). b. 14.61 c. 65.39 to 94.61

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16 57. Many people who bought X-Game gaming systems over the holidays have complained that the systems they purchased were defective. In a sample of 1200 units sold, 18 units were defective. a. Determine a 95% confidence interval for the percentage of defective systems in the population. b. If 1.5 million X-Game gaming systems were sold over the holidays, determine an interval for the number of defectives in sales. ANSWER: a. .00812 to .02188 (rounded) b. 12,184 to 32,816 (using unrounded estimates) 58. Two hundred students are enrolled in an Economics class. After the first examination, a random sample of 6 papers was selected. The scores were 65, 75, 89, 71, 70, and 80. a. Determine the standard error of the mean. b. What assumption must be made before we can determine an interval for the mean score of all the students in the class? Explain why. c. Assume the assumption of Part b is met. Provide a 95% confidence interval for the mean score of all the students in the class. ANSWER: a. 3.474 b. Since n is small and σ is estimated from s , we must assume the distribution of all the scores is normal. c. 66.07 to 83.93 (rounded)

ECO 045 - Practice Problems for Final Exam - Part1_Spring 2023

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