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PROBLEM 1. 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. b. c. What are the expected value, standard deviation, and shape of the sampling distribution of the sample mean? What is the probability that these 64 students will spend a combined total of more than $715.21? What is the probability that these 64 students will spend a combined total between $703.59 and $728.45? ANS: a. 10.5 0.363 b. 0.0314 c. 0.0794 PTS: 1 normal TOP: Sampling Distribution 2. A simple random sample of 6 recent graduates revealed the following information about their weekly incomes. Graduates A B C D E F a. b. Weekly Income $250 270 285 240 255 290 What is the expected value of the average weekly income of all the recent graduates? What is the expected value of the standard deviation for the population? ANS: a. $265 b. $20 PTS: 1 TOP: Sampling Distribution 3. The life expectancy in the United States is 75 with a standard deviation of 7 years. A random sample of 49 individuals is selected. a. b. c. d. e. What is the probability that the sample mean will be larger than 77 years? What is the probability that the sample mean will be less than 72.7 years? What is the probability that the sample mean will be between 73.5 and 76 years? What is the probability that the sample mean will be between 72 and 74 years? What is the probability that the sample mean will be larger than 73.46 years? ANS: a. 0.0228 b. 0.0107 c. 0.7745 d. 0.1573 e. 0.9389 PTS: 1 TOP: Sampling Distribution 4. The SAT scores have an average of 1200 with a standard deviation of 60. A sample of 36 scores is selected. a. b. c. d. e. What is the probability that the sample mean will be larger than 1224? What is the probability that the sample mean will be less than 1230? What is the probability that the sample mean will be between 1200 and 1214? What is the probability that the sample mean will be greater than 1200? What is the probability that the sample mean will be larger than 73.46? ANS: a. 0.0082 b. 0.9986 c. 0.4192 d. 0.5 e. 1.0 PTS: 1 5. TOP: Sampling Distribution A simple random sample of 8 employees of a corporation provided the following information. Employee 1 2 3 4 5 6 7 8 Age 25 32 26 40 50 54 22 23 Gender M M M M F M M F a. b. c. Determine the point estimate for the average age of all employees. What is the point estimate for the standard deviation of the population? Determine a point estimate for the proportion of all employees who are female. ANS: a. 34 b. 12.57 c. 0.25 PTS: 1 TOP: Sampling Distribution 6. Starting salaries of a sample of five management majors along with their genders are shown below. Employee 1 2 3 4 5 a. Salary (in $1,000s) 30 28 22 26 19 Gender F M F F M What is the point estimate for the starting salaries of all management majors? b. c. Determine the point estimate for the variance of the population. Determine the point estimate for the proportion of male employees. ANS: a. 25 (thousands) b. 20 (thousands) c. 0.4 PTS: 1 TOP: Sampling Distribution 7. An experimental diet to induce weight loss was followed for one week by a randomly selected group of 12 students with the following results. Student 1 2 3 4 5 6 7 8 9 10 11 12 a. b. c. Loss in Pounds 2.2 2.6 0.4 2.0 0.0 1.8 5.2 3.8 4.2 3.8 1.4 2.6 Find a point estimate for the average amount lost after one week on this diet. Is this an unbiased estimate of the population mean? Explain. Find a point estimate for the variance of the amount lost on this diet. Is this an unbiased estimate of the population variance? Explain. Find a point estimate for the standard deviation of the amount lost on this diet. ANS: a. 2.5; Yes; E( ) = b. 2.389; Yes; E(s2) = 2 c. 1.546 PTS: 1 TOP: Sampling Distribution 8. Below you are given the values obtained from a random sample of 4 observations taken from an infinite population. 32 a. b. c. d. 34 35 39 Find a point estimate for . Is this an unbiased estimate of ? Explain. Find a point estimate for 2. Is this an unbiased estimate of 2? Explain. Find a point estimate for . 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 . ANS: a. 35; Yes; E(x) = b. 8.667; Yes; E(s2) = 2 c. 2.944 d. E( ) = , the standard deviation = 2, and the sampling distribution of is normally distributed if the population is normally distributed. PTS: 1 TOP: Sampling Distribution 9. The following information gives the number of days absent from work for a population of 5 workers at a small factory. Worker A B C D E a. b. c. d. Number of Days Absent 5 7 1 4 8 Find the mean and the standard deviation for the population. Samples of size 2 will be drawn from the population. Use the answers in part a to calculate the expected value and the standard deviation of the sampling distribution of the sample mean. Find all the samples of 2 workers that can be extracted from this population. Choose the samples without replacement. Compute the sample mean for each of the samples in Part c. ANS: a. 5; 2.449 b. 5; 1.5 c. AB, AC, AD, AE, BC, BD, BE, CD, CE, DE d. 6, 3, 4.5, 6.5, 4, 5.5, 7.5, 2.5, 4.5, 6 PTS: 1 TOP: Sampling Distribution 10. MNM Corporation gives each of its employees an aptitude test. The scores on the test are normally distributed with a mean of 75 and a standard deviation of 15. A simple random sample of 25 is taken from a population of 500. a. b. c. d. e. What are the expected value, the standard deviation, and the shape of the sampling distribution of ? What is the probability that the average aptitude test in the sample will be between 70.14 and 82.14? What is the probability that the average aptitude test in the sample will be greater than 82.68? What is the probability that the average aptitude test in the sample will be less than 78.69? Find a value, C, such that P( ANS: a. 75; 3; normal b. 0.9387 c. 0.0052 d. 0.8907 C) = .015. e. 81.51 PTS: 1 TOP: Sampling Distribution 11. Students of a large university spend an average of $5 a day on lunch. The standard deviation of the expenditure is $3. A simple random sample of 36 students is taken. a. b. c. What are the expected value, standard deviation, and shape of the sampling distribution of the sample mean? What is the probability that the sample mean will be at least $4? What is the probability that the sample mean will be at least $5.90? ANS: a. 5.0; 0.5; normal b. 0.9772 c. 0.0359 PTS: 1 TOP: Sampling Distribution 12. The average lifetime of a light bulb is 3,000 hours with a standard deviation of 696 hours. A simple random sample of 36 bulbs is taken. a. b. c. d. What are the expected value, standard deviation, and shape of the sampling distribution of ? What is the probability that the average life in the sample will be between 2,670.56 and 2,809.76 hours? What is the probability that the average life in the sample will be greater than 3,219.24 hours? What is the probability that the average life in the sample will be less than 3,180.96 hours? ANS: a. 3,000; 116; normal b. 0.0482 c. 0.0294 d. 0.9406 PTS: 1 TOP: Sampling Distribution 13. Michael is running for president. The proportion of voters who favor Michael is 0.8. A simple random sample of 100 voters is taken. a. b. c. What are the expected value, standard deviation, and shape of the sampling distribution of ? What is the probability that the number of voters in the sample who will not favor Michael will be between 26 and 30? What is the probability that the number of voters in the sample who will not favor Michael will be more than 16? ANS: a. 0.8; 0.04; normal b. 0.0606 c. 0.8413 PTS: 1 TOP: Sampling Distribution 14. 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. b. c. What are the expected value, standard deviation, and shape of the sampling distribution of ? What is the probability that the proportion of people who will order coffee with their meal is between 0.85 and 0.875? What is the probability that the proportion of people who will order coffee with their meal is at least 0.945? ANS: a. 0.9; 0.025; normal b. 0.1359 c. 0.0359 PTS: 1 15. A random sample of nine telephone calls in an office provided the following information. Call Number 1 2 3 4 5 6 7 8 9 a. b. c. d. e. TOP: Sampling Distribution Duration (In Minutes) 3 8 4 3 5 6 3 5 8 Type of Call local long distance local local long distance local local local local Determine the point estimate for the average duration of all calls. What is the point estimate for the standard deviation of the population? Determine the standard error of the mean. What is the point estimate for the proportion of all calls that were long distance? Determine the standard error of proportion. ANS: a. 5 b. 2 c. 0.67 d. 0.222 e. 0.138 PTS: 1 TOP: Sampling Distribution 16. A random sample of ten examination papers in a course that was given on a pass or fail basis showed the following scores. Paper Number 1 2 3 4 Grade 65 87 92 35 Status Pass Pass Pass Fail 5 6 7 8 9 10 a. b. c. 79 100 48 74 79 91 Pass Pass Fail Pass Pass Pass What is the point estimate for the mean of the population? What is the point estimate for the standard deviation of the population? What is the point estimate for the proportion of all students who passed the course? ANS: a. 75 b. 20.48 c. 0.8 PTS: 1 TOP: Sampling Distribution 17. Consider a population of five weights identical in appearance but weighing 1, 3, 5, 7, and 9 ounces. a. b. c. Determine the mean and the variance of the population. Sampling without replacement from the above population with a sample size of 2 produces ten possible samples. Using the ten sample mean values, determine the mean of the population and the variance of . Compute the standard error of the mean. ANS: a. 5 and 8 b. 5 and 3 c. 1.732 PTS: 1 TOP: Sampling Distribution 18. Consider a population of five families with the following data representing the number of pets in each family. Family A B C D E a. b. c. Number of Pets 2 6 4 3 1 Determine the mean and the variance of the population. There are ten possible samples of size 2 (sampling without replacement). List the 10 possible samples of size 2, and determine the mean of each sample. Using the ten sample mean values, compute the mean and the standard error of the mean. ANS: a. 3.2 and 2.96 b. Possible Samples Sample Means AB AC AD AE BC BD BE CD CE DE c. 4 3 2.5 1.5 5 4.5 3.5 3.5 2.5 2 3.2 and 1.11 PTS: 1 TOP: Sampling Distribution 19. The average weekly earnings of bus drivers in a city are $950 (that is ) with a standard deviation of $45 (that is ). Assume that we select a random sample of 81 bus drivers. a. b. c. Compute the standard error of the mean. What is the probability that the sample mean will be greater than $960? If the population of bus drivers consisted of 400 drivers, what would be the standard error of the mean? ANS: a. 5 b. 0.0228 c. 4.47 PTS: 1 TOP: Sampling Distribution 20. An automotive repair shop has determined that the average service time on an automobile is 2 hours with a standard deviation of 32 minutes. A random sample of 64 services is selected. a. b. What is the probability that the sample of 64 will have a mean service time greater than 114 minutes? Assume the population consists of 400 services. Determine the standard error of the mean. ANS: a. 0.9332 b. 3.67 PTS: 1 TOP: Sampling Distribution 21. There are 8,000 students at the University of Tennessee at Chattanooga. The average age of all the students is 24 years with a standard deviation of 9 years. A random sample of 36 students is selected. a. b. c. Determine the standard error of the mean. What is the probability that the sample mean will be larger than 19.5? What is the probability that the sample mean will be between 25.5 and 27 years? ANS: a. 1.5 b. c. 0.9986 0.1359 PTS: 1 TOP: Sampling Distribution 22. In a local university, 10% of the students live in the dormitories. A random sample of 100 students is selected for a particular study. a. b. What is the probability that the sample proportion (the proportion living in the dormitories) is between 0.172 and 0.178? What is the probability that the sample proportion (the proportion living in the dormitories) is greater than 0.025? ANS: a. 0.0035 b. 0.9938 PTS: 1 TOP: Sampling Distribution 23. A department store has determined that 25% of all their sales are credit sales. A random sample of 75 sales is selected. a. b. c. d. What is the probability that the sample proportion will be greater than 0.34? What is the probability that the sample proportion will be between 0.196 and 0.354? What is the probability that the sample proportion will be less than 0.25? What is the probability that the sample proportion will be less than 0.10? ANS: a. 0.0359 b. 0.8411 c. 0.5 d. 0.0014 PTS: 1 TOP: Sampling Distribution 24. Ten percent of the items produced by a machine are defective. A random sample of 100 items is selected and checked for defects. a. b. c. Determine the standard error of the proportion. What is the probability that the sample will contain more than 2.5% defective units? What is the probability that the sample will contain more than 13% defective units? ANS: a. 0.03 b. 0.9938 c. 0.1587 PTS: 1 TOP: Sampling Distribution 25. There are 500 employees in a firm, 45% are female. A sample of 60 employees is selected randomly. a. Determine the standard error of the proportion. b. What is the probability that the sample proportion (proportion of females) is between 0.40 and 0.55? ANS: a. 0.0603 b. 0.7482 PTS: 1 TOP: Sampling Distribution 26. A new soft drink is being market tested. It is estimated that 60% of consumers will like the new drink. A sample of 96 taste tested the new drink. a. b. c. Determine the standard error of the proportion What is the probability that more than 70.4% of consumers will indicate they like the drink? What is the probability that more than 30% of consumers will indicate they do not like the drink? ANS: a. 0.05 b. 0.0188 c. 0.9772 PTS: 1 TOP: Sampling Distribution 27. A bank has kept records of the checking balances of its customers and determined that the average daily balance of its customers is $300 with a standard deviation of $48. A random sample of 144 checking accounts is selected. a. b. c. d. What is the probability that the sample mean will be more than $306.60? What is the probability that the sample mean will be less than $308? What is the probability that the sample mean will be between $302 and $308? What is the probability that the sample mean will be at least $296? ANS: a. 0.0495 b. 0.9772 c. 0.2857 d. 0.8413 PTS: 1 TOP: Sampling Distribution 28. In a large university, 20% of the students are business majors. A random sample of 100 students is selected, and their majors are recorded. a. b. c. d. Compute the standard error of the proportion. What is the probability that the sample contains at least 12 business majors? What is the probability that the sample contains less than 15 business majors? What is the probability that the sample contains between 12 and 14 business majors? ANS: a. 0.04 b. 0.9772 c. 0.1056 d. 0.044 PTS: 1 TOP: Sampling Distribution 29. A simple random sample of 6 computer programmers in Houston, Texas revealed the sex of the programmers and the following information about their weekly incomes. Programmer A B C D E F a. b. c. Weekly Income $250 270 285 240 255 290 Sex M M F M M F What is the point estimate for the average weekly income of all the computer programmers in Houston? What is the point estimate for the standard deviation of the population? Determine a point estimate for the proportion of all programmers in Houston who are female. ANS: a. 265 b. 20 c. 0.33 PTS: 1 TOP: Sampling Distribution 30. The milk prices (in quarts) from a sample of 9 convenience stores in Chattanooga, Tennessee are shown below. Store# 1 2 3 4 5 6 7 8 9 a. b. Price Per Quart (x) $1.14 $1.19 $1.25 $1.21 $1.17 $1.19 $1.22 $1.24 $1.19 What is the point estimate for the prices of all convenience stores in Chattanooga? What is the point estimate for the standard deviation of the population? ANS: a. $1.20 b. 0.03428 PTS: 1 TOP: Sampling Distribution 31. A random sample of 15 telephone calls in an office showed the duration of each call and whether it was a local or a long distance call. Call Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 a. b. c. Duration (In Minutes) 2 12 10 3 5 6 3 5 8 4 5 4 5 4 9 Type of Call local long distance local local long distance local local local local local local local local local long distance What is the point estimate for the average duration of all calls? What is the point estimate for the standard deviation of the population? What is the point estimate for the proportion of all calls that were long distance? ANS: a. 5.67 b. 2.85 c. 0.20 PTS: 1 TOP: Sampling Distribution 32. An automotive repair shop has determined that the average service time on an automobile is 130 minutes with a standard deviation of 26 minutes. A random sample of 40 automotive services is selected. a. b. c. Compute the standard error of the mean. What is the probability that the sample of 40 automotive services will have a mean service time greater than 136 minutes? Assume the population consists of 400 automotive services. Determine the standard error of the mean. ANS: a. 4.11 b. 0.0721 c. 3.9 PTS: 1 TOP: Sampling Distribution 33. A department store has determined that 25% of all their sales are credit sales. A random sample of 60 sales is selected. a. b. c. What is the sampling distribution of ? What is the probability that the sample proportion will be greater than 0.30? What is the probability that the sample proportion will be between 0.20 to 0.30? ANS: a. E( ) = 0.25 b. 0.1867 c. 0.6266 PTS: 1 Standard Error = .056 (rounded) TOP: Sampling Distribution