"#1 Chapter 5, page 221 #57 A company is planning to interview Internet users to learn how its proposed Web site will be received by different age groups. According to the Census Bureau, 40% of individuals ages 18 to 54 and 12% of individuals age 55 and older use the Internet (Statistical Abstract of the United States, 2000). 1. How many people from the 18?54 age group must be contacted to find an expected number of at least 10 Internet users? 2. How many people from the age group 55 and older must be contacted to find an expected number of at least 10 Internet users? 3. If you contact the number of 18- to 54-year-old people suggested in part (a), what is the standard deviation of the number who will be Internet users? 4. If you contact the number of people age 55 and older suggested in part (b), what is the standard deviation of the number who will be Internet users? #2 Chapter 5, page 221 #61 Cars arrive at a car wash randomly and independently; the probability of an arrival is the same for any two time intervals of equal length. The mean arrival rate is 15 cars per hour. What is the probability that 20 or more cars will arrive during any given hour of operation? #3 Chapter 5, page 222 #63 A regional director responsible for business development in the state of Pennsylvania is concerned about the number of small business failures. If the mean number of small business failures per month is 10, what is the probability that exactly four small businesses will fail during a given month? Assume that the probability of a failure is the same for any two months and that the occurrence or nonoccurrence of a failure in any month is independent of failures in any other month. #4 Chapter 6, page 252 #41 Motorola used the normal distribution to determine the probability of defects and the number of defects expected in a production process. Assume a production process produces items with a mean weight of 10 ounces. Calculate the probability of a defect and the expected number of defects for a 1000-unit production run in the following situations. 1. The process standard deviation is .15, and the process control is set at plus or minus one standard deviation. Units with weights less than 9.85 or greater than 10.15 ounces will be classified as defects. 2. Through process design improvements, the process standard deviation can be reduced to .05. Assume the process control remains the same, with weights less than 9.85 or greater than 10.15 ounces being classified as defects. 3. What is the advantage of reducing process variation, thereby causing process control limits to be at a greater number of standard deviations from the mean? #5 Chapter 6, page 252 #45 Is lack of sleep causing traffic fatalities? A study conducted under the auspices of the National Highway Traffic Safety Administration found that the average number of fatal crashes caused by drowsy drivers each year was 1550 (BusinessWeek, January 26, 2004). Assume the annual number of fatal crashes per year is normally distributed with a standard deviation of 300. 1. What is the probability of fewer than 1000 fatal crashes in a year? 2. What is the probability the number of fatal crashes will be between 1000 and 2000 for a year? 3. For a year to be in the upper 5% with respect to the number of fatal crashes, how many fatal crashes would have to occur? #6 Chapter 6, page 254 #53 The average travel time to work for New York City residents is 36.5 minutes (Time Almanac, 2001). 1. Assume the exponential probability distribution is applicable and show the probability density function for the travel time to work for a typical New Yorker. 2. What is the probability it will take a typical New Yorker between 20 and 40 minutes to travel to work? 3. What is the probability it will take a typical New Yorker more than 40 minutes to travel to work? #7 Chapter 7, page 293 #43 Americans have become increasingly concerned about the rising cost of Medicare. In 1990, the average annual Medicare spending per enrollee was $3267; in 2003, the average annual Medicare spending per enrollee was $6883 (Money, Fall 2003). Suppose you hired a consulting firm to take a sample of fifty 2003 Medicare enrollees to further investigate the nature of expenditures. Assume the population standard deviation for 2003 was $2000. 1. Show the sampling distribution of the mean amount of Medicare spending for a sample of fifty 2003 enrollees. 2. What is the probability the sample mean will be within ?$300 of the population mean? 3. What is the probability the sample mean will be greater than $7500? If the consulting firm tells you the sample mean for the Medicare enrollees they interviewed was $7500, would you question whether they followed correct simple random sampling procedures? Why or why not? #8 Chapter 7, page 293 #47 Three firms carry inventories that differ in size. Firm A's inventory contains 2000 items, firm B's inventory contains 5000 items, and firm C's inventory contains 10,000 items. The population standard deviation for the cost of the items in each firm's inventory is ? = 144. A statistical consultant recommends that each firm take a sample of 50 items from its inventory to provide statistically valid estimates of the average cost per item. Managers of the small firm state that because it has the smallest population, it should be able to make the estimate from a much smaller sample than that required by the larger firms. However, the consultant states that to obtain the same standard error and thus the same precision in the sample results, all firms should use the same sample size regardless of population size. 1. Using the finite population correction factor, compute the standard error for each of the three firms given a sample of size 50. 2. What is the probability that for each firm the sample mean will be within ?25 of the population mean ?? #9 Chapter 7, page 294 #49 A production process is checked periodically by a quality control inspector. The inspector selects simple random samples of 30 finished products and computes the sample mean product weights . If test results over a long period of time show that 5% of the values are over 2.1 pounds and 5% are under 1.9 pounds, what are the mean and the standard deviation for the population of products produced with this process? #10 Chapter 7, page 294 #53 The proportion of individuals insured by the All-Driver Automobile Insurance Company who received at least one traffic ticket during a five-year period is .15. 1. Show the sampling distribution of if a random sample of 150 insured individuals is used to estimate the proportion having received at least one ticket. 2. What is the probability that the sample proportion will be within ?.03 of the population proportion?
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