Chapter 7

A uniform distribution is defined over the interval from 6 to \(10 .\) a. What are the values for a and \(b\) ? b. What is the mean of this uniform distribution? c. What is the standard deviation? d. Show that the total area is 1.00 . e. Find the probability of a value more than 7 . f. Find the probability of a value between 7 and \(9 .\)

A uniform distribution is defined over the interval from 2 to \(5 .\) a. What are the values for \(a\) and \(b\) ? b. What is the mean of this uniform distribution? c. What is the standard deviation? d. Show that the total area is 1.00 . e. Find the probability of a value more than 2.6 . f. Find the probability of a value between 2.9 and 3.7 .

America West Airlines reports the flight time from Los Angeles International Airport to Las Vegas is 1 hour and 5 minutes, or 65 minutes. Suppose the actual flying time is uniformly distributed between 60 and 70 minutes. a. Show a graph of the continuous probability distribution. b. What is the mean flight time? What is the variance of the flight times? c. What is the probability the flight time is less than 68 minutes? d. What is the probability the flight takes more than 64 minutes?

According to the Insurance Institute of America, a family of four spends between \(\$ 400\) and \(\$ 3,800\) per year on all types of insurance. Suppose the money spent is uniformly distributed between these amounts. a. What is the mean amount spent on insurance? b. What is the standard deviation of the amount spent? c. If we select a family at random, what is the probability they spend less than \(\$ 2,000\) per year on insurance per year? d. What is the probability a family spends more than \(\$ 3,000\) per year?

The April rainfall in Flagstaff, Arizona, follows a uniform distribution between 0.5 and 3.00 inches. a. What are the values for \(a\) and \(b\) ? b. What is the mean amount of rainfall for the month? What is the standard deviation? c. What is the probability of less than an inch of rain for the month? d. What is the probability of "exactly" 1.00 inch of rain? e. What is the probability of more than 1.50 inches of rain for the month?

Customers experiencing technical difficulty with their Internet cable hookup may call an 800 number for technical support. It takes the technician between 30 seconds to 10 minutes to resolve the problem. The distribution of this support time follows the uniform distribution. a. What are the values for \(a\) and \(b\) in minutes? b. What is the mean time to resolve the problem? What is the standard deviation of the time? c. What percent of the problems take more than 5 minutes to resolve. d. Suppose we wish to find the middle 50 percent of the problem-solving times. What are the end points of these two times?

Explain what is meant by this statement: "There is not just one normal probability distribution but a 'family' of them."

List the major characteristics of a normal probability distribution.

The mean of a normal probability distribution is \(500 ;\) the standard deviation is \(10 .\) a. About 68 percent of the observations lie between what two values? b. About 95 percent of the observations lie between what two values? c. Practically all of the observations lie between what two values?

The mean of a normal probability distribution is \(60 ;\) the standard deviation is \(5 .\) a. About what percent of the observations lie between 55 and \(65 ?\) b. About what percent of the observations lie between 50 and \(70 ?\) c. About what percent of the observations lie between 45 and \(75 ?\)

The Kamp family has twins, Rob and Rachel. Both Rob and Rachel graduated from college 2 years ago, and each is now earning \(\$ 50,000\) per year. Rachel works in the retail industry, where the mean salary for executives with less than 5 years' experience is \(\$ 35,000\) with a standard deviation of \(\$ 8,000 .\) Rob is an engineer. The mean salary for engineers with less than 5 years' experience is \(\$ 60,000\) with a standard deviation of \(\$ 5,000\). Compute the \(z\) values for both Rob and Rachel and comment on your findings.

A recent article in the Cincinnati Enquirer reported that the mean labor cost to repair a heat pump is \(\$ 90\) with a standard deviation of \(\$ 22 .\) Monte's Plumbing and Heating Service completed repairs on two heat pumps this morning. The labor cost for the first was \(\$ 75\) and it was \(\$ 100\) for the second. Compute \(z\) values for each and comment on your findings.

A normal population has a mean of 20.0 and a standard deviation of \(4.0 .\) a. Compute the \(z\) value associated with 25.0 . b. What proportion of the population is between 20.0 and \(25.0 ?\) c. What proportion of the population is less than \(18.0 ?\)

A normal population has a mean of 12.2 and a standard deviation of \(2.5 .\) a. Compute the \(z\) value associated with 14.3 . b. What proportion of the population is between 12.2 and \(14.3 ?\) C. What proportion of the population is less than \(10.0 ?\)

A recent study of the hourly wages of maintenance crew members for major airlines showed that the mean hourly salary was \(\$ 20.50,\) with a standard deviation of \(\$ 3.50 .\) If we select a crew member at random, what is the probability the crew member earns: a. Between \(\$ 20.50\) and \(\$ 24.00\) per hour? b. More than \(\$ 24.00\) per hour? c. Less than \(\$ 19.00\) per hour?

The mean of a normal distribution is 400 pounds. The standard deviation is 10 pounds. a. What is the area between 415 pounds and the mean of 400 pounds? b. What is the area between the mean and 395 pounds? c. What is the probability of selecting a value at random and discovering that it has a value of less than 395 pounds?

A normal distribution has a mean of 50 and a standard deviation of 4 a. Compute the probability of a value between 44.0 and 55.0 . b. Compute the probability of a value greater than 55.0 . c. Compute the probability of a value between 52.0 and 55.0 .

A normal population has a mean of 80.0 and a standard deviation of \(14.0 .\) a. Compute the probability of a value between 75.0 and 90.0 . b. Compute the probability of a value 75.0 or less. c. Compute the probability of a value between 55.0 and 70.0 .

A cola-dispensing machine is set to dispense on average 7.00 ounces of cola per cup. The standard deviation is 0.10 ounces. The distribution amounts dispensed follows a normal distribution. a. What is the probability that the machine will dispense between 7.10 and 7.25 ounces of cola? b. What is the probability that the machine will dispense 7.25 ounces of cola or more? c. What is the probability that the machine will dispense between 6.80 and 7.25 ounces of cola?

The amounts of money requested on home loan applications at Down River Federal Savings follow the normal distribution, with a mean of \(\$ 70,000\) and a standard deviation of \(\$ 20,000 .\) A loan application is received this morning. What is the probability: a. The amount requested is \(\$ 80,000\) or more? b. The amount requested is between \(\$ 65,000\) and \(\$ 80,000 ?\) c. The amount requested is \(\$ 65,000\) or more?

WNAE, an all-news AM station, finds that the distribution of the lengths of time listeners are tuned to the station follows the normal distribution. The mean of the distribution is 15.0 minutes and the standard deviation is 3.5 minutes. What is the probability that a particular listener will tune in: a. More than 20 minutes? b. For 20 minutes or less? c. Between 10 and 12 minutes?

The mean starting salary for college graduates in the spring of 2005 was \(\$ 36,280 .\) Assume that the distribution of starting salaries follows the normal distribution with a standard deviation of \(\$ 3,300 .\) What percent of the graduates have starting salaries: a. Between \(\$ 35,000\) and \(\$ 40,000 ?\) b. More than \(\$ 45,000 ?\) c. Between \(\$ 40,000\) and \(\$ 45,000 ?\)

A normal distribution has a mean of 50 and a standard deviation of \(4 .\) Determine the value below which 95 percent of the observations will occur.

A normal distribution has a mean of 80 and a standard deviation of \(14 .\) Determine the value above which 80 percent of the values will occur.

The amounts dispensed by a cola machine follow the normal distribution with a mean of 7 ounces and a standard deviation of 0.10 ounces per cüp. How much cola is dispensed in the largest 1 percent of the cups?

Refer to Exercise \(20,\) where the amount requested for home loans followed the normal distribution with a mean of \(\$ 70,000\) and a standard deviation of \(\$ 20,000\). a. How much is requested on the largest 3 percent of the loans? b. How much is requested on the smallest 10 percent of the loans?

Assume that the mean hourly cost to operate a commercial airplane follows the normal distribution with a mean \(\$ 2,100\) per hour and a standard deviation of \(\$ 250 .\) What is the operating cost for the lowest 3 percent of the airplanes?

The monthly sales of mufflers in the Richmond, Virginia, area follow the normal distribution with a mean of 1,200 and a standard deviation of \(225 .\) The manufacturer would like to establish inventory levels such that there is only a 5 percent chance of running out of stock. Where should the manufacturer set the inventory levels?

The amount of cola in a 12 -ounce can is uniformly distributed between 11.96 ounces and 12.05 ounces. a. What is the mean amount per can? b. What is the standard deviation amount per can? c. What is the probability of selecting a can of cola and finding it has less than 12 ounces? d. What is the probability of selecting a can of cola and finding it has more than 11.98 ounces? e. What is the probability of selecting a can of cola and finding it has more than 11.00 ounces?

A tube of Listerine Tartar Control toothpaste contains 4.2 ounces. As people use the toothpaste, the amount remaining in any tube is random. Assume the: amount of toothpaste left in the tube follows a uniform distribution. From this information, we can determine the following information about the amount remaining in a toothpaste tube without invading anyone's privacy. a. How much toothpaste would you expect to be remaining in the tube? b. What is the standard deviation of the amount remaining in the tube? c. What is the likelihood there is less than 3.0 ounces remaining in the tube? d. What is the probability there is more than 1.5 ounces remaining in the tube?

Many retail stores offer their own credit cards. At the time of the credit application the customer is given a 10 percent discount on the purchase. The time required for the credit application process follows a uniform distribution with the times ranging from 4 minutes to 10 minutes. a. What is the mean time for the application process? b. What is the standard deviation of the process time? c. What is the likelihood a particular application will take less than 6 minutes? d. What is the likelihood an application will take more than 5 minutes?

The times patrons at the Grande Dunes Hotel in the Bahamas spend waiting for an elevator follows a uniform distribution between 0 and 3.5 minutes. a. Show that the area under the curve is 1.00 . b. How long does the typical patron wait for elevator service? c. What is the standard deviation of the waiting time? d. What percent of the patrons wait for less than a minute? e. What percent of the patrons wait more than 2 minutes?

A recent report in USA Today indicated a typical family of four spends \(\$ 490\) per month on food. Assume the distribution of food expenditures for a family of four follows the normal distribution, with a mean of \(\$ 490\) and a standard deviation of \(\$ 90 .\) a. What percent of the families spend more than \(\$ 30\) but less than \(\$ 490\) per month on food? b. What percent of the families spend less than \(\$ 430\) per month on food? c. What percent spend between \(\$ 430\) and \(\$ 600\) per month on food? d. What percent spend between \(\$ 500\) and \(\$ 600\) per month on food?

A study of long distance phone calls made from the corporate offices of the Pepsi Bottling Group, Inc., in Somers, New York, showed the calls follow the normal distribution. The mean length of time per call was 4.2 minutes and the standard deviation was 0.60 minutes. a. What fraction of the calls last between 4.2 and 5 minutes? b. What fraction of the calls last more than 5 minutes? c. What fraction of the calls last between 5 and 6 minutes? d. What fraction of the calls last between 4 and 6 minutes? e. As part of her report to the president, the Director of Communications would like to report the length of the longest (in duration) 4 percent of the calls. What is this time?

Shaver Manufacturing, Inc. offers dental insurance to its employees. A recent study by the Human Resource Director shows the annual cost per employee per year followed the normal distribution, with a mean of \(\$ 1,280\) and a standard deviation of \(\$ 420\) per year. a. What fraction of the employees cost more than \(\$ 1,500\) per year for dental expenses? b. What fraction of the employees cost between \(\$ 1,500\) and \(\$ 2,000\) per year? c. Estimate the percent that did not have any dental expense. d. What was the cost for the 10 percent of employees who incurred the highest dental expense?

The annual commissions earned by sales representatives of Machine Products, Inc. a manufacturer of light machinery, follow the normal distribution. The mean yearly amount earned is \(\$ 40,000\) and the standard deviation is \(\$ 5,000\). a. What percent of the sales representatives earn more than \(\$ 42,000\) per year? b. What percent of the sales representatives earn between \(\$ 32,000\) and \(\$ 42,000 ?\) c. What percent of the sales representatives earn between \(\$ 32,000\) and \(\$ 35,000 ?\) d. The sales manager wants to award the sales representatives who earn the largest commissions a bonus of \(\$ 1,000 .\) He can award a bonus to 20 percent of the representatives. What is the cutoff point between those who earn a bonus and those who do not?

According to the South Dakota Department of Health, the mean number of hours of TV viewing per week is higher among adult women than men. A recent study showed women spent an average of 34 hours per week watching \(\mathrm{TV}\) and men 29 hours per week (www.state.sd.us/DOH/Nutrition/TV.pdt). Assume that the distribution of hours watched follows the normal distribution for both groups, and that the standard deviation among the women is 4.5 hours and it is 5.1 hours for the men. a. What percent of the women watch TV less than 40 hours per week? b. What percent of the men watch TV more than 25 hours per week? c. How many hours of TV do the one percent of women who watch the most TV per week watch? Find the comparable value for the men.

According to a government study among adults in the 25 - to 34 -year age group, the mean amount spent per year on reading and entertainment is \(\$ 1,994\) (www. infoplease.com/ipa/A0908759.html. Assume that the distribution of the amounts spent follows the normal distribution with a standard deviation of \(\$ 450 .\) a. What percent of the adults spend more than \(\$ 2,500\) per year on reading and entertainment? b. What percent spend between \(\$ 2,500\) and \(\$ 3,000\) per year on reading and entertainment? c. What percent spend less than \(\$ 1,000\) per year on reading and entertainment?

The weights of cans of Monarch pears follow the normal distribution with a mean of 1,000 grams and a standard deviation of 50 grams. Calculate the percentage of the cans that weigh: a. Less than 860 grams. b. Between 1,055 and \(1 ; 100\) grams. c. Between 860 and 1,055 grams.

The number of passengers on the Carnival Sensation during one-week cruises in the Caribbean follows the normal distribution. The mean number of passengers per cruise is 1,820 and the standard deviation is \(120 .\) a. What percent of the cruises will have between 1,820 and 1,970 passengers? b. What percent of the cruises will have 1,970 passengers or more? c. What percent of the cruises will have 1,600 or fewer passengers? d. How many passengers are on the cruises with the fewest 25 percent of passengers?

Management at Gordon Electronics is considering adopting a bonus system to increase production. One suggestion is to pay a bonus on the highest 5 percent of production based on past experience. Past records indicate weekly production follows the normal distribution. The mean of this distribution is 4,000 units per week and the standard deviation is 60 units per week. If the bonus is paid on the upper 5 percent of production, the bonus will be paid on how many units or more?

Fast Service Truck Lines uses the Ford Super Duty F-750 exclusively. Management made a study of the maintenance costs and determined the number of miles traveled during the year followed the normal distribution. The mean of the distribution was 60,000 miles and the standard deviation 2,000 miles. a. What percent of the Ford Super Duty F-750s logged 65,200 miles or more? b. What percent of the trucks logged more than 57,060 but less than 58,280 miles? c. What percent of the Fords traveled 62,000 miles or less during the year? d. Is it reasonable to conclude that any of the trucks were driven more than 70,000 miles? Explain.

Best Electronics, Inc. offers a "no hassle" returns policy. The number of items returned per day follows the normal distribution. The mean number of customer returns is 10.3 per day and the standard deviation is 2.25 per day. a. In what percent of the days are there 8 or fewer customers returning items? b. In what percent of the days are between 12 and 14 customers returning items? c. Is there any chance of a day with no returns?

The current model Boeing 737 has a capacity of 189 passengers. Suppose Delta Airlines uses this equipment for its Atlanta to Houston flights. The distribution of the number of seats sold for the Atlanta to Houston flights follows the normal distribution with a mean of 155 seats and a standard deviation of 15 seats. a. What is the likelihood of selling more than 134 seats? b. What is the likelihood of selling less than 173 seats? c. What is the likelihood of selling more than 134 seats but less than 173 seats? d. What percent of the time would Delta be able to sell more seats than there are seats actually available?

The goal at U.S. airports handling international flights is to clear these flights within 45 minutes. Let's interpret this to mean that 95 percent of the flights are cleared in 45 minutes, so 5 percent of the flights take longer to clear. Let's also assume that the distribution is approximately normal. a. If the standard deviation of the time to clear an international flight is 5 minutes, what is the mean time to clear a flight? b. Suppose the standard deviation is 10 minutes, not the 5 minutes suggested in part a. What is the new mean? c. A customer has 30 minutes from the time her flight landed to catch her limousine. Assuming a standard deviation of 10 minutes, what is the likelihood that she will be cleared in time?

The funds dispensed at the ATM machine located near the checkout line at the Kroger's in Union, Kentucky, follows a normal distribution with a mean of \(\$ 4,200\) per day and a standard deviation of \(\$ 720\) per day. The machine is programmed to notify the nearby bank if the amount dispensed is very low (less than \(\$ 2,500)\) or very high (more than \(\$ 6,000)\). a. What percent of the days will the bank be notified because the amount dispensed is very low? b. What percent of the time will the bank be notified because the amount dispensed is high? c. What percent of the time will the bank not be notified regarding the amount of funds dispersed?

The weights of canned hams processed at the Henline Ham Company follow the normal distribution, with a mean of 9.20 pounds and a standard deviation of 0.25 pounds. The label weight is given as 9.00 pounds. a. What proportion of the hams actually weigh less than the amount claimed on the label? b. The owner, Glen Henline, is considering two proposals to reduce the proportion of hams below label weight. He can increase the mean weight to 9.25 and leave the standard deviation the same, or he can leave the mean weight at 9.20 and reduce the standard deviation from 0.25 pounds to \(0.15 .\) Which change would you recommend?

Most four-year automobile leases allow up to 60,000 miles. If the lessee goes beyond this amount, a penalty of 20 cents per mile is added to the lease cost. Suppose the distribution of miles driven on four-year leases follows the normal distribution. The mean is 52,000 miles and the standard deviation is 5,000 miles. a. What percent of the leases will yield a penalty because of excess mileage? b. If the automobile company wanted to change the terms of the lease so that 25 percent of the leases went over the limit, where should the new upper limit be set? c. One definition of a low-mileage car is one that is 4 years old and has been driven less than 45,000 miles. What percent of the cars returned are considered low-mileage?

The price of shares of Bank of Florida at the end of trading each. day for the last year followed the normal distribution. Assume there were 240 trading days in the year. The mean price was \(\$ 42.00\) per share and the standard deviation was \(\$ 2.25\) per share. a. What percent of the days was the price over \(\$ 45.00 ?\) How many days would you estimate? b. What percent of the days was the price between \(\$ 38.00\) and \(\$ 40.00 ?\) c. What was the stock's price on the highest 15 percent of days?

The annual sales of romance novels follow the normal distribution. However, the mean and the standard deviation are unknown. Forty percent of the time sales are more than 470,000 , and 10 percent of the time sales are more than \(500,000 .\) What are the mean and the standard deviation?