Mathematics

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{Exercise 6.17}

1. The mean cost of domestic airfares in the United States rose to an all-time high of $385 per ticket (Bureau of Transportation Statistics website, November 2, 2012). Airfares were based on the total ticket value, which consisted of the price charged by the airlines plus any additional taxes and fees. Assume domestic airfares are normally distributed with a standard deviation of $110.

a. What is the probability that a domestic airfare is $550 or more (to 4 decimals)?

b. What is the probability that a domestic airfare is $250 or less (to 4 decimals)?

c. What if the probability that a domestic airfare is between $300 and $500 (to 4 decimals)?

d. What is the cost for the 3% highest domestic airfares? (rounded to nearest dollar) $ or

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{Exercise 6.18 (Algorithmic)}

2. The average return for large-cap domestic stock funds over the three years 2009–2011 was 14.4%. Assume the three-year returns were normally distributed across funds with a standard deviation of 4.6%.

a. What is the probability an individual large-cap domestic stock fund had a three-year return of at least 20% (to 4 decimals)?

b. What is the probability an individual large-cap domestic stock fund had a three-year return of 10% or less (to 4 decimals)?

c. How big does the return have to be to put a domestic stock fund in the top 10% for the three-year period (to 2 decimals)? %

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{Exercise 6.23 (Algorithmic)}

3. The time needed to complete a final examination in a particular college course is normally distributed with a mean of 79 minutes and a standard deviation of 8 minutes. Answer the following questions.

a. What is the probability of completing the exam in one hour or less (to 4 decimals)?

b. What is the probability that a student will complete the exam in more than 60 minutes but less than 75 minutes (to 4 decimals)?

c. Assume that the class has 60 students and that the examination period is 90 minutes in length. How many students do you expect will be unable to complete the exam in the allotted time (to 2 decimals)?

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4. {Exercise 6.9}

A random variable is normally distributed with a mean of μ = 50 and a standard deviation of σ = 5.

a. What is the probability that the random variable will assume a value between 45 and 55 (to 3 decimals)?

b. What is the probability that the random variable will assume a value between 40 and 60 (to 3 decimals)?

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6. Delta Airlines quotes a flight time of 2 hours, 5 minutes for its flights from Cincinnati to Tampa. Suppose we believe that actual flight times are uniformly distributed between 2 hours and 2 hours, 20 minutes.

a. Which of the following graphs accurately represents the probability density function for flight time in minutes? http://cnow.apps.ng.cengage.com/ilrn/books/aseb06h/images/ch06/6.3.jpg

b. What is the probability that the flight will be no more than 5 minutes late (to 2 decimals)?

c. What is the probability that the flight will be more than 10 minutes late (to 2 decimals)?

d. What is the expected flight time, in minutes?

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7. According to Salary Wizard, the average base salary for a brand manager in Houston, Texas, is $88,592 and the average base salary for a brand manager in Los Angeles, California, is $97,417 (Salary Wizard website, February 27, 2008). Assume that salaries are normally distributed, the standard deviation for brand managers in Houston is $19,900, and the standard deviation for brand managers in Los Angeles is $21,800. Use the book table to calculate probabilities and z-values

a. What is the probability that a brand manager in Houston has a base salary in excess of $100,000? Round your answer to four decimal places.

b. What is the probability that a brand manager in Los Angeles has a base salary in excess of $100,000? Round your answer to four decimal places.

c. What is the probability that a brand manager in Los Angeles has a base salary of less than $75,000? Round your answer to four decimal places.

d. How much would a brand manager in Los Angeles have to make in order to have a higher salary than 99% of the brand managers in Houston? $

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8. Collina’s Italian Café in Houston, Texas, advertises that carryout orders take about 25 minutes (Collina’s website, February 27, 2008). Assume that the time required for a carryout order to be ready for customer pickup has an exponential distribution with a mean of 25 minutes. Round your answers to 4 decimal places.

a. What is the probability that a carryout order will be ready within 20 minutes?

b. If a customer arrives 30 minutes after placing an order, what is the probability that the order will not be ready?

c. A particular customer lives 15 minutes from Collina’s Italian Café. If the customer places a telephone order at 5:20 P.M., what is the probability that the customer can drive to the café, pick up the order, and return home by 6:00 P.M.?

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9. The port of South Louisiana, located along 54 miles of the Mississippi River between New Orleans and Baton Rouge, is the largest bulk cargo port in the world. The U.S. Army Corps of Engineers reports that the port handles a mean of 4.5 million tons of cargo per week (USA Today, September 25, 2012). Assume that the number of tons of cargo handled per week is normally distributed with a standard deviation of .82 million tons.

a. What is the probability that the port handles less than 5 million tons of cargo per week (to 4 decimals)?

b. What is the probability that the port handles 3 or more million tons of cargo per week (to 4 decimals)?

c. What is the probability that the port handles between 3 million and 4 million tons of cargo per week (to 4 decimals)?

d. Assume that 85% of the time the port can handle the weekly cargo volume without extending operating hours. What is the number of tons of cargo per week that will require the port to extend its operating hours (to 2 decimals)?

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10. Suppose we are interested in bidding on a piece of land and we know one other bidder is interested. The seller announced that the highest bid in excess of $10,000 will be accepted. Assume that the competitor’s bid x is a random variable that is uniformly distributed between $10,000 and $15,000.

If required, round your answers to two decimal places.

a. Suppose you bid $12,000. What is the probability that your bid will be accepted (to 2 decimals)?

b. Suppose you bid $14,000. What is the probability that your bid will be accepted (to 2 decimals)?

c. What amount should you bid to maximize the probability that you get the property? $

d. Suppose you know someone who is willing to pay you $16,000 for the property. Would you consider bidding less than the amount in part (c)?

Why or why not?

The input in the box below will not be graded, but may be reviewed and considered by your instructor.

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11. 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 7 ounces.

a. The process standard deviation is 0.1, and the process control is set at plus or minus 1.5 standard deviations. Units with weights less than 6.85 or greater than 7.15 ounces will be classified as defects. What is the probability of a defect (to 4 decimals)? In a production run of 1000 parts, how many defects would be found (to 0 decimals)?

b. Through process design improvements, the process standard deviation can be reduced to 0.05. Assume the process control remains the same, with weights less than 6.85 or greater than 7.15 ounces being classified as defects. What is the probability of a defect (rounded to 4 decimals; getting the exact answer, although not necessary, will require Excel)? In a production run of 1000 parts, how many defects would be found (to 0 decimals)?

c. What is the advantage of reducing process variation?

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12. A business executive, transferred from Chicago to Atlanta, needs to sell her house in Chicago quickly. The executive’s employer has offered to buy the house for $210,000, but the offer expires at the end of the week. The executive does not currently have a better offer, but can afford to leave the house on the market for another month. From conversations with her realtor, the executive believes the price she will get by leaving the house on the market for another month is uniformly distributed between $200,000 and $225,000.

a. If she leaves the house on the market for another month, what is the probability density function for the sales price? Note: x is in thousands of dollars. http://cnow.apps.ng.cengage.com/ilrn/books/aseb06h/images/ch06/6.39.png

b. If she leaves it on the market for another month, what is the probability she will get at least $215,000 for the house? (to 2 decimals)

c. If she leaves it on the market for another month, what is the probability she will get less than $210,000? (to 2 decimals)

d. What is the expected selling price of the house if the executive waits one month (in dollars)? Should the executive leave the house on the market for another month?

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13. In October 2012, Apple introduced a much smaller variant of the Apple iPad, known as the iPad Mini. Weighing less than 11 ounces, it was about 50% lighter than the standard iPad. Battery tests for the iPad Mini showed a mean life of 10.25 hours (The Wall Street Journal, October 31, 2012). Assume that battery life of the iPad Mini is uniformly distributed between 8.5 and 12 hours.

a. Give a mathematical expression for the probability density function of battery life.

a.

http://cnow.apps.ng.cengage.com/ilrn/books/aseb06h/images/ch06/aseb06h_ch06_05a.png

b.

http://cnow.apps.ng.cengage.com/ilrn/books/aseb06h/images/ch06/aseb06h_ch06_05b.png

c.

http://cnow.apps.ng.cengage.com/ilrn/books/aseb06h/images/ch06/aseb06h_ch06_05c.png

The correct answer is:

b. What is the probability that the battery life for an iPad Mini will be 10 hours or less (to 4 decimals)?

c. What is the probability that the battery life for an iPad Mini will be at least 11 hours (to 4 decimals)?

d. What is the probability that the battery life for an iPad Mini will be between 9.5 and 11.5 hours (to 4 decimals)?

e. In a shipment of 100 iPad Minis, how many should have a battery life of at least 9 hours (to nearest whole value)?

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14. A person must score in the upper 2% of the population on an IQ test to qualify for membership in Mensa, the international high-IQ society. There are 110,000 Mensa members in 100 countries throughout the world (Mensa International website, January 8, 2013). If IQ scores are normally distributed with a mean of 100 and a standard deviation of 15, what score must a person have to qualify for Mensa (0 decimals)?

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15. Given that z is a standard normal random variable, compute the following probabilities (to 4 decimals).

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