a. Show the graph of the probability density function.b. Compute P(x = 1.25).
c. Compute P(1.0 ≤ x ≤ 1.25).
d. Compute P(1.20 < x < 1.5).
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6.2 The random variable x is known to be uniformly distributed between 10 and 20.
a. Show the graph of the probability density function.
b. Compute P(x < 15).
c. Compute P(12 ≤ x ≤ 18).
d. Compute E(x).
e. Compute Var(x).
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6.3 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. Show the graph of the probability density function for flight time.
b. What is the probability that the flight will be no more than 5 minutes late?
c. What is the probability that the flight will be more than 10 minutes late?
d. What is the expected flight time?
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6.4 Most computer languages include a function that can be used to generate random numbers. In Excel, the RAND function can be used to generate random numbers between 0 and 1. If we let x denote a random number generated using RAND, then x is a continuous random variable with the following probability density function....
a. Graph the probability density function.
b. What is the probability of generating a random number between .25 and .75?
c. What is the probability of generating a random number with a value less than or equal to .30?
d. What is the probability of generating a random number with a value greater than .60?
e. Generate 50 random numbers by entering = RAND() into 50 cells of an Excel worksheet.
f. Compute the mean and standard deviation for the random numbers in part (e).
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6.5 The driving distance for the top 100 golfers on the PGA tour is between 284.7 and 310.6 yards (Golfweek, March 29, 2003). Assume that the driving distance for these golfers is uniformly distributed over this interval.
a. Give a mathematical expression for the probability density function of driving distance.
b. What is the probability the driving distance for one of these golfers is less than 290 yards?
c. What is the probability the driving distance for one of these golfers is at least 300 yards?
d. What is the probability the driving distance for one of these golfers is between 290 and 305 yards?
e. How many of these golfers drive the ball at least 290 yards?
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6.6
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6.7 Suppose we are interested in bidding on a piece of land and we know one other bidder is interested.1 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.
a. Suppose you bid $12,000. What is the probability that your bid will be accepted?
b. Suppose you bid $14,000. What is the probability that your bid will be accepted?
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?
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6.8 Using Figure 6.4 as a guide, sketch a normal curve for a random variable x that has a mean of μ = 100 and a standard deviation of σ = 10. Label the horizontal axis with values of 70, 80, 90, 100, 110, 120, and 130.
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6.9 A random variable is normally distributed with a mean of μ = 50 and a standard deviation of σ = 5.
a. Sketch a normal curve for the probability density function. Label the horizontal axis with values of 35, 40, 45, 50, 55, 60, and 65. Figure 6.4 shows that the normal curve almost touches the horizontal axis at three standard deviations below and at three standard deviations above the mean (in this case at 35 and 65).
b. What is the probability the random variable will assume a value between 45 and 55?
c. What is the probability the random variable will assume a value between 40 and 60?
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6.10 Draw a graph for the standard normal distribution. Label the horizontal axis at values of −3, −2, −1, 0, 1, 2, and 3. Then use the table of probabilities for the standard normal distribution inside the front cover of the text to compute the following probabilities.
a. P(z ≤ 1.5)
b. P(z ≤ 1)
c. P(1 ≤ z ≤ 1.5)
d. P(0 < z < 2.5)
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6.11 Given that z is a standard normal random variable, compute the following probabilities.
a. P(z ≤ −1.0)
b. P(z ≥ −1)
c. P(z ≥ −1.5)
d. P(−2.5 ≤ z)
e. P(−3 < z ≤ 0)
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6.12 Given that z is a standard normal random variable, compute the following probabilities.
a. P(0 ≤ z ≤ .83)
b. P(−1.57 ≤ z ≤ 0)
c. P(z > .44)
d. P(z ≥ −.23)
e. P(z < 1.20)
f. P(z ≥ −.71)
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6.13 Given that z is a standard normal random variable, compute the following probabilities.
a. P(−1.98 ≤ z ≤ .49)
b. P(.52 ≤ z ≤ 1.22)
c. P(−1.75 ≤ z ≤ −1.04)
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6.14 Given that z is a standard normal random variable, find z for each situation.
a. The area to the left of z is .9750.
b. The area between 0 and z is .4750.
c. The area to the left of z is .7291.
d. The area to the right of z is .1314.
e. The area to the left of z is .6700.
f. The area to the right of z is .3300.
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6.15 Given that z is a standard normal random variable, find z for each situation.
a. The area to the left of z is .2119.
b. The area between −z and z is .9030.
c. The area between −z and z is .2052.
d. The area to the left of z is .9948.
e. The area to the right of z is .6915.
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6.16 Given that z is a standard normal random variable, find z for each situation.
a. The area to the right of z is .01.
b. The area to the right of z is .025.
c. The area to the right of z is .05.
d. The area to the right of z is .10.
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6.17 For borrowers with good credit scores, the mean debt for revolving and installment accounts is $15,015 (BusinessWeek, March 20, 2006). Assume the standard deviation is $3540 and that debt amounts are normally distributed.
a. What is the probability that the debt for a randomly selected borrower with good credit is more than $18,000?
b. What is the probability that the debt for a randomly selected borrower with good credit is less than $10,000?
c. What is the probability that the debt for a randomly selected borrower with good credit is between $12,000 and $18,000?
d. What is the probability that the debt for a randomly selected borrower with good credit is no more than $14,000?
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6.18 The average stock price for companies making up the S&P 500 is $30, and the standard deviation is $8.20 (BusinessWeek, Special Annual Issue, Spring 2003). Assume the stock prices are normally distributed.
a. What is the probability a company will have a stock price of at least $40?
b. What is the probability a company will have a stock price no higher than $20?
c. How high does a stock price have to be to put a company in the top 10%?
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6.19
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6.20 In January 2003, the American worker spent an average of 77 hours logged on to the Internet while at work (CNBC, March 15, 2003). Assume the population mean is 77 hours, the times are normally distributed, and that the standard deviation is 20 hours.
a. What is the probability that in January 2003 a randomly selected worker spent fewer than 50 hours logged on to the Internet?
b. What percentage of workers spent more than 100 hours in January 2003 logged on to the Internet?
c. A person is classified as a heavy user if he or she is in the upper 20% of usage. In January 2003, how many hours did a worker have to be logged on to the Internet to be considered a heavy user?
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6.21 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 (U.S. Airways Attaché, September 2000). 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?
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6.22 The mean hourly pay rate for financial managers in the East North Central region is $32.62, and the standard deviation is $2.32 (Bureau of Labor Statistics, September 2005). Assume that pay rates are normally distributed.
a. What is the probability a financial manager earns between $30 and $35 per hour?
b. How high must the hourly rate be to put a financial manager in the top 10% with respect to pay?
c. For a randomly selected financial manager, what is the probability the manager earned less than $28 per hour?
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6.23 The time needed to complete a final examination in a particular college course is normally distributed with a mean of 80 minutes and a standard deviation of 10 minutes. Answer the following questions.
a. What is the probability of completing the exam in one hour or less?
b. What is the probability that a student will complete the exam in more than 60 minutes but less than 75 minutes?
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?
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6.24 Trading volume on the New York Stock Exchange is heaviest during the first half hour (early morning) and last half hour (late afternoon) of the trading day. The early morning trading volumes (millions of shares) for 13 days in January and February are shown here (Barron’s, January 23, 2006; February 13, 2006; and February 27, 2006)....The probability distribution of trading volume is approximately normal.
a. Compute the mean and standard deviation to use as estimates of the population mean and standard deviation.
b. What is the probability that, on a randomly selected day, the early morning trading volume will be less than 180 million shares?
c. What is the probability that, on a randomly selected day, the early morning trading volume will exceed 230 million shares?
d. How many shares would have to be traded for the early morning trading volume on a particular day to be among the busiest 5% of days?
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6.25 According to the Sleep Foundation, the average night’s sleep is 6.8 hours (Fortune, March 20, 2006). Assume the standard deviation is .6 hours and that the probability distribution is normal.
a. What is the probability that a randomly selected person sleeps more than 8 hours?
b. What is the probability that a randomly selected person sleeps 6 hours or less?
c. Doctors suggest getting between 7 and 9 hours of sleep each night. What percentage of the population gets this much sleep?
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6.26 A binomial probability distribution has p = .20 and n = 100.
a. What are the mean and standard deviation?
b. Is this situation one in which binomial probabilities can be approximated by the normal probability distribution? Explain.
c. What is the probability of exactly 24 successes?
d. What is the probability of 18 to 22 successes?
e. What is the probability of 15 or fewer successes?
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6.27 Assume a binomial probability distribution has p = .60 and n = 200.
a. What are the mean and standard deviation?
b. Is this situation one in which binomial probabilities can be approximated by the normal probability distribution? Explain.
c. What is the probability of 100 to 110 successes?
d. What is the probability of 130 or more successes?
e. What is the advantage of using the normal probability distribution to approximate the binomial probabilities? Use part (d) to explain the advantage.
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6.28
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6.29
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6.30 When you sign up for a credit card, do you read the contract carefully? In a FindLaw.com survey, individuals were asked, “How closely do you read a contract for a credit card?” (USA Today, October 16, 2003). The findings were that 44% read every word, 33% read enough to understand the contract, 11% just glance at it, and 4% don’t read it at all.
a. For a sample of 500 people, how many would you expect to say that they read every word of a credit card contract?
b. For a sample of 500 people, what is the probability that 200 or fewer will say they read every word of a credit card contract?
c. For a sample of 500 people, what is the probability that at least 15 say they don’t read credit card contracts?
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6.31 A Myrtle Beach resort hotel has 120 rooms. In the spring months, hotel room occupancy is approximately 75%.
a. What is the probability that at least half of the rooms are occupied on a given day?
b. What is the probability that 100 or more rooms are occupied on a given day?
c. What is the probability that 80 or fewer rooms are occupied on a given day?
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6.32 Consider the following exponential probability density function....
a. Find P(x ≤ 6).
b. Find P(x ≤ 4).
c. Find P(x ≥ 6).
d. Find P(4 ≤ x ≤ 6).
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6.33 Consider the following exponential probability density function....
a. Write the formula for P(x ≤ x0).
b. Find P(x ≤ 2).
c. Find P(x ≥ 3).
d. Find P(x ≤ 5).
e. Find P(2 ≤ x ≤ 5).
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6.34 The time required to pass through security screening at the airport can be annoying to travelers. The mean wait time during peak periods at Cincinnati/Northern Kentucky International Airport is 12.1 minutes (The Cincinnati Enquirer, February 2, 2006). Assume the time to pass through security screening follows an exponential distribution.
a. What is the probability it will take less than 10 minutes to pass through security screening during a peak period?
b. What is the probability it will take more than 20 minutes to pass through security screening during a peak period?
c. What is the probability it will take between 10 and 20 minutes to pass through security screening during a peak period?
d. It is 8:00 A.M. (a peak period) and you just entered the security line. To catch your plane you must be at the gate within 30 minutes. If it takes 12 minutes from the time you clear security until you reach your gate, what is the probability you will miss your flight?
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6.35 The time between arrivals of vehicles at a particular intersection follows an exponential probability distribution with a mean of 12 seconds.
a. Sketch this exponential probability distribution.
b. What is the probability that the arrival time between vehicles is 12 seconds or less?
c. What is the probability that the arrival time between vehicles is 6 seconds or less?
d. What is the probability of 30 or more seconds between vehicle arrivals?
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6.36
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6.37
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6.38 Do interruptions while you are working reduce your productivity? According to a University of California-Irvine study, businesspeople are interrupted at the rate of approximately ... times per hour (Fortune, March 20, 2006). Suppose the number of interruptions follows a Poisson probability distribution.
a. Show the probability distribution for the time between interruptions.
b. What is the probability a businessperson will have no interruptions during a 15-minute period?
c. What is the probability that the next interruption will occur within 10 minutes for a particular businessperson?
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6.39 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 mathematical expression for the probability density function of the sales price?
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?
c. If she leaves it on the market for another month, what is the probability she will get less than $210,000?
d. Should the executive leave the house on the market for another month? Why or why not?
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6.40 The U.S. Bureau of Labor Statistics reports that the average annual expenditure on food and drink for all families is $5700 (Money, December 2003). Assume that annual expenditure on food and drink is normally distributed and that the standard deviation is $1500.
a. What is the range of expenditures of the 10% of families with the lowest annual spending on food and drink?
b. What percentage of families spend more than $7000 annually on food and drink?
c. What is the range of expenditures for the 5% of families with the highest annual spending on food and drink?
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6.41 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.
a. 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.
b. 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.
c. 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?
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6.42 The average annual amount American households spend for daily transportation is $6312 (Money, August 2001). Assume that the amount spent is normally distributed.
a. Suppose you learn that 5% of American households spend less than $1000 for daily transportation. What is the standard deviation of the amount spent?
b. What is the probability that a household spends between $4000 and $6000?
c. What is the range of spending for the 3% of households with the highest daily transportation cost?
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6.43 Condé Nast Traveler publishes a Gold List of the top hotels all over the world. The Broadmoor Hotel in Colorado Springs contains 700 rooms and is on the 2004 Gold List (Condé Nast Traveler, January 2004). Suppose Broadmoor’s marketing group forecasts a mean demand of 670 rooms for the coming weekend. Assume that demand for the upcoming weekend is normally distributed with a standard deviation of 30.
a. What is the probability all the hotel’s rooms will be rented?
b. What is the probability 50 or more rooms will not be rented?
c. Would you recommend the hotel consider offering a promotion to increase demand? What considerations would be important?
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6.44 Ward Doering Auto Sales is considering offering a special service contract that will cover the total cost of any service work required on leased vehicles. From experience, the company manager estimates that yearly service costs are approximately normally distributed, with a mean of $150 and a standard deviation of $25.
a. If the company offers the service contract to customers for a yearly charge of $200, what is the probability that any one customer’s service costs will exceed the contract price of $200?
b. What is Ward’s expected profit per service contract?
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6.45 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.
a. What is the probability of fewer than 1000 fatal crashes in a year?
b. What is the probability the number of fatal crashes will be between 1000 and 2000 for a year?
c. 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?
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6.46 Assume that the test scores from a college admissions test are normally distributed, with a mean of 450 and a standard deviation of 100.
a. What percentage of the people taking the test score between 400 and 500?
b. Suppose someone receives a score of 630. What percentage of the people taking the test score better? What percentage score worse?
c. If a particular university will not admit anyone scoring below 480, what percentage of the persons taking the test would be acceptable to the university?
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6.47
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6.48 A machine fills containers with a particular product. The standard deviation of filling weights is known from past data to be .6 ounce. If only 2% of the containers hold less than 18 ounces, what is the mean filling weight for the machine? That is, what must μ equal? Assume the filling weights have a normal distribution.
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6.49 Consider a multiple-choice examination with 50 questions. Each question has four possible answers. Assume that a student who has done the homework and attended lectures has a 75% probability of answering any question correctly.
a. A student must answer 43 or more questions correctly to obtain a grade of A. What percentage of the students who have done their homework and attended lectures will obtain a grade of A on this multiple-choice examination?
b. A student who answers 35 to 39 questions correctly will receive a grade of C. What percentage of students who have done their homework and attended lectures will obtain a grade of C on this multiple-choice examination?
c. A student must answer 30 or more questions correctly to pass the examination. What percentage of the students who have done their homework and attended lectures will pass the examination?
d. Assume that a student has not attended class and has not done the homework for the course. Furthermore, assume that the student will simply guess at the answer to each question. What is the probability that this student will answer 30 or more questions correctly and pass the examination?
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6.50 A blackjack player at a Las Vegas casino learned that the house will provide a free room if play is for four hours at an average bet of $50. The player’s strategy provides a probability of .49 of winning on any one hand, and the player knows that there are 60 hands per hour. Suppose the player plays for four hours at a bet of $50 per hand.
a. What is the player’s expected payoff?
b. What is the probability the player loses $1000 or more?
c. What is the probability the player wins?
d. Suppose the player starts with $1500. What is the probability of going broke?
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6.51 The time in minutes for which a student uses a computer terminal at the computer center of a major university follows an exponential probability distribution with a mean of 36 minutes. Assume a student arrives at the terminal just as another student is beginning to work on the terminal.
a. What is the probability that the wait for the second student will be 15 minutes or less?
b. What is the probability that the wait for the second student will be between 15 and 45 minutes?
c. What is the probability that the second student will have to wait an hour or more?
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6.52 The website for the Bed and Breakfast Inns of North America gets approximately seven visitors per minute (Time, September 2001). Suppose the number of website visitors per minute follows a Poisson probability distribution.
a. What is the mean time between visits to the website?
b. Show the exponential probability density function for the time between website visits.
c. What is the probability no one will access the website in a 1-minute period?
d. What is the probability no one will access the website in a 12-second period?
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6.53
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6.54 The time (in minutes) between telephone calls at an insurance claims office has the following exponential probability distribution....
a. What is the mean time between telephone calls?
b. What is the probability of having 30 seconds or less between telephone calls?
c. What is the probability of having 1 minute or less between telephone calls?
d. What is the probability of having 5 or more minutes without a telephone call?
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