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).
Get solution
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 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.
b. What is the probability that the battery life for an iPad Mini will be 10 hours or less?
c. What is the probability that the battery life for an iPad Mini will be at least 11 hours?
d. What is the probability that the battery life for an iPad Mini will be between 9.5 and 11.5 hours?
e. In a shipment of 100 iPad Minis, how many should have a battery life of at least 9 hours?
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6.6 A Gallup Daily Tracking Survey found that the mean daily discretionary spending by Americans earning over $90,000 per year was $136 per day (USA Today, July 30, 2012). The discretionary spending excluded home purchases, vehicle purchases, and regular monthly bills. Let x = the discretionary spending per day and assume that a uniform probability density function applies with f (x) = .00625 for a ≤ x ≤ b.
a. Find the values of a and b for the probability density function.
b. What is the probability that consumers in this group have daily discretionary spending between $100 and $200?
c. What is the probability that consumers in this group have daily discretionary spending of $150 or more?
d. What is the probability that consumers in this group have daily discretionary spending of $80 or less?
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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 that the random variable will assume a value between 45 and 55?
c. What is the probability that 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 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 isa 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 isa 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 isa 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 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?
b. What is the probability that a domestic airfare is $250 or less?
c. What if the probability that a domestic airfare is between $300 and $500?
d. What is the cost for the 3% highest domestic airfares?
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6.18 The average return for large-cap domestic stock funds over the three years 2009–2011 was 14.4% (AAII Journal, February, 2012). Assume the three-year returns were normally distributed across funds with a standard deviation of 4.4%.
a. What is the probability an individual large-cap domestic stock fund had a three-year return of at least 20%?
b. What is the probability an individual large-cap domestic stock fund had a three-year return of 10% or less?
c. How big does the return have to be to put a domestic stock fund in the top 10% for the three-year period?
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6.19
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6.20 The average price for a gallon of gasoline in the United States is $3.73 and in Russia it is $3.40 (Bloomberg Businessweek, March 5–March 11, 2012). Assume these averages are the population means in the two countries and that the probability distributions are normally distributed with a standard deviation of $.25 in the United States and a standard deviation of $.20 in Russia.
a. What is the probability that a randomly selected gas station in the United States charges less than $3.50 per gallon?
b. What percentage of the gas stations in Russia charge less than $3.50 per gallon?
c. What is the probability that a randomly selected gas station in Russia charged more than the mean price in the United States?
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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. 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?
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6.22 Television viewing reached a new high when the Nielsen Company reported a mean daily viewing time of 8.35 hours per household (USA Today, November 11, 2009). Use a normal probability distribution with a standard deviation of 2.5 hours to answer the following questions about daily television viewing per household.
a. What is the probability that a household views television between 5 and 10 hours a day?
b. How many hours of television viewing must a household have in order to be in the top 3% of all television viewing households?
c. What is the probability that a household views television more than 3 hours a day?
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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 The American Automobile Association (AAA) reported that families planning to travel over the Labor Day weekend would spend an average of $749 (The Associated Press, August 12, 2012). Assume that the amount spent is normally distributed with a standard deviation of $225.
a. What the probability of family expenses for the weekend being less that $400?
b. What is the probability of family expenses for the weekend being $800 or more?
c. What is the probability that family expenses for the weekend will be between $500 and $1000?
d. What would the Labor Day weekend expenses have to be for the 5% of the families with the most expensive travel plans?
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6.25 New York City is the most expensive city in the United States for lodging. The mean hotel room rate is $204 per night (USA Today, April 30, 2012). Assume that room rates are normally distributed with a standard deviation of $55.
a. What is the probability that a hotel room costs $225 or more per night?
b. What is the probability that a hotel room costs less than $140 per night?
c. What is the probability that a hotel room costs between $200 and $300 per night?
d. What is the cost of the 20% most expensive hotel rooms in New York City?
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6.26 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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6.27 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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6.28 Battery life between charges for the Motorola Droid Razr Maxx is 20 hours when the primary use is talk time (The Wall Street Journal, March 7, 2012). The battery life drops to 7 hours when the phone is primarily used for Internet applications over cellular. Assume that the battery life in both cases follows an exponential distribution.
a. Show the probability density function for battery life for the Droid Razr Maxx phone when its primary use is talk time.
b. What is the probability that the battery charge for a randomly selected Droid Razr Maxx phone will last no more than 15 hours when its primary use is talk time?
c. What is the probability that the battery charge for a randomly selected Droid Razr Maxx phone will last more than 20 hours when its primary use is talk time?
d. What is the probability that the battery charge for a randomly selected Droid Razr Maxx phone will last no more than 5 hours when its primary use is Internet applications?
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6.29 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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6.30 Comcast Corporation is the largest cable television company, the second largest Internet service provider, and the fourth largest telephone service provider in the United States. Generally known for quality and reliable service, the company periodically experiences unexpected service interruptions. On January 14, 2014, such an interruption occurred for the Comcast customers living in southwest Florida. When customers called the Comcast office, a recorded message told them that the company was aware of the service outage and that it was anticipated that service would be restored in two hours. Assume that two hours is the mean time to do the repair and that the repair time has an exponential probability distribution.
a. What is the probability that the cable service will be repaired in one hour or less?
b. What is the probability that the repair will take between one hour and two hours?
c. For a customer who calls the Comcast office at 1:00 P.M., what is the probability that the cable service will not be repaired by 5:00 P.M.?
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6.31
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6.32 The Boston Fire Department receives 911 calls at a mean rate of 1.6 calls per hour (Mass.gov website, November 2012). Suppose the number of calls per hour follows a Poisson probability distribution.
a. What is the mean time between 911 calls to the Boston Fire Department in minutes?
b. Using the mean in part (a), show the probability density function for the time between 911 calls in minutes.
c. What is the probability that there will be less than one hour between 911 calls?
d. What is the probability that there will be 30 minutes or more between 911 calls?
e. What is the probability that there will be more than 5 minutes, but less than 20 minutes between 911 calls?
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6.33 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 that 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 that 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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6.34 The NCAA estimates that the yearly value of a full athletic scholarship at in-state public universities is $19,000 (The Wall Street Journal, March 12, 2012). Assume the scholarship value is normally distributed with a standard deviation of $2100.
a. For the 10% of athletic scholarships of least value, how much are they worth?
b. What percentage of athletic scholarships are valued at $22,000 or more?
c. For the 3% of athletic scholarships that are most valuable, how much are they worth?
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6.35 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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6.36 During early 2012, economic hardship was stretching the limits of France’s welfare system. One indicator of the level of hardship was the increase in the number of people bringing items to a Paris pawnbroker; the number of people bringing items to the pawnbroker had increased to 658 per day (Bloomberg Businessweek, March 5–March 11, 2012). Assume the number of people bringing items to the pawnshop per day in 2012 is normally distributed with a mean of 658.
a. Suppose you learn that on 3% of the days, 610 or fewer people brought items to the pawnshop. What is the standard deviation of the number of people bringing items to the pawnshop per day?
b. On any given day, what is the probability that between 600 and 700 people bring items to the pawnshop?
c. How many people bring items to the pawnshop on the busiest 3% of days?
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6.37 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?
b. What is the probability that the port handles 3 or more million tons of cargo per week?
c. What is the probability that the port handles between 3 million and 4 million tons of cargo per week?
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?
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6.38 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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6.39
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6.40 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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6.41
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6.42 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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6.43
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6.44 A website for bed and breakfast inns gets approximately seven visitors per minute. 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 that no one will access the website in a 1-minute period?
d. What is the probability that no one will access the website in a 12-second period?
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6.45
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6.46 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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