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Questions and Answers of
Statistics
Three grades of jet fuel are blended to be used on long commercial flights. Each grade cost about the same and has about the same ignition or burn rate (gal/min) in a jet engine. The question is how
This problem will be referred to in the study of control charts (Section 6.1). In the binomial probability distribution, let the number of trials be n = 3, and let the probability of success be p =
In the western United States, there are many dry-land wheat farms that depend on winter snow and spring rain to produce good crops. About 65% of the years, there is enough moisture to produce a good
Only about 70% of all donated human blood can be used in hospitals. The remaining 30% cannot be used because of various infections in the blood. Suppose a blood bank has 10 newly donated pints of
Consider a multinomial experiment. This means the following:1. The trials are independent and repeated under identical conditions.2. The outcomes of each trial falls into exactly one of k ≥ 2
Aircraft inspectors (who specialize in mechanical engineering) report wing cracks in aircraft as nonexistent, detectable (but still functional), or critical (needs immediate repair). For a particular
Consider a binomial distribution with n = 10 trials and the probability of success on a single trial p = 0.05.(a) Is the distribution skewed left, skewed right, or symmetric?(b) Compute the expected
In Hawaii, January is a favorite month for surfing since 60% of the days have a surf of at least 6 feet (Reference: Hawaii Data Book, Robert C. Schmitt). You work day shifts in a Honolulu hospital
Old Friends Information Service is a California company that is in the business of finding addresses of long-lost friends. Old Friends claims to have a 70% success rate (Source: The Wall Street
A tidal wave or tsunami is usually the result of an earthquake in the Pacific Rim, often 1000 or more miles from Hawaii. Tsunamis are rare but dangerous. Many tsunamis are small and do little
USA Today reported that about 20% of all people in the United States are illiterate. Suppose you interview seven people at random off a city street.(a) Make a histogram showing the probability
The Wall Street Journal reported that approximately 25% of the people who are told a product is improved will believe that it is, in fact, improved. The remaining 75% believe that this is just hype
USA Today reports that about 25% of all prison parolees become repeat offenders. Alice is a social worker whose job is to counsel people on parole. Let us say success means a person does not become a
The probability that a single radar station will detect an enemy plane is 0.65.(a) Quota Problem How many such stations are required for 98% certainty that an enemy plane flying over will be detected
Have you ever tried to get out of jury duty? About 25% of those called will find an excuse (work, poor health, travel out of town, etc.) to avoid jury duty.(a) If 12 people are called for jury
The Denver Post reported that a recent audit of Los Angeles 911 calls showed that 85% were not emergencies. Suppose the 911operators in Los Angeles have just received four calls.(a) What is the
The FBI Standard Survey of Crimes shows that for about 80% of all property crimes (burglary, larceny, car theft, etc.), the criminals are never found and the case is never solved (Source: True Odds,
A large bank vault has several automatic burglar alarms. The probability is 0.55 that a single alarm will detect a burglar.(a) How many such alarms should be used for 99% certainty that a burglar
In Japanese criminal trials, about 95% of the defendants are found guilty. In the United States, about 60% of the defendants are found guilty in criminal trials (Source: The Book of Risks, by Larry
We now have the tools to solve the chapter Focus Problem. In the book A Guide to the Development and Use of the Myers–Briggs Type Indicators by Myers and McCaully, it was reported that
The owners of a motel in Florida have noticed that in the long run, about 40% of the people who stop and inquire about a room for the night actually rent a room.(a) Quota Problem How many inquiries
Let r be a binomial random variable representing the number of successes out of n trials.(a) Explain why the sample space for r consists of the set {0, 1, 2, . . . , n} and why the sum of the
Consider a binomial distribution with n = 10 trials and the probability of success on a single trial p = 0.85.(a) Is the distribution skewed left, skewed right, or symmetric?(b) Compute the expected
Bob is a recent law school graduate who intends to take the state bar exam. According to the National Conference on Bar Examiners, about 57% of all people who take the state bar exam pass (Source:
On the leeward side of the island of Oahu, in the small village of Nanakuli, about 80% of the residents are of Hawaiian ancestry (Source: The Honolulu Advertiser). Let n = 1, 2, 3, . . represent the
Approximately 3.6% of all (untreated) Jonathan apples had bitter pit in a study conducted by the botanists Ratkowsky and Martin (Source: Australian Journal of Agricultural Research, Vol. 25, pp.
At Fontaine Lake Camp on Lake Athabasca in northern Canada, history shows that about 30% of the guests catch lake trout over 20 pounds on a 4-day fishing trip (Source: Athabasca Fishing Lodges,
At Burnt Mesa Pueblo, in one of the archaeological excavation sites, the artifact density (number of prehistoric artifacts per 10 liters of sediment) was 1.5 (Source: Bandelier Archaeological
In his doctoral thesis, L. A. Beckel (University of Minnesota, 1982) studied the social behavior of river otters during the mating season. An important role in the bonding process of river otters is
The Denver Post reported that, on average, a large shopping center has had an incident of shoplifting caught by security once every 3 hours. The shopping center is open from 10A.M. to 9 P.M. (11
USA Today reported that the U.S. (annual) birthrate is about 16 per 1000 people, and the death rate is about 8 per 1000 people.(a) Explain why the Poisson probability distribution would be a good
Henry Petroski is a professor of civil engineering at Duke University. In his book To Engineer Is Human: The Role of Failure in Successful Design, Professor Petroski says that up to 95% of all
Much of Trail Ridge Road in Rocky Mountain National Park is over 12,000 feet high. Although it is a beautiful drive in summer months, in winter the road is closed because of severe weather
USA Today reported that Parkeld, California, is dubbed the world’s earthquake capital because it sits on top of the notorious San Andreas fault. Since 1857, Parkeld has had a major earthquake
Jim is a real estate agent who sells large commercial buildings. Because his commission is so large on a single sale, he does not need to sell many buildings to make a good living. History shows that
The Honolulu Advertiser stated that in Honolulu there was an average of 661 burglaries per 100,000 households in a given year. In the Kohola Drive neighborhood there are 316 homes. Let r = number of
Harper's Index reported that the number of Orange County, California convicted drunk drivers whose sentence included a tour of the morgue was 569, of which only 1 became a repeat offender.(a) Suppose
USA Today reported that for all airlines, the number of lost bags wasMay: 6.02 per 1000 passengers December: 12.78 per 1000 passengersNote: A passenger could lose more than one bag.(a) Let r = number
Risk and Odds in Everyday Life, by James Burke, reports that the probability a police officer will be killed in the line of duty is 0.5% (or less).(a) In a police precinct with 175 officers, let
Risk and Odds in Everyday Life, by James Burke, reports that only 2% of all local franchises are business failures. A Colorado Springs shopping complex has 137 franchises (restaurants, print shops,
(a) For n = 100, p = 0.02, and r = 2, compute P(r) using the formula for the binomial distribution and your calculator:P (r) = Cn, rpr(1 – p)n–r(b) For n = 100, p = 0.02, and r = 2, estimate
Pyramid Lake is located in Nevada on the Paiute Indian Reservation. This lake is famous for large cutthroat trout. The mean number of trout (large and small) caught from a boat is 0.667 fish per
In western Kansas, the summer density of hailstorms is estimated at about 2.1 storms per 5 square miles. In most cases, a hailstorm damages only a relatively small area in a square mile (Reference:
Suppose you have binomial trials for which the probability of success on each trial is p and the probability of failure is q = 1 – p. Let k be a fixed whole number greater than or equal to 1. Let
Susan is a sales representative who has a history of making a successful sale from about 80% of her sales contacts. If she makes 12 successful sales this week, Susan will get a bonus. Let n be a
Blood type A occurs in about 41% of the population (Reference: Laboratory and Diagnostic Tests by F. Fischbach). A clinic needs 3 pints of type A blood. A donor usually gives a pint of blood. Let n
If you enjoy a little abstract thinking, you may want to derive the formula for the negative binomial probability distribution. Use the notation of Problem 30. Consider two events, A and B.A = {event
Given a binomial experiment with n = 150 trials and probability of success on a single trial p = 0.06, find the value of l and then use the Poisson distribution to estimate the probability of r ( 2
Susan is taking Western Civilization this semester on a pass/fail basis. The department teaching the course has a history of passing 77% of the students in Western Civilization each term. Let n = 1,
In Hawaii shark attacks are very rare. Furthermore, about 55% of all shark attacks are serious, but not fatal (Reference: Sharks of Hawaii, Dr. Leighton Taylor, University of Hawaii Press). You are
Suppose you are a (junk) bond broker who buys only bonds that have a 50% chance of default. You want a portfolio with at least five bonds that do not default. You can dispose of the other bonds in
A person with a cough is a persona non grata on airplanes, elevators, or at the theater. In theaters especially, the irritation level rises with each muffled explosion. According to Dr. Brian Carlin,
Flying over the western states with mountainous terrain in a small aircraft is 40% riskier than flying over similar distances in flatter portions of the nation, according to a General Accounting
Records over the past year show that 1 out of 350 loans made by Mammon Bank have defaulted. Find the probability that 2 or more out of 300 loans will default. Is it appropriate to use the Poisson
In Hawaii, the rate of motor vehicle theft is 551 thefts per 100,000 vehicles (Reference: U.S. Department of Justice, Federal Bureau of Investigation). A large parking structure in Honolulu has
An experiment consists of tossing a coin a specified number of times and recording the outcomes.(a) What is the probability that the first head will occur on the second trial? Does this probability
Cathy is planning to take the Certified Public Accountant Examination (CPA exam). Records kept by the college of business from which she graduated indicate that 83% of the students who graduated pass
Yellowstone Park Medical Services (YPMS) provides emergency health care for park visitors. Such health care includes treatment for everything from indigestion and sunburn to more serious injuries. A
Tri-County Bank is a small independent bank in central Wyoming. This is a rural bank that makes loans on items as small as horses and pickup trucks to items as large as ranch land. Total monthly loan
The manager of Motel 11 has 316 rooms in Palo Alto, California. From observation over a long period of time, she knows that on an average night, 268 rooms will be rented. The long-term standard
The visibility standard index (VSI) is a measure of Denver air pollution that is reported each day in the Denver Post. The index ranges from 0 (excellent air quality) to 200 (very bad air quality).
The Poisson distribution (Section 5.4) gives the probability for the number of occurrences for a “rare” event. Now, let x be a random variable that represents the waiting time between rare
Another application for exponential distributions is supply/demand problems. The operator of a pumping station in a small Wyoming town has observed that demand for water on a typical summer afternoon
Consider a binomial setting in which "neutral" is defined to be a success. So, p = P(success) = P(10 ( x < 20). Suppose n = 65 geomagnetic readings are taken. Let r be a binomial random variable
Please give a careful but brief answer to each of the following questions.(a) What is a population? How do you get a simple random sample? Give examples.(b) What is a sample statistic? What is a
Workers at a large toxic cleanup project are concerned that their white blood cell counts may have been reduced. Let x be a random variable that represents white blood cell count per cubic millimeter
Write a brief but complete essay in which you describe the probability distributions you have studied so far. Which apply to discrete random variables? Which apply to continuous random variables?
Suppose a "hot spot" is a site with a reading of 40 or higher.(a) In a binomial setting, let us call success a "hot spot." Use Table A to find p = P(success) 5 P(40 ( x) for a single trial.(b)
Suppose an archaeologist is looking for geomagnetic "hot spots" in an unexplored region of Tara. As in Problem 8, we have a binomial setting where success is a "hot spot." In this case, the
Attendance at large exhibition shows in Denver averages about 8000 people per day, with standard deviation of about 500. Assume that the daily attendance figures follow a normal distribution.(a) What
Most exhibition shows open in the morning and close in the late evening. A study of Saturday arrival times showed that the average arrival time was 3 hours and 48 minutes after the doors opened, and
The amount of money spent weekly on cleaning, maintenance, and repairs at a large restaurant was observed over a long period of time to be approximately normally distributed, with mean m = $615 and
Suppose you want to eat lunch at a popular restaurant. The restaurant does not take reservations, so there is usually a waiting time before you can be seated. Let x represent the length of time
Suppose an x distribution has mean ( = 5. Consider two corresponding x distributions, the first based on samples of size n = 49 and the second based on samples of size n = 81.(a) What is the value of
Let x be a random variable that represents checkout time (time spent in the actual checkout process) in minutes in the express lane of a large grocery. Based on a consumer survey, the mean of the x
The taxi and takeoff time for commercial jets is a random variable x with a mean of 8.5 minutes and a standard deviation of 2.5 minutes. Assume that the distribution of taxi and takeoff times is
The highest dunes at Great Sand Dunes National Monument can exceed the highest dunes in the Great Sahara, extending over 700 feet in height. However, like all sand dunes, they tend to move around in
Suppose x has a distribution with m = 72 and s – 8.(a) If random samples of size n 5 16 are selected, can we say anything about the x distribution of sample means?(b) If the original x distribution
We have studied two approximations to the binomial, the normal approximation and the Poisson approximation (See Section 5.4). Write a brief but complete essay in which you discuss and summarize the
Many people consider their smart phone to be essential! Communication, news, Internet, entertainment, photos, and just keeping current are all conveniently possible with a smart phone. However, the
Hydraulic pressure in the main cylinder of the landing gear of a commercial jet is very important for a safe landing. If the pressure is not high enough, the landing gear may not lower properly. If
The personnel office at a large electronics firm regularly schedules job interviews and maintains records of the interviews. From the past records, they have found that the length of a first
In a combined study of northern pike, cutthroat trout, rainbow trout, and lake trout, it was found that 26 out of 855 sh died when caught and released using barbless hooks on ‑ies or lures. All
A random sample of 328 medical doctors showed that 171 have a solo practice (Source: Practice Patterns of General Internal Medicine, American Medical Association).(a) Let p represent the proportion
Inorganic phosphorous is a naturally occurring element in all plants and animals, with concentrations increasing progressively up the food chain (fruit < vegetables < cereals < nuts < corpse).
Please see the setting and reference in Problem 11. Independent random samples from two regions (not those cited in Problem 11) gave the following phosphorous measurements (in ppm). Assume the
For large U.S. companies, what percentage of their total income comes from foreign sales? A random sample of technology companies (IBM, Hewlett-Packard, Intel, and others) gave the following
Independent random samples of professional football and basketball players gave the following information (References: Sports Encyclopedia of Pro Football and Official NBA Basketball Encyclopedia).
Independent random samples of professional football and basketball players gave the following information (References: Sports Encyclopedia of Pro Football and Official NBA Basketball Encyclopedia).
The following data represent petal lengths (in cm) for independent random samples of two species of iris (Reference: E. Anderson, Bulletin American Iris Society). Note: These data are also available
The U.S. Geological Survey compiled historical data about Old Faithful Geyser (Yellowstone National Park) from 1870 to 1987. Some of these data are published in the book The Story of Old Faithful, by
“Parental Sensitivity to Infant Cues: Similarities and Differences Between Mothers and Fathers” by M. V. Graham (Journal of Pediatric Nursing, Vol. 8, No. 6) reports a study of parental empathy
S. C. Jett is a professor of geography at the University of California, Davis. He and a colleague, V. E. Spencer, are experts on modern Navajo culture and geography. The following information is
“Unknown cultural affiliations and loss of identity at high elevations.” These words are used to propose the hypothesis that archaeological sites tend to lose their identity as altitude
David E. Brown is an expert in wildlife conservation. In his book The Wolf in the Southwest: The Making of an Endangered Species (University of Arizona Press), he lists the following weights of adult
At Community Hospital, the burn center is experimenting with a new plasma compress treatment. A random sample of n1 = 316 patients with minor burns received the plasma compress treatment. Of these
Female undergraduates in randomized groups of 15 took part in a self-esteem study (“There’s More to Self-Esteem than Whether It Is High or Low: The Importance of Stability of Self-Esteem,” by
What about sample size? If we want a confidence interval with maximal margin of error E and level of confidence c, then Section 7.1 shows us which formulas to apply for a single mean m and
What about the sample size n for confidence intervals for the difference of proportions p1 – p2? Let us make the following assumptions: equal sample sizes n = n1 = n2 and all four quantities
Given x1 and x2 distributions that are normal or approximately normal with unknown σ1 and σ2, the value of t corresponding to x̅1 – x̅2 has a distribution that is approximated by a
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