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mathematics
statistics

Questions and Answers of Statistics

Refer to the Journal of Education and Human Development (Vol. 3, 2009) investigation of plagiarism among English-as-a-second language (ESL) students, Exercise. Recall that of six ESL students taking
Refer to the Journal of Behavioral Decision Making (Jan., 2007) study of how guilty feelings impact decisions, Exercise. Recall that 57 students were assigned to a guilty state through a
Imagine that you are purchasing small lots of a manufactured product. If it is very costly to test a single item, it may be desirable to test a sample of items from the lot instead of testing every
Refer to the Journal of Engineering, Computing and Architecture (Vol. 3., 2009) study of cell phone handoff behavior, Exercise 3.58 (p. 134). Recall that a “handoff” describes the process of a
The Journal of Business & Economic Statistics (July 2000) presented a case in which a charge of gender discrimination was filed against the U.S. Postal Service. At the time, there were 302 U.S.
Which of the following describe discrete random variables, and which describe continuous random variables? a. The length of time that an exercise physiologist’s pro-gram takes to elevate her
For each of the following examples, decide whether x is a binomial random variable and explain your decision: a. A manufacturer of computer chips randomly selects 100 chips from each hour’s
Given that x is a hypergeometric random variable, compute p(x) for each of the following cases: a. N = 8, n = 5, r = 3, x = 2 b. N = 6, n = 2, r = 2, x = 2 c. N = 5, n = 4, r = 4, x = 3
Refer to Exercise 5.3. Find the following probabilities:a. P(10 ≤ x ≤ 25)b. P(20 < x < 30)c. P(x ≥ 25)d. P(x ≤ 10)e. P(x ≤ 25)f. P(20.5 ≤ x ≤ 25.5)
Refer to Exercise 5.5. Find the value of a that makes each of the following probability statements true:a. P(x ≥ a) = .5b. P(x ≤ a) = .2c. P(x ≤ a) = 0d. P(2.5 ≤ x ≤ a) = .5
The random variable x is best described by a uniform probability distribution with mean 50 and standard deviation 5. Find c, d, and f(x). Graph the probability distribution.
According to Brighton Webs LTD, a British company that specializes in data analysis, the arrival time of requests to a Web server within each hour can be modeled by a uniform distribution
The data set listed here was created in the MINITAB random-number generator. Construct a relative frequency histogram for the data (saved in the RANUNI file). Except for the expected variation in
Social networks involve interactions (connections) between members of the network. Sociologists define network density as the ratio of actual network connections to the number of possible one-to-one
A bus is scheduled to stop at a certain bus stop every half hour on the hour and the half hour. At the end of the day, buses still stop after every 30 minutes, but because delays often occur earlier
The reliability of a piece of equipment is frequently defined to be the probability p that the equipment performs its intended function success fully for a given period under specific conditions
Find the area under the standard normal distribution between the following pairs of z -scores: a. z = –2.00 and z = 0 b. z = –1.00 and z = 0 c. z = –1.69 and z = 0 d. z = –.58 and z = 0
Find the following probabilities for the standard normal random variable z: a. P(z > 1.46) b. P(z < – 1.56) c. P(.67 ≤ z ≤ 2.41) d. P(–1.96 ≤ z 6 < – .33) e. P(z ≥ 0) f. P(–2.33
Find a value z0 of the standard normal random variable z such thata. P(z ≤ z0) = .0401b. P(−z0 ≤ z ≤ z0) = .95c. P(−z0 ≤ z ≤ z0) = .90d. P(−z0 ≤ z ≤ z0) = .8740e. P(−z0 ≤ z
Suppose the random variable x is best described by a normal distribution with μ = 25 and σ = 5. Find the z -score that corresponds to each of the following x values: a. x = 25 b. x = 30 c. x =
Suppose x is a normally distributed random variable with μ = 30 and σ = 8. Find a value x0 of the random variable x such that a. P(x ≥ x0) = .5 b. P(x < x0) = .025 c. P(x > x0) = .10 d. P(x >
Suppose x is a normally distributed random variable with mean 100 and standard deviation 8. Draw a rough graph of the distribution of x. Locate m and the interval μ ± 2σ on the graph. Find the
The random variable x has a normal distribution with μ = 300 and σ = 30. a. Find the probability that x assumes a value more than two standard deviations from its mean. More than three standard
Miraculin—a protein naturally produced in a rare tropical fruit—can convert a sour taste into a sweet taste. Consequently, miraculin has the potential to be an alternative low-calorie sweetener.
Casino gaming yields over $35 billion in revenue each year in the United States. In Chance (Spring 2005), University of Denver statistician R. C. Hannum discussed the business of casino gaming and
Refer to the Aquatic Biology (Vol. 9, 2010) study of green sea turtles inhabiting the Grand Cayman South Sound lagoon, Exercise. Researchers discovered that the curved carapace (shell) length of
When attempting to score a goal in soccer, where should you aim your shot? Should you aim for a goalpost (as some soccer coaches teach), the middle of the goal, or some other target? To answer these
Almost all companies utilize some type of year-end performance review for their employees. Human Resources (HR) at the University of Texas Health Science Center provides guidelines for supervisors
A group of Florida State University psychologists examined the effects of alcohol on the reactions of people to a threat (Journal of Abnormal Psychology, Vol. 107, 1998). After obtaining a specified
The characteristics of an industrial filling process in which an expensive liquid is injected into a container were investigated in the Journal of Quality Technology (July 1999). The quantity
In the Journal of the International Association for Shell and Spatial Structures (April 2004), Japanese environmental researchers studied the performance of truss-and-frame structures subjected to
Give four methods for determining whether the sample data come from a normal population.
What is a normal probability plot and how is it used?
Consider a sample data set with the following summary statistics: s = 95, QL = 72 and QU = 195. a. Calculate IQR. b. Calculate IQR/s. c. Is the value of IQR/s approximately equal to 1.3? What does
Examine the following sample data saved in the LM5_59 file.a. Construct a stem-and-leaf plot to assess whether the data are from an approximately normal distribution. b. Find the values of QL, QU,
Examine the following sample data saved in the LM5_60 file.a. Construct a stem-and-leaf plot to assess whether the data are from an approximately normal distribution.b. Compute s for the sample
Understanding the characteristics of rock masses, especially the nature of the fractures, are essential when building dams and power plants. The shear strength of rock fractures was investigated in
Scientists at GlaxoSmithKline Medicines Research Center used high-performance liquid chromatography (HPLC) to determine the amount of drug in a tablet produced by the company (Analytical Chemistry,
An evaluation of the habitats of endangered salmon species was performed in Conservation Ecology (December 2003). The researchers identified 734 sites (habitats) for Chinook, coho, or steelhead
Refer to The Sport Journal (Winter 2007) article on a new method for ranking the driving performance of PGA golfers, presented in Exercise.In exerciseRecall that the method incorporates a golfer's
Refer to the data on the Jan. 2010 sanitation scores for 186 cruise ships, presented in Exercise. In exerciseShip Name Sanitation ScoreAdventure of the Seas ......... 98AID Aaura ............
Refer to the Southern Economic Journal (Apr. 2008) rankings of Ph.D. programs in economics at 129 colleges and universities, Exercise. Recall that the number of publications published by faculty
Why do we need a correction for continuity when approximating a binomial probability with the normal distribution?
Assume that x is a binomial random variable with n and p as specified in parts a–f that follows. For which cases would it be appropriate to use a normal distribution to approximate the binomial
Assume that x is a binomial random variable with n = 1,000 and p = .50. Find each of the following probabilities: a. P(x > 500) b. P(490 ≤ x < 500) c. P(x > 550)
Refer to Exercise and the Associated Press/Petside.com poll that revealed that half of all pet owners would get their next dog or cat from a shelter (USA Today, May 12, 2010). Consider a random
According to recent studies, 1% of all patients who undergo laser surgery (i.e., LASIK) to correct their vision have serious post-laser vision problems (All About Vision, 2006). In a random sample of
In Exercise, you learned that 32% of all births in the United States occur by Caesarian section each year (National Vital Statistics Reports, Mar. 2010). In a random sample of 1,000 births this year,
Refer to the Cell (May 14, 2010) study of the ability of a mouse to recognize the odor of a potential predator, Exercise. You learned that 40% of lab mice cells exposed to chemically produced major
In Exercise, you learned that some researchers believe that one in every three women has been a victim of domestic abuse (Domestic Vidence: Incidence and Prevalance study, Sep–Dec., 2005). a. For
The percentage of fat in the bodies of American men is an approximately normal random variable with mean equal to 15% and standard deviation equal to 2%. a. If these values were used to describe the
According to Health Affairs Oct. 28, 2004, the median time a patient waits to see a doctor in a typical U.S. emergency room is 30 minutes. On a day when 150 patients visit the emergency room, what is
Use Table V in Appendix A to determine the value of e -a/u or each of the following cases: a. θ = 1, α = 1 b. θ = 1, α = 2.5 c. θ = .4, α = 3 d. θ = .2, α = .3
Suppose x has an exponential distribution with θ = 2.5. Find the following probabilities: a. P(x ≤ 4) b. P(x > 5) c. P(x ≤ 2) d. P(x > 3)
The random variable x can be adequately approximated by an exponential probability distribution with θ = 2. Find the probability that x assumes a value a. More than three standard deviations from
In NASCAR races such as the Daytona 500, 43 drivers start the race; however, about 10% of the cars do not finish due to the failure of critical parts. University of Portland professors conducted a
The optimal scheduling of preventative maintenance tests of some (but not all) of n independently operating components was developed in Reliability Engineering and System Safety (Jan. 2006). The time
Refer to the Conservation Ecology (Dec. 2003) study on the causes of fragmentation in 54 South American forests, presented in Exercise. Recall that the cause is classified as either anthropogenic or
In Reliability Ques (March 2004), the exponential distribution was used to model the lengths of life of CD-ROM drives in a two-drive system. The two CD-ROM drives operate independently, and at least
An essential tool in the monitoring of the quality of a manufactured product is acceptance sampling. An acceptance sampling plan involves knowing the distribution of the life length of the item
Consider the continuous random variables that follow. Give the probability distribution (uniform, normal, or exponential) that is likely to best approximate the distribution of the random variable:
Assume that x is a random variable best described by a uniform distribution with c=40 and d=70. a. Find f(x). b. Find the mean and standard deviation of x. c. Graph the probability distribution
Find a z -score, say, z0, such that a. P(z ≤ z0) = .8708 b. P(z ≥ z0) = .0526 c. P(z ≤ z0) = .5 d. P(-z0 ≤ z ≤ z0) = .8164 e. P(z ≥ z0) = .8023 f. P(z ≥ z0) = .0041
The random variable x has a normal distribution with μ = 40 and σ2 = 36. Find a value of x, say, x0, such that a. P(x ≥ x0) = .5 b. P(x ≤ x0) = .9911 c. P(x ≤ x0) = .0028 d. P(x ≥ x0) =
Assume that x has an exponential distribution with θ = 3. Find a. P(x ≤ 1) b. P(x > 1) c. P(x = 1) d. P(x ≤ 6) e. P(2 ≤ x ≤ 10)
Researchers at the University of California-Berkeley have designed, built, and tested a switched-capacitor circuit for generating random signals (International Journal of Circuit Theory and
Refer to the Chance (Winter 2001) study of students who paid a private tutor to help them improve their SAT scores, presented in Exercise. The table summarizing the changes in both the
Refer to the U.S. Army Corps of Engineers data on contaminated fish in the Tennessee River, presented in Example. Recall that one of the variables measured for each captured fish is weight (in
In Exercise, you learned that 50% of pet owners will get their next dog or cat from a Shelter (USA Today, May 12, 2010). In a random sample of 200 pet owners, let x be the number who will get their
The length of time between arrivals at a hospital clinic has an approximately exponential probability distribution. Suppose the mean time between arrivals for patients at a clinic is 4 minutes. a.
The frequency distribution shown in the accompanying table depicts the property and marine losses incurred by a large oil company over a two-year period. In the insurance business, each
Refer to the Teaching Psychology (May 1998) study of how external clues influence performance, presented in Exercise. Recall that two different forms of a midterm psychology examination were given,
On the basis of data from the National Center for Health Statistics, N. Wetzel used the normal distribution to model the length of gestation for pregnant U.S. women (Chance, Spring 2001). Gestation
In Exercise, you learned that 60% of parents with young children condone spanking their child as a regular form of punishment (Tampa Tribune, Oct. 5, 2000). A child psychologist with 150 parent
A physical-fitness association is including the mile run in its secondary-school fitness test for students. The time for this event is approximately normally distributed with a mean of 450 seconds
The physical fitness of a patient is often measured by the patient’s maximum oxygen uptake (recorded in milliliters per kilogram, ml/kg). The mean maximum oxygen uptake for cardiac patients who
Geologists have successfully used statistical models to evaluate the nature of sedimentary deposits (called facies) in reservoirs. One of the models’ key parameters is the proportion P of facies
A. K. Shah published a simple approximation for areas under the normal curve in The American Statistician (Feb. 1985). Shah showed that the area A under the standard normal curve between 0 and z isa.
An infestation of a certain species of caterpillar called the spruce budworm can cause extensive damage to the timberlands of the northern United States. It is known that an outbreak of this type of
An article in IEEE Transactions (Mar. 1990) gave an example of a flexible manufacturing system with four machines operating independently. The repair rates for the machines (i.e., the time, in hours,
The number of serious accidents in a manufacturing plant has (approximately) a Poisson probability distribution with a mean of two serious accidents per month. It can be shown that if x, the number
The net weight per bag of a certain brand of corn chips is listed as 10 ounces. The weight of chips actually dispensed in each bag by an automated machine, when operating to specifications, is a
Simulate sampling from the population described in Exercise 6.3 by marking the values of x, one on each of four identical coins (or poker chips, etc.). Place the coins (marked 0, 2, 4, and 6) into a
Refer to Exercise 6.5 and find E(x) = Î¼. Then use the sampling distribution of x-bar found in Exercise 6.5 to find the expected value of x-bar. Note that E(x-bar) = Î¼.In
In Example, we use the computer to generate 1,000 samples, each containing n = 11 observations, from a uniform distribution over the interval from 150 to 200. Now use the computer to generate 500
Consider the following probability distribution:a. Find μ and σ2. b. Find the sampling distribution of the sample mean x-bar for a random sample of n = 2 measurements from this distribution.c.
Consider the following probability distribution:a. Find μ.b. For a random sample of n = 3 observations from this distribution, find the sampling distribution of the sample mean.c. Find the sampling
Refer to Exercise 6.5.a. Show that x is an unbiased estimator of Î¼.b. Find Ïƒ2x-bar.c. Find the probability that x-bar will fall within 2Ïƒ x-bar of Î¼.
Refer to Exercise 6.7, in which we found the sampling distribution of the sample median. Is the median an unbiased estimator of the population mean Î¼?
Suppose a random sample of n measurements is selected from a population with mean μ = 100 and variance σ2 = 100. For each of the following values of n, give the mean and standard deviation of the
Suppose a random sample of n = 25 measurements is selected from a population with mean μ and standard deviation σ. For each of the following values of μ and σ, give the values of µ and σ. a.
A random sample of n = 64 observations is drawn from a population with a mean equal to 20 and standard deviation equal to 16.a. Give the mean and standard deviation of the (repeated) sampling
A random sample of n = 100 observations is selected from a population with μ = 30 and σ = 16. a. Find μx-bar and σx-bar. b. Describe the shape of the sampling distribution of x-bar. c. Find
A random sample of n = 900 observations is selected from a population with μ = 100 and σ = 10. a. What are the largest and smallest values of x-bar that you would expect to see? b. How far, at
In Exercise, you learned that phishing describes an attempt to extract personal/financial information from unsuspecting people through fraudulent e-mail. Data from an actual phishing attack against
Refer to The American Statistician (May 2001) study of female students who suffer from bulimia, presented in Exercise. Recall that each student completed a questionnaire from which a “fear of
Refer to the Aquatic Biology (Vol. 9, 2010) study of green sea turtles inhabiting the Grand Cayman South Sound lagoon, Exercise. Research shows that the curved carapace (shell) lengths of these
Refer to the American Mineralogist (October 2009) study of the evolution of uranium minerals in the Earth’s crust, Exercise. Recall that researchers estimate that the trace amount of uranium x in
According to the American Automobile Association (AAA), the average cost of a gallon of regular unleaded fuel at gas stations in August 2010 was $2.78 (AAA Fuel Gauge Report). Assume that the
Refer to the Applied Psychology in Criminal Justice (Sept. 2009) investigation of the personality characteristics of drug dealers, Exercise. Convicted drug dealers were scored on the Wanting

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