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fundamentals of statistics

Questions and Answers of Fundamentals Of Statistics

12 Random numbers. 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
17 Soft-drink dispenser. The manager of a local soft-drink bottling company believes that when a new beveragedispensing machine is set to dispense 7 ounces, it in fact dispenses an amount x at random
Find the probability that a standard normal random variable exceeds 1.64; that is, find P(z 7 1.64).
Find the probability that a standard normal random variable exceeds 1.96 in absolute value.
Find the probability that a standard normal random variable lies to the left of .67.
Find the value of z —call it z0 —in the standard normal distribution that will be exceeded only 10% of the time. That is, find z0 such that P(z Ú z0) = .10.
Find the value of z0 such that 95% of the standard normal z values lie between -z0 and +z0;that is, find P(-z0 … z … z0) = .95.
20 Describe the shape of a normal probability distribution.
21 What is the name given to a normal distribution when m = 0 and s = 1?
22 If x has a normal distribution with mean m and standard deviation s, describe the distribution of z = (x - m)>s.
23 Find the area under the standard normal probability distribution between the following pairs of z -scores:a. z = 0 and z = 2.00b. z = 0 and z = 1.00c. z = 0 and z = 3d. z = 0 and z = .58
25 Find each of the following probabilities for a standard normal random variable z :a. P(z = 1)b. P(z … 1)c. P(z 6 1)d. P(z 7 1)e. P(-1 … z … 1)f. P(-2 … z … 2)g. P(-2.16 … z … .55)h.
26 Find the following probabilities for the standard normal random variable z :a. P(z 7 1.46)b. P(z 6 -1.56)c. P(.67 … z … 2.41)d. P(-1.96 … z 6 -.33)e. P(z Ú 0)f. P(-2.33 6 z 6 1.50)g. P(z Ú
28 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 … 0)
29 Find a value z0 of the standard normal random variable z such thata. P(z Ú z0) = .05b. P(z Ú z0) = .025c. P(z … z0) = .025d. P(z Ú z0) = .10e. P(z 7 z0) = .10
30 Suppose the random variable x is best described by a normal distribution with m = 25 and s = 5. Find the z -score that corresponds to each of the following x values:a. x = 25b. x = 30c. x = 37.5d.
31 Suppose x is a normally distributed random variable with m = 11 and s = 2. Find each of the following:a. P(10 … x … 12)b. P(6 … x … 10)c. P(13 … x … 16)d. P(7.8 … x … 12.6)e. P(x
32 Suppose x is a normally distributed random variable with m = 30 and s = 8. Find a value x0 of the random variable x such thata. P(x Ú x0) = .5b. P(x 6 x0) = .025c. P(x 7 x0) = .10d. P(x 7 x0) =
33 Refer to Exercise 32. Find x0 such thata. 10% of the values of x are less than x0.b. 80% of the values of x are less than x0.c. 1% of the values of x are greater than x0.
34 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 m { 2s on the graph. Find the
35 The random variable x has a normal distribution with standard deviation 25. It is known that the probability that x exceeds 150 is .90. Find the mean m of the probability distribution.
38 Tomato as a taste modifier. 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
39 Most powerful American women. Fortune (Oct. 16, 2008)published a list of the 50 most powerful women in America.The data on age (in years) of each woman is stored in the WPOWER50 file. The ages in
41 Transmission delays in wireless technology . Resource Reservation Protocol (RSVP) was originally designed to establish signaling links for stationary networks. In Mobile Networks and Applications
42 Shell lengths of sea turtles. In an Aquatic Biology (Vol.9, 2010) study of green sea turtles inhabiting the Grand Cayman South Sound lagoon, r esearchers discovered that the curved carapace
43 NHTSA crash safety tests. Data for the National Highway Traffic Safety Administration (NHTSA) crash test data for new cars is saved in the CRASH file. One of the variables measured is the severity
50 Industrial filling process. 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
The EPA mileage ratings on 100 cars are reproduced in Table 2 . Recall that these data are saved in the EPAGAS file. Numerical and graphical descriptive measures for the data are shown on the MINITAB
53 Why is it important to check whether the sample data come from a normal population?
54 Give four methods for determining whether the sample data come from a normal population.
55 If a population data set is normally distributed, what is the proportion of measurements you would expect to fall within the following intervals? 2. b. 20 c. 3
56 What is a normal probability plot and how is it used?
59 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,
61 Most powerful American women. Refer to the Fortune(Oct. 16, 2008) list of the 50 most powerful women in America. In Exercise 39 , you assumed that the ages (in years) of these women are
64 Drug content assessment. Scientists at GlaxoSmithKline Medicines Research Center used high-performance liquid chromatography (HPLC) to determine the amount of drug in a tablet produced by the
65 Estimating glacier elevations. Digital elevation models(DEMs) are now used to estimate elevations and slopes of remote regions. In Arctic, Antarctic, and Alpine Research(May 2004), geographers
67 Baseball batting averages. Major League Baseball (MLB)has two leagues: the American League (AL), which utilizes the designated hitter (DH) to bat for the pitcher, and the National League (NL),
68 Ranking the driving performance of professional golfers.The Sport Journal (Winter 2007) published an article on a new method for ranking the driving performance of PGA golfers . Recall that the
69 NHTSA crash tests. Refer to the National Highway Traffic Safety Administration (NHTSA) crash test data for new cars. In Exercise 43 , you assumed that the driver’s head injury rating is
70 Cruise ship sanitation scores. Refer to the data on the Jan. 2010 sanitation scores for 186 cruise ships. The data are saved in the SHIPSANIT file. Assess whether the sanitation scores are
71 Laptop use in middle school. In the American Secondary Education (Fall 2009) study of student usage of laptops at a middle school, x , the number of minutes per day each student used his or her
72 Ranking Ph.D. programs in economics. In to the Southern Economic Journal (Apr. 2008) rankings of Ph.D. programs in economics at 129 colleges and universities, the number of publications published
73 For large n (say, n = 100 ), why is it advantageous to use the normal distribution to approximate a binomial probability?
74 Why do we need a correction for continuity when approximating a binomial probability with the normal distribution?
75 Suppose x is a binomial random variable with p = .4 and n = 25.a. Would it be appropriate to approximate the probability distribution of x with a normal distribution?Explain.b. Assuming that a
77 Assume that x is a binomial random variable with n = 25 and p = .5. Use Table II of Appendix: Tables and the normal approximation to find the exact and approximate values, respectively, of the
78 Assume that x is a binomial random variable with n = 1,000 and p = .50. Find each of the following probabilities:a. P(x 7 500)b. P(490 … x 6 500)c. P(x 7 550)
79 Assume that x is a binomial random variable with n = 100 and p = .40. Use a normal approximation to find the following:a. P(x … 35)b. P(40 … x … 50)c. P(x Ú 38)
80 Where will you get your next pet? An Associated Press/Petside.com poll revealed that half of all pet owners would get their next dog or cat from a shelter ( USAToday, May 12, 2010). Consider a
81 Analysis of bottled water. In a the Scientific American (July 2003) report on whether bottled water is really purified water, the Natural Resources Defense Council found that 25% of bottled-water
82 LASIK surgery complications. 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
83 Melanoma deaths. According to the American Cancer Society , melanoma, a form of skin cancer, kills 15% of Americans who suffer from the disease each year. Consider a sample of 10,000 melanoma
84 Caesarian birth study. 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,
86 Chemical signals of mice. Cell (May 14, 2010) ran a study of the ability of a mouse to recognize the odor of a potential predator . You learned that 40% of lab mice cells exposed to chemically
87 Defects in semiconductor wafers. The computer chips in notebook and laptop computers are produced from semiconductor wafers. Certain semiconductor wafers are exposed to an environment that
88 Victims of domestic abuse. 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 a
89 Hotel guest satisfaction. A 2009 North American Hotel Guest Satisfaction Index Study determined that the probability of a hotel guest participating in the hotel’s “green”conservation program
90 Body fat in men. 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
91 Luggage inspection at Newark airport. New Jersey Business reports that Newark International Airport’s terminal handles an average of 3,000 international passengers an hour, but is capable of
93 What are the characteristics of an exponential random variable?
94 The exponential distribution is often called the _________ distribution. (Fill in the blank.)
95 The random variables x and y have exponential distributions with u = 3 and u = 1, respectively. Using Table V in Appendix: Tables , carefully plot both distributions on the same set of axes.
96 Use Table V in Appendix: Tables to determine the value of e-a/u for each of the following cases:a. u = 1, a = 1b. u = 1, a = 2.5c. u = .4, a = 3d. u = .2, a = .3
97 Suppose x has an exponential distribution with u = 1. Find the following probabilities:a. P(x 7 1)b. P(x … 3)c. P(x 7 1.5)d. P(x … 5)
98 Suppose x has an exponential distribution with u = 2.5.Find the following probabilities:a. P(x … 4)b. P(x 7 5)c. P(x … 2)d. P(x 7 3)
99 Suppose the random variable x has an exponential probability distribution with u = 1. Find the mean and standard deviation of x . Find the probability that x will assume a value within the
100 The random variable x can be adequately approximated by an exponential probability distribution with u = 2. Find the probability that x assumes a valuea. More than three standard deviations from
101 Lead in metal shredder residue. On the basis of data collected from metal shredders across the nation, the amount x of extractable lead in metal shredder residue has an approximate exponential
103 Phishing attacks to e-mail accounts. Chance (Summer 2007) published an article on phishing attacks at a company.P hishing describes an attempt to extract personal/financial information through
105 NHL overtime games. In the National Hockey League(NHL), games that are tied at the end of three periods are sent into “sudden-death” overtime. In overtime, the team to score the first goal
106 Forest fragmentation study. Conservation Ecology (Dec.2003) conducted a study on the causes of fragmentation in 54 South American forests . Recall that the cause is classified as either
107 Ship-to-shore transfer times. Lack of port facilities or shallow water may require cargo on a large ship to be transferred to a pier in smaller craft. The smaller craft may have to cycle back and
111 Length of life of a halogen bulb. For a certain type of halogen light bulb, an old bulb that has been in use for a while tends to have a longer life than a new bulb. Let x represent the life (in
113 Identify the type of continuous random variable—uniform, normal, or exponential—described by each of the following probability density functions: a. f(x)=(e)/7, x>0 b. f(x) 1/20, 5 < x < 25 =
114 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
115 Find the following probabilities for the standard normal random variable z :a. P(z … 2.1)b. P(z Ú 2.1)c. P(z Ú -1.65)d. P(-2.13 … z … -.41)e. P(-1.45 … z … 2.15)f. P(z … -1.43)
116 Find a z -score, say, z0, such thata. P(z … z0) = .8708b. P(z Ú z0) = .0526c. P(z … z0) = .5d. P(-z0 … z … z0) = .8164e. P(z Ú z0) = .8023f. P(z Ú z0) = .0041
117 The random variable x has a normal distribution with m = 70 and s = 10. Find the following probabilities:a. P(x … 75)b. P(x Ú 90)c. P(60 … x … 75)d. P(x 7 75)e. P(x = 75)f. P(x … 95)
118 The random variable x has a normal distribution with m = 40 and s2 = 36. Find a value of x , say, x0, such thata. P(x Ú x0) = .5b. P(x … x0) = .9911c. P(x … x0) = .0028d. P(x Ú x0) =
120 Assume that x has an exponential distribution with u = 3. Finda. P(x … 1)b. P(x 7 1)c. P(x = 1)d. P(x … 6)e. P(2 … x … 10)
121 Galaxy velocity study. Astronomical Journal (July 1995)did a study of galaxy velocities . A histogram of the velocities of 103 galaxies located in a particular cluster named A2142 is reproduced
122 Trajectory of an electrical circuit. Researchers at the University of California–Berkeley have designed, built, and tested a switched-capacitor circuit for generating random signals (
124 Improving SAT scores. Chance (Winter 2001) performed a study of students who paid a private tutor to help them improve their SAT scores. The table summarizing the changes in both the
126 Weights of contaminated fish. In the U.S. Army Corps of Engineers data on contaminated fish in the Tennessee River, one of the variables measured for each captured fish is weight (in grams). The
127 Sickle-cell anemia. Eight percent of the African-American population is known to carry the trait for sickle-cell anemia (Sickle Cell-Information Center, Atlanta, GA). If 1,000 African-Americans
128 Where will you get your next pet? 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
129 Tropical island temperatures. Records indicate that the daily high January temperatures on a tropical island tend to have a uniform distribution over the interval from 75F to 90F. A tourist
134 Comparison of exam scores: red versus blue exam. In a Teaching Psychology (May 1998) study of how external clues influence performance, two different forms of a midterm psychology examination
135 Waiting for an elevator. The manager of a large department store with three floors reports that the time a customer on the second floor must wait for an elevator has a uniform distribution
137 Galaxy velocity study. Recall The Astronomical Journal(July 1995) study of galaxy velocity, Exercise 121 . The observed velocity of a galaxy located in galaxy cluster A2142 was found to have a
138 Parents who condone spanking. 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
139 Fungi in beech forest trees. In an Applied Ecology and Environmental Research (Vol. 1, 2003) study of beech trees damaged by fungi, researchers found that 25% of the beech trees in east central
140 Mile run times. 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
141 Wear-out of used display panels. Wear-out failure time of electronic components is often assumed to have a normal distribution. Can the normal distribution be applied to the wear-out of used
145 Normal curve approximation. 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
146 Spruce budworm infestation. 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
147 Maximum time to take a test. A professor believes that if a class is allowed to work on an examination as long as desired, the times spent by the students would be approximately normal with mean
148 Machine repair times. 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
153 IQs and The Bell Curve . In their controversial book The Bell Curve (Free Press, 1994), Professors Richard J.Herrnstein (a Harvard psychologist who died while the book was in production) and
For large values of n , the computational effort involved in working with the binomial probability distribution is considerable.Fortunately, in many instances the normal distribution provides a good

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