According to Table II, the area under the standard normal curve that lies to the left of 1.96 is 0.975. Without further consulting Table II, determine the area under the standard normal curve that lies to the left of −1.96. Explain your reasoning.
> Frustrated passengers, congested streets, time schedules, and air and noise pollution are just some of the physical and social pressures that lead many urban bus drivers to retire prematurely with disabilities such as coronary heart disease and stomach d
> A variable is normally distributed with mean 68 and standard deviation 10. Find the percentage of all possible values of the variable that a. lie between 73 and 80. b. are at least 75. c. are at most 90.
> Assume that the variable under consideration has a density curve. Note that the answers required here may be only approximately correct. The area under the density curve that lies between 15 and 20 is 0.414. What percentage of all possible observations o
> A variable is normally distributed with mean 6 and standard deviation 2. Find the percentage of all possible values of the variable that a. lie between 1 and 7. b. exceed 5. c. are less than 4.
> State the empirical rule as specialized to variables.
> Explain why the percentage of all possible observations of a normally distributed variable that lie within two standard deviations to either side of the mean equals the area under the standard normal curve between −2 and 2.
> Briefly, for a normally distributed variable, how do you obtain the percentage of all possible observations that lie within a specified range?
> Let 0
> In an experiment reported by J. Singer and D. Andrade in the article “Regression Models for the Analysis of Pretest/Posttest Data” (Biometrics, Vol. 53, pp. 729–735), the effect of using either a conventional or experimental (hugger) toothbrush was inves
> In this section, we mentioned that the total area under any curve representing the distribution of a variable equals 1. Explain why.
> Complete the following table.
> Is an extreme observation necessarily an outlier? Explain your answer.
> Illustrate your work with graphs. Determine the two z-scores that divide the area under the standard normal curve into a middle 0.99 area and two outside 0.005 areas.
> Illustrate your work with graphs. Determine the two z-scores that divide the area under the standard normal curve into a middle 0.90 area and two outside 0.05 areas.
> Illustrate your work with graphs. Obtain the following z-scores. a. z0.20 b. z0.06
> Assume that the variable under consideration has a density curve. Note that the answers required here may be only approximately correct. The area under the density curve that lies between 30 and 40 is 0.832. What percentage of all possible observations o
> Illustrate your work with graphs. Find the following z-scores. a. z0.03 b. z0.005
> Illustrate your work with graphs. Determine z0.015.
> Illustrate your work with graphs. Determine z0.33.
> Illustrate your work with graphs. Obtain the z-score that has area 0.70 to its right.
> Two different options are under consideration for comparing the lifetimes of four brands of flashlight battery, using 20 flashlights. a. One option is to randomly divide 20 flashlights into four groups of 5 flashlights each and then randomly assign each
> Illustrate your work with graphs. Obtain the z-score that has an area of 0.95 to its right.
> Explain why the minimum and maximum observations are added to the three quartiles to describe better the variation in a data set.
> Illustrate your work with graphs. Obtain the z-score that has area 0.80 to its left under the standard normal curve.
> Illustrate your work with graphs. Find the z-score that has an area of 0.75 to its left under the standard normal curve.
> Illustrate your work with graphs. Determine the z-score for which the area under the standard normal curve to its left is 0.01.
> Illustrate your work with graphs. Obtain the z-score for which the area under the standard normal curve to its left is 0.025.
> The total area under the following standard normal curve is divided into eight regions. a. Determine the area of each region. b. Complete the following table.
> Assume that the variable under consideration has a density curve. Note that the answers required here may be only approximately correct. The area under the density curve that lies to the right of 15 is 0.324. What percentage of all possible observations
> In each part, find the area under the standard normal curve that lies between the specified z-scores, sketch a standard normal curve, and shade the area of interest. a. −1 and 1 b. −2 and 2 c. −3 and 3
> Use Table II to obtain each shaded area under the standard normal curve.
> Use Table II to obtain each shaded area under the standard normal curve.
> In the paper “Outcomes at School Age After Postnatal Dexamethasone Therapy for Lung Disease of Prematurity” (New England Journal of Medicine, Vol. 350, No. 13, pp. 1304–1313), T. Yeh et al. studied the outcomes at school age in children who had participa
> Explain what each symbol represents. a. Σ b. n c. x¯
> Sketch a standard normal curve and shade the area of interest. Find the area under the standard normal curve that lies a. either to the left of −1 or to the right of 2. b. either to the left of −2.51 or to the right of −1.
> Sketch a standard normal curve and shade the area of interest. Find the area under the standard normal curve that lies a. either to the left of −2.12 or to the right of 1.67. b. either to the left of 0.63 or to the right of 1.54.
> Sketch a standard normal curve and shade the area of interest. Determine the area under the standard normal curve that lies between a. −0.88 and 2.24. b. −2.5 and −2. c. 1.48 and 2.72. d. −5.1 and 1.
> Sketch a standard normal curve and shade the area of interest. Determine the area under the standard normal curve that lies between a. −2.18 and 1.44. b. −2 and −1.5. c. 0.59 and 1.51. d. 1.1 and 4.2.
> Sketch a standard normal curve and shade the area of interest. Find the area under the standard normal curve that lies to the right of a. 2.02. b. −0.56. c. −4.
> Sketch a standard normal curve and shade the area of interest. Find the area under the standard normal curve that lies to the right of a. −1.07. b. 0.6. c. 0. d. 4.2.
> Sketch a standard normal curve and shade the area of interest. Determine the area under the standard normal curve that lies to the left of a. −0.87. b. 3.56. c. 5.12.
> Assume that the variable under consideration has a density curve. Note that the answers required here may be only approximately correct. The area under the density curve that lies to the left of 10 is 0.654. What percentage of all possible observations o
> Sketch a standard normal curve and shade the area of interest. Determine the area under the standard normal curve that lies to the left of a. 2.24. b. −1.56. c. 0. d. −4.
> The area under the standard normal curve that lies to the left of a z-score is always strictly between ____ and ____.
> Identify an advantage that the median and interquartile range have over the mean and standard deviation, respectively.
> In a study by A. Elliot et al., titled “Women’s Use of Red Clothing as a Sexual Signal in Intersexual Interaction” (Journal of Experimental Social Psychology, Vol. 49, Issue 3, pp. 599–602), women were studied to determine the effect of apparel color cho
> Explain how Table II is used to determine the area under the standard normal curve that lies a. to the left of a specified z-score. b. to the right of a specified z-score. c. between two specified z-scores.
> Why is the standard normal curve sometimes referred to as the z-curve?
> Property 4 of Key Fact 6.5 states that most of the area under the standard normal curve lies between −3 and 3. Use Table II to determine precisely the percentage of the area under the standard normal curve that lies between −3 and 3.
> According to Table II, the area under the standard normal curve that lies to the left of 0.43 is 0.6664. Without further consulting Table II, determine the area under the standard normal curve that lies to the right of 0.43. Explain your reasoning.
> According to Table II, the area under the standard normal curve that lies to the left of −2.08 is 0.0188. Without further consulting Table II, determine the area under the standard normal curve that lies to the right of 2.08. Explain your reasoning.
> Without consulting Table II, explain why the area under the standard normal curve that lies to the right of 0 is 0.5.
> With which normal distribution is the standard normal curve associated?
> Assume that the variable under consideration has a density curve. Note that the answers required here may be only approximately correct. The percentage of all possible observations of the variable that lie to the right of 4 equals the area under its dens
> Identify by name three important groups of percentiles.
> In the paper, “Delayed Metamorphosis of a Tropical Reef Fish (Acanthurus triostegus): A Field Experiment” (Marine Ecology Progress Series, Vol. 176, pp. 25–38), M. McCormick studied larval duration of the convict surgeonfish, a common tropical reef fish.
> In a study by P. M. West titled “The Lion’s Mane” (American Scientist, Vol. 93, No. 3, pp. 226–236), the effects of the mane of a male lion as a signal of quality to mates and rivals was explored. Four life-sized dummies of male lions provided a tool for
> Refer to the simulation of human gestation periods discussed. a. Sketch the normal curve for human gestation periods. b. Simulate 1000 human gestation periods. c. Approximately what values would you expect for the sample mean and sample standard deviati
> Students in an introductory statistics course at the U.S. Air Force Academy participated in Nabisco’s “Chips Ahoy! 1,000 Chips Challenge” by confirming that there were at least 1000 chips in every 18-ounce bag of cookies that they examined. As part of th
> From the U.S. Census Bureau, in the document International Data Base, we obtained data on the total fertility rates for women in various countries. Those data are presented on the WeissStats site. The total fertility rate gives the average number of chil
> Each year, thousands of high school students bound for college take the Scholastic Assessment Test (SAT). This test measures the verbal and mathematical abilities of prospective college students. Student scores are reported on a scale that ranges from a
> A classic study by F. Thorndike on the number of calls to a wrong number appeared in the paper “Applications of Poisson’s Probability Summation” (Bell Systems Technical Journal, Vol. 5, pp. 604–624). The study examined the number of calls to a wrong numb
> In the paper “Cloudiness: Note on a Novel Case of Frequency” (Proceedings of the Royal Society of London, Vol. 62, pp. 287–290), K. Pearson examined data on daily degree of cloudiness, on a scale of 0 to 10, at Breslau (Wroclaw), Poland, during the decad
> The National Center for Health Statistics publishes information about birth rates (per 1000 population) in the document National Vital Statistics Report. The following table provides a frequency distribution for birth rates during one year for the 50 sta
> From the document National Vital Statistics Reports, a publication of the National Center for Health Statistics, we obtained the following frequency distribution for the ages of women who became mothers during one year. a. Obtain a relative-frequency his
> A data set consists of 2m2 − 1 zeros, one −m, and one m. a. Compute x¯ and s for this data set. b. How many standard deviations from the mean is the observation m? c. Assuming that m ≥ 4, what percentage of the observations lie within three standard devi
> From the paper “Effects of Chronic Nitrate Exposure on Gonad Growth in Green Sea Urchin Strongylocentrotus droebachiensis” (Aquaculture, Vol. 242, No. 1–4, pp. 357–363) by S. Siikavuopio et al., we found that weights of adult green sea urchins are normal
> Assume that the variable under consideration has a density curve. Note that the answers required here may be only approximately correct. The percentage of all possible observations of the variable that lie between 7 and 12 equals the area under its densi
> In a classic study, described by F. Yates in The Design and Analysis of Factorial Experiments (Commonwealth Bureau of Soils, Technical Communication No. 35), the effect on oat yield was compared for three different varieties of oats and four different co
> As reported in Runner’s World magazine, the times of the finishers in the New York City 10-km run are normally distributed with mean 61 minutes and standard deviation 9 minutes. Let x denote finishing time for finishers in this race. a. Sketch the distri
> According to the National Health and Nutrition Examination Survey, published by the National Center for Health Statistics, the serum (noncellular portion of blood) total cholesterol level of U.S. females 20 years old or older is normally distributed with
> One of the larger species of tarantulas is the Grammostola mollicoma, whose common name is the Brazilian giant tawny red. A tarantula has two body parts. The anterior part of the body is covered above by a shell, or carapace. From a recent article by F.
> Refer to Example. a. The area under the normal curve with parameters μ = 64.4 and σ = 2.4 that lies to the left of 61 is 0.0783. Use this information to estimate the percentage of female students who are shorter than 61 inches. b. Use the relative-freque
> Refer to Example. a. Use the relative-frequency distribution in Table 6.1 to obtain the percentage of female students who are between 60 and 65 inches tall. b. Use your answer from part (a) to estimate the area under the normal curve having parameters μ
> Desert Samaritan Hospital in Mesa, Arizona, keeps records of its emergency-room traffic. Beginning at 6:00 P.M. on any given day, the elapsed time, in hours, until the first patient arrives is a variable with density curve y = 6.9e−6.9x for x > 0, and y
> The loss, in millions of dollars, due to a fire in a commercial building is a variable with density curve y = 1 − x/2 for 0 < x < 2, and y = 0 otherwise. Using the fact that the area of a triangle equals one-half its base times its height, we find that t
> How many standard deviations to either side of the mean must we go to ensure that, for any data set, at least 95% of the observations lie within?
> A petri dish is a small, shallow dish of thin glass or plastic, used especially for cultures in bacteriology. A 2-inch-radius petri dish, containing nutrients upon which bacteria can multiply, is smeared with a uniform suspension of bacteria. Subsequentl
> A commuter train arrives punctually at a station every half hour. Each morning, a commuter named John leaves his house and casually strolls to the train station. The time, in minutes, that John waits for the train is a variable with density curve y = 1/3
> A variable has the density curve whose equation is y = 1 for 0 < x < 1, and y = 0 otherwise. a. Graph the density curve of this variable. b. Show that the area under this density curve to the left of any number x between 0 and 1 equals x. What percentage
> A driver’s ability to detect highway signs is an important consideration in highway safety. In his dissertation, Highway Construction Safety and the Aging Driver, S. Younes investigated the distance at which drivers can first detect highway caution signs
> For a variable with a density curve, what is the relationship between the percentage of all possible observations of the variable that lie within any specified range and the corresponding area under its density curve.
> A variable has the density curve whose equation is y = 2x for 0 < x < 1, and y = 0 otherwise. a. Graph the density curve of this variable. b. Show that the area under this density curve to the left of any number x between 0 and 1 equals x 2. What percent
> The area under a particular normal curve between 10 and 15 is 0.6874. A normally distributed variable has the same mean and standard deviation as the parameters for this normal curve. What percentage of all possible observations of the variable lie betwe
> The area under a particular normal curve to the left of 105 is 0.6227. A normally distributed variable has the same mean and standard deviation as the parameters for this normal curve. What percentage of all possible observations of the variable lie to t
> For a normally distributed variable, what is the relationship between the percentage of all possible observations that lie to the right of 7 and the area under the associated normal curve to the right of 7? What if the variable is only approximately norm
> For a normally distributed variable, what is the relationship between the percentage of all possible observations that lie between 2 and 3 and the area under the associated normal curve between 2 and 3? What if the variable is only approximately normally
> How many standard deviations to either side of the mean must we go to ensure that, for any data set, at least 99% of the observations lie within?
> Sketch the normal distribution with a. μ = −2 and σ = 2. b. μ = −2 and σ = 1/2. c. μ = 0 and σ = 2.
> Sketch the normal distribution with a. μ = 3 and σ = 3. b. μ = 1 and σ = 3. c. μ = 3 and σ = 1.
> What are the parameters for a normal curve?
> True or false: The mean of a normal distribution has no effect on its spread. Explain your answer.
> Supermarkets are interested in strategies to increase temporarily the unit sales of a product. In one study, researchers compared the effect of display type and price on unit sales for a particular product. The following display types and pricing schemes
> As you learned on page 450, recent research by R. Brame et al. indicates that almost a third of Americans have been arrested by age 23, and this figure excludes arrests for minor traffic violations. Now that you have studied inferences for proportions, w
> Consider two normal distributions, one with mean −4 and standard deviation 3, and the other with mean −4 and standard deviation 6. Answer true or false to each statement and explain your answers. a. The two normal distributions have the same spread. b. T
> State the two basic properties of every density curve.
> Consider two normal distributions, one with mean −4 and standard deviation 3, and the other with mean 6 and standard deviation 3. Answer true or false to each statement and explain your answers. a. The two normal distributions have the same spread. b. Th
> Which normal distribution has a wider spread: the one with mean 1 and standard deviation 2 or the one with mean 2 and standard deviation 1? Explain your answer.
> Identify two methods for obtaining a simple random sample.
> Two normally distributed variables have the same means and the same standard deviations. What can you say about their distributions? Explain your answer.
