Identify one reason why the complementation rule is useful.
> Relative to the population mean, what happens to the possible sample means for samples of the same size as the sample size increases? Explain the relevance of this property in estimating a population mean by a sample mean.
> Provide two synonyms for “the distribution of all possible sample means for samples of a given size.”
> What is the sampling distribution of a statistic? Why is it important?
> A variable is said to be uniformly distributed or to have a uniform distribution with parameters a and b if its distribution has the shape of the horizontal line segment with equation y = 1/(b − a), for a < x < b. The mean and standard deviation of such
> The Athletic Coping Skills Inventory (ACSI) is a test to measure psychological skills believed to influence athletic performance. Researchers E. Estanol et al. studied the relationship between ACSI scores and eating disorders in dancers in the article “M
> 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
> Suppose that you have constructed a stem-and-leaf diagram and discover that it is only moderately useful because there are too few stems. How can you remedy the problem?
> A paint manufacturer in Pittsburgh claims that his paint will last an average of 5 years. Assuming that paint life is normally distributed and has a standard deviation of 0.5 year, answer the following questions: a. Suppose that you paint one house with
> The American Council of Life Insurers provides information about life insurance in force per covered family in the Life Insurers Fact Book. Assume that the standard deviation of life insurance in force is $50,900. a. Determine the probability that the sa
> In the article “Drinking Glucose Improves Listening Span in Students Who Miss Breakfast” (Educational Research, Vol. 43, No. 2, pp. 201–207), authors N. Morris and P. Sarll explored the relationship between students who skip breakfast and their performan
> Refer to Problem 12. a. Find the percentage of all samples of four pygmy-possums that have mean weights within 0.225 g of the population mean weight of 8.5 g. b. Obtain the probability that the mean weight of four randomly selected pygmy-possums will be
> The foraging behavior of the western pygmy-possum was investigated in the article “Strategies of a Small Nectarivorous Marsupial, the Western Pygmy-Possum, in Response to Seasonal Variation in Food Availability” (Journal of Mammalogy, Vol. 96, No. 6, pp.
> Repeat Problem 10, assuming that the number of hours worked by female marketing and advertising managers is normally distributed. Data from Problem 10: In the article “How Hours of Work Affect Occupational Earnings” (Monthly Labor Review, Vol. 121), D.
> In the article “How Hours of Work Affect Occupational Earnings” (Monthly Labor Review, Vol. 121), D. Hecker discussed the number of hours actually worked as opposed to the number of hours paid for. The study examines both full-time men and full-time wome
> Define sampling error.
> Explain the relationship between percentages for a normally distributed variable and areas under the corresponding normal curve.
> Answer true or false to each statement. Give reasons for your answers. a. Two variables that have the same mean and standard deviation have the same distribution. b. Two normally distributed variables that have the same mean and standard deviation have t
> Suppose that you have a data set that contains a large number of observations. Which graphical display is generally preferable: a histogram or a stem-and-leaf diagram? Explain your answer.
> Define a. normally distributed variable. b. normally distributed population. c. parameters for a normal curve.
> State two of the main reasons for studying the normal distribution.
> A variable has the density curve with equation y = 1 − x/2 for 0 < x < 2, 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 2 equals x − x 2/4. What
> Assume that the variable under consideration has a density curve. Note that the answers required here may be only approximately correct. The area under a density curve that lies between 5 and 6 is 0.728. What percentage of all possible observations of th
> In an issue of National Mortgage News, a special report was published on publicly traded mortgage industry companies. A sample of 25 mortgage industry companies had the following numbers of employees. a. Obtain a normal probability plot of the data. b. U
> Researchers M. Kroll et al. studied the influence of paternity on rates of mortality and development in eggs and larvae of Northwest Atlantic cod in the article, “Paternal Effects on Early Life History Traits in Northwest Atlantic Cod, Gadus morhua” (Jou
> Assume that the variable under consideration has a density curve. Note that the answers required here may be only approximately correct. The area under a density curve that lies to the left of 60 is 0.364. What percentage of all possible observations of
> Refer to Problem 28, and fill in the following blanks. a. Approximately 68% of students who took the verbal portion of the GRE scored between ___ and ___. b. Approximately 95% of students who took the verbal portion of the GRE scored between ___ and ___.
> The Graduate Record Examination (GRE) is a standardized test that students usually take before entering graduate school. According to the document GRE Guide to the Use of Scores, a publication of the Educational Testing Service, the scores on the verbal
> The study “Intrathecal Sufentanil versus Fentanyl for Lower Limb Surgeries – A Randomized Controlled Trial” (Journal of Anaesthesiology Clinical Pharmacology, Vol. 27, Issue 1, pp. 67–73) by P. Motiani et al. compares two different agents, intrathecal su
> Discuss the relative advantages and disadvantages of stem-andleaf diagrams versus frequency histograms.
> The WONDER database, maintained by the Centers for Disease Control and Prevention, provides a single point of access to a wide variety of reports and numeric public health data. From that database, we obtained the following data for one year’s birth weig
> In 1903, K. Pearson and A. Lee published a paper entitled “On the Laws of Inheritance in Man. I. Inheritance of Physical Characters” (Biometrika, Vol. 2, pp. 357–462). From information presented in that paper, forearm length of men, measured from the elb
> A coffee machine is supposed to dispense 6 fluid ounces (fl oz) of coffee into a paper cup. In reality, the amounts dispensed vary from cup to cup. In fact, the amount dispensed, in fl oz, is a variable with density curve y = 2 for 5.75 < x < 6.25, and y
> For the standard normal curve, find the z-score(s) a. that has area 0.30 to its left. b. that has area 0.10 to its right. c. z0.025, z0.05, z0.01, and z0.005. d. that divide the area under the curve into a middle 0.99 area and two outside 0.005 areas.
> Determine and sketch the area under the standard normal curve that lies a. to the left of −3.02. b. to the right of 0.61. c. between 1.11 and 2.75. d. between −2.06 and 5.02. e. between −4.11 and −1.5. f. either to the left of 1 or to the right of 3.
> According to Table II, the area under the standard normal curve that lies to the left of 1.05 is 0.8531. Without further reference to Table II, determine the area under the standard normal curve that lies a. to the right of 1.05. b. to the left of −1.05
> Sketch the normal curve having the parameters a. μ = −1 and σ = 2. b. μ = 3 and σ = 2. c. μ = −1 and σ = 0.5.
> 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 a variable that lie between 25 and 50 equals the area under its densit
> If you observe the values of a normally distributed variable for a sample, a normal probability plot should be roughly .
> Roughly speaking, what are the normal scores corresponding to a sample of observations?
> For data that are grouped in classes based on more than a single value, lower class limits (or cutpoints) are used on the horizontal axis of a histogram for depicting the classes. Class marks (or midpoints) can also be used, in which case each bar is cen
> State the empirical rule for variables.
> What does the symbol zα signify?
> Explain how to use Table II to determine the z-score that has a specified area to its a. left under the standard normal curve. b. right under the standard normal curve.
> Explain how to use Table II 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.
> What key fact permits you to determine percentages for a normally distributed variable by first converting to z-scores and then determining the corresponding area under the standard normal curve?
> Consider the normal curves that have the parameters μ = 1.5 and σ = 3; μ = 1.5 and σ = 6.2; μ = −2.7 and σ = 3; μ = 0 and σ = 1. a. Which curve has the largest spread? b. Which curves are centered at the same place? c. Which curves have the same spread?
> Answer true or false to each statement. Explain your answers. a. Two normal distributions that have the same mean are centered at the same place, regardless of the relationship between their standard deviations. b. Two normal distributions that have the
> Identify the distribution of the standardized version of a normally distributed variable.
> What is a density curve, and why are such curves important?
> Of the variables you have studied so far, which type yields non numerical data?
> Answer true or false to each statement and explain your answers. a. For any two events, the probability that one or the other of the events occurs equals the sum of the two individual probabilities. b. For any event, the probability that it occurs equals
> Suppose that E is an event. Use probability notation to represent a. the probability that event E occurs. b. the probability that event E occurs is 0.436.
> What does it mean for two or more events to be mutuallyexclusive?
> Identify a commonly used graphical technique for portraying events and relationships among events.
> Refer to Problem 40. a. Draw a probability histogram for the random variable X. b. The selection of the four households was done without replacement. Strictly speaking, then, why is the probability distribution that you obtained in Problem 40(b) only app
> According to JAVMA News, a publication of the American Veterinary Medical Association, roughly 60% of U.S. households own one or more pets. Four U.S. households are selected at random. Use Table VII in Appendix A to solve the following problems. a. Find
> Decide which of these numbers could not possibly be probabilities. Explain your answers. a. 0.047 b. −0.047 c. 3.5 d. 1/3.5
> In the game of soccer, a penalty kick is a direct free kick, taken from 12 yards out from the goal on the penalty mark. According to the article “Penalty Kicks in Soccer: An Empirical Analysis of Shooting Strategies and Goalkeeper’s Preferences” (Soccer
> Use the binomial probability formula. Compare your results.
> Refer to the probability distribution displayed in the table in Problem 36. a. Find the mean of the random variable Y . b. On average, how many lines are busy? c. Compute the standard deviation of Y . d. Construct a probability histogram for Y ; locate t
> Explain the advantages and disadvantages of frequency histograms versus frequency distributions.
> An accounting office has six incoming telephone lines. The probability distribution of the number of busy lines, Y, is as follows. Use random-variable notation to express each of the following events. The number of busy lines is a. exactly four. b. at le
> According to the Arizona State University Enrollment Summary, a frequency distribution for the number of undergraduate students attending Arizona State University (ASU) in the Fall 2012 semester, by class level, is as shown in the following table. Here,
> Consider the events (not J ), (H & I), (H or K), and (H & K) discussed in Problem 31. a. Find the probability of each of those four events, using the f/N rule. b. Compute P(J ), using the complementation rule and your answer for P(not J ) from part (a).
> Refer to Problems 30 and 31. a. Use the second column of Table 5.21 and the f/N rule to compute the probability of each of the events H, I, J, and K. b. Express each of the events H, I, J , and K in terms of the mutually exclusive events displayed. c. Co
> For the following groups of events, determine which are mutually exclusive. a. H and I b. I and K c. H and (not J ) d. H, (not J ), and K
> A federal individual income tax return is selected at random. Let H = event the return shows an AGI between $20K and $100K, I = event the return shows an AGI of less than $50K, J = event the return shows an AGI of less than $100K, and K = event the retur
> The Internal Revenue Service compiles data on income tax returns and summarizes its findings in Statistics of Income. The first two columns of Table 5.21 show a frequency distribution (number of returns) for adjusted gross income (AGI) from federal indiv
> What meaning is given to the probability of an event by the frequentist interpretation of probability?
> The Television Bureau of Advertising publishes a report titled TV Basics for the purpose of providing information to help advertisers make the most effective and efficient use of local and national spot television advertisements. The following table give
> Name three common discrete probability distributions other than the binomial distribution.
> Explain the difference between a frequency histogram and a relative-frequency histogram.
> Suppose that a simple random sample of size n is taken from a finite population in which the proportion of members having a specified attribute is p. Let X be the number of members sampled that have the specified attribute. a. If the sampling is done wit
> The following are two probability histograms of binomial distributions. For each, specify whether the success probability is less than, equal to, or greater than 0.5.
> The game of craps is played by rolling two balanced dice. A first roll of a sum of 7 or 11 wins; and a first roll of a sum of 2, 3, or 12 loses. To win with any other first sum, that sum must be repeated before a sum of 7 is thrown. It can be shown that
> In 10 Bernoulli trials, how many outcomes contain exactly three successes?
> What is the relationship between Bernoulli trials and the binomial distribution?
> List the three requirements for repeated trials of an experiment to constitute Bernoulli trials.
> Determine the value of each binomial coefficient.
> Determine 0!, 3!, 4!, and 7!.
> Regarding the equal-likelihood model, a. what is it? b. how are probabilities computed?
> Two random variables, X and Y , have standard deviations 2.4 and 3.6, respectively. Which one is more likely to take a value close to its mean? Explain your answer.
> We used slightly different methods for determining the “middle” of a class with limit grouping and cutpoint grouping. Identify the methods and the corresponding terminologies.
> A random variable X has mean 3.6. If you make a large number of repeated independent observations of the random variable X, the average value of those observations will be approximately ___.
> A random variable X equals 2 with probability 0.386. a. Use probability notation to express that fact. b. If you make repeated independent observations of the random variable X, in approximately what percentage of those observations will you observe the
> If you sum the probabilities of the possible values of a discrete random variable, the result always equals ___.
> How do you graphically portray the probability distribution of a discrete random variable?
> What does the probability distribution of a discrete random variable tell you?
> Fill in the blanks. a. A is a quantitative variable whose value depends on chance___. b. A discrete random variable is a random variable whose possible values ____.
> A and B are events such that P(A) = 0.2, P(B) = 0.6, and P(A & B) = 0.1. Find P(A or B).
> E is an event and P(not E) = 0.4. Find P(E).
> A, B, and C are mutually exclusive events such that P(A) = 0.2, P(B) = 0.6, and P(C) = 0.1. Find P(A or B or C).
> Why is probability theory important to statistics?
> For quantitative data, we examined three types of grouping: single-value grouping, limit grouping, and cut point grouping. For each type of data given, decide which of these three grouping types is usually best. Explain your answers. a. Continuous data d
> In an on-line press release, ABCNews.com reported that “. . . 73 percent of Americans. . . favor a law that would require every gun sold in the United States to be test-fired first, so law enforcement would have its fingerprint in case it were ever used
> Based on the least-squares criterion, the line that best fits a set of data points is the one with the ____ possible sum of squared errors.
> Regarding the variables in a regression analysis, a. what is the independent variable called? b. what is the dependent variable called?
> Identify one use of a regression equation.
> What kind of plot is useful for deciding whether finding a regression line for a set of data points is reasonable?
> Explain your answers. If a line has a positive slope, y-values on the line decrease as the x-values decrease.
