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statistics informed decisions using data
Understanding Basic Statistics 7th Edition Charles Henry Brase, Corrinne Pellillo Brase - Solutions
Critical Thinking: Boxplots and Confi dence Intervals The distribution of heights of 18-year-old men in the United States is approximately normal, with mean 68 inches and standard deviation 3 inches (U.S. Census Bureau). In Minitab, we can simulate the drawing of random samples of size 20 from this
Finance: P/E Ratio The price of a share of stock divided by a company’s estimated future earnings per share is called the P/E ratio. High P/E ratios usually indicate “growth” stocks, or maybe stocks that are simply overpriced.Low P/E ratios indicate “value’’ stocks or bargain stocks. A
Baseball: Home Run Percentage The home run percentage is the number of home runs per 100 times at bat. A random sample of 43 professional baseball players gave the following data for home run percentages (Reference: The Baseball Encyclopedia, Macmillan).1.6 2.4 1.2 6.6 2.3 0.0 1.8 2.5 6.5 1.8 2.7
Statistical Literacy For a binomial experiment with r successes out of n trials, what value do we use as a point estimate for the probability of success p on a single trial?
Statistical Literacy In order to use a normal distribution to compute confi -dence intervals for p, what conditions on np and nq need to be satisfi ed?
Critical Thinking Results of a poll of a random sample of 3003 American adults showed that 20% did not know that caffeine contributes to dehydration.The poll was conducted for the Nutrition Information Center and had a margin of error of 1.4%.(a) Does the margin of error take into account any
Critical Thinking You want to conduct a survey to determine the proportion of people who favor a proposed tax policy. How does increasing the sample size affect the size of the margin of error?
Critical Thinking Jerry tested 30 laptop computers owned by classmates enrolled in a large computer science class and discovered that 22 were infected with keystroke-tracking spyware. Is it appropriate for Jerry to use his data to estimate the proportion of all laptops infected with such spyware?
Critical Thinking: Brain Teaser A requirement for using the normal distribution to approximate the pˆ distribution is that both np 5 and nq 5.Since we usually do not know p, we estimate p by pˆ and q by qˆ 5 1 pˆ. Then we require that npˆ 5 and nqˆ 5.Show that the conditions npˆ 5 and nqˆ
Basic Computation: Confi dence Interval for p Consider n 5 100 binomial trials with r 5 30 successes.(a) Check Requirements Is it appropriate to use a normal distribution to approximate the pˆ distribution?(b) Find a 90% confi dence interval for the population proportion of successes p.(c)
Basic Computation: Confi dence Interval for p Consider n 5 200 binomial trials with r 5 80 successes.(a) Check Requirements Is it appropriate to use a normal distribution to approximate the pˆ distribution?(b) Find a 95% confi dence interval for the population proportion of successes p.(c)
Basic Computation: Sample Size What is the minimal sample size needed for a 95% confi dence interval to have a maximal margin of error of 0.1(a) if a preliminary estimate for p is 0.25?(b) if there is no preliminary estimate for p?
Basic Computation: Sample Size What is the minimal sample size needed for a 99% confi dence interval to have a maximal margin of error of 0.06(a) if a preliminary estimate for p is 0.8?(b) if there is no preliminary estimate for p?
Myers–Briggs: Actors Isabel Myers was a pioneer in the study of personality types. The following information is taken from A Guide to the Development and Use of the Myers–Briggs Type Indicator by Myers and McCaulley(Consulting Psychologists Press). In a random sample of 62 professional actors,
Myers–Briggs: Judges In a random sample of 519 judges, it was found that 285 were introverts. (See reference in Problem 11.)(a) Let p represent the proportion of all judges who are introverts. Find a point estimate for p.(b) Find a 99% confi dence interval for p. Give a brief interpretation of
Navajo Lifestyle: Traditional Hogans A random sample of 5222 permanent dwellings on the entire Navajo Indian Reservation showed that 1619 were traditional Navajo hogans (Navajo Architecture: Forms, History, Distributions by Jett and Spencer, University of Arizona Press).(a) Let p be the proportion
Archaeology: Pottery Santa Fe black-on-white is a type of pottery commonly found at archaeological excavations in Bandelier National Monument. At one excavation site a sample of 592 potsherds was found, of which 360 were identifi ed as Santa Fe black-on-white (Bandelier Archaeological Excavation
Health Care: Colorado Physicians A random sample of 5792 physicians in Colorado showed that 3139 provide at least some charity care (i.e., treat poor people at no cost). These data are based on information from State Health Care Data: Utilization, Spending, and Characteristics (American Medical
Law Enforcement: Escaped Convicts Case studies showed that out of 10,351 convicts who escaped from U.S. prisons, only 7867 were recaptured (The Book of Odds by Shook and Shook, Signet).(a) Let p represent the proportion of all escaped convicts who will eventually be recaptured. Find a point
Fishing: Barbless Hooks In a combined study of northern pike, cutthroat trout, rainbow trout, and lake trout, it was found that 26 out of 855 fi sh died when caught and released using barbless hooks on fl ies or lures. All hooks were removed from the fi sh (Source: A National Symposium on Catch and
Focus Problem: Trick or Treat In a survey of a random sample of 35 households in the Cherry Creek neighborhood of Denver, it was found that 11 households turned out the lights and pretended not to be home on Halloween.(a) Compute a 90% confi dence interval for p, the proportion of all housholds in
Marketing: Customer Loyalty In a marketing survey, a random sample of 730 women shoppers revealed that 628 remained loyal to their favorite supermarket during the past year (i.e., did not switch stores) (Source: Trends in the United States: Consumer Attitudes and the Supermarket, The Research
Marketing: Bargain Hunters In a marketing survey, a random sample of 1001 supermarket shoppers revealed that 273 always stock up on an item when they fi nd that item at a real bargain price. (See reference in Problem 19.)(a) Let p represent the proportion of all supermarket shoppers who always
Lifestyle: Smoking In a survey of 1000 large corporations, 250 said that, given a choice between a job candidate who smokes and an equally qualifi ed nonsmoker, the nonsmoker would get the job (USA Today).(a) Let p represent the proportion of all corporations preferring a nonsmoking candidate. Find
Opinion Poll: Crime and Violence A New York Times/CBS poll asked the question, “What do you think is the most important problem facing this country today?” Nineteen percent of the respondents answered, “Crime and violence.”The margin of sampling error was plus or minus 3 percentage points.
Medical: Blood Type A random sample of medical fi les is used to estimate the proportion p of all people who have blood type B.(a) If you have no preliminary estimate for p, how many medical fi les should you include in a random sample in order to be 85% sure that the point estimate pˆ will be
Business: Phone Contact How hard is it to reach a businessperson by phone?Let p be the proportion of calls to businesspeople for which the caller reaches the person being called on the first try.(a) If you have no preliminary estimate for p, how many business phone calls should you include in a
Campus Life: Coeds What percentage of your campus student body is female? Let p be the proportion of women students on your campus.(a) If no preliminary study is made to estimate p, how large a sample is needed to be 99% sure that a point estimate pˆ will be within a distance of 0.05 from p?(b)
Small Business: Bankruptcy The National Council of Small Businesses is interested in the proportion of small businesses that declared Chapter 11 bankruptcy last year. Since there are so many small businesses, the National Council intends to estimate the proportion from a random sample. Let p be the
Brain Teaser: Algebra Why do we use 1/4 in place of p 1 p in formula(22) for sample size when the probability of success p is unknown?(a) Show that p 1 p 5 1/4 p 1/2 2.(b) Why is p 1 p never greater than 1/4?
Expand Your Knowledge: Plus Four Confi dence Interval for a Single Proportion One of the technical diffi culties that arises in the computation of confi dence intervals for a single proportion is that the exact formula for the maximal margin of error requires knowledge of the population proportion
Critical Thinking If you have a 99% confi dence interval for m based on a simple random sample,(a) is it correct to say that the probability that m is in the specifi ed interval is 99%? Explain.(b) is it correct to say that in the long run, if you computed many, many confi dence intervals using the
Expand Your Knowledge: Two Confi dence Intervals What happens if we want several confi dence intervals to hold at the same time (concurrently)? Do we still have the same level of confi dence we had for each individual interval?(a) Suppose we have two independent random variables x1 and x2 with
Statistical Literacy Discuss each of the following topics in class or review the topics on your own. Then write a brief but complete essay in which you answer the following questions.(a) What is a null hypothesis H0?(b) What is an alternate hypothesis H1?(c) What is a type I error? a type II
Statistical Literacy In a statistical test, we have a choice of a left-tailed test, a right-tailed test, or a two-tailed test. Is it the null hypothesis or the alternate hypothesis that determines which type of test is used? Explain your answer.
Statistical Literacy If we fail to reject (i.e., “accept”) the null hypothesis, does this mean that we have proved it to be true beyond all doubt? Explain your answer.
Statistical Literacy If we reject the null hypothesis, does this mean that we have proved it to be false beyond all doubt? Explain your answer.
Statistical Literacy What terminology do we use for the probability of rejecting the null hypothesis when it is true? What symbol do we use for this probability? Is this the probability of a type I or a type II error?
Statistical Literacy What terminology do we use for the probability of rejecting the null hypothesis when it is, in fact, false?
Statistical Literacy If the P-value in a statistical test is greater than the level of signifi cance for the test, do we reject or fail to reject H0?
Statistical Literacy If the P-value in a statistical test is less than or equal to the level of signifi cance for the test, do we reject or fail to reject H0?
Statistical Literacy Suppose the P-value in a right-tailed test is 0.0092. Based on the same population, sample, and null hypothesis, what is the P-value for a corresponding two-tailed test?
Statistical Literacy Suppose the P-value in a two-tailed test is 0.0134. Based on the same population, sample, and null hypothesis, and assuming the test statistic z is negative, what is the P-value for a corresponding left-tailed test?
Basic Computation: Setting Hypotheses Suppose you want to test the claim that a population mean equals 40.(a) State the null hypothesis.(b) State the alternate hypothesis if you have no information regarding how the population mean might differ from 40.(c) State the alternate hypothesis if you
Basic Computation: Setting Hypotheses Suppose you want to test the claim that a population mean equals 30.(a) State the null hypothesis.(b) State the alternate hypothesis if you have no information regarding how the population mean might differ from 30.(c) State the alternate hypothesis if you
Basic Computation: Find Test Statistic, Corresponding P-value, and Conclude Test A random sample of size 20 from a normal distribution with s 5 4 produced a sample mean of 8.(a) Check Requirements Is the x distribution normal? Explain.(b) Compute the sample test statistic z under the null
Basic Computation: Find the Test Statistic, Corresponding P-value, and Conclude Test A random sample of size 16 from a normal distribution with s 5 3 produced a sample mean of 4.5.(a) Check Requirements Is the x distribution normal? Explain.(b) Compute the sample test statistic z under the null
Veterinary Science: Colts The body weight of a healthy 3-month-old colt should be about m 5 60 kg (Source: The Merck Veterinary Manual, a standard reference manual used in most veterinary colleges).(a) If you want to set up a statistical test to challenge the claim that m 5 60 kg, what would you
Marketing: Shopping Time How much customers buy is a direct result of how much time they spend in a store. A study of average shopping times in a large national housewares store gave the following information (Source: Why We Buy: The Science of Shopping by P. Underhill):Women with female companion:
Meteorology: Storms Weatherwise magazine is published in association with the American Meteorological Society. Volume 46, Number 6 has a rating system to classify Nor’easter storms that frequently hit New England states and can cause much damage near the ocean coast. A severe storm has an average
Chrysler Concorde: Acceleration Consumer Reports stated that the mean time for a Chrysler Concorde to go from 0 to 60 miles per hour is 8.7 seconds.(a) If you want to set up a statistical test to challenge the claim of 8.7 seconds, what would you use for the null hypothesis?(b) The town of
Dividend Yield: Australian Bank Stocks Let x be a random variable representing dividend yield of Australian bank stocks. We may assume that x has a normal distribution with s 5 2.4%. A random sample of 10 Australian bank stocks gave the following yields.5.7 4.8 6.0 4.9 4.0 3.4 6.5 7.1 5.3 6.1 The
Glucose Level: Horses Gentle Ben is a Morgan horse at a Colorado dude ranch. Over the past 8 weeks, a veterinarian took the following glucose readings from this horse (in mg/100 ml).93 88 82 105 99 110 84 89 The sample mean is x < 93.8. Let x be a random variable representing glucose readings taken
Ecology: Hummingbirds Bill Alther is a zoologist who studies Anna’s hummingbird(Calypte anna) (Reference: Hummingbirds by K. Long and W. Alther).Suppose that in a remote part of the Grand Canyon, a random sample of six of these birds was caught, weighed, and released. The weights (in grams) were
Finance: P/E of Stocks The price-to-earnings (P/E) ratio is an important tool in fi nancial work. A random sample of 14 large U.S. banks (J.P. Morgan, Bank of America, and others) gave the following P/E ratios (Reference: Forbes).24 16 22 14 12 13 17 22 15 19 23 13 11 18 The sample mean is x <
Insurance: Hail Damage Nationally, about 11% of the total U.S. wheat crop is destroyed each year by hail (Reference: Agricultural Statistics, U.S.Department of Agriculture). An insurance company is studying wheat hail damage claims in Weld County, Colorado. A random sample of 16 claims in Weld
Medical: Red Blood Cell Volume Total blood volume (in ml) per body weight(in kg) is important in medical research. For healthy adults, the red blood cell volume mean is about m 5 28 ml/kg (Reference: Laboratory and Diagnostic Tests by F. Fischbach). Red blood cell volume that is too low or too high
Statistical Literacy For the same sample data and null hypothesis, how does the P-value for a two-tailed test of m compare to that for a one-tailed test?
Statistical Literacy To test m for an x distribution that is mound-shaped using sample size n 30, how do you decide whether to use the normal or the Student’s t distribution?
Statistical Literacy When using the Student’s t distribution to test m, what value do you use for the degrees of freedom?
Critical Thinking Consider a test for m. If the P-value is such that you can reject H0 at the 5% level of signifi cance, can you always reject H0 at the 1%level of signifi cance? Explain.
Critical Thinking Consider a test for m. If the P-value is such that you can reject H0 for a 5 0.01, can you always reject H0 for a 5 0.05? Explain.
Critical Thinking If sample data is such that for a one-tailed test of m you can reject H0 at the 1% level of signifi cance, can you always reject H0 for a twotailed test at the same level of signifi cance? Explain.
Basic Computation: P-value Corresponding to t Value For a Student’s t distribution with d.f. 5 10 and t 5 2.930,(a) fi nd an interval containing the corresponding P-value for a two-tailed test.(b) fi nd an interval containing the corresponding P-value for a right-tailed test.
Basic Computation: P-value Corresponding to t Value For a Student’s t distribution with d.f. 5 16 and t 5 1.830,(a) fi nd an interval containing the corresponding P-value for a two-tailed test.(b) fi nd an interval containing the corresponding P-value for a left-tailed test.
Basic Computation: Testing m, s Unknown A random sample of 25 values is drawn from a mound-shaped and symmetrical distribution. The sample mean is 10 and the sample standard deviation is 2.Use a level of signifi cance of 0.05 to conduct a two-tailed test of the claim that the population mean is
Basic Computation: Testing m, s Unknown A random sample has 49 values.The sample mean is 8.5 and the sample standard deviation is 1.5. Use a level of signifi cance of 0.01 to conduct a left-tailed test of the claim that the population mean is 9.2.(a) Check Requirements Is it appropriate to use a
Meteorology: Storms Weatherwise is a magazine published by the American Meteorological Society. One issue gives a rating system used to classify Nor’easter storms that frequently hit New England and can cause much damage near the ocean. A severe storm has an average peak wave height of m 5 16.4
Medical: Blood Plasma Let x be a random variable that represents the pH of arterial plasma (i.e., acidity of the blood). For healthy adults, the mean of the x distribution is m 5 7.4 (Reference: The Merck Manual, a commonly used reference in medical schools and nursing programs). A new drug for
Wildlife: Coyotes A random sample of 46 adult coyotes in a region of northern Minnesota showed the average age to be x 5 2.05 years, with sample standard deviation s 5 0.82 years (based on information from the book Coyotes: Biology, Behavior and Management by M. Bekoff, Academic Press). However, it
Fishing: Trout Pyramid Lake is on the Paiute Indian Reservation in Nevada.The lake is famous for cutthroat trout. Suppose a friend tells you that the average length of trout caught in Pyramid Lake is m 5 19 inches. However, the Creel Survey (published by the Pyramid Lake Paiute Tribe Fisheries
Investing: Stocks Socially conscious investors screen out stocks of alcohol and tobacco makers, fi rms with poor environmental records, and companies with poor labor practices. Some examples of “good,” socially conscious companies are Johnson and Johnson, Dell Computers, Bank of America, and
Agriculture: Ground Water Unfortunately, arsenic occurs naturally in some ground water (Reference: Union Carbide Technical Report K/UR-1). A mean arsenic level of m 5 8.0 parts per billion (ppb) is considered safe for agricultural use. A well in Texas is used to water cotton crops. This well is
Medical: Red Blood Cell Count Let x be a random variable that represents red blood cell (RBC) count in millions of cells per cubic millimeter of whole blood. Then x has a distribution that is approximately normal. For the population of healthy female adults, the mean of the x distribution is about
Medical: Hemoglobin Count Let x be a random variable that represents hemoglobin count (HC) in grams per 100 milliliters of whole blood. Then x has a distribution that is approximately normal, with population mean of about 14 for healthy adult women (see reference in Problem 17). Suppose that a
Ski Patrol: Avalanches Snow avalanches can be a real problem for travelers in the western United States and Canada. A very common type of avalanche is called the slab avalanche. These have been studied extensively by David McClung, a professor of civil engineering at the University of British
Longevity: Honolulu USA Today reported that the state with the longest mean life span is Hawaii, where the population mean life span is 77 years.A random sample of 20 obituary notices in the Honolulu Advertizer gave the following information about life span (in years) of Honolulu residents:72 68 81
Fishing: Atlantic Salmon Homser Lake, Oregon, has an Atlantic salmon catch and release program that has been very successful. The average fi sherman’s catch has been m 5 8.8 Atlantic salmon per day (Source: National Symposium on Catch and Release Fishing, Humboldt State University).Suppose that a
Archaeology: Tree Rings Tree-ring dating from archaeological excavation sites is used in conjunction with other chronologic evidence to estimate occupation dates of prehistoric Indian ruins in the southwestern United States. It is thought that Burnt Mesa Pueblo was occupied around 1300 a.d. (based
Critical Thinking: One-Tailed versus Two-Tailed Tests(a) For the same data and null hypothesis, is the P-value of a one-tailed test(right or left) larger or smaller than that of a two-tailed test? Explain.(b) For the same data, null hypothesis, and level of signifi cance, is it possible that a
Critical Thinking: Comparing Hypothesis Tests with U.S. Courtroom System Compare statistical testing with legal methods used in a U.S. court setting. Then discuss the following topics in class or consider the topics on your own. Please write a brief but complete essay in which you answer the
Expand Your Knowledge: Confi dence Intervals and Two-Tailed Hypothesis Tests Is there a relationship between confi dence intervals and two-tailed hypothesis tests? Let c be the level of confi dence used to construct a confi dence interval from sample data. Let a be the level of signifi cance for a
Confi dence Intervals and Two-Tailed Hypothesis Tests Change the null hypotheses of Problem 25 to H0: m 5 21 and H1: m ? 21.Repeat parts (a) and (b).
Critical Region Method: Standard Normal Solve Problem 11 using the critical region method of testing (i.e., traditional method). Compare your conclusion with the conclusion obtained by using the P-value method. Are they the same?
Critical Region Method: Student’s t Table 4 of the Appendix gives critical values for the Student’s t distribution. Use an appropriate d.f. as the row header.For a right-tailed test, the column header is the value of a found in the one-tail area row. For a left-tailed test, the column header is
Critical Region Method: Student’s t Solve Problem 13 using the critical region method of testing. Hint: See Problem 28.Compare your conclusion with the conclusion obtained by using the P-value method. Are they the same?
Critical Region Method: Student’s t Solve Problem 14 using the critical region method of testing. Hint: See Problem 28.Compare your conclusion with the conclusion obtained by using the P-value method. Are they the same?
Statistical Literacy To use the normal distribution to test a proportion p, the conditions np 5 and nq 5 must be satisfi ed. Does the value of p come from H0, or is it estimated by using pˆ from the sample?
Statistical Literacy Consider a binomial experiment with n trials and r successes. For a test for a proportion p, what is the formula for the sample test statistic? Describe each symbol used in the formula.
Critical Thinking In general, if sample data are such that the null hypothesis is rejected at the a 5 1% level of signifi cance based on a two-tailed test, is H0 also rejected at the a 5 1% level of signifi cance for a corresponding onetailed test? Explain.
Critical Thinking An article in a newspaper states that the proportion of traffi c accidents involving road rage is higher this year than it was last year, when it was 15%. Reconstruct the information of the study in terms of a hypothesis test. Discuss possible hypotheses, possible issues about the
Basic Computation: Testing p A random sample of 30 binomials trials resulted in 12 successes. Test the claim that the population proportion of successes does not equal 0.50. Use a level of signifi cance of 0.05.(a) Check Requirements Can a normal distribution be used for the pˆdistribution?
Basic Computation: Testing p A random sample of 60 binomials trials resulted in 18 successes. Test the claim that the population proportion of successes exceeds 18%. Use a level of signifi cance of 0.01.(a) Check Requirements Can a normal distribution be used for the pˆ distribution?Explain.(b)
Focus Problem: Benford’s Law Please read the Focus Problem at the beginning of this chapter. Recall that Benford’s Law claims that numbers chosen from very large data fi les tend to have “1” as the fi rst nonzero digit disproportionately often. In fact, research has shown that if you
Let p represent the population proportion of all numbers in the corporate fi le that have a fi rst nonzero digit of 1.i. Test the claim that p is less than 0.301. Use a 5 0.01.ii. If p is in fact less than 0.301, would it make you suspect that there are not enough numbers in the data fi le with
Focus Problem: Benford’s Law Again, suppose you are the auditor for a very large corporation. The revenue fi le contains millions of numbers in a large computer data bank (see Problem 7). You draw a random sample of n 5 228 numbers from this fi le and r 5 92 have a fi rst nonzero digit of
Let p represent the population proportion of all numbers in the computer fi le that have a leading digit of 1.i. Test the claim that p is more than 0.301. Use a 5 0.01.ii. If p is in fact larger than 0.301, it would seem there are too many numbers in the fi le with leading 1’s. Could this
Sociology: Crime Rate Is the national crime rate really going down? Some sociologists say yes! They say that the reason for the decline in crime rates in the 1980s and 1990s is demographics. It seems that the population is aging, and older people commit fewer crimes. According to the FBI and the
College Athletics: Graduation Rate Women athletes at the University of Colorado, Boulder, have a long-term graduation rate of 67% (Source: Chronicle of Higher Education). Over the past several years, a random sample of 38 women athletes at the school showed that 21 eventually graduated. Does this
Highway Accidents: DUI The U.S. Department of Transportation, National Highway Traffi c Safety Administration, reported that 77% of all fatally injured automobile drivers were intoxicated. A random sample of 27 records of automobile driver fatalities in Kit Carson County, Colorado, showed that 15
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