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essentials of statistics

The Basic Practice Of Statistics 5th Edition David S Moore - Solutions

The brain responds to sound. The usual way to study the brain’s response to ST EP sounds is to have subjects listen to “pure tones.” The response to recognizable sounds may differ. To compare responses, researchers anesthetized macaque monkeys.They fed pure tones and also monkey calls
Ancient air, continued. Do the data of Exercise 17.7 give good reason to think ST EP that the percent of nitrogen in the air during the Cretaceous era was different from the present 78.1%? Carry out a test of significance, following the four-step process as illustrated in Example 17.3.
Is it significant? The one-sample t statistic from a sample of n = 25 observations for the two-sided test of H0: μ = 64 Ha: μ = 64 has the value t = 1.12.(a) What are the degrees of freedom for t?(b) Locate the two critical values t∗ from Table C that bracket t. What are the two-sided P-values
Is it significant? The one-sample t statistic for testing H0: μ = 0 Ha: μ > 0 from a sample of n = 15 observations has the value t = 1.82.(a) What are the degrees of freedom for this statistic?(b) Give the two critical values t∗ from Table C that bracket t. What are the one-sided P-values for
Ancient air. The composition of the earth’s atmosphere may have changed over ST EP time. To try to discover the nature of the atmosphere long ago, we can examine the gas in bubbles inside ancient amber. Amber is tree resin that has hardened and been trapped in rocks. The gas in bubbles within
To gamble or not to gamble. Our decisions depend on how the options are presented to us. Here’s an experiment that illustrates this phenomenon. Tell 20 subjects that they have been given $50 but can’t keep it all. Then present them with a long series of choices between bets they can make with
Critical values. What critical value t∗ from TableCwould you use for a confidence interval for the mean of the population in each of the following situations?(a) A 95% confidence interval based on n = 10 observations.(b) A 99% confidence interval from an SRS of 20 observations.(c) A 90%
More critical values. You have an SRS of size 25 and calculate the one-sample t statistic. What is the critical value t∗ such that(a) t has probability 0.025 to the right of t∗?(b) t has probability 0.75 to the left of t∗?
Critical values. Use Table C or software to find(a) the critical value for a one-sided test with level α = 0.05 based on the t(5)distribution.(b) the critical value for a 98% confidence interval based on the t(21) distribution.
Is that light moving? When two lights close together blink alternately, we“see” one light moving back and forth if the time between blinks is short.What is the longest interval of time between blinks that preserves the illusion of motion? Ask subjects to turn a knob that slows the blinking
Travel time to work. A study of commuting times reports the travel times to work of a random sample of 20 employed adults in New York State. The mean is x = 31.25 minutes and the standard deviation is s = 21.88 minutes. What is the standard error of the mean?c Oote Boe Photography/Alamy
Two types of error. Your company markets a computerized medical diagnostic program used to evaluate thousands of people. The program scans the results of routine medical tests (pulse rate, blood tests, etc.) and refers the case to a doctor if there is evidence of a medical problem. The program
How power behaves. Another approach to improving the unsatisfactory power APPLET ••• of the test in Exercise 15.13 is to improve the measurement process. That is, use a measurement process that is less variable. Use the Power of a Test applet to find the power of the test in Exercise 15.13 in
How power behaves. In the setting of Exercise 15.13, use the Power of a Test APPLET ••• applet to find the power in each of the following circumstances. Be sure to set the applet to the two-sided alternative.(a) Standard deviation σ = 0.2, significance level α = 0.05, alternative μ =5.1,
Thinking about power. Answer these questions in the setting of the previous exercise about measuring the conductivity of a liquid.(a) You could get higher power against the same alternative with the same α by changing the number of measurements you make. Should you make more measurements or fewer
What is power? You manufacture and sell a liquid product whose electrical conductivity is supposed to be 5. You plan to make 6 measurements of the conductivity of each lot of product. You know that the standard deviation of your measurements is σ = 0.2. If the product meets specifications, the
Number skills of young men. Suppose that scores of men aged 21 to 25 years on the quantitative part of the National Assessment of Educational Progress (NAEP)test follow a Normal distribution with standard deviation σ = 60. You want to estimate the mean score within ±10 with 90% confidence. How
Body mass index of young women. Example 14.1 (page 360) assumed that the body mass index (BMI) of all American young women follows a Normal distribution with standard deviation σ = 7.5. How large a sample would be needed to estimate the mean BMI μ in this population to within ±1 with 95%
Searching for ESP. A researcher looking for evidence of extrasensory perception(ESP) tests 500 subjects. Four of these subjects do significantly better (P < 0.01)than random guessing.(a) You can’t conclude that these four people have ESP. Why not?(b) What should the researcher now do to test
Confidence intervals help. Give a 95% confidence interval for the mean pH μfor each sample size in the previous exercise. The intervals, unlike the P-values, give a clear picture of what mean pH values are plausible for each sample.
Detecting acid rain. Emissions of sulfur dioxide by industry set off chemical changes in the atmosphere that result in “acid rain.”The acidity of liquids is measured by pH on a scale of 0 to 14. Distilled water has pH 7.0, and lower pH values indicate acidity. Normal rain is somewhat acidic, so
Is it significant? In the absence of special preparation SAT mathematics (SATM)scores in recent years have varied Normally with mean μ = 518 and σ = 114. Fifty students go through a rigorous training program designed to raise their SATM scores by improving their mathematics skills. Either by hand
Is your food safe? “Do you feel confident or not confident that the food available at most grocery stores is safe to eat?”When a Gallup Poll asked this question, 87%of the sample said they were confident.3 Gallup announced the poll’s margin of error for 95% confidence as ±3 percentage
Sample size and margin of error. Example 14.1 (page 360) described NHANES survey data on the body mass index (BMI) of 654 young women. The mean BMI in the sample was x = 26.8. We treated these data as an SRS from a Normally distributed population with standard deviation σ = 7.5.(a) Suppose that we
Confidence level and margin of error. Example 14.1 (page 360) described NHANES survey data on the body mass index (BMI) of 654 young women. The mean BMI in the sample was x = 26.8. We treated these data as an SRS from a Normally distributed population with standard deviation σ = 7.5.(a) Give three
Sampling shoppers. A marketing consultant observes 50 consecutive shoppers at a supermarket, recording how much each shopper spends in the store. Suggest some reasons why it may be risky to act as if 50 consecutive shoppers at a particular time are an SRS of all shoppers at this store.
Running red lights.Asurvey of licensed drivers inquired about running red lights.One question asked, “Of every ten motorists who run a red light, about how many do you think will be caught?” The mean result for 880 respondents was x = 1.92 and the standard deviation was s = 1.83.2 For this
Rate that movie.Aprofessor interested in the opinions of college-age adults about a new hit movie asks the 25 students in her course on documentary filmmaking to rate the entertainment value of the movie on a scale of 0 to 5. Which of the following is the most important reason why a confidence
Testing a random number generator. A random number generator is supposed to produce random numbers that are uniformly distributed on the interval from 0 to 1. If this is true, the numbers generated come from a population with μ = 0.5 and σ = 0.2887. A command to generate 100 random numbers gives
Significance froma table.Atest of H0: μ = 1 against Ha: μ = 1 has test statistic z = 1.776. Is this test significant at the 5% level (α = 0.05)? Is it significant at the 1% level?
Significance froma table.Atest of H0: μ = 1 against Ha: μ > 1 has test statistic z = 1.776. Is this test significant at the 5% level (α = 0.05)? Is it significant at the 1% level (α = 0.01)?
Reading a computer screen. Does the use of fancy type fonts slow down the ST EP reading of text on a computer screen? Adults can read four paragraphs of text in an average time of 22 seconds in the common Times New Roman font. Ask 25 adults to read this text in the ornate font named Gigi. Here are
Measuring conductivity. Here are 6 measurements of the electrical conductivity ST EP of a liquid:5.32 4.88 5.10 4.73 5.15 4.75 The liquid is supposed to have conductivity 5. Do the measurements give good evidence that the true conductivity is not 5?The 6 measurements are an SRS from the population
The z statistic. Published reports of research work are terse. They often report just a test statistic and P-value. For example, the conclusion of Example 14.9 might be stated as “(z = −1.09, P = 0.2758).” Find the values of the one-sample z statistic needed to complete these conclusions:(a)
Measuring conductivity. Exercise 14.7 describes 6 measurements of the elec-APPLET ••• trical conductivity of a liquid. You stated the null and alternative hypotheses in Exercise 14.9.(a) One set of measurements has mean conductivity x = 4.98. Enter this x, along with the other required
Student attitudes. Exercise 14.6 describes a study of the attitudes of older college APPLET ••• students. You stated the null and alternative hypotheses in Exercise 14.8.(a) One sample of 25 students had mean SSHA score x = 118.6. Enter this x, along with the other required information, into
Protecting long-distance runners. A randomized comparative experiment compared vitamin C with a placebo as protection against respiratory infections after running a very long distance. The report of the study said:5 Sixty-eight percent of the runners in the placebo group reported the development of
Job satisfaction: find the P-value. The P-value in Example 14.8 is the probability(taking the null hypothesis μ = 0 to be true) that x takes a value at least as far from 0 as 17.(a) What is the sampling distribution of x when μ = 0? This distribution appears in Figure 14.7.(b) Do a Normal
Sweetening colas: find the P-value. The P-value for the first cola in Example 14.7 is the probability (taking the null hypothesis μ = 0 to be true) that x takes a value at least as large as 0.3.(a) What is the sampling distribution of x when μ = 0? This distribution appears in Figure 14.6.(b) Do
Stating hypotheses. In planning a study of the birth weights of babies whose mothers did not see a doctor before delivery, a researcher states the hypotheses as H0: x = 1000 grams Ha: x < 1000 grams What’s wrong with this?
Women’s heights. The average height of 18-year-old American women is 64.2 inches. You wonder whether the mean height of this year’s female graduates from your local high school is different from the national average. You measure an SRS of 78 female graduates and find that x = 63.1 inches. What
Grading a teaching assistant. The examinations in a large accounting class are scaled after grading so that the mean score is 50. The professor thinks that one teaching assistant is a poor teacher and suspects that his students have a lower mean score than the class as a whole. The TA’s students
Measuring conductivity. State the null and alternative hypotheses for the study of electrical conductivity described in Exercise 14.7. (Is the alternative hypothesis one-sided or two-sided?)
Student attitudes. State the null and alternative hypotheses for the study of older students’ attitudes described in Exercise 14.6. (Is the alternative hypothesis onesided or two-sided?)
Measuring conductivity. The National Institute of Standards and Technology(NIST) supplies a “standard liquid” whose electrical conductivity is supposed to be exactly 5. Is there reason to think that the true conductivity of a shipment of this liquid is not 5? To find out, NIST measures the
Student attitudes. The Survey of Study Habits and Attitudes (SSHA) is a psychological test that measures students’ study habits and attitude toward school.Scores range from 0 to 200. The mean score for college students is about 115, and the standard deviation is about 30. A teacher suspects that
IQ test scores. Here are the IQ test scores of 31 seventh-grade girls in a Midwest ST EP school district:4 114 100 104 89 102 91 114 114 103 105 108 130 120 132 111 128 118 119 86 72 111 103 74 112 107 103 98 96 112 112 93(a) These 31 girls are an SRS of all seventh-grade girls in the school
Measuring conductivity. The National Institute of Standards and Technology ST EP(NIST) supplies “standard materials” whose physical properties are supposed to be known. For example, you can buy from NIST a liquid whose electrical conductivity is supposed to be 5. (The units for conductivity are
Find a critical value. The critical value z∗ for confidence level 97.5% is not in Table C. Use software or Table A of standard Normal probabilities to find z∗.Include in your answer a sketch like Figure 14.3 with C = 0.975 and your critical value z∗ marked on the axis.
Confidence intervals in action. The idea of an 80% confidence interval is that•••APPLET in 80% of all samples the method produces an interval that captures the true parameter value. That’s not high enough confidence for practical use, but 80% hits and 20% misses make it easy to see how a
Number skills of young men. The National Assessment of Educational Progress(NAEP) gave a test of basic arithmetic and the ability to apply it in everyday life to a sample of 840 men 21 to 25 years of age.2 Scores range from 0 to 500; for example, someone with a score of 325 can determine the price
Checking for survey errors. One way of checking the effect of undercoverage, nonresponse, and other sources of error in a sample survey is to compare the sample with known facts about the population. About 12% of American adults are black.The number X of blacks in random samples of 1500 adults
College admissions. A small liberal arts college would like to have an entering class of 415 students next year. Past experience shows that about 27% of the students admitted will decide to attend. The college therefore plans to admit 1535 students. Suppose that students make their decisions
Using Benford’s law. According to Benford’s law (Example 10.7, page 271) the probability that the first digit of the amount of a randomly chosen invoice is a 1 or a 2 is 0.477. You examine 90 invoices from a vendor and find that 29 have first digits 1 or 2. If Benford’s law holds, the count
Proofreading. Return to the proofreading setting of Exercise 13.5.(a) If X is the number of word errors missed, what is the distribution of X? If Y is the number of word errors caught, what is the distribution of Y ?(b) What is the mean number of errors caught? What is the mean number of errors
Random digit dialing. When an opinion poll calls residential telephone numbers at random, only 20% of the calls reach a live person. You watch the random digit dialing machine make 15 calls.Randomness turns silver to bronze After many charges of favoritism by judges, the rules for scoring
Google does binomial. Point yourWeb browser to www.google.com. Instead of searching the Web or looking for images, you can request a calculation in the Search box.(a) Enter 5 choose 2 and click Search. What does Google return?(b) You see that Google calculates the binomial coefficient “5 choose
Random digit dialing. When an opinion poll calls residential telephone numbers at random, only 20% of the calls reach a live person. You watch the random digit dialing machine make 15 calls.(a) What is the probability that exactly 3 calls reach a person?(b) What is the probability that at most 3
Proofreading. Typing errors in a text are either nonword errors (as when “the” is typed as “teh”) or word errors that result in a real but incorrect word. Spell-checking software will catch nonword errors but not word errors. Human proofreaders catch 70% of word errors. You ask a fellow
Teens feel the heat. Opinion polls find that 63% of American teens say that their parents put at least some pressure on them to get into a good college.2 If you take an SRS of 1000 teens, what is the approximate distribution of the number in your sample who say they feel at least some pressure from
Computer instruction. A student studies binomial distributions using computerassisted instruction. After the lesson, the computer presents 10 problems. The student solves each problem and enters her answer. The computer gives additional instruction between problems if the answer is wrong. The count
Random digit dialing. When an opinion poll calls residential telephone numbers at random, only 20% of the calls reach a live person. You watch the random dialing machine make calls. X is the number of calls until the first live person answers.
Random digit dialing. When an opinion poll calls residential telephone numbers at random, only 20% of the calls reach a live person. You watch the random dialing machine make 15 calls. X is the number that reach a live person.
False HIV positives. Continue your work from Exercise 12.14. What is the probability that a person has the antibody, given that the test is positive? (Your result illustrates a fact that is important when considering proposals for widespread testing for HIV, illegal drugs, or agents of biological
Peanut and tree nut allergies. Continue your work from Exercise 12.13. What is the conditional probability that exactly 1 of the people will be allergic to peanuts or tree nuts, given that at least 1 of the 5 people suffers from one of these allergies?Politically correct
Testing for HIV. Enzyme immunoassay tests are used to screen blood specimens for the presence of antibodies to HIV, the virus that causes AIDS. Antibodies indicate the presence of the virus. The test is quite accurate but is not always correct. Here are approximate probabilities of positive and
Peanut and tree nut allergies. About 1% of the American population is allergic ST EP to peanuts or tree nuts.6 Choose 5 individuals at random and let the random variable X be the number in this sample who are allergic to peanuts or tree nuts. The possible values X can take are 0, 1, 2, 3, 4, and 5.
Independent? The Clemson University Fact Book for 2007 shows that 123 of the university’s 338 assistant professors were women, along with 76 of the 263 associate professors and 73 of the 375 full professors.(a) What is the probability that a randomly chosen Clemson professor is a woman?(b) What
The probability of a flush. A poker player holds a flush when all 5 cards in the hand belong to the same suit (clubs, diamonds, hearts, or spades).We will find the probability of a flush when 5 cards are dealt. Remember that a deck contains 52 cards, 13 of each suit, and that when the deck is well
Teens online.We saw in Example 12.8 that 93% of teenagers are online and that ST EP 55% of online teens have posted a profile on a social networking site. Of online teens with a profile, 76% have placed comments on a friend’s blog. What percent of all teens are online, have a profile, and comment
At the gym. Suppose that 10% of adults belong to health clubs, and 40% of these health club members go to the club at least twice a week. What percent of all adults go to a health club at least twice a week? Write the information given in terms of probabilities and use the general multiplication
Computer games. Here is the distribution of computer games sold by type of game:4 Game type Probability Strategy 0.354 Role playing 0.139 Family entertainment 0.127 Shooters 0.109 Children’s 0.057 Other 0.214 What is the conditional probability that a computer game is a role-playing game, given
Distance learning. In the setting of Exercise 12.5, what is the conditional probability that a student is local, given that he or she is less than 25 years old?
College degrees. In the setting of Exercise 12.4, what is the conditional probability that a degree is earned by a woman, given that it is a bachelor’s degree?
Distance learning. A study of the students taking distance learning courses at a university finds that they are mostly older students not living in the university town. Choose a distance learning student at random. Let Abe the event that the student is 25 years old or older and B the event that the
College degrees. Of all college degrees awarded in the United States, 50% are bachelor’s degrees, 59% are earned by women, and 29% are bachelor’s degrees earned by women. Make a Venn diagram and use it to answer these questions.(a) What percent of all degrees are earned by men?(b) What percent
Lost Internet sites. Internet sites often vanish or move, so that references to them can’t be followed. In fact, 13% of Internet sites referenced in major scientific journals are lost within two years after publication.2 If a paper contains seven Internet references, what is the probability that
Common names. The Census Bureau says that the 10 most common names in the United States are (in order) Smith, Johnson, Williams, Brown, Jones, Miller, Davis, Garcia, Rodriguez, and Wilson. These names account for 9.6% of all U.S.residents. Out of curiosity, you look at the authors of the textbooks
Older college students. Government data show that 8% of adults are full-time college students and that 30% of adults are age 55 or older. Nonetheless, we can’t conclude that, because (0.08)(0.30) = 0.024, about 2.4% of adults are college students 55 or older. Why not?
Detecting gypsy moths. The gypsy moth is a serious threat to oak and aspen trees.Astate agriculture department places traps throughout the state to detect the moths. When traps are checked periodically, the mean number of moths trapped is only 0.5, but some traps have several moths. The
What does the central limit theorem say? Asked what the central limit theorem says, a student replies, “As you take larger and larger samples from a population, the histogram of the sample values looks more and more Normal.” Is the student right? Explain your answer.Bruce Coleman/Alamy
Measurements in the lab. Juan makes a measurement in a chemistry laboratory and records the result in his lab report. The standard deviation of students’ lab measurements is σ = 10 milligrams. Juan repeats the measurement 3 times and records the mean x of his 3 measurements.(a) What is the
Larger sample, more accurate estimate. Suppose that in fact the blood cholesterol level of all men aged 20 to 34 follows the Normal distribution with meanμ = 188 milligrams per deciliter (mg/dl) and standard deviation σ = 41 mg/dl.(a) Choose an SRS of 100 men from this population. What is the
A sample of young men. A government sample survey plans to measure the blood cholesterol level of an SRS of men aged 20 to 34. The researchers will report the mean x from their sample as an estimate of the mean cholesterol level μ in this population.(a) Explain to someone who knows no statistics
Generating a sampling distribution. Let’s illustrate the idea of a sampling distribution in the case of a very small sample from a very small population. The population is the scores of 10 students on an exam:Student 0 1 2 3 4 5 6 7 8 9 Score 82 62 80 58 72 73 65 66 74 62 The parameter of
Sampling distribution versus population distribution. During World War II, 12,000 able-bodied male undergraduates at the University of Illnois participated in required physical training. Each student ran a timed mile. Their times followed the Normal distribution with mean 7.11 minutes and standard
Insurance. The idea of insurance is that we all face risks that are unlikely but carry high cost. Think of a fire destroying your home. Insurance spreads the risk:we all pay a small amount, and the insurance policy pays a large amount to those few of us whose homes burn down. An insurance company
The law of large numbers made visible. Roll two balanced dice and count the APPLET ••• spots on the up faces. The probability model appears in Example 10.5 (page 267).You can see that this distribution is symmetric with 7 as its center, so it’s no surprise that the mean is μ = 7. This is
Ancient projectile points. Most of what we know about North America before Columbus comes from artifacts such as fragments of clay pottery and stone projectile points. Locations and cultures can be distinguished by the types of artifacts found. At one site in North Carolina, 82% of the projectile
Florida voters. Florida has played a key role in recent presidential elections. Voter registration records show that 41% of Florida voters are registered as Democrats and 37% as Republicans. (Most of the others did not choose a party.) To test a random digit dialing device, you use it to call 250
Effects of caffeine. How does caffeine affect our bodies? In a matched pairs experiment, subjects pushed a button as quickly as they could after taking a caffeine pill and also after taking a placebo pill. The mean pushes per minute were 283 for the placebo and 311 for caffeine. Is each of the
Winning the ACC tournament. The annual Atlantic Coast Conference men’s basketball tournament has temporarily taken Joe’s mind off the Chicago Cubs. He says to himself, “I think that Duke has probability 0.2 of winning. Clemson’s probability is half of Duke’s and North Carolina’s
Will you have an accident? The probability that a randomly chosen driver will be involved in an accident in the next year is about 0.2. This is based on the proportion of millions of drivers who have accidents. “Accident” includes things like crumpling a fender in your own driveway, not just
Running a mile. A study of 12,000 able-bodied male students at the University of Illinois found that their times for the mile run were approximately Normal with mean 7.11 minutes and standard deviation 0.74 minute.6 Choose a student at random from this group and call his time for the mile Y .
Grades in a statistics course. North Carolina State University posts the grade distributions for its courses online.5 Students in Statistics 101 in the Fall 2007 semester received 26% A’s, 42% B’s, 20% C’s, 10% D’s, and 2% F’s. Choose a Statistics 101 student at random. To “choose at
Iowa Test scores. The Normal distribution with mean μ = 6.8 and standard deviationσ = 1.6 is a good description of the Iowa Test vocabulary scores of seventhgrade students in Gary, Indiana. This is a continuous probability model for the score of a randomly chosen student. Figure 3.1 (page 68)
Adding random numbers. Generate two random numbers between 0 and 1 and take X to be their sum. The sum X can take any value between 0 and 2. The density curve of X is the triangle shown in Figure 10.7.0 1 2 Height = 1 FIGURE 10.7 The density curve for the sum of two random numbers, for Exercise
Random numbers. Let Y be a random number between 0 and 1 produced by the idealized random number generator described in Example 10.8 and Figure 10.4.Find the following probabilities:(a) P(Y ≤ 0.4)(b) P(Y < 0.4)(c) P(0.3 ≤ Y ≤ 0.5)
Working out. Choose a person aged 19 to 25 years at random and ask, “In the past seven days, how many times did you go to an exercise or fitness center or work out?”Call the response X for short. Based on a large sample survey, here is a probability model for the answer you will get:4 Days 0 1
Benford’s law. The first digit of a randomly chosen expense account claim follows Benford’s law (Example 10.7). Consider the events A = {first digit is 7 or greater}B = {first digit is odd}(a) What outcomes make up the event A? What is P(A)?(b) What outcomes make up the event B? What is

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