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nonparametric statistical inference

Statistical Methods For The Social Sciences 4th Edition Barbara Finlay, Alan Agresti - Solutions

5.29. The 2004 GSS asked respondents how many sex partners they had in the previous 12 months. Software reports:Variable N Mean StDev SE Mean 95.0'/. CI partners 2198 1.130 1.063 0.0227 (1.09, 1.18)(a) Interpret the confidence interval reported.(b) Based on these results, explain why the
5.27. The GSS has asked subjects, "How long have you lived in the city, town or community where you live now?" The responses of 1415 subjects in one
5.26. In response to the GSS question in 2006 about the number of hours daily spent watching TV. the responses by the 15 subjects who identified themselves as Buddhist were 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 3, 4. 4. 5.(a) Estimate the mean, standard deviation, and standard error.(b) Construct a 95%
5.25. The 2004 GSS asked, "On the average day about how many hours do you personally watch television?" Software reports;Variable N Mean SE Mean 95.07. CI TVH0URS 892 2.76 0.08 (2.60, 2.93)What's wrong with the interpretation, "In the long run, 95% of the time subjects watched between 2.60 and 2.93
5.24. Example 5.5 (page 120) analyzed data from a study that compared therapies for anorexia. For the 17 girls who received the family therapy, the changes in weight during the study were 11,11,6,9,14, -3,0,7,22, -5, -4.13,13,9.4.6,11.(a) Verify that y = 7.29 and x = 7.18 pounds.(b) Verify that the
5.22. A GSS asked, "What do you think is the ideal number of children for a family to have?" The 497 females who responded had a median of 2, mean of 3.02, and standard deviation of 1.81.(a) Report the point estimate of the population mean.(b) Find and interpret the standard error of the sample
5.21. The 2004 GSS asked male respondents how many female partners they have had sex with since their eighteenth birthday. The median = 6 and mode = 1 (16.0% of the sample). A computer printout summarizes other results:Variable n Mean StDev SE Mean 95.07. CI NUMW0MEN 1007 24.745 52.554 1.656 (21.5,
5.20. Find and interpret the 95% confidence interval for jtr. ify = 70 and x = 10, based on a sample size of(a) 5(b) 20.
5.19. Report the t-score that multiplies by the standard error to form a(a) 95% confidence interval with 5 observations(b) 95% confidence interval with 15 observations(c) 95% confidence interval with 25 observations(d) 95% confidence interval with df = 25(e) 99% confidence interval with df = 25
5.12. In the 2006 GSS, respondents were asked whether they favored or opposed the death penalty for people convicted of murder. Software shows results:x n Sample prop 95.0% CI 1885 2815 0.6696 (0.652, 0.687)Here, x refers to the number of the respondents who were in favor.(a) Show how to obtain the
5.11. State the z-score used in finding a confidence interval for a proportion with confidence level(a) 0.98(b) 0.90(c) 0.50(d) 0.9973.
5.9. In 2006, the Florida Poll conducted by Florida International University asked whether current environmental regulations arc too strict or not too strict. Of 1200 respondents, 229 said they were too strict. Find and interpret a (a) 95%, (b) 99%confidence interval for a relevant parameter at the
5.4. A national survey conducted in July 2006 by Pew Forum on Religion & Public Life asked whether the subject favored allowing homosexual couples to enter into civil unions—legal agreements that would give them many of the same rights as married couples. Of 2003 adults interviewed, 54% said yes,
5.2. In response to the GSS question in 2006 about the number of hours daily spent watching TV, the responses by the seven subjects who identified themselves as Hindus were 2, 3, 2,1,0,1, 4, 3.(a) Find a point estimate of the population mean for Hindus.(b) The margin of error for this point
5.1. Of 577,006 people involved in motor vehicle accidents in Florida in a recent year, 412,878 were wearing scat belts (Source: Florida Department of Highway Safety and Motor Vehicles). Find a point estimate of the population proportion of Florida motorists wearing seat belts.
4.57. The standard error formula ay = a/Jn treats the population size N as infinitely large relative to the sample size n. The formula for ay for a finite population size N is IN - n (
4.56. The curve for a normal distribution with mean p and standard deviation a has mathematical formula f(y) = (>'-M)2/(2^2)_•J2tti ira Show that this curve is symmetric, by showing that for any constantc, the curve has the same value ni y - p + c as at y = /x —c. (The integral of/(y) for y
4.55. The standard deviation of a discrete probability distribution is" = VSCv - m) P{y)-(a) Suppose y = 1 with probability 0.50 and y = 0 with probability 0.50, such as in Example 4.8(page 91). Show that a = 0.50.(b) Suppose y = 1 with probability tt and y = 0 with probability 1 - tt, where tt
4.54. Lake Wobegon Junior College admits students only if they score above 400 on a standardized achievement test. Applicants from group A have a mean of 500 and a standard deviation of 100 on this test, and applicants from group B have a mean of 450 and a standard deviation of 100. Both
4.50. For an election exit poll for a Senatorial election, find the standard error of the sample proportion voting for a candidate for whom the population proportion is 0.50, when n = 100,1000, and 10,000.In each case, predict an interval within which the sample proportion is almost certain to
4.49. In an exit poll of 1336 voters in the 2006 Senatorial election in New York State, 67% said they voted for Hillary Clinton. Based on this information, would you be willing to predict the winner of the election? Explain your reasoning.
4.48. For a normal distribution, show that(a) The upper quartile equals /x + 0.67a.(b) According to the 1.5(IQR) criterion, an outlier is an observation falling more than 2.7 standard deviations below or above the mean, and this happens for only 0.7% of the data.
4.47. The Palestinian Central Bureau of Statistics(www.pcbs.gov.ps) asked mothers of age 20-24 about the ideal number of children. For those living on the Gaza Strip, the probability distribution is approximately P(l) = 0.01, P(2) = 0.10, P{3) = 0.09, P(4) = 0.31, P{5) = 0.19, and P(6 or more) =
4.46. (a) Which distribution docs the sample data distribution tend to resemble more closely—the sampling distribution or the population distribution? Explain.(b) Explain carefully the difference between a sample data distribution and the sampling?
4.45. (Class Exercise) For a single toss of a coin, let y = 1 for a head and y = 0 for a tail. This simulates the vote in an election with two equally-preferred candidates.(a) Construct the probability distribution for y, and find its mean.(b) The coin is flipped ten times, yielding six heads and
4.44. (Class Exercise) Table 4.5 provides the ages of all 50 heads of households in a small Nova Scotian fishing village. The distribution of these ages is characterized by/x = 47.18 and a = 14.74.(a) Construct a stem-and-Ieaf plot of the population distribution.(b) Using a random number table,
4.43. (Class Exercise) Refer to Exercises 1.11 and 1.12(page 8). Using the population defined by your class or using the studentsurvey, the instructor will select a variable,such as weekly time watching television.(a) Construct a histogram or stcm-and-lcaf plot of the population distribution of the
4.41. You can use an applet on a computer or on the Internet to repeatedly generate random samples from artificial populations and analyze them to study the properties of statistical methods. To try this, go to www.prenhall.com/agresti and use the sampling distribution applet. Select binary for the
4.40. Refer to the previous exercise.(a) Describe the sampling distribution of y for a random sample of size n -(b) Describe the sampling distribution of y if you sample all 50,000 residents.
4.39. Sunshine City was designed to attract retired people. Its current population of 50,000 residents has a mean age of 60 years and a standard deviation of 16 years. The distribution of ages is skewed to the left, reflecting the predominance of older individuals. A random sample of 100 residents
4.36. The distribution of family size in a particular tribal society is skewed to the right, with /x = 5.2 and a - 3.0. These values are unknown to an anthropologist, who samples families to estimate mean family size. For a random sample of36 families,she gets a mean of 4.6 and a standard deviation
4.34. A study plans to sample randomly 100 government records of farms in Ontario to estimate the mean 4.35. According to the U.S. Census Bureau, in 2000 the number of people in a household had a mean of2.6 and a standard deviation of 1.5. Suppose the Census Bureau instead had estimated this mean
4.33. The scores on the Psychomotor Development Index (PD1), a scale of infant development, are approximately normal with mean 100 and standard deviation 15.(a) An infant is selected at random. Find the probability that PDI is below 90.(b) A study uses a random sample of 25 infants.Specify the
4.32. According to recent General Social Surveys (variable PARTNERS), in the United States the distribution ofy = number ofsex partners you have had in the past 12 months has a mean of about 1.1 and a standard deviation of about 1.1. Suppose these arc the population mean and standard deviation.(a)
4.31. Referto Exercise 4.6. The mean and standard deviation of the probability distribution for the lottery winnings y arc p, = 0.10 and a = 316.23. Suppose you play the lottery 1 million times. Let y denote your average winnings.(a) Find the mean and standard error of the sampling distribution of
4.29. An exit poll of 2293 voters in the 2006 Ohio Senatorial election indicated that 44% voted for the Republican candidate, Mike DeWine, and 56% voted for the Democratic candidate, Sherrod Brown.(a) If actually 50% of the population voted for DeWine, find the standard error ofthe sample
4.28. The probability distribution associated with the outcome of rolling a balanced die has probability 1/6 attached to each integer, {1,2,3,4,5,6}. Let(yi^yi) denote the outcomes for rolling the die twice.(a) Enumerate the 36 possible (yi,^) pairs (e.g.,(2.1) represents a 2 followed by a 1).(b)
4.26. Five students, Ann, Betty, Clint, Douglas, and Edward, arc rated equally qualified for admission to law school, ahead of other applicants. However, all but two positions have been filled for the entering class. Since the admissions committee can admit only two more students, it decides to
4.25. An energy study in Gainesville, Florida, found that in March 2006, household use of electricity had a mean of 673 and a standard deviation of 556 kWh(kilowatt-hours).(a) If the distribution were normal, what percentage of the households had use above 1000 kWh?(b) Do you think the distribution
4.23. For an SAT distribution (/x = 500, rr = 100) and an ACT distribution (/x = 21, cr = 4.7), which score is relatively higher, SAT = 600 or ACT = 29?Explain.
4.21. Suppose that the weekly use of gasoline for motor travel by adults in North America is approximately normally distributed, with a mean of 16 gallons and a standard deviation of 5 gallons.(a) What proportion of adults use more than 20 gallons per week?(b) Assuming that the standard deviation
4.20. For 5459 pregnant women using Aarhus University Hospital in Denmark in a two-year period who reported information on length of gestation until birth, the mean was 281.9 days, with standard deviation 11.4 days.-1 A baby is classified as premature if the gestation time is 258 days or less.(a)
4.19. The Mental Development Index (MDI) ofthe Bayley Scales of Infant Development is a standardized measure used in studies with high-risk infants. It has approximately a normal distribution with a mean of 100 and a standard deviation of 16.(a) What proportion of children have a MDI of at least
4.17. Mensa is a society of high-lQ people whose members have a score on an IQ test atthe 98th percentile or higher.(a) How many standard deviations above the mean is the 98th percentile?(b) For the normal IQ distribution with mean 100 and standard deviation 16, whatisthe IQ score for the 98th
4.15. What proportion of a normal distribution falls in the following ranges?(a) Above a z-score of 2.10(b) Below a z-score of-2.10(c) Above a z-score of —2.10(d) Between z-scores of -2.10 and 2.10
4.14. If z is the positive number such that the interval from fx - zo-\o fx + z
4.13. Show that if z is the number such that the interval from fx - zcr Xo ix + zu contains 90% of a normal distribution, then ^ + zcr equals the 95th percentile.
4.12. Find the z-values corresponding to the(a) 90th.(b) 95th,(c) 98th, and
4.11. Find the z-value such that for a normal distribution the interval from /t - zcr to jtx + zo- contains(a) 50%(b) 90%(c) 95%(d) 98%(e) 99% of the probability.
4.10. Find the z-value for which the probability that a normal variable exceeds fx + zo- equals(a) 0.01(b) 0.025(c) 0.05(d) 0.10(e) 0.25(f) 0.50
4.9. For a normally distributed variable, verify that the probability between:(a) fi - a and /x + a equals 0.68(b) fx — 1.96(r and/x + L96(t equals 0.95(c) /x — 3o- and /x + 3(r equals 0.997(d) /x — 0.67(7 and /x + 0.67(7 equals 0.50
4.8. For a normal distribution, find the probability that an observation falls(a) At least one standard deviation above the mean(b) At least one standard deviation below the mean(c) At least 0.67 standard deviations above the mean
4.7. Let y be the outcome of selecting a single digit from a random number table.(a) Construct the probability distribution for y.(This type of distribution is called a uniform distribution because of the uniform spread of probabilities across the possible outcomes.)(b) Find the mean ofthis
4.5. Let y denote the number of people known personally who were victims of homicide within the past 12 months. According to results from recent General Social Surveys,for a randomly chosen person in the U.S. the probability distribution of y is approximately: P(0) = 0.91,P(1) = 0.06.P(2) =0.02,
4.4. Let y — number of languages in which a person is fluent. According to Statistics Canada, for residents of Canada this has probability distribution P(0) = 0.02, P(l) = 0.81, P{2) =0.17, with negligible probability for higher values of y.(a) Is y a discrete or a continuous variable? Why?(b)
4.3. In 2000, the GSS asked subjects whether they are a member of an environmental group (variable GRNGROUP) and whether they would be very willing to pay much higher prices to protect the environment (variable GRNPRICE). Table 4.4 shows results.(a) Explain why 96/1117 = 0.086 estimates the
3.81. The leastsquares property of the mean states that the data fall closer to y than to any other numberc, in the sense that the sum of squares of deviations of the data about their mean is smaller than the sum of squares of their deviations about c.That is, 2(y/ - y)2< 'Ziyi - c)2.Ifyou have
3.80. The Russian mathematician Tchebysheff proved that for any A: > 1, the proportion of observations that fall more than k standard deviations from the mean can be no greater than \/k~. This holds for any distribution, notjust bell-shaped ones,(a) Find the upper bound for the proportion of
- >y«-
3.79. Showthat S(y,- — y) must equal 0 for any collection of observations yi,y2^
3.78. For a sample with mean y, adding a constant c to each observation changes the mean to y +c, and the standard deviation s is unchanged. Multiplying each observation by c changes the mean to cy and the standard deviation to |c|s.(a) Scores on a difficult exam have a mean of 57 and a standard
3.77. Refer to Table 3.2 (page 34). Explain why the mean of these 50 observations is not necessarily the same as the violent crime rate for the entire U.S. population.
3.76. Ten people are randomly selected in Florida and another ten people are randomly selected in Alabama. Table 3.18 provides summary information on mean income. The mean is higher in Alabama both in rural areas and in urban areas.Which state has the larger overall mean income?(The reason for this
3.75. A teacher summarizes grades on the midterm exam by Min = 26,01 - 67, Median - 80.Q3 = 87, Max = 100, Mean = 76. Mode = 100, Standard dev. = 76, IOR = 20.She incorrectly recorded one of these. Which one do you think it was? Why?
3.71. Answer true or false to the following:(a) The mean, median, and mode can never all be the same.(b) The mean is always one of the data points.(c) The median is the same as the second quartile and the 50th pcrccntilc.(d) For 67 sentences for murder recently imposed using U.S.
3.70. To measure variability, why is(a) The standard deviation s usually preferred over the range?(b) The 1QR sometimes preferred to .v?
3.65. The fertility rate (mean number of children per adult woman) varies in Western European?
3.64. A U.S. Federal Reserve study in 2000 indicated that for those families with annual incomes above $100,000, their median net worth was about$500,000 both in 1995 and in 1998, but their mean net worth rose from $1.4 million in 1995 to $1.7 million in 1998. A newspaper story about this said that
3.53. What is the difference between the descriptive measures symbolized by(a) y and /x?(b) s and rr?
3.51. For the data for OECD nations in Table 3.11 in Exercise 3.6, use software to construct a scatterplot relating y - carbon dioxide emissions and x = GDP.(a) Based on this plot, would you expect the correlation between these variables to be positive or negative? Why?
3.46. The United Nations publication Energy Statistics Yearbook (unstats.un.org/unsd/energy) lists consumption of energy. For the 25 nations that made up the EU in 2006, the energy values (in kilograms per capita) had a mean of 4998 and a standard deviation of 1786.(a) Italy had a value of 4222.
3.41. In 2004, the five-numbersummary forthe statewide percentage ofpeople without health insurance had a minimum of 8.9% (Minnesota), Ql = 11.6, Med = 14.2, (93 = 17.0, and maximum of 25.0%(Texas) {Statistical Abstract of the United States, 2006).(a) Sketch a box plot.
3.40. Infant mortality rates (number of infant deaths, per 1000 live births) are reported by the UN.In their 2006 report, the values for Africa had a five-number summary of min — 54,Q1 - 76, median - 81, Q3 = 101, max = 154.The values for Western Europe had a five-number summary of min = 3,Q I =
3.39. For the student survey data on number of times a week reading a newspaper, referred to in Exercise 1.11, Figure 3.20 shows a computer printout of the stem-and-leaf plot and the box plot.(a) From the box plot, identify the minimum, lower quartile, median, upper quartile, and maximum.(b)
3.38. The January 2007 unemployment rates of the 27 countries in the European Union ranged from 3.2 (Denmark) to 12.6 (Poland), with lower quartile = 5.0, median = 6.7, upper quartile =7.9, mean = 6.7, and standard deviation = 2.2.Sketch a box plot, labeling which of these values are used in the
3.36. For each of the following variables, indicate whether you would expect its relative frequency histogram to be bell shaped, U shaped, skewed to the right, or skewed to the left.(a) Exam score of easy exam, with mean = 88, standard deviation = 10, minimum = 65, lower quartile = 77, median = 85,
3.34. During a recent semester at the University of Florida, computer usage3 of students having accounts on a mainframe computer was summarized by a mean of 1921 and a standard deviation of 11,495 kilobytes of drive usage.(a) Docs the Empirical Rule apply to this distribution? Wh}'?(b) The
3.29. For all homes in Gainesville, Florida, the residential electrical consumption2 for the year 2006 had a mean of 10,449 and a standard deviation of 7489 kilowatt-hours (kWh). The maximum usage was 336,240 kWh.(a) What shape do you expect this distribution to have? Why?(b) Do you expect this
3.28. According to the U.S. Census Bureau, the U.S.nationwide median selling price of homes sold in 2005 was $184,100. Which of the following is the most plausible value for the standard deviation:(a) -15,000, (b) 1,000, (c) 10,000, (d) 60,000, (e)1,000,000? Why?
3.26. The first exam in your Statistics course is graded on a scale of 0 to 100, and the mean is 76. Which value is most plausible for the standard deviation:-20, 0, 10, or 50? Why?
3.16. According to the U.S. Bureau of the Census, 2005 American Community Survey, the median earnings in the past 12 months was $32,168 for females and$41,965 for males, whereas the mean was $39,890 for females and $56,724 for males.(a) Docs this suggest that the distribution of income for each
3.14. Table 3.13 summarizes responses of 2333 subjects in the 2006 General Social Survey to the question,"About how often did you have sex during the last 12 months?"(a) Report the median and the mode. Interpret.(b) Treat this scale in a quantitative manner by assigning the scores 0, 0.1, 1.0, 2.5,
3.8. Global warming seems largely a result of human activity that produces carbon dioxide emissions and other greenhouse gases. The Human DevelopmentReport2005, published by the United Nations Development Programme, reported per capita emissions in 2002 for the eight largest countries in population
3.6. The OECD (Organization for Economic Cooperation and Development) consists of advanced, industrialized countries that accept the principles of representative democracy and a free market economy. Table 3.11 shows UN data for OECD nations on several variables: gross domestic product (GDP, per
3.3. A teacher shows her class the scores on the midterm exam in the stem-and-leaf plot:6 15 8 8 7101136779 8 I 1223334677789 91011234458(a) Identify the number of students and the minimum and maximum scores.(b) Sketch a histogram with four intervals.
2.40. An interval scale for which ratios are valid is called a ratio scale. Such scales have a well-defined 0 point, so, for instance, one can regard the value 20 as twice the quantity of the value 10. Explain why annual income is measured on a ratio scale, but temperature (in Fahrenheit or
2.38. An interviewer stands at an entrance to a popular shopping mall and conducts interviews. True or false: Because we cannot predict who will be interviewed, the sample obtained is an example of a random sample. Explain.2.39. In a recent Miss America beauty pageant, television viewers could cast
2.37. A recent GSS asked subjects if they supported legalizing abortion in each of seven different circumstances. The percentage who supported legalization varied between 45% (if the woman wants it for any reason) to 92% (if the woman's health is seriously endangered by the pregnancy). This
2.36. Crosson (1994, p. 168) described an analysis of published medical studies involving treatments for heart attacks. In the studies having randomization and strong controls for bias, the new therapy provided improved treatment 9% of the time. In studies without randomization or other controlsfor
2.35. Ifwe use random numbers to take a simple random sample of 50 students from the 20,000 students at a university,(a) It is impossible to get the random number 11111, because it is not a random sequence.(b) If we get 20001 for the first random number, for the second random number that number is
2.33. To sample residents of registered nursing homes in Yorkshire, UK, 1 construct a list of all nursing homes in the county, which I numberfrom 1 to 110.Beginning randomly, I choose every tenth home on the list, ending up with 11 homes. 1 then obtain lists of residents from those 11 homes, and 1
2.32. Find an example of results of an Internet poll. Do you trustthe results ofthe poll? If not, explain why not.
2.31. In a cluster random sample with equal-sized clusters, every subject has the same chance ofselection.However, the sample is not a simple random sample. Explain why not.
2.30. With a total sample ofsize 100, we wantto compare Native Americans to other Americans on the percentage favoring legalized gambling. Why might it be useful to take a disproportional stratified random sample?
2.29. In a systematic random sample, every subject has the same chance of selection, but the sample is not a simple random sample. Explain why.
2.28. A content analysis of a daily newspaperstudies the percentage of newspaper space devoted to news about entertainment. The sampling frame consists of the daily editions of the newspaper for the previous year. What potential problem might there be in using a systematic sample with skip number
2.27. In each of the following situations, evaluate whether the method of sample selection is appropriate for obtaining information about the population of interest. How would you improve the sample design?(a) A newspaper wants to determine whether its readers believe that government expenditures
2.26. When the Yankclovich polling organization asked,6"Should laws be passed to eliminate all possibilities of special interests giving huge sums of money to candidates?" 80% of the sample answered yes. When they posed the question,"Should laws be passed to prohibit interest groups from
2.25. With quota samplinga researcherstands at a street corner and conducts interviews until obtaining a quota representing the relative sizes of various groups in the population. For instance, the quota might be 50 factory workers, 100 housewives, 60 elderly people, 30 blacks, and so forth. Is
2.24. You are directing a study to determine the factors that relate to good academic performance at your school.(a) Describe how you might select a sample of 100 students for the study.(b) List some variables that you would measure.For each, provide the scale you would use to measure it, and

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