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Statistical Methods For The Social Sciences 3rd Edition Alan Agresti, Barbara Finlay - Solutions

48. Refer to Problein 3.10a) Using the data set without D.C., find the quarules and the interquartile rangeb) According to the definition of an outlier in terms of the IQR, are any of the observations outliers?c) Construct a box plot for the distribution.d) Repeat the analyses, including the D.C.
49. What is the difference between the descriptive measures symbolized bya) Y and ?b) s and ?
50. For the WWW data file (Problem 1.7), use computer software to conduct graphical and numerical suminaries fora) distance from hoine town,b) weekly hours of TV watching.c) weekly number of times reading a newspaper, andd) number of HIV-AIDS victims known. Describe the shapes of the distributions,
51. Refer to the data file yon created in Problem 1.7 For variables chosen by your instructor, conduct descriptive statistical analyses Prepare a report, interpreting and summanzing your findings.
52. Refer to the data in Table 9.1 on poverty rates. Using methods of this chapter, summarize these data Prepare a report, graphically displaying the data and summarizing the central tendency and variation. In your report, discuss whether there any outliers, and if there are, analyze their
53. The number of therapeutic abortions in 1988 in Canada, per 100 live births, is shown in Table 3.17 Using methods of this chapter, present a descriptive statistical analysis of these data, interpreting your results. TABLE 3.17 Alberta 150 Bntish Columbia 255 Manitoba 16.6 New Brunswick 4.9
54. Refer to Problem 3.19. Table 3.18 shows the HDI ratings for African countries Using graphical and numerical methods of this chapter, summarize HDI for these countries, and compare to the disinbution of HDI for Central American countries.
55. Obtain data on statewide murder rates from the latest edition of Statistical Abstract of the United Statesa) Analyze the data using the graphical and numerical methods of this chapter.b) Use graphical and numencal methods to compare the muiden rate distribution to the one for the data in Table
56. During the strike of professional baseball players in 1994, two quite different numbers were reported for the central tendency of players annual salanes. One was $1.2 million and the other was $500.000. One of these was the median and one was the mean. Which value do you think was the incan?
57 In 1986. the US. Federal Reserve sampled about 4000 households to estimate overall net worth of a family. Excluding some outliers of extremely wealthy individuals. they reported the summaries $44,000 and $145.000 One of these was the mean. and one was the median. Which do you think was the
58. According to a recent report from the U.S. National Center for Health Statistics. for males with age 25-34 years. 2% of their heights are 64 inches or less. 8% are 66 inclies or less. 27% are 68 inches or less, 39% are 69 inches or less, 54% arc 70 inches or less. 68% are 71 inches or less. 80%
59. Grade point averages of graduating seniors at the University of Rochester are approxi- mately bell-shaped in distribution ranging from 20 to 4.0 with a mean of about 3.0. Us- ing the fact that all or nearly all measurements for this form of distribution occur within three standard deviations of
60. In Canada in 1981. for the categories Catholic. Protestant, Eastern Orthodox. Jewish. None. Other for religious affiliation, the relative frequencies were 47 3%, 41.2%. 15%. 1.2%. 7.3%, 1.5% (Canada Year Book, 1992).a) The median religion is Protestant.b) The distribution is bimodalc) Only 2.7%
61. The 1991 General Social Survey asked whether having sex before manage is always wiong, almost always wrong, wrong only sometimes, not wrong at all. The response counts in these four categories were 274, 98. 186. 435. This distribution isa) Skewed to the right.b) Approximately bell-shaped.c)
62 Ten families are randomly selected in Florida and another ten families are randomly se- lected in Alabaina Table 3.19 provides summary information on incan family income The mean is higher in Alabama both in rural areas and in mban areas Which state has the large overall mean income" (The reason
63 Refer to Problem 3.10. Explain why the mean of these 50 measurements is not necessarily the same as the violent crime rate for the entire U.S. population.
64 The mican and standard deviation of a sample may change if data are rescaled For a sain- ple 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 changes the mean to c and the standard deviation
65. The results of the study described in Problem 3.34 are to be reported in a French news- paper. The ten measurements are converted to kilometer units (1 inile 1.6 kilometers). Report the incan and standard deviation of the converted incasurements. (Hinr You do not need to convert the original
66 *. The crude death rate is the number of deaths in a year. per size of the population. mul- tiplied by 1000 According to the U.S. Bureau of the Census in 1995 Mexico had a crude death rate of 4.6 (i.e.4.6 deaths per 1000 population) while the United States had a crude death rate of 84. Explain
67 "The sample means fork sets of data with sample sizes m are Y. Y...Y. Show that the overall saniple mean for the combined data set is Interpret as a weighted average of Y. .....Interpret as a weighted average of Y. .....
68. Show that E();-) must equal 0 for any collection of measureinents Y, Y....Y-
69. The Russian inathematician Tchebysheff proved that for any real number > 1, the pro- portion of the measurements that fall more than & standard deviations from the mean can be no greater than i 'k Moreover, this holds for any disinbution. not just bell-shaped oncsa) Find the upper bound for the
70. The least squares property of the mean states that the data fall closer to than to any other real 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 aboutc. That is, - 197 < - 0 If you have studied
1 Let the number of people you have known personally that have committed suicide within the past 12 months According to recent General Social Surveys, the probabilities for the potential values of Y are approximately P(0)=90, P(1) = .08. P(2) = .02.a) Construct a probability disuibution and its
2 Let = number of people you lave known personally who came down with AIDS Sup- pose Table 3.9 refers to the population of interesta) Construct the probability distribution for Yb) Find the mcan of the distribution of Y.
3. Five students, Ann. Betty. Clint. Douglas, and Edward. are rated equally qualified for ad- mission to law school. ancad of other applicants However, all but two positions have been filled for the entering class Since the admissions cominittee can admit only two more students, it decides to
4. Refer to the previous exercise. Construct the probability distribution of X = number of females admitted. when three of the five students are randomly selected for admission. Find the mean of the distribution
5 Let be the outcome of selecting a single digit from a random number table.a) List the possible values for Yb) Construct the probability distribution for) (This type of distribution is called a uni- form distribution. because of the uniform spread of probabilities across the possible out-
6. In a statewide lottery, one can buy a ticket for $1. With probability .0000001. one wins a million dollars (S1,000,000). and with probability .9999999 one wins nothing (SO).a) Let Y denote the winnings from buying one ticket. Construct the probability distribu- tion for Y. Show that the mean of
7. For a normal distribution. find the probability that an observation isa) At least one standard deviation above the mean.b) At least one standard deviation below the mean.c) At least .67 standard deviations above the meand) At least 2.33 standard deviations above the mean.
8. For a normally distributed variable, venly that the probability betweena) - and equals .68b) -1.960 and + 1.960 equals .95. c)-30 and -30 equals.997. -d) 670 and .670 equals.50
9 Find the -value for which the probability that a normal variable exceeds so equalsa) .01b) .025 c).05 d).10e) .25 1).50
10. Find the z-value such that the interval from -zo to +20 containsa) 50%.b) 90%.c) 95%,d) 98%.e) 99% of the probability for a nonnal distribution.
11. Find the z-values corresponding to thea) 90th,b) 95th,c) 98th. andd) 99th percentiles of a normal distribution
12. Show that if is the number such that the interval from 0 to +co contains 90% of a normal distribution, then u+zo equals the 95th percentile.
13. If is the positive number such that the interval from u-go to + contains 50% of a normal distribution, then which percentile isa) +b) zo? Find this value of :. Using this result, explain how to find the upper quartile, lower quartile, and interquartile range of a normal distribution.
14. An observation is .50 standard deviations below the mean on a normally distributed vari- able. What proportion of the data fall below that observation? Above it?
15 What proportion of a normal distribution falls in the following rangesa) 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-scores of -2.10 and 2 10.
16. Find the z-score for the number that is less than ouly 1% of the values of a nonnal distn- bution
17. Find the z-scores for the lower and upper quartiles of a normal distribution.
18. According to Current Population Reports, self-employed individuals in the United States in 1990 worked an average of 44.6 hours per week. with a standard deviation of 14.5. Assuming this variable is approximately normally distributed, find the proportion who averaged more than 40 hours per week.
19. The Mental Development Index (MDI) of the Bayley Scales of Infant Development is a standardized measure used in longitudinal follow-up of high-risk infants. It has approxi- mately a normal distribution with a mean of 100 and a standard deviation of 16.a) What proportion of children have a MDI
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 2819 days. with standard deviation 114 days. A baby is classified as premature if the gestation ume is 258 days or less (British
21 Suppose that the weekly use of gasoline for motor travel by adults in North Amenca 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) Find and interpret the lower and upper
22. In 1992, the murder rates (per 100,000 residents) for 74 major US. citics had a can of 20.8 and a standard deviation of 14.6 (Statistical Abstract of the United States. 1994)a) Could the distribution of murder rates be well approximated by a normal distribution? Why or why not?b) One of these
23. Suppose that monthly rental rates for one-bedroom unfurnished apartments in Ann Arbor, Michigan, are approximately normal in distribution, with a niean of $750 and a standard deviation of $150a) What proportion of the rental rates are at least $1000 per month?b) What proportion of rates are
24 Suppose that property taxes on homes in Gainesville. Florida. are approximately nonnal in distribution, with a mean of $1400 and a standard deviation of $600 The property tax for one particular home is $1700.a) Find the z-score corresponding to that valucb) What proportion of the property taxes
25. An energy study in Gainesville, Florida. found that in a typical month household use of electricity has a mean of 800 and a standard deviation of 500 (kilowatt-hours). Each ob- servation is converted to a z-score. Each household with az-score greater than 1.0 (i.e.. electricity use more than
26. Refer to Problem 4.3 For randomly selecting two of the five students, construct the sam- pling distribution of the proportion of the students selected who are female What is the probability that the proportion equals 0?
27 Construct the sampling distribution of the proportion of heads. for hipping a balanced coina) Onceb) Twice. (Hint The possible saniples are (H. Hi. (H. T) (T. H). (T. T).)c) Three times (Hint: There are 8 possible samples.)d) Four tines (Hint: There are 16 possible samples!e) Describe how the
28 The probability distribution associated with the outcome of rolling a balanced die has probability 1/6 attached to each integer, 11, 2.3 4.5.6). Consider the experiment of rolling a balanced die twicea) Enumerate the 36 possible pairs of numbers, taking into account the order of rolling (eg
29. According to Current Population: Reports, tire population distribution of number of years of education for self-employed individuals in the United States in 1990 had a mean of 13.6 and a standard deviation of 30 Find the mean and standard deviation of the sampling distribution of F for a random
30 Refer to Problem 4.6. The mean and standard deviation of the probability distribution for the lottery winnings are =.10 and 316.23. Suppose you play the lottery 1 million times Let denote your average winnings.a) Find the inean and standard deviation of the distribution of .b) How likely is it
31. Suppose that weekly incomes of migrant workers in California have a mean of $250 and a standard deviation of $75.a) If the distribution is normal, find the probability that a migrant worker has income over $300.b) For a random sample of nine inigrant workers. report the sampling distribution of
32. According to recent General Social Surveys, in the United States the distribution of Y = number of sex partners you have had in the past 12 months has a mean of about 1.0 and a standard deviation of about 1.0.a) Does Y have a normal distribution? Explain.b) For a random sample of 100 adults.
33 According to Statistical Abstract of the United States, 1995, household size has a mean of 2.6 and a standard deviation of 15. For a random sample of 225 homes, find the prob- ability that the sample mean household size falls within.1 of the population nean
34. The distribution of family size in a particular tribal society is skewed to the right, with = 5.2 and a = 3.0. These values are unknown to an anthropologist, who takes a sample of families in this society in order to estimate mean family size. Let Y denote the sample mean family size she
35. The scores on the Psychoinotor Development Index (PDI), a scale of infant development, are approximately nonnally distributed with mean 100 and standard deviation 15. An in- fant is selected at randoin.a) Find the probability that the infant's PDI score is at least 100b) Find the probability
36 Refer to the previous exercise. A study uses a random sample of 225 infants.a) Describe the sampling distribution of the sample mean PDI,b) Find the probability that the sample mean exceeds 103.c) Find the probability that the sample mean falls between 97 and 103d) Find the z-score froin the
37. Refer to the previous two exercisesa) Sketch the population distribution for PDIb) Superimpose, on the sketch of part (a), a sketch of the sampling distribution of the sample mean, for a random sample of 225 infantsc) Superimpose, on the above sketches. a sketch of the sampling distribution of
38. We need an estimate of the mean acreage of Canadian fanns. We plan to measure acreage for a random sample of 100 farms. Residis from an earlier study suggest that 200 acres is a reasonable guess for the standard deviation of farm size. Find the probability that the sample mean acreage falls
39. An executive in a savings and loan association decided to estimate the inean amount of money lent to individuals for financing higher education in the past year From past ex- perience, she believes that $240 is a reasonable guess for the standard deviation of the distribution of loan amounts.
40. A study investigating the relationships among voting patterns, pohtical opinions, and age takes a random sample of 100 individuals in a typical precinct.a) If the standard deviation of the ages of all individuals in the precinct is = 15. find the probability that the mean age of the individuals
41. (Class Exercise) Refer to Problem 1.7. Using the population defined by your class or us- ing the WWW data set, the instructor will select a variable, such as weekly time watching televisiona) Construct a histogram or stem and leaf plot of the population distribution of the vari- able for the
42. (Class Exercise) Table 4.4 provides the ages of all 50 heads of households in a sinall Nova Scotian fishmg village. The distribution of these ages is characterized by j = 47.18 and = 14.74.a) Construct a stein and leaf plot of the population distribution of the ages of all heads of householdsb)
43 (Class Exercise) For a single loss of a coin, let Y=1 for a head and y = 0 for a taila) Assuming the coin is balanced, construct the probability distribution for Y. and calcu late its meanb) The coin is flipped ten times. yielding six heads and four tails. Construct the sample relative frequency
44. (Class Exercise) The previous exercise assumes the coin is balanced. Suppose instead that the probability of a head equals 75 and the probability of a tail equals .25a) Construct the probability distribution of the outcome of a single flip, and calculate its mean.b) Each student in the class
45. (Class Exercise) Each student should bring ten coins to class The observation for each coin is its age. the difference between the current year and the year on the coin.a) Using all the students' observations, the class should construct a stem and leaf plot of the ages. What is its shape?b) Now
46a) Which distribution does the sample distribution tend to resemble inore closely-the sampling distribution of the population distribution? Explain.b) Explain carefully the difference between a distribution of sample measurements and the sampling distribution of . Illustrate your answer for a
47. The standard error of a statistic describesa) The standard deviation of the sampling distribution of that statistic.b) The standard deviation of the sample measurements.c) How close that statistic is likely to fall to the parameter that it estimates.d) The variability in the values of the
48. The Central Limit Theorem implies thata) All variables have approximately bell-shaped sample distributions if a random sample contains at least 30 observations.b) Population disuibutions are nonnal whenever the population size is large.c) For large random samples. the sampling distribution of Y
49. The numbers in the body of Table A provide tail probabilitiesa) For the normal distribution.b) For any symmetric distribution.e) For any distributiond) Only for the standard normal distribution
50. True or False: As the sample size increases. the standard error of the sainpling distribution of increases.
51. * Lake Wobegon Junior College adinits students only if they score above 400 on a stand- ardized achievement lest Applicants from group A have a incan of 500 and a standard deviation of 100 on this test, and applicants from group B have a mean of 450 and a stand- ard deviation of 100 Both
52. For a normal distribution, show thata) The upper quarnle equals + 670.b) The interquartile range, JQR. is related to by IQR = 1.350.c) For sample data that are approximately normal, an outlier (according to its definition for box plots) is an observation falling more than 2,7 standard
53. Sunshine City was designed to attract retired people. and 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 randoin sample of 100
54. Refer to Example 4.7 Compute the standard error when n=10, 100, 1000 and 10,000. In each case. use the Empirical Rule to predict an interval within which the sample pro- portion is almost certain to fall. Notice that the interval shrinks in width as the sample size increases This is a
55. * The variance of a discrete probability distribution is 0 =0- P(3)a) Suppose Y = I with probability .5 and Y=0 with probability 5, such as in Example 4.7 on preference for Democratic and Republican candidates Using each formula show that o25. and hence the standard deviation = .50.b) Suppose 1
56. + Refer to the formula for the variance of a probability distribution in the previous ex- crcise. Find the standard deviation for the distribution in Problem 4.1. Can one use the Empincal Rule to interpret this standard deviation? Explain.
57 The curve for a nonnal distribution with mean and standard deviation o has mathe- matical fonnula f(x)= I 2x0 (The integral of this function with respect to between +zo and co equals the tail prob- ability tabulated in Table A). Show that this curve is symmetric: that is. for any constantc. the
58 + The standard error formula o; = a/ actually treats the population size N as infinitely large relative to the sample size n. The formula for y for a finite population size N is IN-n The term (Na)/(N-1) is called the finite population correction. =a) When 300 students are selected from a college
1 The numbers of children in six families selected at random from 1870 census records in a southern town are 1. 4. 0. 0, 6, 7. Calculate and interpret point estimates of the population mean and standard deviation.
2 Find and interpret the 95% confidence interval for , if = 70 and 3 = 10. based on a sample size of (a) n = 25. (b) = 100.
3. In the General Social Survey respondents are asked to rate their poliucal views on a seven- point scale, where I = extremely liberal. 4 = moderate. and 7 = extremely conservative. For the 1994 survey. a computer printout reports N 2879 Mean 4.171 Std Dev 1.390 Std Err 0.0259a) Construct and
4. Refer to Example 51 and the responses about the number of male sex partners. Of the 1431 subjects responding with a positve number, the mean was 4.8 and the standard de- viation was 6.2.a) Verify that the 95% confidence interval for that population mean equals (4.5.5.1).b) What statistical
5. Refer to the data in Table 3.13 regarding the subjects' number of sex partners in the last 12 months A computer printout shows the following information. No. Cases 637 Mean 1.314 SD 5.418 SE of Mean 0.215a) Let u denote the population mean of this distribution. Report a point estimate of and a
6. A survey is taken to estimate the mean annual family income for families living in public housing in Chicago. Foi a random sample of 30 families. the annual incomes (in hundreds of dollars) are as follows 83 90 77 100 83 86 10% 64 78 92 73 122 96 60 85 70 139 56 94 84 11) 93 120 70 92 100 124 59
7. A hospital administrator wants to estimate the mean length of stay for all inpatients who use that hospital. Based on a systematic random sample of 100 records of patients for the previous year, she reports that "The sample mean was 5.3 In repeated random samples of this size, the sample mean
8. Find the z-score used in the formula for a confidence interval with confidence coefficienta) .98.b) .90.c) .50.d) .9973.
9. For the WWW data on number of times a week reading a newspaper. referred to in Prob- lein 1.7. the observations for the 60 University of Flonda social science students have a mean of 4.] and standard deviation of 3.0a) For the population that this sample represents, construct and interpret a 95%
10. Recent General Social Surveys asked subjects. "How long have you lived in the city, town or community where you live now? The possible responses were (less than one year. 1. 2,3,4. The responses of 1415 subjects had a inode of "less than one year," a median of 16 years, a mean of 20.3 and a
11. In the 1994 General Social Survey. responses of 1964 subjects to the question. "On the average day, about how many hours do you personally watch television had a mean of 2.8 The standard error of the mean was .05.a) Calculate a 98% confidence interval for the mean daily time spent watching
12 A study is conducted of the geographic distribution of the residences of the employees at a large factory. in order to determine the suitability of initiating busing to that factory. One variable considered is the distance the employee lives from the factory For a randomi sample of 100
13 The 1991 General Social Survey asked. "During the last year. did anyone take something from you by using force-such as a stickup. mugging, or threat Of 98 subjects, 17 answered yes and 970 answered noa) Find the point estimate of the proportion of the population who were victims of this type.b)
14. In the 1991 General Social Survey. respondents were asked whether people convicted of murder should receive the death penalty. 1078 responded yes and 336 responded no. Con- struct a 99% confidence interval for the proportion of American adults who would answer yes. Interpret. Can you conclude
15. The 1994 General Social Survey asked subjects "Do you believe there is a life after death" Of 1958 respondents. 1425 answered ves. A report based on these data stated that "73% of Americans believe in life afies death. The margin of error for this result is plus or minus 2. Explain how the
16. An anthropologist wants to estimate the proportion of children in a tribe in the Philip- pines who die before reaching adulthood. For families she knew who had children born between 1970 and 1975, 3 of 30 children died before reaching adulthood.a) Construct a 90% confidence interval for the
17. A study in 1991 by the US National Center for Health Statistics provided a point estimate of 25 5% for the percentage of adult Americans who were currently smokers. The sample size was 42,000. Assuming that this sample has the charactenstics of a random sample. construct and interpret a 99%
18. Refer to the previous problem. The same study provided a point estimate of 5.5% for the percentage of Americans who averaged two or more alcoholic drinks per daya) Construct and interpret a 99% confidence interval for the proportion of the population who drink that muchb) Compare the width of
19. A national television network samples 1400 voters after each has cast a vote in a state gubernatorial election Of these 1400 voters, 742 claim to have voted for the Democratic candidate and 658 for the Republican candidate. There are only two candidates in the election.a) Assuming that each

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