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statistical sampling to auditing
Statistical Methods For The Social Sciences 3rd Edition Alan Agresti, Barbara Finlay - Solutions
18. height. X = length of left leg, and X = length of right leg are measured for a sample of 100 adults The model E(Y) = a + BX + X is fitted to the data. and neither Ho: B =0 nor Ho. 2 =0 is rejected.a) Does this imply that length of leg is not a good predictor of height? Why?b) Does this imply
19. Refer to Tables 11.5 and 11.7. Note that X1 and X2 lose their significance after entering the interaction term, even though that term is not significanta) Explain why this happensb) Rerun the interaction inodel, after centering the predictor scores about their mean. that is. take X = X, -
20 Sketch the following mathematical functions on the sanic set of axes, for values of X be- tween 0 and 4.a) = 10 + 4Xc) f = 10+ 4X - Xe) = 10-4X+X g) = 10(1.50)*b) = 10+4X+Xd) = 10-4Xf) = 10-4X - X h) = 10(.50)*
21 Refer to the housing data in Table 9.4 The quadratic model relating selling price to size of house has fit -2.04+49.345 +6.745.a) Interpret the coefficients of this equation. What shape does it have?b) Find the predicted selling price for homes with (i) S=1. (ii) S-2, tiii) S = 3. Explain why
22 Table 14.9 shows results of fitting models to the housing data in Table 94. using number of bedrooms as a predictor of selling price.a) For the bivariate model with number of bedrooms as a linear predictor, interpret the estimated coefficients, and interpret r.b) Report and interpret the P-value
23. Refer to the previous exercise. Fit the exponential regression model, and interpret. Fhid the predicted selling price when the number of bedrooms equals (i) 2, (ii) 3, (iii) 4. and compare to the predictions with the other models.
24. Refer to Table 9.13 Let y = birth rate and X = women's economic activitya) Fit the straight-line regression model. Interpret coefficients.b) Fit the quadratic regression model, and interpret parameters.c) Using the quadratic fil. find the X-value at which predicted birth rate takes its minimum
25. Refer to the previous example. Fit the exponential regression model, and interpret the parameter estimates Describe the estimated effect of a 10 unit increase in economic ac- tivity.
26. Refer to Example 9 1. Table 91. and Figure 9.4. Though inference may not be relevant for these data, we use them to illustrate how a single observation can be highly influential in detennining whether to assume a nonlinear relationshipa) Using all 51 observations. fit the quadratic model
27. Refer to the previous example Fit the exponential regression model to the full data set, and interprct ifte parameter estimates.
28. Table 14 10 shows the results of fitting two models to 54 observations on Y = mental health score. X; degree of social interaction, and XSES The vanables X, and X2 are measured on scales of 0-100. and larget Y-scores represent better mental health. The variable symbol X12 represents X, and X1
29. Refer to Problem 14.12.a) What does the plot of the residuals against the predictor suggest?b) Using a gamma GLM, fit the linear model Compare the parameter estimates and the standard error of the slope to results obtained using least squares. Interpret.c) Compare the two fits to the least
30. Refer to Example 14.4 on birth rates. To allow for greater variation at higher values of the mean, use the gamma GLM with identity link.a) Fit the gamma linear model. Compare estimates to those obtained using least squares (i.e., the GLM with normal random component)b) Fit the gamma quadratic
31. Draw rough sketches of the following mathematical functions on the same set of axes, for values of X between 0 and 35. a)=5(1.02). (= would population size in billions X years from now, if there is a 2% rate of growth every year.) ==b) 5(1.04). (What does this represent?)
1. Distinguish between description and inference as two purposes for using statistical meth- ods. Illustrate the distinction using an example
2 Give an example of a situation in which descriptive statistics would be helpful but infer- ential statistical methods would not be needed.
3. The Environmental Protection Agency (EPA) uses a few new automobiles of each brand every year to collect data on pollution entission and gasoline mileage performance For a particular brand, identify the (a) subject. (b) sample. (c) population.
4.a) Distinguish between a statistic and a parameter.b) An article in a Florida newspaper (The Gainesville Sun) in 1995 reported that 66.5% of Flondians believe that state governinent should not restrict access to abortion. Is 66.5% most likely the value of a statistic, or of a parameter? Why?
5 The student government at the University of Wisconsin conducts a study about alcohol abuse among students. One hundred of the 40.000 members of the student body are sani- pled and asked to complete a questionnaire One question asked is "On how many days in the past week did you consume at least
6. The Current Population Survey of about 60,000 households in the United States in 1992 indicated that 10.3% of whites, 31 0% of blacks, and 26 7% of Hispanics in the United States have annual income below the poverty level (Statistical Abstract of the United States. 1994). 3) Are these numbers
7. Your instructor will help the class create a data file consisting of the values for class mem- hers of characteristics such as GE = gender. AG= age in years. HI=high school GPA (on a four-point scale), CO= college GPA, DH = distance (in miles) of the campus from your home town, DR distance (un
8. A sociologist is interested in estimating the average age at marnage for women in New England in the early eighteenth century. She finds within her state archives reasonably complete marriage records for a large Puritan village for the years 1700-1730. She then takes a sample of those records,
9. The 1994 General Social Survey of adult Americans asked subjects whether astrology- the study of star signs-has some scientific truth. Of 1245 sampled subjects who had an opinion, 651 responded definitely or probably true. and 594 responded definitely or probably not true The proportion
10. Look at a few recent issues of a major social science journal, such as American Socio- logical Review or American Political Science Review. About what proportion of the articles seem to use statistics? Find some examples of descriptive statistics.
11. Find out what statistical software is available to you while taking this course, either on PCs or workstations in a computer lab or on an institution-wide mainframe computer Find out how to access the software. enter data and print any files you create As an ex- excisc, create a data file using
1. Explain the difference betweena) Discrete and continuous variablesb) Qualitative and quantitative variablesc) Noininal and ordinal variables Why do these distinctions matter for statistical analysis"
2 Identify each of the following variables as qualitative or quantitativea) Number of pets in familyb) County of residencec) Choice of auto (domestic or import)d) Distance (in miles) commute to workc) Choice of diet (vegetarian. nonvegetarian)f) Time spent in previous month browsing the World Wide
3. Which scule of measurement (nominal, ordinal. oi interval) is most appropriate fora) Attitude toward legalization of marijuana (favor. neutral, oppose)b) Gender (male, female)c) Number of siblings (0, 1.2, ..)d) Political party affiliation (Democrat. Republican, Independent)e) Religious
4. Which scale of measureinent is most appropriate for each of the following vanables?a) Occupation (plumber. teacher secretary, etc.)b) Occupational status (blue collar, white collar)c) Social status (lower. riddle, upper class)d) Statewide murder rate (number murders per 1000 population)e) County
5 Which scale of measurement is most appropriate for "attained education" when it is meas- ured asa) Number of years (0. 1. 2. 3....)b) Giade level (elementary school, middle school, high school, college, graduate school)c) School type (public school, private school)
6. Give an example of a variable that isa) Qualitativeb) Quantitativee) Ordinal scaled) Categoricale) Discretef) Continuous g) Quantitative and discrete
7 A survey asks subjects to rate five issues according to their importance in determining voting intention for U.S. senator, using the scale very important, somewhat important, unimportant. The issues are foreign policy, uneinployinent, inflation, the anus race, and civil rights The evaluations can
8. Which of the following variables are continuous when the measurements are as fine as possible?a) Age of motherb) Number of children in familyc) Income of spoused) Population of citiese) Latitude and longitude of citiesf) Distance of home from place of employment
9. Which of the following variables could theoretically be measured on a continuous scale?a) Method of contraception used|b) Length of ume of residence in a statec) Task completion timed) Intelligencee) Authoritarianismf) Alienation g) State of residence
10. A class bas 50 students. Use the column of the first two digits in the random number table (Table 2.1) to select a simple random sample of three students. If the students are numbered 1 to 50, what are the numbers of the three students selected?
11. In the 1995 Miss America beauty pageant, television viewers could cast their vote on whether to cancel the swimwear parade by phoning a nuniber the network provided About I million viewers called and registered their opinion, of whom 79% said they wanted to see the contestants dressed as
12. In fall 1995, the BBC in Britain requested viewers to call the network and indicate their favorite poem. Of more than 7500 callers, more than twice as many voted for Rudyard Kipling's If than for any other poem. The BBC then reported that this was the clear fa- vorite. Was their sample a
13. A local telephone directory has 300 pages with 120 names per page.a) Explain how you would choose a simple randoin sample of 5 names. Using the second column of Table 21, seleci 5 numbers to identify subjects for the sample.b) Select five nombers to identify subjects for a systematic sample of
14 Refer to the preceding problem Is cluster sampling applicable? How could it be cared out, and what would be the advantages and disadvantages?
15. Refer to the data file created in Problem 1.7. For each variable in the data set indicate whether it isa) Qualitative or quantitativeb) Nominal, ordinal. or interval
16 Repeat the previous exercise for the World Wide Web (WWW) data set (Problem 1.7).
17. Table 22 shows the result of the 1992 Presidential election along with the vote predicted by several organizations in the days before the election The sample sizes were typically about 2000. The percentages for cach poll do not sum to 100 because of voters report- ing as undecided What factors
18 A simple random sample of size n is one in which (select the best response)a) Every nth member is selected from the populationb) Each possible sample of size n has the same chance of being selected.c) There is exactly the same proportion of women in the sample as is in the population.d) One
19 You are assigned to direct a study on your campus to determine the factors that relate to strong academic performance at the school.a) Describe how you might select a sample of 100 students for the study.b) List some variables that you would measure in your study For cacli, provide the scale you
20 Quota sampling is a method used when. for example, an interviewer stands at a street corner and conducts interviews until obtaining a quota in various categories representing the relative sizes of various groups in the population of interest For instance, the quota might be 50 factory workers,
21 in 1995 in the United Kingdoin. the Equality Code used by the legal profession added a section to make members more aware of the dangers of sexual harassment It states that "esearch for the Bar found that over 40 percent of lemale junior tenants said they had encountered sexual harassment during
22. When the Yankelovich polling organization asked, "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 contributing
23 Crosson (1994, p 168) describes an analysis of published medical studies involving treat- ments for heart attacks. In the studics having randomization and strong controls for bias. the new therapy provided improved treatment 9% of the time. In studies without random- ization or other controls
24 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 governincnt expendi- tures
25. Refer to the previous exercise. Repeat for the following scenarios:a) A principal in a large high school is interested in student attitudes toward a proposed general achievement test to determine whether a student should graduate. She lists all of the first-period classes, assigning a number to
26 You plan to sample from the 5000 students at your college in order to compare the pro- portions of men and women who believe that women should have the right to an abortion.a) Explain how you would proceed. if you want a simple random sample of 100 studentsb) How would you proceed if you want a
27 In a systematic random sample. every subject has the same chance of selection. but the sample is not a simple random sample. Explain why, by showing that every possible saru- ple of size n is not equally likely.
28. 1 need to collect data for a sample of residents of registered nursing homes in my state. 1 obtain from the state a list of all nursing homes, which I number from 1 to 317 Beginning randomly, I choose every tenth home on the list. ending up with 31 homes. I then obtain lists of residents from
1 According to the Bureau of the Census (Current Population Reports). in 1994 in the United States there were 23.6 million households with one person, 31.2 million with two persons, 169 million with three persons, 15.1 million with four persons, 6.7 million with five per- sons, 2.2 million with six
2 According to News America Syndicate in 1986 the number of followers of the world's major religions were 835 million for Christianity, 420 million for Islan, 322 million for Hinduism. 300 million for Confucianism, 210 million for Buddhism. 79 million for Shinto. 50 inillion for Taoism. and 12
3. Refer to Table 3.1. Use software to construct a histogram for these data. using its default method of forming intervals Describe the shape of the distribution, and construct the corresponding relative fiequency distribution.
4. Table 311 shows the number (in millions) of the foreign-born population of the United States in 1990. by place of birth.a) Construct a relative liequency distribution.b) Plot the data in a bar graph.c) Is "Place of birth" quantitative or qualitative" How, if at all, can you describe these data
5 A researcher in an alcoholism treatment center. interested in summarizing the length of stay in the center for first-time patients, randomly selects ten records of individuals insu- tutionalized within the previous two years. The lengths of stay in the center. in days. are as follows. 11. 6. 20.
6 The 1994 General Social Survey asked respondents "How often do you read the newspa- per? The possible responses were (every day, a few times a week once a week, less than once a week. never). and the counts in those categories were (969, 452. 261, 196, 76).a) Identify the median response.b)
7 Table 3.12 summanzes responses of 1250 subjects in the 1991 General Social Survey to the question. "About how often did you have sex during the last 12 months?"a) Construct a bar graph, and interpretb) Report the median and the mode. Interpret.c) Treat this scale in a quantitative manner by
8. The 1991 General Social Survey asked respondents, "How many sex partners have you had in the last 12 months?" Table 3.13 shows results for 637 respondents.a) Calculate and interpret the median and the mode.TABLE 3.12 How Often Had Sex Frequency Not at all 292 Once or twice 99 About once a month
9. For 1992, the statewide muumber of abortions per 1000 wonen 15 to 44 years of age, for states in the Pacific region of the United States, were: Washington, 33, Oregon, 16: Cal- ifornia. 304; Alaska. 2; and Hawaii, 11 (Statistical Abstract of the United States. 1994).a) Calculate the meanb)
10. For 1993. Table 91 in Chapter 9 shows data on the statewide violent crime rate per 100,000 population. In this exercise. do not use the observation for D.Ca) Using the intervals 0-100, 100-200, 200-300, and so forth, tally the frequencies and construct a frequency distributionb) Find the
11. Refer to the preceding problem. Table 3.14 shows part of a computer printout for analyz- ing the data, the first colunin refers to the entire data set, and the second column deletes the observation for D.Ca) Report and interpret the mean and median of the first set of crime rates. Explain what
12. In 1992 in the United States, the median family income was $38 909 for white families. $21.161 for black families, and $23.901 for Hispanic families (C.S Bureau of the Census. Current Population Reports. P-60-184) In constant 1992 dollars, the median family in- comes in 1975 were $35.619 for
13 Table 3.15 shows 1994 female economic acu vity for countries in South America.a) Construct a back-to-back stem and leaf plot of these values contrasted with those from Eastern Europe in Table 3 6. What is your interpretation?b) Compare the means for the two sets of nations, and interpret.e)
14 According to the US Bureau of the Census. Current Population Reports, in 1994 the median household income was $32.368 for whites and $18,660 for blacks, whereas the mean household income was $40.708 for whites and 525,409 for blacks Does this sug- TABLE 3.15 Country Activity Argentina 38 Ecuador
15 Refer to the previous exercise. The results refer to 57.9 million white households and 8.0 million black households.a) Find the overall mean income.b) If the mean income equals $30,291 for 59 million Hispanic families. find the overall mean to the three groups combined.
16. For towns with population size 2500 to 4599 in the US. Northeast in 1994. the mean salary of chiefs of police was $37.527, and the median was $30.500 (The Municipal Year Book 1995. Washington. D.C. International City/County Management Association. 1995) Docs this suggest that the distribution
17. According to the National Association of Home Builders. the US nationwide median selling price of hoines sold in 1995 was $118,000.a) Would you expect the mean to be larger, smaller. or equal to $118.000? Explainb) Which of the following is the most plausible value for the standard deviation:
18. The 1990 General Social Survey asked respondents, "During the past 12 months, how many people have you known personally that were victims of homicide." Table 3.16 show's a computer printout from analyzing responses for 1370 subjects TABLE 3.16 VICTIMS Frequency Percent 0 1244 90.8 1 81 5.9 2 27
19 The Human Development Index (HDI) has three components life expectancy at birth. cducational attainment, and income. It ranges from 0 to 1, with higher values representing greater development. In 1992, the HDI ratings for eight Central American countries were .884 for Belize, .884 for Costa
20 According to Statistical Abstract of the United States, 1995, average salary (in dollars) of secondary school classroom teachers in 1994 in the Unued States varied among states with a five-number summary of: 100% Max 51,700 (Connecticut) 38,500 75% Q3 50% Med 33,900 258 Q1 29,800 04 Min 25,300
21 Consider the data in Table 3.8 on the number of people you know who have committed suicide. The mean equals .145, and the standard deviation equals .457 Froin the results reported in the table, what percentage of measurements fall within one standard deviation of the mean? Is the Empirical Rule
22 Why is the median sometimes preferred over the mean as a measure of central tendency? Give an example to illustrate your answer.
23. Why is the mean sometimes preferred over the median? Give an example to illustrate your answer
24 Give an example of a vanable for which the mode applies, but not the mean or median.
25. A group of high school students takes an exain The mean score for the boys is 65. and the median is 75. Both the mean and the median score for the girls is 70. How can you explain the large difference between the two sunumary measures for the boy's"
26. During the spring semester of 1995 at the University of Flonda, coinputer usage of stu- dents having accounts on a mainframe computer at the university was summarized by a mean of 1921 and a standard deviation of 11,495 kilobytes of drive usage.a) Does the Empirical Rule apply to this
27. Refer to Problem 3.26. The five-number summary of these data was minimum = 4, Q1= 256, median = 530. Q3 = 1105, and maximum = 320,000. What does this suggest about the shape of the distribution? Why?
28 Residential electrical consumption in March 1994 in Gainesville. Florida, had a mean of 780 and a standard deviation of 506 kilowatt-hours (Kwh). The minimum usage was 3 Kwh and the maximum was 9390 Kwh. (Data supplied by N. Todd Kamboot. Gainesville Regional Utilities.)a) What shape do you
29. Residential water consumption in March 1994 in Gainesville, Flonda, had a mean of 7.1 and a standard deviation of 6.2 (thousand gallons) What shape do you expect this distn- bution to have? Why" (Data supplied by N Todd Kamhoot. Gainesville Regional Utili- ties.)
30 For each of the following. sketch roughly what you expect a histograin to look like, and explain whether the mean or the median would be greater. Also sketch box plots for cases (a) and (c) that are consistent with the histograms.a) The selling price of new homes in 1997b) The number of children
31 For each of the following vanables. indicate whether you would expect its relative fre- quency histogram to be bell-shaped, U-shaped, skewed to the nght, or skewed to the left. For parts (a), (b), and (g). sketch a box plot that would be plausible for that variablea) Exain scoic (scores fall
32. Give examples of social science variables having a distribution that you would expect to bea) Approximately syinmetricb) Skewed to the rightc) Skewed to the leftd) Bimodale) Skewed to the right, with a mode and median of 0 but a positive mean
33. A recent Roper organization survey asked. "How far have environmental protection laws and regulations gone?" For the possible responses not far enough. about right, and too far, the percentages of responses were 51%, 33%. and 16%a) Which response is the mode?b) Can you compute a meau oi a
34. A company conducts a study of the number of miles traveled asing public transportation by its employees during a typical day. A random sample of ten employees yields the fol- lowing values (in miles): 0.0.4, 0.0.0. 10.0.6.0a) Calculate and interpret the mean, median, mode, range, variance, and
35. To measure variation:a) Why is the standard deviations usually prefened over the range?b) The IQR is sometimes preferred to when there are some extreme outliers Why?
36 In the mid-1980s. the General Social Survey asked respondents how many close friends they had. For a sample of size 1467. the mean was 7.4 and the standard deviation was 110. The distribution had a median of 5 and a mode of 4 Based on these statistics, what would you surmise about the shape of
37. In 1994 the General Social Survey asked, "On the average day, about how many hours do you personally watch television" Of 1964 responses, the mode was 2, the inedian was 2, the inean was 28, and the standard deviation was 24. Based on these statistics. a hal would you surmise about the shape of
38 For an exam given to a class, the students' scores ranged from 35 to 98, with a incan of 74. Which of the following is the most realistic value for the standard deviation? 1.12. 60, -10" Why"
39 The sample mean for a data set equals 80 Which of the following is an impossible value for the standard deviation? 200,0 -20.
40 According to a recent report from the US National Center for Health Statistics, feinales with age between 25 and 34 years have a bell-shaped distribution on height, with mean of 65 inches and standard deviation of 35 inchesa) Give an interval within which about 95% of the heights fall.b) What is
41 In a large northern city, monthly payinents to people on welfare last year were observed to have approximately a bell shape with mean $700 and standard deviation $100 Give a range of values within which all or nearly all the payments fell.
42 For the WWW data on number of times a week reading a newspaper, referred to in Prob- lem 1.7, Figure 3.18 shows a computer printout of the stein and lead plot and the box plot.a) From the box plot, identity the minimuin. lower quarule. median, upper quartile. and maxinuinb) Identify these five
43 Suppose the distribution of the prices of new homes built in the United States in 1996 was approximately bell-shaped, with a mean of $120.000 and a standard deviation of $40,000a) Describe the distribution using properties of the standard deviationb) If your new house was priced half a standard
44 In 1993, the five-number sunniary for the statewide percentage of people without health insurance had a minimum of 8.7% (Wisconsin). Q1 = 11 9, Med = 13.4, Q3 = 17.8, and maximum of 23 9% (Louisiana) (Statistical Abstract of the United States, 1995).a) Construct a box plot for these data.b) Do
45 Refer to Problem 3.20. Construct a box plot for these data Are there any apparent out- liers?
46 The distribution of high school graduation rates in the United States in 1993 had a mini- anum value of 64.3 (Mississippi), lower quartile of 73 9, median of 76.75. upper quartile of 80.1. and maximum value of 86.6 (Alaska) (Statistical Abstract of the United States, 1995)a) Report and interpret
47. In your library, find the percentage of the vote that Bill Clinton received in each state in the 1996 presidential election.a) Prepare a stein and leaf plot. Are there any apparent oudiers?b) Construct a box plot Are there any outliers?c) Construct back-to-back stem and leaf plots or
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