ABRAHAM DE MOIVRE: PAVING THE WAY FOR PROPORTION INFERENCES
13.48 Party and Class. Refer to Table 13.12
a. If you have not done Exercise 13.47, group the bivariate data for the two variables into a contingency table.
b. Determine the conditional distribution of class level within each political party affiliation.
c. Are the variables “political party affiliation” and “class level”
for this population of night-school students associated? Ex- plain your answer.
d. Without doing any further calculation, determine the marginal distribution of class level.
e. Without doing further calculation, respond true or false to the following statement and explain your answer: “The con- ditional distributions of political party affiliation within class levels are identical to each other and to the marginal distribu- tion of political party affiliation.”
13.49 AIDS Cases. According to theCenters for Disease Con- trol and PreventionpublicationHIV/AIDS Surveillance Report, Vol. 19, the number of AIDS cases in the United States in 2007, by gender and race, is as shown in the following contingency table.
Race
Gender
Male Female Total
White 10,563 12,534
Black 14,247 7,196 Other 6,471
Total 42,496
a. How many cells does this contingency table have?
b. Fill in the missing entries.
c. What was the total number of AIDS cases in the United States in 2007?
d. How many AIDS cases were blacks?
e. How many AIDS cases were males?
f. How many AIDS cases were white females?
13.50 Vehicles in Use. As reported by theMotor Vehicle Manu- facturers Association of the United StatesinMotor Vehicle Facts and Figures, the number of cars and trucks in use by age are as shown in the following contingency table. Frequencies are in millions.
a. How many cells does this contingency table have?
b. Fill in the missing entries.
c. What is the total number of cars and trucks in use?
d. How many vehicles are trucks?
Age(yr)
Type
Car Truck Total
Under 6 46.2 27.8 74.0
6–8 26.9 40.0
9–11 23.3 10.7
12 & over 26.8 18.6 45.4 Total 123.2
e. How many vehicles are between 6 and 8 years old?
f. How many vehicles are trucks that are between 9 and 11 years old?
13.51 Education of Prisoners. In the article “Education and Correctional Populations” (Bureau of Justice Statistics Special Report, NCJ 195670), C. Harlow examined the educational at- tainment of prisoners by type of prison facility. The following contingency table was adapted from Table 1 of the article. Fre- quencies are in thousands, rounded to the nearest hundred.
Educationalattainment
Prison facility
State Federal Local Total 8th grade
149.9 10.6 66.0 226.5
or less Some high
269.1 12.9 168.2 450.2
school
GED 300.8 20.1 71.0 391.9
High school
216.4 24.0 130.4 370.8
diploma
Postsecondary 95.0 14.0 51.9 160.9
College grad
25.3 7.2 16.1 48.6
or more
Total 1056.5 88.8 503.6 1648.9
How many prisoners a. are in state facilities?
b. have at least a college education?
c. are in federal facilities and have at most an 8th-grade edu- cation?
d. are in federal facilities or have at most an 8th-grade education?
e. in local facilities have a postsecondary educational attain- ment?
f. with a postsecondary educational attainment are in local faci- lities?
g. are not in federal facilities?
13.52 U.S. Hospitals. TheAmerican Hospital Associationpub- lishes information about U.S. hospitals and nursing homes in Hospital Statistics. The following contingency table provides a cross-classification of U.S. hospitals and nursing homes by type of facility and number of beds.
In the following questions, the termhospitalrefers to either a hospital or nursing home.
a. How many hospitals have at least 75 beds?
b. How many hospitals are psychiatric facilities?
Facility
Number of beds
24 or fewer 25–74 75 or more Total
General 260 1586 3557 5403
Psychiatric 24 242 471 737
Chronic 1 3 22 26
Tuberculosis 0 2 2 4
Other 25 177 208 410
Total 310 2010 4260 6580
c. How many hospitals are psychiatric facilities with at least 75 beds?
d. How many hospitals either are psychiatric facilities or have at least 75 beds?
e. How many general facilities have between 25 and 74 beds?
f. How many hospitals with between 25 and 74 beds are chronic facilities?
g. How many hospitals have more than 24 beds?
13.53 Farms. The U.S. Department of Agriculture, National Agricultural Statistics Service, publishes information about U.S. farms inCensus of Agriculture. A joint frequency distribu- tion for number of farms, by acreage and tenure of operator, is provided in the following contingency table. Frequencies are in thousands.
Acreage
Tenure of operator Full Part
owner owner Tenant Total
Under 50 64 41
50–179 487 131 41 659
180–499 203 389
500–999 54 91 17 162
1000 & over 46 112 18 176
Total 1429 551
a. Fill in the six missing entries.
b. How many cells does this contingency table have?
c. How many farms have under 50 acres?
d. How many farms are tenant operated?
e. How many farms are operated by part owners and have be- tween 500 acres and 999 acres, inclusive?
f. How many farms are not full-owner operated?
g. How many tenant-operated farms have 180 acres or more?
13.54 Housing Units. The U.S. Census Bureaupublishes in- formation about housing units inAmerican Housing Survey for the United States. The following table cross-classifies occupied housing units by number of persons and tenure of occupier. The frequencies are in thousands.
a. How many occupied housing units are occupied by exactly three persons?
b. How many occupied housing units are owner occupied?
c. How many occupied housing units are rented and have seven or more persons in them?
Persons
Tenure Owner Renter 1 16,686 13,310
2 27,356 9,369
3 12,173 5,349
4 11,639 4,073
5 5,159 1,830
6 1,720 716
7+ 914 398
d. How many occupied housing units are occupied by more than one person?
e. How many occupied housing units are either owner occupied or have only one person in them?
13.55 AIDS Cases. Refer to Exercise 13.49. For AIDS cases in the United States in 2007, answer the following questions:
a. Find and interpret the conditional distribution of gender by race.
b. Find and interpret the marginal distribution of gender.
c. Are the variables “gender” and “race” associated? Explain your answer.
d. What percentage of AIDS cases were females?
e. What percentage of AIDS cases among whites were females?
f. Without doing further calculations, respond true or false to the following statement and explain your answer: “The condi- tional distributions of race by gender are not identical.”
g. Find and interpret the marginal distribution of race and the conditional distributions of race by gender.
13.56 Vehicles in Use. Refer to Exercise 13.50. Here, the term
“vehicle” refers to either a U.S. car or truck currently in use.
a. Determine the conditional distribution of age group for each type of vehicle.
b. Determine the marginal distribution of age group for vehicles.
c. Are the variables “type” and “age group” for vehicles associ- ated? Explain your answer.
d. Find the percentage of vehicles under 6 years old.
e. Find the percentage of cars under 6 years old.
f. Without doing any further calculations, respond true or false to the following statement and explain your answer: “The con- ditional distributions of type of vehicle within age groups are not identical.”
g. Determine and interpret the marginal distribution of type of vehicle and the conditional distributions of type of vehicle within age groups.
13.57 Education of Prisoners. Refer to Exercise 13.51.
a. Find the conditional distribution of educational attainment within each type of prison facility.
b. Does an association exist between educational attainment and type of prison facility for prisoners? Explain your answer.
c. Determine the marginal distribution of educational attainment for prisoners.
d. Construct a segmented bar graph for the conditional distribu- tions of educational attainment and marginal distribution of educational attainment that you obtained in parts (a) and (c), respectively. Interpret the graph in light of your answer to part (b).
602 CHAPTER 13 Chi-Square Procedures
e. Without doing any further calculations, respond true or false to the following statement and explain your answer: “The con- ditional distributions of facility type within educational attain- ment categories are identical.”
f. Determine the marginal distribution of facility type and the conditional distributions of facility type within educational at- tainment categories.
g. Find the percentage of prisoners who are in federal facilities.
h. Find the percentage of prisoners with at most an 8th-grade ed- ucation who are in federal facilities.
i. Find the percentage of prisoners in federal facilities who have at most an 8th-grade education.
13.58 U.S. Hospitals. Refer to Exercise 13.52.
a. Determine the conditional distribution of number of beds within each facility type.
b. Does an association exist between facility type and number of beds for U.S. hospitals? Explain your answer.
c. Determine the marginal distribution of number of beds for U.S. hospitals.
d. Construct a segmented bar graph for the conditional distribu- tions and marginal distribution of number of beds. Interpret the graph in light of your answer to part (b).
e. Without doing any further calculations, respond true or false to the following statement and explain your answer: “The con- ditional distributions of facility type within number-of-beds categories are identical.”
f. Obtain the marginal distribution of facility type and the con- ditional distributions of facility type within number-of-beds categories.
g. What percentage of hospitals are general facilities?
h. What percentage of hospitals that have at least 75 beds are general facilities?
i. What percentage of general facilities have at least 75 beds?
Working with Large Data Sets
In each of Exercises13.59–13.61, use the technology of your choice to solve the specified problems.
13.59 Governors. The National Governors Associationpub- lishes information on U.S. governors in Governors’ Political Affiliations & Terms of Office. Based on that document, we obtained the data on region of residence and political party given on the WeissStats CD.
a. Group the bivariate data for these two variables into a contin- gency table.
b. Determine the conditional distribution of region within each party and the marginal distribution of region.
c. Determine the conditional distribution of party within each re- gion and the marginal distribution of party.
d. Are the variables “region” and “party” for U.S. governors as- sociated? Explain your answer.
13.60 Motorcycle Accidents. TheScottish Executive,Analyti- cal Services Division Transport Statistics, compiles information on motorcycle accidents in Scotland. During one year, data on the number of motorcycle accidents, by day of the week and type of road (built-up or non built-up), are as presented on the WeissStats CD.
a. Group the bivariate data for these two variables into a contin- gency table.
b. Determine the conditional distribution of day of the week within each type-of-road category and the marginal distribu- tion of day of the week.
c. Determine the conditional distribution of type of road within each day of the week and the marginal distribution of type of road.
d. Does an association exist between the variables “day of the week” and “type of road” for these motorcycle accidents? Ex- plain your answer.
13.61 Senators. TheU.S. Congress, Joint Committee on Print- ing, provides information on the composition of Congress inCon- gressional Directory. On the WeissStats CD, we present data on party and class for the senators in the 111th Congress.
a. Group the bivariate data for these two variables into a contin- gency table.
b. Determine the conditional distribution of party within each class and the marginal distribution of party.
c. Determine the conditional distribution of class within each party and the marginal distribution of class.
d. Are the variables “party” and “class” for U.S. senators in the 111th Congress associated? Explain your answer.
Extending the Concepts and Skills
13.62 In this exercise, you are to consider two variables,xandy, defined on a hypothetical population. Following are the condi- tional distributions of the variableycorresponding to each value of the variablex.
y
x
A B C Total
0 0.316 0.316 0.316 1 0.422 0.422 0.422 2 0.211 0.211 0.211 3 0.047 0.047 0.047 4 0.004 0.004 0.004 Total 1.000 1.000 1.000
a. Are the variablesxandyassociated? Explain your answer.
b. Determine the marginal distribution ofy.
c. Can you determine the marginal distribution of x? Explain your answer.
13.63 Age and Gender. TheU.S. Census Bureaupublishes cen- sus data on the resident population of the United States inCurrent Population Reports. According to that document, 7.3% of male residents are in the age group 20–24 years.
a. If no association exists between age group and gender, what percentage of the resident population would be in the age group 20–24 years? Explain your answer.
b. If no association exists between age group and gender, what percentage of female residents would be in the age group 20–24 years? Explain your answer.
c. There are about 153 million female residents of the United States. If no association exists between age group and gender, how many female residents would there be in the age group 20–24 years?
d. In fact, there are some 10.2 million female residents in the age group 20–24 years. Given this number and your answer to part (c), what do you conclude?