Homework Answers -- Bivariate Frequency Distribution
For each problem, identify the theory, the null hypothesis, answer the questions, and report the effect size
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Problem 1: The students in the Corps of Cadets are faced with a challenge, a pass-fail assignment. Of the 56 people in the class, (comprised of 24 females and 32 males) 11 females failed and 23 males failed. Create the tables necessary to inform the community about the difference in the dependent variable (pass-fail) between the two levels of the independent variable (male and female cadets), and compute MOE to analyze if gender improved the predictability of pass fail rate among cadets.
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What is the probability of randomly selecting a female cadet who failed from the population?
What is the best prediction of whether a cadet (male and female) passed the assignment?
Compare male cadets who passed to female cadets who passed.
What is the magnitude of effect of gender?
Problem 1 answer:
What is the theory? Females are higher achievers than are males. N = 56
What is the null hypothesis? There is not a statistically significant difference in cadets who “pass” among females and males between the observed and the expected count.
Females are passing at a greater rate (54%) than are males (28%). Using the independent variable of student type (male or female) improved the predictability of pass fail by 9%.
Females Males Total
Count Pass 13 9 (22)
Percent Pass 54% 28%
Count Fail 11 23 (34)
Percent Fail 46% 72%
Total (24) (32) (56)
λ = E1 - E2 / E1
λ = 22 - 20 / 22
λ = 2 / 22
λ = .09 = 9%
What is the probability of randomly selecting a female cadet who failed from the population?
46%
What is the best prediction of whether a cadet (male and female) passed the assignment?
Most cadets fail
Compare male cadets who passed to female cadets who passed.
Females 54%, Males 28%
What is the magnitude of effect of gender?
λ = .09 = 9%
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Problem 2: A group of experienced police officers is under the impression that Latino drivers are less likely to exceed the speed limit than Caucasian drivers. They set up an observation point on a stretch of highway and observed 5600 drivers over a two-week period (3200 Caucasian and 2400 Latino). Of that total, 1100 drivers were ticketed for speeding (700 were Caucasian and 400 were Latino). Create a bivariate frequency distribution comparing the dependent variable (ticketed - not ticketed) between the two levels of the independent variable (Caucasian and Latino drivers), and compute MOE to determine if race improves the predictability of which drivers are more likely to speed between those two ethnicities.
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What is the best prediction of whether a driver will be ticketed for speeding?
What is the probability of randomly selecting a Latino driver who was ticketed for speeding?
Compare ticketing among Caucasian drivers and Latino drivers.
What is the magnitude of effect of race.
Problem 2 answer:
What is the theory? Latino drivers are less likely to exceed the speed limit than Caucasian drivers. N = 5600
What is the null hypothesis? There is not a statistically significant difference in drivers who “speed” among Latino and Caucasian drivers between the observed and the expected count.
There is no improvement in predictability through knowledge of a driver’s ethnicity.
Caucasian Latino Total
Count "Ticket" 700 400 (1100)
Percent "Ticket" 22% 17%
Count "No Ticket" 2500 2000 (4500)
Percent "No Ticket" 78% 83%
Total (3200) (2400) (5600)
λ = E1 - E2 / E1
λ = 1100 - 1100 / 1100
λ = 0 / 1100
λ = -0-
What is the best prediction of whether a driver will be ticketed for speeding? Most don't get tickets
What is the probability of randomly selecting a Latino driver who was ticketed for speeding? 17%
Compare ticketing among Caucasian drivers and Latino drivers. 22% Caucasian get tickets, 17% Latinos get tickets
What is the magnitude of effect of race. 0%
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Problem 3: A new infant behavior analysis device has demonstrated the ability to foretell the potential for criminality among newborns in suburban and inner city hospitals. Out of 78 suburb babies, 23 were positively identified, and 55 were negative. Inner-city positives numbered 90, with 51 negative. The sample totaled 219. Which neighborhood is more likely to produce criminals (dependent variable = Positive or Negative ID, independent variable = Inner-city or Suburbs)? Does knowledge about which neighborhood the baby is from improve the ability to predict whether or not a given individual will be a criminal?
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What is the best prediction of whether a newborn will be a criminal?
What is the probability of randomly selecting a Suburban infant who is criminal?
Compare criminality among Inner-city and Suburban infants
What is the magnitude of effect of environment on criminality?
Problem 3 answer:
What is the theory? Suburb babies are less likely to become criminals than their inner-city counterparts. N = 219
What is the null hypothesis? There is not a statistically significant difference in persons who become offenders among those born in the Suburbs and Inner-city, between the observed and the expected count.
The percentage of criminals produced by the inner-city appears to be higher than the percentage of criminals produced by the suburbs (64% vs. 29%). Knowledge about which neighborhood the baby is from does improve the ability to predict whether or not a given individual will be a criminal λ = .302.
Inner-city Subs Total
Count "Positive" 90 23 (113)
Percent "Positive" 64% 29%
Count "Negative" 51 55 (106)
Percent "Negative" 36% 71%
Total (141) (78) (219)
λ = E1 - E2 / E1
λ = 106 - 74 / 106
λ = 32 / 106
λ = .302 = 30%
What is the best prediction of whether a newborn will be a criminal? Most are Criminals
What is the probability of randomly selecting a Suburban infant who is criminal? 29%
Compare criminality among Inner-city and Suburban infants
64% Inter city are crims 29% of subbies are crims
What is the magnitude of effect of environment on criminality? 30%
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Problem 4: Two specialty units with the Memphis police department (Independent variable = the Narcotics unit and the Vice squad) are in competition to find which unit is more efficient at making arrests. Arrest efficiency is determined by whether or not a conviction is obtained (dependent variable = conviction or dismissed). The two divisions produced 926 arrests last year (vice = 447, Narc = 479). Vice scored 306 convictions, Narcotics scored 204).
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What is the best prediction of whether an arrest will end up as a conviction?
What is the probability of randomly selecting a vice arrest who got convicted?
Compare convictions among Narcotics and Vice?
What is the magnitude of effect of the arresting unit on conviction?
What is the theory? Narcotics officers are less likely to get a conviction than vice officers. N = 926
What is the null hypothesis? There is not a statistically significant difference in the number of convictions among Vice and Narcotics arrests, between the observed and the expected count.
The bivariate percentages reveal that Vice had the higher efficiency rating (68% vs. 43%). Using unit type as a predictor improved the predictability of convictions by 17%.
"Narcotics" "Vice" Total
Count "Convictions" 204 306 (510)
Percent "Convictions" 43% 68%
Count "Dismissals" 275 141 (416) E2 345
Percent "Dismissals" 57% 32%
Total (479) (447) (926)
λ = E1 - E2 / E1
λ = 416 - 345 / 416
λ = 71 / 416
λ = .171 = 17%
What is the best prediction of whether an arrest will end up as a conviction?
Most are convicted
What is the probability of randomly selecting a vice arrest who got convicted?
68%
Compare convictions among Narcotics and Vice?
Narcotics 43%, Vice 68%
What is the magnitude of effect of the arresting unit on conviction?
λ = 17%
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Problem 5: Are Asian immigrant teens more likely to join gangs in America than Latino teens? A social worker believes so, and asks Asian immigrant and Latino teens if they had been solicited to become members of an ethnic gang (Dependent variable = solicited / not solicited, independent variable = Asian / Latino). Of 326 Asian immigrant teens 228 had been solicited. Latino teens numbered 78 in the affirmative category out of 185.
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What is the best prediction of whether a youth will be solicited to join a gang?
What is the probability of randomly selecting a Latino who got solicited to join a gang?
Compare solicitations among Asian and Latino teens?
Does knowing race improve ability to predict solicitations?
What is the magnitude of effect of race on gang solicitations?
Problem 5 answer:
What is the theory? Asians will be more likely to be solicited than Latinos. N = 511
What is the null hypothesis? There is not a statistically significant difference in the number of solicitations among Latino and Asian teens, between the observed and the expected count.
The bivariate percentages reveal that Asians were more vigorously sought after for gang membership (70% vs. 42%). Using unit type as a predictor improved the predictability of convictions by 14%.
Asian Latino Total E1 205
Count "Solicited" 228 78 (306)
98
Percent "Solicited" 70% 42% + 78
Count "Not Solicited" 98 107 (205) E2 176
Percent "Not Solicited" 30% 58%
Total (326) (185) (511)
λ = E1 - E2 / E1
λ = 205 - 176 / 205
λ = 29 / 205
λ = .141 = 14%
What is the best prediction of whether a youth will be solicited to join a gang?
Most are solicited
What is the probability of randomly selecting a Latino who got solicited to join a gang?
42%
Compare solicitations among Asian and Latino teens?
Asians 70%, Latinos 42%
Does knowing race improve ability to predict solicitations?
Yes
What is the magnitude of effect of race on gang solicitations?
λ = 14%
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Problem 6: The eligible driving population in millions (which were persons age 16 and over), counting only African Americans and Caucasians, was 181.8 (Blacks 24.5, White 157.3). The actual licensed drivers in that group was 158.9 (Black = 18.1, White = 140.8). Create bivariate distribution tables (freq and %) for two nominal variables (dependent variable = license / no license, independent variable = African American / Caucasian) and determine Lambda.
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What is the best prediction of whether an eligible driver will be a licensed driver?
Most are not licensed
What is the probability of randomly selecting a African Americans who will be a licensed driver?
74%
Compare licensing among African Americans and Caucasians.
African American 74%, Caucasian 90%
What is the magnitude of effect of race on licensing?
λ = 0%
Problem 6 answer:
What is the theory? Caucasians will be more likely be licensed than African Americans.
What is the null hypothesis? There is not a statistically significant difference in the number of licensees among African Americans or Caucasians, between the observed and the expected count.
Although it appears that race is responsible for the difference in licensing, the Magnitude of effect (MOE) is -0-, because knowing a persons race provides no improvement in predictability in whether or not they are licensed to drive.
African American Caucasian Total E1 22.9
Count "License 18.1 140.8 (158.9)
6.4
Percent "License 74% 90% +16.5
Count "No License" 6.4 16.5 (22.9) E2 22.9
Percent "No License" 26% 10%
Total (24.5) (157.3) (181.8)
λ = E1 - E2 / E1
λ = 22.9 - 22.9 / 22.9
λ = 0 / 22.9
λ = .000 = 0%
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Problem 7: Among African Americans and Caucasian drivers (Black = 18.1, White = 140.8), police accomplished 17.1 vehicle stops during the year (White = 14.9, Black = 2.2). The phenomenon commonly forwarded by sociologists is that police initiate vehicle stops for the crime “driving while Black” (DWB). Conduct a study to ascertain if the data supports or dispels the phenomenon. Create bivariate distribution tables (freq and %) for two nominal variables to measure the difference in the dependent variable (stopped / not stopped) between the two levels of the independent variable (African American / Caucasian) and compute MOE.
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What is the best prediction of whether a driver will be stopped?
Most are not stopped
Compare vehicle stops among African American drivers and Caucasian drivers?
African American 12%, Caucasian 11%
What is the magnitude of effect of race on vehicle stops?
λ = 0%
Problem 7 answer:
What is the theory? Africans Americans are more likely to be stopped than Caucasians.
N = 158.9
What is the null hypothesis? There is not a statistically significant difference in the number of vehicle stops among African American or Caucasian drivers, between the observed and the expected count.
The difference in vehicle stops among African Americans and Caucasians is so slight that there is no improvement in prediction of vehicle stop between ethnicities. Moe = -0-.
African American Caucasian Total E1 17.1
Count "Stopped" 2.2 14.9 (17.1)
Percent "Stopped" 12% 11% 2.2 + 14.9
Count "Not Stopped" 15.9 125.9 (141.8) E2 17.1
Percent "Not Stopped" 88% 89%
Total (18.1) (140.8) (158.9)
λ = E1 - E2 / E1
λ = 17.1 - 17.1 / 17.1
λ = 0 / 17.1
λ = 0.0 = 0%
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Problem 8: Of the total eligible drivers equaling 186.2 million (Male = 95.4, Female = 90.8), there were a total of 78.9 vehicle stops (Male = 50.4, Female = 28.5). You are in charge of determining if there is a disparity among the independent variable (males and females) concerning the dependent variable (traffic stops)? Create bivariate distribution tables (freq and %) for two nominal variables and compute MOE.
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What is the best prediction of whether a male or female driver will be stopped?
Most are not stopped
Compare vehicle stops among male and female drivers?
Female 31% Male 53%
What is the magnitude of effect of gender on vehicle stops?
λ = .068 = 7%
Problem 8 answer:
What is the theory? Males are stopped more than Females. N = 186.2
What is the null hypothesis? There is not a statistically significant difference in the number of vehicle stops among Male or Female drivers, between the observed and the expected count.
There is a big difference in the percent of vehicle stops among male and female drivers (male = 53%, female = 31%). The magnitude of effect (MOE) is weak MOE = 6%.
Male Female Total E1 78.9
Count "Stopped" 50.4 28.5 (78.9)
45
Percent "Stopped" 53% 31% + 28.5
Count "Not Stopped" 45 62.3 (107.3) E2 73.5
Percent "Not Stopped" 47% 69%
Total (95.4) (90.8) (186.2)
λ = E1 - E2 / E1
λ = 78.9 - 73.5 / 78.9
λ = 5.4 / 78.9
λ = .068 = 7%
Group Study 2
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It is widely believed that police dogs, although effective at apprehending criminals, leave the organization they are attached to at greater liability for lawsuits if the dog injures the arrestee during the apprehension process. Defenders of the canine unit typically site the number of law suits against units who do not use dogs, claiming that the presence of a dog does not necessarily make a lawsuit more likely than arrests without a dog. A random sample of departments nationwide reveal that 251 arrests made with a dog produced 100 lawsuits, 151 had no lawsuits. Out of 207 arrests with a no dog, 150 had lawsuits and 57 did not. Ascertain if the independent variable (dog / no dog) puts the unit at greater risk on the dependent variable (lawsuit / no lawsuit) and compute the Magnitude of Effect (MOE).
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Answer:
What is the theory? Arrests that involve a police dog are less likely to result in a lawsuit than those where no dog is present.
What is the null hypothesis? There is not a statistically significant difference in the number of lawsuits among arrests made with or without a police dog, between the observed and the expected count. N = 458
There is a difference in the ability to predict whether or not a lawsuit will occur based on if the dog was with the unit or not. Prediction was improved by 25%.
"Dog" "No Dog" Total E1 208
Count "Lawsuit" 100 150 (250)
100
Percent "Lawsuit" 40% 72% + 57
Count "No Lawsuit" 151 57 (208) E2 157
Percent "No Lawsuit" 60% 28%
Total (251) (207) (458)
λ = (E1 - E2) / E1
λ = (208 - 157) / 208
λ = 51 / 208
λ = 0.245 = 25%